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Preface: Introduction to Time Series Analysis
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Chapter 0: Advanced Time Series Analysis
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1. General Introduction
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2. Time Series Forecasting
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3. Evaluation Metrics for Forecast Accuracy
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Mean Absolute Error
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Root Mean Squared Error
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Mean Absolute Percentage Error
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Symmetric Mean Absolute Percentage Error
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<div class="section" id="chapter-0-advanced-time-series-analysis">
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<h1>Chapter 0: Advanced Time Series Analysis<a class="headerlink" href="#chapter-0-advanced-time-series-analysis" title="Permalink to this headline">¶</a></h1>
|
||||
<p>Organized by: Benjur Emmanuel L. Borja, Sebastian C. Ibañez and Maria Eloisa M. Ventura</p>
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<p>This notebook introduces us to the concept of time series, forecasting, and different fundamentals that we will use across the different chapters of our discussion. Specifically, this notebook will discuss:</p>
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<ol class="simple">
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||||
<li><p>Time Series</p></li>
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||||
<li><p>Forecasting</p></li>
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<li><p>Stochastic Processes</p></li>
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</ol>
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<p>The contents of this notebook is compiled from the following references:</p>
|
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<ul class="simple">
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||||
<li><p>C.Monterola, <em>“Notebook 10 Time Series Forecasting Method — ARIMA MSDS2021”</em></p></li>
|
||||
<li><p>C.Monterola, <em>“Advanced Time Series Analysis - Recurrent Neural Networks — PhDinDS2023”</em></p></li>
|
||||
<li><p>J.Valenzuela, <em>“Advanced Time Series Analysis — PhDinDS2023”</em></p></li>
|
||||
</ul>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Required Libraries</span>
|
||||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">numpy.random</span> <span class="k">as</span> <span class="nn">rng</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">import</span> <span class="nn">scipy.stats</span> <span class="k">as</span> <span class="nn">st</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">import</span> <span class="nn">statsmodels.graphics.tsaplots</span> <span class="k">as</span> <span class="nn">tg</span>
|
||||
<span class="o">%</span><span class="k">matplotlib</span> inline
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="general-introduction">
|
||||
<h2>1. General Introduction<a class="headerlink" href="#general-introduction" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Most of us would have heard about the new buzz in the market i.e. Cryptocurrency. Many of us would have invested in their coins too. But, is investing money in such a volatile currency safe? How can we make sure that investing in these coins now would surely generate a healthy profit in the future? We can’t be sure but we can surely generate an approximate value based on the previous prices. Time series models is one way to predict them.</p>
|
||||
<div class="section" id="what-is-a-time-series">
|
||||
<h3>What is a time series?<a class="headerlink" href="#what-is-a-time-series" title="Permalink to this headline">¶</a></h3>
|
||||
<p>A time series is a series of indexed values, where each value is an <em>outcome</em> of a random variable. In other words, a time series is one realization of a corresponding process.</p>
|
||||
<p>An example of a time series is the weather timeseries dataset recorded at the Weather Station at the Max Planck Institute for Biogeochemistry in Jena, Germany. In this dataset, 14 different quantities (such air temperature, atmospheric pressure, humidity, wind direction, and so on) were recorded every 10 minutes, over several years. The original data goes back to 2003, but this example is limited to data from 2009–2016. This dataset is perfect for learning to work with numerical timeseries. We will be using this dataset to build our models. Here are the actual variables in the data:</p>
|
||||
<ul class="simple">
|
||||
<li><p>Date Time</p></li>
|
||||
<li><p>p (mbar) atmospheric pressure</p></li>
|
||||
<li><p>T (degC) temperature</p></li>
|
||||
<li><p>Tpot (K) potential temperature</p></li>
|
||||
<li><p>Tdew (degC) dew point temperature</p></li>
|
||||
<li><p>rh (%) relative humidity</p></li>
|
||||
<li><p>VPmax (mbar) saturation water vapor pressure</p></li>
|
||||
<li><p>VPact (mbar) actual water vapor pressure</p></li>
|
||||
<li><p>VPdef (mbar) water vapor pressure deficit</p></li>
|
||||
<li><p>sh (g/kg) specific humidity</p></li>
|
||||
<li><p>H2OC (mmol/mol) water vapor concentration</p></li>
|
||||
<li><p>rho (g/<span class="math notranslate nohighlight">\(m^3\)</span>) air density</p></li>
|
||||
<li><p>wv (m/s) wind velocity</p></li>
|
||||
<li><p>max. wv (m/s) maximum wind velocity</p></li>
|
||||
<li><p>wd (deg) wind direction</p></li>
|
||||
</ul>
|
||||
<p>Below is the temperature data from this dataset which we will be using frequently in the following chapters:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">data</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">'../data/jena_climate_2009_2016.csv'</span><span class="p">)</span>
|
||||
<span class="n">data</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">:]</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
<span class="n">temp</span> <span class="o">=</span> <span class="n">data</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="c1">#temperature (in degrees Celsius)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">temp</span><span class="p">)),</span> <span class="n">temp</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'Temperature</span><span class="se">\n</span><span class="s1"> (degree Celcius)'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'Time (every 10 mins)'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_5_0.png" src="../_images/00_Introduction_5_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>Time series is anything that has been recorded sequentially over a period of time – regular or irregular. In any case, the goal is to see if the observed patterns can help us <em><strong>forecast</strong></em> a period of time into the future.</p>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="time-series-forecasting">
|
||||
<h2>2. Time Series Forecasting<a class="headerlink" href="#time-series-forecasting" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Forecasting is predicting future values of our time series given all available information such as historical data and any possible variables that can influence the data. Besides Crypto Currencies, there are multiple important areas where time series forecasting is used for example : forecasting Sales, Call Volume in a Call Center, Solar activity, Ocean tides, Stock market behaviour, and many others. Assume the Manager of a hotel wants to predict how many visitors should he expect next year to accordingly adjust the hotel’s inventories and make a reasonable guess of the hotel’s revenue. Based on the data of the previous years/months/days, (S)he can use time series forecasting and get an approximate value of the visitors. Forecasted value of visitors will help the hotel to manage the resources and plan things accordingly.</p>
|
||||
<p>There are a lot of things to consider before we forecast our time-series:</p>
|
||||
<ol class="simple">
|
||||
<li><p>The kind of data (Trends, Seasonality, etc)</p></li>
|
||||
<li><p>Factors that influence it (Multivariate Analysis, Causality, etc)</p></li>
|
||||
<li><p>The amount of data available</p></li>
|
||||
</ol>
|
||||
<p>Through out the following Chapters, we will discuss different forecasting methods how they are used. Each method have their own sets of assumptions and limitations, but the general idea is we assume that there is a fundamental mathematical process that defines our time-series. Understanding these mathematical processes helps us find the best method or algorithm (and the corresponding hyperparameters) to better forecast our time-series.</p>
|
||||
<p>In the next section, we’ll discuss the first formulation of time-series data sets, <em><strong>Stochastic Processes</strong></em>.</p>
|
||||
</div>
|
||||
<div class="section" id="stochastic-processes">
|
||||
<h2>3. Stochastic Processes<a class="headerlink" href="#stochastic-processes" title="Permalink to this headline">¶</a></h2>
|
||||
<p>A stochastic, or random process (process hereon) is a series of indexed random variables. In the context of time series analysis, we’ll use <span class="math notranslate nohighlight">\(t = 0, 1, 2, ...\)</span>, (discrete) time, as an index.</p>
|
||||
<p><span class="math notranslate nohighlight">\(\lbrace \mathbf{X_0}, \mathbf{X_1}, \mathbf{X_2}, ...\rbrace\)</span> denotes a process where <span class="math notranslate nohighlight">\(\mathbf{X_0}, \mathbf{X_1}, \mathbf{X_2}, ...\)</span> are random variables.</p>
|
||||
<p>A time series, in this context, is a series of indexed values, where each value is an <em>outcome</em> of a random variable. In other words, a time series is one realization of a corresponding process.</p>
|
||||
<p><strong>Example 1: Bernoulli trials.</strong> Recall that a Bernoulli trial has two possible outcomes or realizations. Let’s take them to be <span class="math notranslate nohighlight">\(\{-1, +1\}\)</span>. Let’s generate a few values:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">bern_outcomes</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
|
||||
<span class="n">size</span> <span class="o">=</span> <span class="mi">100</span>
|
||||
<span class="c1"># Generate 100 Bernoulli trials</span>
|
||||
<span class="n">flips</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">bern_outcomes</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
|
||||
<span class="n">series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">flips</span><span class="p">)</span>
|
||||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span> <span class="c1"># here we implicitly index them; let the index stand for time</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_8_0.png" src="../_images/00_Introduction_8_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>Example 2: Gaussian white noise.</strong> A Gaussian white noise process generates a time series in which each value was drawn from a Gaussian distribution with zero mean and unit variance: <span class="math notranslate nohighlight">\(\bf{X_t} \sim Gaussian(0, 1)\)</span>:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">size</span> <span class="o">=</span> <span class="mi">1000</span>
|
||||
<span class="n">wn</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
|
||||
<span class="n">wn_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">wn</span><span class="p">)</span>
|
||||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">wn_series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_10_0.png" src="../_images/00_Introduction_10_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>Gaussian white noise has the properties of being <em>independent</em> and <em>identically-distributed</em>, which leads some key properties for linear time series, to be brought up below.</p>
|
||||
<p><strong>Example 3: Brownian motion in one dimension.</strong> One-dimensional Brownian motion, or Wiener process, can be thought of as the cumulative sum of white noise time series. The value of the series at <span class="math notranslate nohighlight">\(t\)</span> is a sum of all preceding values, and can be thought of as a random walker’s displacement at time <span class="math notranslate nohighlight">\(t\)</span>: <span class="math notranslate nohighlight">\(X_t = X_{t_1} + X_{t-2} + ... = \sum_{i=0}^{t}{X_i}\)</span>.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">size</span> <span class="o">=</span> <span class="mi">100000</span>
|
||||
<span class="n">bms</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">bms_msq</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">):</span>
|
||||
<span class="n">wn</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
|
||||
<span class="n">bm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">wn</span><span class="p">)</span>
|
||||
<span class="n">bms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
|
||||
<span class="n">bms_msq</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">bm</span><span class="p">])</span>
|
||||
<span class="n">bm_ave</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">bms</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">bm_msq</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">bms_msq</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||||
|
||||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">bms_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">bm_ave</span><span class="p">)</span>
|
||||
<span class="n">bms_msq_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">bm_msq</span><span class="p">)</span>
|
||||
<span class="n">bms_series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s2">"Displacement"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
|
||||
<span class="n">bms_msq_series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s2">"MSD"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_12_0.png" src="../_images/00_Introduction_12_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>Example 4: Moving average.</strong> Given a white noise process (see above), a moving average process of order (<span class="math notranslate nohighlight">\(q\)</span>) can be generated by taking a weighted sum of the present and the previous <span class="math notranslate nohighlight">\(q\)</span> values of the white noise at each time <span class="math notranslate nohighlight">\(t\)</span>:</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = \epsilon_t + a_1\epsilon_{t-1} + ... + a_q\epsilon_{t-q}\)</span>,</p>
|
||||
<p>where <span class="math notranslate nohighlight">\(\{\epsilon_t\} \sim WN(0, \sigma^2)\)</span>. We write <span class="math notranslate nohighlight">\(\{X_t\} \sim MA(q)\)</span>. The <span class="math notranslate nohighlight">\(MA\)</span> process is an example of a <em>linear process.</em></p>
|
||||
<p>Let’s consider an <span class="math notranslate nohighlight">\(MA(2)\)</span> process with <span class="math notranslate nohighlight">\(a_1 = 0.5\)</span> and <span class="math notranslate nohighlight">\(a_2 = 0.25\)</span>. Then</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = \epsilon_t + 0.5\epsilon_{t-1} + 0.25\epsilon_{t-2}\)</span>.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">ma2</span><span class="p">(</span><span class="n">wn</span><span class="p">):</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="c1"># First element is just epsilon_t</span>
|
||||
<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">wn</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
<span class="c1"># For the second element, let's take just the first coefficient (admittedly a judgment call)</span>
|
||||
<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">wn</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">wn</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
<span class="c1"># For the rest, use the full equation</span>
|
||||
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">wn</span><span class="p">)):</span>
|
||||
<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">wn</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">wn</span><span class="p">[</span><span class="n">t</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.25</span><span class="o">*</span><span class="n">wn</span><span class="p">[</span><span class="n">t</span> <span class="o">-</span> <span class="mi">2</span><span class="p">])</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||||
|
||||
<span class="n">size</span> <span class="o">=</span> <span class="mi">1000</span>
|
||||
<span class="n">wn</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
|
||||
<span class="n">ma2_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span>
|
||||
<span class="n">ma2</span><span class="p">(</span><span class="n">wn</span><span class="p">)</span>
|
||||
<span class="p">)</span>
|
||||
<span class="n">wn_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">wn</span><span class="p">)</span>
|
||||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">wn_series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s2">"WN(0,1)"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
|
||||
<span class="n">ma2_series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"MA(2)"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_14_0.png" src="../_images/00_Introduction_14_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p><strong>Exercise 1: Autoregressive process.</strong> Given a white noise process (see above), a time series from an autoregressive process of order <span class="math notranslate nohighlight">\(p\)</span> can be generated by taking a weighted sum of the previous <span class="math notranslate nohighlight">\(p\)</span> values of the time series, plus white noise at each time <span class="math notranslate nohighlight">\(t\)</span>:</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = b_{1}X_{t-1} + ... + b_{p}X_{t-p} + \epsilon_{t}\)</span>,</p>
|
||||
<p>where <span class="math notranslate nohighlight">\(\{\epsilon_t\} \sim WN(0, \sigma^2)\)</span>. We write <span class="math notranslate nohighlight">\(\{X_t\} \sim MA(q)\)</span>.</p>
|
||||
<p>Given an <span class="math notranslate nohighlight">\(AR(2)\)</span> process with <span class="math notranslate nohighlight">\(b_1 = 0.6, b_2 = -0.2\)</span>, generate and plot a time series, similar to the <span class="math notranslate nohighlight">\(MA(1)\)</span> example above.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">ar2</span><span class="p">(</span><span class="n">wn</span><span class="p">):</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="c1"># First element is just epsilon_t</span>
|
||||
<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">wn</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
<span class="c1"># For the second element, let's take just the first coefficient (admittedly a judgment call)</span>
|
||||
<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mf">0.6</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">wn</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||||
<span class="c1"># For the rest, use the full equation</span>
|
||||
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">wn</span><span class="p">)):</span>
|
||||
<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">wn</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.6</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">t</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="o">-</span><span class="mf">0.2</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">t</span> <span class="o">-</span> <span class="mi">2</span><span class="p">])</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||||
|
||||
<span class="n">size</span> <span class="o">=</span> <span class="mi">1000</span>
|
||||
<span class="n">wn</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
|
||||
<span class="n">ar2_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span>
|
||||
<span class="n">ar2</span><span class="p">(</span><span class="n">wn</span><span class="p">)</span>
|
||||
<span class="p">)</span>
|
||||
<span class="n">wn_series</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">wn</span><span class="p">)</span>
|
||||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">wn_series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="s2">"WN(0,1)"</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
|
||||
<span class="n">ar2_series</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"AR(2)"</span><span class="p">,</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_16_0.png" src="../_images/00_Introduction_16_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="stationarity">
|
||||
<h3>Stationarity<a class="headerlink" href="#stationarity" title="Permalink to this headline">¶</a></h3>
|
||||
<p>Consider a process <span class="math notranslate nohighlight">\(\{X_t\}\)</span>, with corresponding joint distribution <span class="math notranslate nohighlight">\(P(X_{t_1 + \tau}, X_{t_2 + \tau}, ..., X_{t_n + \tau})\)</span> at times <span class="math notranslate nohighlight">\(t_1 + \tau, t_2 + \tau, ..., t_n + \tau\)</span>.</p>
|
||||
<p>This process is <em>stationary</em> if:</p>
|
||||
<p><span class="math notranslate nohighlight">\(P(X_{t_1 + \tau}, X_{t_2 + \tau}, ..., X_{t_n + \tau}) = P(X_{t_1}, X_{t_2}, ..., X_{t_n})\)</span> for <span class="math notranslate nohighlight">\(\tau, t_1 ... t_n\)</span> and <span class="math notranslate nohighlight">\(n\)</span>.</p>
|
||||
<p>Essentially, the statistics of a stationary process is invariant to time-translation.</p>
|
||||
<p><span class="math notranslate nohighlight">\(\{X_t\}\)</span> is <em>weakly stationary</em> if:</p>
|
||||
<p>(i) <span class="math notranslate nohighlight">\(\mu(X_t) = \mu(X_{t + \tau}) = \mu\)</span> for all <span class="math notranslate nohighlight">\(\tau\)</span>,<br />
|
||||
(ii) <span class="math notranslate nohighlight">\(\Gamma(t_1, t_2) = \Gamma(t_1 - t_2)\)</span>, and <br />
|
||||
(iii) <span class="math notranslate nohighlight">\(E[|X_t|^2] < \infty\)</span>.</p>
|
||||
<p>In the above, <span class="math notranslate nohighlight">\(\Gamma(t_1, t_2)\)</span> is the <em>autocovariance</em>, defined by:</p>
|
||||
<p><span class="math notranslate nohighlight">\(\Gamma(t_1, t_2) = cov~(t_1, t_2) = E\left[(X_{t_1} - \mu(X_{t_1}))(X_{t_2} - \mu(X_{t_2}))\right]\)</span></p>
|
||||
<p>Weak stationarity implies that the lag, <span class="math notranslate nohighlight">\(\tau\)</span>, can be used to parametrize the autocovariance, leading to:</p>
|
||||
<p><span class="math notranslate nohighlight">\(\Gamma(\tau) = E\left[(X_t - \mu)(X_{t + \tau} - \mu)\right]\)</span>.</p>
|
||||
<p>Normalizing the autocovariance by the standard deviations at <span class="math notranslate nohighlight">\(t_1\)</span> and <span class="math notranslate nohighlight">\(t_2\)</span> yields the <em>autocorrelation</em>:</p>
|
||||
<p><span class="math notranslate nohighlight">\(P(t_1, t_2) = \frac{cov~(t_1, t_2)}{\sigma_{t_1}\sigma_{t_2}} = \frac{E\left[(X_{t_1} - \mu(X_{t_1}))(X_{t_2} - \mu(X_{t_2}))\right]}{\sigma_{t_1}\sigma_{t_2}}\)</span></p>
|
||||
<p>And for a weak stationary process:</p>
|
||||
<p><span class="math notranslate nohighlight">\(P(\tau) = \frac{E\left[(X_t - \mu)(X_{t + \tau} - \mu)\right]}{\sigma^2}\)</span></p>
|
||||
<p>For Gaussian white noise, weak stationarity implies stationarity, something not always valid for other processes.</p>
|
||||
</div>
|
||||
<div class="section" id="autocovariance-of-a-stationary-process">
|
||||
<h3>Autocovariance of a stationary process<a class="headerlink" href="#autocovariance-of-a-stationary-process" title="Permalink to this headline">¶</a></h3>
|
||||
<p>The autocovariance matrix of a stationary process has a particular form. Letting <span class="math notranslate nohighlight">\(\tau = |t_1 - t_2|\)</span> be the <em>lag</em>, for <span class="math notranslate nohighlight">\(n\)</span> successive observations from the time series <span class="math notranslate nohighlight">\({X_t}\)</span>, the autocovariance matrix is symmetric and positive-definite, given by:</p>
|
||||
<p><span class="math notranslate nohighlight">\(
|
||||
\Gamma_n = \begin{pmatrix}
|
||||
\gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n-1} \\
|
||||
\gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n-2} \\
|
||||
\gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n-3} \\
|
||||
\vdots & \vdots & \vdots & \ddots & \vdots \\
|
||||
\gamma_{n-1} & \gamma_{n-2} & \cdots & \gamma_{n-3} & \gamma_0
|
||||
\end{pmatrix}
|
||||
\)</span></p>
|
||||
<p>Since for a stationary process the variance at zero lag is constant, the <em>autocorrelation matrix</em> is given by:</p>
|
||||
<p><span class="math notranslate nohighlight">\(
|
||||
P_n = \sigma^2\Gamma_n = \begin{pmatrix}
|
||||
1 & \rho_1 & \rho_2 & \cdots & \rho_{n-1} \\
|
||||
\rho_1 & 1 & \rho_1 & \cdots & \rho_{n-2} \\
|
||||
\rho_2 & \rho_1 & 1 & \cdots & \rho_{n-3} \\
|
||||
\vdots & \vdots & \vdots & \ddots & \vdots \\
|
||||
\rho_{n-1} & \rho_{n-2} & \cdots & \rho_{n-3} & 1
|
||||
\end{pmatrix}
|
||||
\)</span></p>
|
||||
<p>The <em>autocovariance function</em>, <span class="math notranslate nohighlight">\(\Gamma(\tau)\)</span>, can simply be read off from the diagonals of the autocovariance matrix; likewise with the <em>autocorrelation function</em>.</p>
|
||||
<p><strong>Q.</strong> What is the autocorrelation function for the Gaussian white noise process with zero mean and unit variance?</p>
|
||||
<p><strong>A.</strong> Since the values are iid, the values are uncorrelated, thus:</p>
|
||||
<p><span class="math notranslate nohighlight">\(P(\tau) = \delta_{\tau,0}\)</span></p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Plotting:</span>
|
||||
<span class="n">tg</span><span class="o">.</span><span class="n">plot_acf</span><span class="p">(</span><span class="n">wn_series</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_20_0.png" src="../_images/00_Introduction_20_0.png" />
|
||||
<img alt="../_images/00_Introduction_20_1.png" src="../_images/00_Introduction_20_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Plotting:</span>
|
||||
<span class="n">tg</span><span class="o">.</span><span class="n">plot_acf</span><span class="p">(</span><span class="n">wn</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_21_0.png" src="../_images/00_Introduction_21_0.png" />
|
||||
<img alt="../_images/00_Introduction_21_1.png" src="../_images/00_Introduction_21_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>Autocorrelations for other time series above:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">tg</span><span class="o">.</span><span class="n">plot_acf</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_23_0.png" src="../_images/00_Introduction_23_0.png" />
|
||||
<img alt="../_images/00_Introduction_23_1.png" src="../_images/00_Introduction_23_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">tg</span><span class="o">.</span><span class="n">plot_acf</span><span class="p">(</span><span class="n">ar2_series</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/00_Introduction_24_0.png" src="../_images/00_Introduction_24_0.png" />
|
||||
<img alt="../_images/00_Introduction_24_1.png" src="../_images/00_Introduction_24_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>What is the autocorrelation of the <span class="math notranslate nohighlight">\(AR(2)\)</span> process above?</p>
|
||||
</div>
|
||||
<div class="section" id="moving-average-process">
|
||||
<h3>Moving Average Process<a class="headerlink" href="#moving-average-process" title="Permalink to this headline">¶</a></h3>
|
||||
<p>From above, <span class="math notranslate nohighlight">\({X_t}\)</span> is a <em>moving average</em> process of order <span class="math notranslate nohighlight">\(q\)</span> if <span class="math notranslate nohighlight">\(X_t\)</span> follows:</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = \epsilon_t + a_1\epsilon_{t-1} + ... + a_q\epsilon_{t-q}\)</span>.</p>
|
||||
<p>Let’s use the backward shift, or lag operator <span class="math notranslate nohighlight">\(\bf{B}\)</span>:</p>
|
||||
<p><span class="math notranslate nohighlight">\(\mathbf{B} X_t = X_{t-1}\)</span> <br />
|
||||
<span class="math notranslate nohighlight">\(\mathbf{B}^2 X_t = X_{t-2}\)</span></p>
|
||||
<p>and so on. We can rewrite the above as:</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = \epsilon_t + a_1\mathbf{B}\epsilon_t + ... + a_q\mathbf{B}^q\epsilon_{t}\)</span></p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = \theta(\mathbf{B})\epsilon_t\)</span>, where</p>
|
||||
<p><span class="math notranslate nohighlight">\(\theta(\mathbf{B}) = 1 + a_1\mathbf{B} + ... + a_q\mathbf{B}^q\)</span>, with an associated polynomial</p>
|
||||
<p><span class="math notranslate nohighlight">\(\theta(z) = 1 + a_1z + ... + a_qz^q = \sum_{j=0}^q{a_jz^j}\)</span>.</p>
|
||||
</div>
|
||||
<div class="section" id="autoregressive-process">
|
||||
<h3>Autoregressive Process<a class="headerlink" href="#autoregressive-process" title="Permalink to this headline">¶</a></h3>
|
||||
<p>From above, <span class="math notranslate nohighlight">\({X_t}\)</span> is an <em>autoregressive</em> process of order <span class="math notranslate nohighlight">\(p\)</span> if <span class="math notranslate nohighlight">\(X_t\)</span> follows:</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = b_{1}X_{t-1} + ... + b_{p}X_{t-p} + \epsilon_{t}\)</span>.</p>
|
||||
<p>Rewriting and using <span class="math notranslate nohighlight">\(\mathbf{B}\)</span>:</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t - b_{1}X_{t-1}~ - ...~ - b_{p}X_{t-p} = \epsilon_{t}\)</span></p>
|
||||
<p><span class="math notranslate nohighlight">\((1 - b_1\mathbf{B}~ - ...~ - b_p\mathbf{B}^p)X_t = \epsilon_{t}\)</span></p>
|
||||
<p><span class="math notranslate nohighlight">\(\phi(\mathbf{B})X_t = \epsilon_{t}\)</span>, where</p>
|
||||
<p><span class="math notranslate nohighlight">\(\phi(\mathbf{B}) = 1 - b_1\mathbf{B}~ - ... - b_p\mathbf{B}^p\)</span>, and associated polynomial</p>
|
||||
<p><span class="math notranslate nohighlight">\(\phi(z) = 1 - b_1z~ - ... - b_pz^p = \sum_{j=0}^{p}{b_jz^j}\)</span>.</p>
|
||||
</div>
|
||||
<div class="section" id="autoregressive-moving-average-process">
|
||||
<h3>Autoregressive Moving Average Process<a class="headerlink" href="#autoregressive-moving-average-process" title="Permalink to this headline">¶</a></h3>
|
||||
<p>Combining the two yields the <em>autoregressive moving average</em> process of order <span class="math notranslate nohighlight">\((p, q)\)</span>:</p>
|
||||
<p><span class="math notranslate nohighlight">\(X_t = \epsilon_t + a_1\epsilon_{t-1} + ... + a_q\epsilon_{t-q} + b_{1}X_{t-1} + ... + b_{p}X_{t-p}\)</span></p>
|
||||
<p>This can be rewritten as:</p>
|
||||
<p><span class="math notranslate nohighlight">\(\phi(\mathbf{B})X_t = \theta(\mathbf{B})\epsilon_t\)</span>,</p>
|
||||
<p>with the characteristic polynomials as defined above.</p>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="evaluation-metrics-for-forecast-accuracy">
|
||||
<h2>3. Evaluation Metrics for Forecast Accuracy<a class="headerlink" href="#evaluation-metrics-for-forecast-accuracy" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Forecasting is one of the most common inference tasks in time series analysis. In order to properly gauge the performance of a time series model, it is common practice to divide the dataset into two parts: training and test data. Model parameters are estimated using training data, then the models are used to generate forecasts that are evaluated against the test data.</p>
|
||||
<p>Error statistics come in different flavors, each with their own advantages and disadvantages.</p>
|
||||
<div class="section" id="mean-absolute-error">
|
||||
<h3>Mean Absolute Error<a class="headerlink" href="#mean-absolute-error" title="Permalink to this headline">¶</a></h3>
|
||||
<p>The most commonly used metric is the <b><i>mean absolute error</i></b> or <b>MAE</b>. Its main advantage is that is easy to understand and compute. We note that the metric is scale-dependent, meaning that it cannot be compared to other time series that have different units. In addition, methods that minimize MAE generate forecasts that lead to the median.</p>
|
||||
<div class="amsmath math notranslate nohighlight">
|
||||
\[\begin{align*}
|
||||
\text{MAE} &= \frac{1}{n}\sum_{t=1}^{n} |y_t - \hat{y}_t|
|
||||
\end{align*}\]</div>
|
||||
</div>
|
||||
<div class="section" id="root-mean-squared-error">
|
||||
<h3>Root Mean Squared Error<a class="headerlink" href="#root-mean-squared-error" title="Permalink to this headline">¶</a></h3>
|
||||
<p>Another commonly used metric is the <b><i>root mean squared error</i></b> or <b>RMSE</b>, although it is more difficult to interpret. Methods that minimize RMSE generate forecasts that lead to the mean and, similar to MAE, RMSE is scale-dependent.</p>
|
||||
<div class="amsmath math notranslate nohighlight">
|
||||
\[\begin{align*}
|
||||
\text{RMSE} &= \sqrt{\frac{1}{n}\sum_{t=1}^{n} (y_t - \hat{y}_t)^2}
|
||||
\end{align*}\]</div>
|
||||
</div>
|
||||
<div class="section" id="mean-absolute-percentage-error">
|
||||
<h3>Mean Absolute Percentage Error<a class="headerlink" href="#mean-absolute-percentage-error" title="Permalink to this headline">¶</a></h3>
|
||||
<p>In order to make comparisons between time series that have different units, one can use percentage errors. One frequently used percentage-based metric is the <b><i>mean absolute percentage error</i></b> or <b>MAPE</b>. Unfortunately, the main disadvantage of percentage errors is that they may be infinite or undefined when <span class="math notranslate nohighlight">\(y_t\)</span> is zero or close to zero.</p>
|
||||
<div class="amsmath math notranslate nohighlight">
|
||||
\[\begin{align*}
|
||||
\text{MAPE} &= \frac{1}{n}\sum_{t=1}^{n} \bigg|\frac{y_t - \hat{y}_t}{y_t}\bigg|
|
||||
\end{align*}\]</div>
|
||||
</div>
|
||||
<div class="section" id="symmetric-mean-absolute-percentage-error">
|
||||
<h3>Symmetric Mean Absolute Percentage Error<a class="headerlink" href="#symmetric-mean-absolute-percentage-error" title="Permalink to this headline">¶</a></h3>
|
||||
<p>One particular disadvantage of MAPE is that it puts a larger penalty on negative errors compared to positive errors. This leads to an alternative percentage-based metric called <b><i>symmetric mean absolute percentage error</i></b> or <b>SMAPE</b>. However, calculating the metric can still be unstable if <span class="math notranslate nohighlight">\(y_t\)</span> and <span class="math notranslate nohighlight">\(\hat{y}_t\)</span> are both close to zero.</p>
|
||||
<div class="amsmath math notranslate nohighlight">
|
||||
\[\begin{align*}
|
||||
\text{SMAPE} &= \frac{1}{n}\sum_{t=1}^{n} \frac{|\hat{y}_t - y_t|}{|y_t| + |\hat{y}_t|}
|
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<h1>Advanced Time Series Analysis<a class="headerlink" href="#advanced-time-series-analysis" title="Permalink to this headline">¶</a></h1>
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<p>This notebook introduces us to the concept of time series, fore-casting, and different fundamentals that we will use across the different chapters of our discussion. Specifically, this notebook will discuss:</p>
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<h1>Chapter 1: AutoRegressive Integrated Moving Average (ARIMA)<a class="headerlink" href="#chapter-1-autoregressive-integrated-moving-average-arima" title="Permalink to this headline">¶</a></h1>
|
||||
<p>In this notebook, we will introduce our first approach to time-series forecasting which is <strong>ARIMA</strong> or AutoRegressive Integrated Moving Average. ARIMA, or AutoRegressive Integrated Moving Average, is a set of models that explains a time series using its own previous values given by the lags (<strong>A</strong>uto<strong>R</strong>egressive) and lagged errors (<strong>M</strong>oving <strong>A</strong>verage) while considering stationarity corrected by differencing (oppossite of <strong>I</strong>ntegration.) In other words, ARIMA assumes that the time series is described by autocorrelations in the data rather than trends and seasonality.</p>
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<h1>Chapter 2: Linear, Trend, and Momentum Forecasting<a class="headerlink" href="#chapter-2-linear-trend-and-momentum-forecasting" title="Permalink to this headline">¶</a></h1>
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<p>In this chapter we introduce basic tools on forecasting, which utilize simple algebraic formula. In the previous chapter, ARIMA was discussed where the future values of a time series are forecasted using its past or lagged values. It was shown that ARIMA can only be applied after removing the trend and seasonality of the data. We note however that for some forecasting tools, the trend is relevant and is part of the formula for prediction. In this work, forecasting will be demonstrated while making use of the relationships and trends in the data.</p>
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<p>In the first half of this notebook, we demonstrate forecasting by fitting time series data with linear regression. For the second half, we demonstrate that by using the trends of the time series data such as moving averages, we can predict the possible future direction of the trend using momentum forecasting.</p>
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<h1>Chapter 3: Vector Autoregressive Methods<a class="headerlink" href="#chapter-3-vector-autoregressive-methods" title="Permalink to this headline">¶</a></h1>
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<p>Previously, we have introduced the classical approaches in forecasting single/univariate time series like the
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Autoregressive-Moving-Average (ARIMA) model and the simple linear regression model. We learned that stationarity
|
||||
is a condition that is necessary when using ARIMA while this need not be imposed when using the linear regression model.
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In this notebook, we extend the forecasting problem to a more generalized framework where we deal with
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<strong>multivariate time series</strong>–time series which has more than one time-dependent variable. More specifically,
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we introduce <strong>vector autoregressive (VAR)</strong> models and show how they can be used in forecasting mutivariate time series.</p>
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<li><p>Multivariate Time Series model</p>
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<ul>
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<li><p>Motivation</p></li>
|
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<li><p>Univariate VS Multivariate Time Series</p></li>
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|
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<li><p>Foundations</p></li>
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<li><p>Vector Autoregressive (VAR) Models</p>
|
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|
||||
<li><p>VAR(1) model</p></li>
|
||||
<li><p>VAR(<em>p</em>) model</p></li>
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<li><p>Structural Analysis</p>
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<li><p>Impulse Response Function</p></li>
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<li><p>Forecast Error Variance Decomposition</p></li>
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<h1>Chapter 4: Granger Causality Test<a class="headerlink" href="#chapter-4-granger-causality-test" title="Permalink to this headline">¶</a></h1>
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<p>In the first three chapters, we discussed the classical methods for both univariate and multivariate time series forecasting. We now introduce the notion of causality and its implications on time series analysis in general. We also describe a test for the linear VAR model discussed in the previous chapter.</p>
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<h1>Chapter 5: Empirical Dynamic Modeling (Simplex and SMap Projections)<a class="headerlink" href="#chapter-5-empirical-dynamic-modeling-simplex-and-smap-projections" title="Permalink to this headline">¶</a></h1>
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<p>In the previous sections, we looked at the different methods to characterize a time-series and other statistical operations that we can execute to perform predictions. Many of these methods involve calculating for the models that would best fit the time series, extracting the optimal parameters that would describe the data with the least error possible. However, many real world processes exhibit nonlinear, complex, dynamic characteristics, necessitating the need of other methods that can accommodate as such.</p>
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<p>In this section, we will introduce and discuss methods that uses empirical models instead of complex, parametized, and hypothesized equations. Using raw time series data, we will try to reconstruct the underlying mechanisms that might be too complex, noisy, or dynamic to be captured by equations. This method proposes a altenatively more flexible approach in working and predicting with dynamic systems.</p>
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<h1>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h1>
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<p>In the previous chapters we talked about Simplex Projection, a forecasting technique that looks for similar trends in the past to forecast the future by computing for nearest neighbors on an embedding. In this chapter, we discuss Convergent Cross Mapping (CCM) also formulated by <a class="reference external" href="https://science.sciencemag.org/content/338/6106/496">Sugihara et al., 2012</a> as a methodology that uses ideas from Simplex Projection to identify causality between variables in a complex dynamical system (e.g. ecosystem) using just time series data.</p>
|
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<h2>causal-ccm Package<a class="headerlink" href="#causal-ccm-package" title="Permalink to this headline">¶</a></h2>
|
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<p><code class="docutils literal notranslate"><span class="pre">ccm_sugihara.ipynb</span></code> explains the CCM methodology in detail. If you wish to apply this in your own projects, install the framework using <code class="docutils literal notranslate"><span class="pre">pip</span> <span class="pre">install</span> <span class="pre">causal-ccm</span></code>. See <code class="docutils literal notranslate"><span class="pre">using_causal_ccm_package.ipynb</span></code> notebook for details how to use.</p>
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Install CCM and Load Packages
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<div class="section" id="applying-ccm-using-causal-ccm-package">
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<h1>Applying CCM Using causal-ccm package<a class="headerlink" href="#applying-ccm-using-causal-ccm-package" title="Permalink to this headline">¶</a></h1>
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<p>In this section we show how we can use <code class="docutils literal notranslate"><span class="pre">causal-ccm</span></code> package to infer causality.</p>
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<h2>Install CCM and Load Packages<a class="headerlink" href="#install-ccm-and-load-packages" title="Permalink to this headline">¶</a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># !pip install causal-ccm</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">causal_ccm.causal_ccm</span> <span class="kn">import</span> <span class="n">ccm</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
|
||||
<span class="c1">### Just to remove warnings to prettify the notebook. </span>
|
||||
<span class="kn">import</span> <span class="nn">warnings</span>
|
||||
<span class="n">warnings</span><span class="o">.</span><span class="n">filterwarnings</span><span class="p">(</span><span class="s2">"ignore"</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="load-dataset">
|
||||
<h2>Load Dataset<a class="headerlink" href="#load-dataset" title="Permalink to this headline">¶</a></h2>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">'../data/Ipo_dataset.csv'</span><span class="p">)</span>
|
||||
<span class="n">df</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_html"><div>
|
||||
<style scoped>
|
||||
.dataframe tbody tr th:only-of-type {
|
||||
vertical-align: middle;
|
||||
}
|
||||
|
||||
.dataframe tbody tr th {
|
||||
vertical-align: top;
|
||||
}
|
||||
|
||||
.dataframe thead th {
|
||||
text-align: right;
|
||||
}
|
||||
</style>
|
||||
<table border="1" class="dataframe">
|
||||
<thead>
|
||||
<tr style="text-align: right;">
|
||||
<th></th>
|
||||
<th>Time</th>
|
||||
<th>Rain</th>
|
||||
<th>ONI</th>
|
||||
<th>NIA</th>
|
||||
<th>Dam</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<th>0</th>
|
||||
<td>0</td>
|
||||
<td>0.0</td>
|
||||
<td>-0.7</td>
|
||||
<td>38.225693</td>
|
||||
<td>100.70</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>1</th>
|
||||
<td>1</td>
|
||||
<td>0.0</td>
|
||||
<td>-0.7</td>
|
||||
<td>57.996530</td>
|
||||
<td>100.63</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>2</th>
|
||||
<td>2</td>
|
||||
<td>0.0</td>
|
||||
<td>-0.7</td>
|
||||
<td>49.119213</td>
|
||||
<td>100.56</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>3</th>
|
||||
<td>3</td>
|
||||
<td>0.0</td>
|
||||
<td>-0.7</td>
|
||||
<td>47.034720</td>
|
||||
<td>100.55</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>4</th>
|
||||
<td>4</td>
|
||||
<td>0.0</td>
|
||||
<td>-0.7</td>
|
||||
<td>42.223380</td>
|
||||
<td>100.48</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div></div></div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># get the variables we want to check causality on</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s1">'Rain'</span><span class="p">]</span>
|
||||
<span class="n">Y</span> <span class="o">=</span> <span class="n">df</span><span class="p">[</span><span class="s1">'Dam'</span><span class="p">]</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="run-ccm">
|
||||
<h2>Run CCM<a class="headerlink" href="#run-ccm" title="Permalink to this headline">¶</a></h2>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># inputs</span>
|
||||
<span class="n">tau</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># time lag</span>
|
||||
<span class="n">E</span> <span class="o">=</span> <span class="mi">2</span> <span class="c1"># shadow manifold embedding dimensions</span>
|
||||
<span class="n">L</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="c1"># length of time period to consider</span>
|
||||
|
||||
<span class="c1"># if we wish to check if X causes Y</span>
|
||||
<span class="n">ccm1</span> <span class="o">=</span> <span class="n">ccm</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">tau</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">L</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># visualize sample cross mapping</span>
|
||||
<span class="n">ccm1</span><span class="o">.</span><span class="n">visualize_cross_mapping</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/using_causal_ccm_package_9_0.png" src="../_images/using_causal_ccm_package_9_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># causality X -> Y</span>
|
||||
<span class="c1"># returns: (correlation ("strength" of causality), p-value(significance))</span>
|
||||
<span class="n">corr_</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">ccm1</span><span class="o">.</span><span class="n">causality</span><span class="p">()</span>
|
||||
<span class="n">corr_</span><span class="p">,</span> <span class="n">p</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(0.2775286246309841, 5.4283283439476714e-126)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># visualize correlation of X->Y</span>
|
||||
<span class="c1"># stronger correlation = stronger causal relationship</span>
|
||||
<span class="n">ccm1</span><span class="o">.</span><span class="n">plot_ccm_correls</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="../_images/using_causal_ccm_package_11_0.png" src="../_images/using_causal_ccm_package_11_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># checking convergence</span>
|
||||
<span class="c1"># Looking at "convergence"</span>
|
||||
<span class="n">L_range</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="mi">1000</span><span class="p">)</span> <span class="c1"># L values to test</span>
|
||||
<span class="n">tau</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">E</span> <span class="o">=</span> <span class="mi">2</span>
|
||||
|
||||
<span class="n">Xhat_My</span><span class="p">,</span> <span class="n">Yhat_Mx</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span> <span class="c1"># correlation list</span>
|
||||
<span class="k">for</span> <span class="n">L</span> <span class="ow">in</span> <span class="n">L_range</span><span class="p">:</span>
|
||||
<span class="n">ccm_XY</span> <span class="o">=</span> <span class="n">ccm</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">tau</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">L</span><span class="p">)</span> <span class="c1"># define new ccm object # Testing for X -> Y</span>
|
||||
<span class="n">ccm_YX</span> <span class="o">=</span> <span class="n">ccm</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">tau</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">L</span><span class="p">)</span> <span class="c1"># define new ccm object # Testing for Y -> X </span>
|
||||
<span class="n">Xhat_My</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ccm_XY</span><span class="o">.</span><span class="n">causality</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>
|
||||
<span class="n">Yhat_Mx</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ccm_YX</span><span class="o">.</span><span class="n">causality</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'X->Y r'</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">Xhat_My</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">2</span><span class="p">),</span> <span class="s1">'p value'</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">ccm_XY</span><span class="o">.</span><span class="n">causality</span><span class="p">()[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">4</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Y->X r'</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">Yhat_Mx</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">2</span><span class="p">),</span> <span class="s1">'p value'</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">ccm_YX</span><span class="o">.</span><span class="n">causality</span><span class="p">()[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">4</span><span class="p">))</span>
|
||||
|
||||
<span class="c1"># plot convergence as L->inf. Convergence is necessary to conclude causality</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">L_range</span><span class="p">,</span> <span class="n">Xhat_My</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'$\hat</span><span class="si">{X}</span><span class="s1">(t)|M_y$'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">L_range</span><span class="p">,</span> <span class="n">Yhat_Mx</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'$\hat</span><span class="si">{Y}</span><span class="s1">(t)|M_x$'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'L'</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'correl'</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">prop</span><span class="o">=</span><span class="p">{</span><span class="s1">'size'</span><span class="p">:</span> <span class="mi">12</span><span class="p">})</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>X->Y r 0.31 p value 0.0
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Y->X r 0.02 p value 0.0775
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><matplotlib.legend.Legend at 0x7fad1e31db50>
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="../_images/using_causal_ccm_package_12_3.png" src="../_images/using_causal_ccm_package_12_3.png" />
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="interpretation">
|
||||
<h2>Interpretation<a class="headerlink" href="#interpretation" title="Permalink to this headline">¶</a></h2>
|
||||
<p>From the charts above, we find significant convergent cross mapping for the effect of rain <span class="math notranslate nohighlight">\(X\)</span> on dam levels <span class="math notranslate nohighlight">\(Y\)</span>. Note that for <span class="math notranslate nohighlight">\(X -> Y\)</span> cases, p-value is 0.0 which means the causality is significant while for <span class="math notranslate nohighlight">\(Y -> X\)</span>, p value is > 0.05. We can then say that rain drives dam levels but dam levels do not drive rain.</p>
|
||||
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|
||||
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|
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|
||||
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||||
<a class='left-prev' id="prev-link" href="ccm_sugihara.html" title="previous page">Chapter 6: Convergent Cross Mapping</a>
|
||||
<a class='right-next' id="next-link" href="../07_CrosscorrelationsFourierTransformandWaveletTransform/07_CrosscorrelationsFourierTransformandWaveletTransform.html" title="next page">Chapter 7: Cross-Correlations, Fourier Transform, and Wavelet Transform</a>
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<h1>Chapter 7: Cross-Correlations, Fourier Transform, and Wavelet Transform<a class="headerlink" href="#chapter-7-cross-correlations-fourier-transform-and-wavelet-transform" title="Permalink to this headline">¶</a></h1>
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<p>Gaining a deeper undersanding of causality, we look at time series forecasting through another lens.</p>
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<p>In this chapter, we will take a different approach to how we analzye time series that is complementary to forecasting. Previously, methods of explaining such as the Granger Causality, S-mapping, and Cross-mapping, focused on the time domain - the values of the time series when measured over time or in its phase space. While both are useful for many tasks, it can be often useful to transform these time domain measurements to unearth patterns which are difficult to tease out. Specifically, we want to look at the frequency domain to both analyze the dynamics and perform pre-processing techniques that may be used to modify real-world datasets.</p>
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<p>We will be analyzing the dynamics of time series not exactly to make forecasts, but to understand them in terms of their frequencies in complement to the previous methods of causality and explainability presented.</p>
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M5 Dataset
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<h1>Chapter 8: Winningest Methods in Time Series Forecasting<a class="headerlink" href="#chapter-8-winningest-methods-in-time-series-forecasting" title="Permalink to this headline">¶</a></h1>
|
||||
<p>In previous sections, we examined several models used in time series forecasting such as ARIMA, VAR, and Exponential Smoothing methods. While the main advantage of traditional statistical methods is their ability to perform more sophisticated inference tasks directly (e.g. hypothesis testing on parameters, causality testing), they usually lack predictive power because of their rigid assumptions. That is not to say that they are necessarily inferior when it comes to forecasting, but rather they are typically used as performance benchmarks.</p>
|
||||
<p>In this section, we demonstrate several of the fundamental ideas and approaches used in the recently concluded M5 Competition where challengers from all over the world competed in building time series forecasting models for both accuracy and uncertainty prediction tasks. Specifically, we explore the machine learning model that majority of the competition’s winners utilized: LightGBM, a tree-based gradient boosting framework designed for speed and efficiency.</p>
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<h2>M5 Dataset<a class="headerlink" href="#m5-dataset" title="Permalink to this headline">¶</a></h2>
|
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<p>The M5 dataset consists of Walmart sales data. Specifically, daily unit sales of 3,049 products, classified in 3 product categories (Hobbies, Foods, and Household), and 7 product departments in which the previously mentioned categories are disaggregated. The products are sold across 10 stores, located in 3 States (California, Texas, and Wisconsin).</p>
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Chapter 6: Convergent Cross Mapping
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Applying CCM Using causal-ccm package
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Chapter 7: Cross-Correlations, Fourier Transform, and Wavelet Transform
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Chapter 8: Winningest Methods in Time Series Forecasting
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Hyperparameter Tuning (Supplementary Notebook)
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Temperature Forecasting (Supplementary Notebook)
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Hyperparameter Tuning
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Tree Depth
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Learning Rate
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More than one variable…
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Final Model
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<div class="section" id="hyperparameter-tuning-supplementary-notebook">
|
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<h1>Hyperparameter Tuning (Supplementary Notebook)<a class="headerlink" href="#hyperparameter-tuning-supplementary-notebook" title="Permalink to this headline">¶</a></h1>
|
||||
<p>This notebook explores a grid search with repeated k-fold cross validation scheme for tuning the hyperparameters of the LightGBM model used in forecasting the M5 dataset. In general, the techniques used below can be also be adapted for other forecasting models, whether they be classical statistical models or machine learning methods.</p>
|
||||
<p>Prepared by: Sebastian C. Ibañez</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">import</span> <span class="nn">lightgbm</span> <span class="k">as</span> <span class="nn">lgb</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">RepeatedKFold</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">15</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
|
||||
|
||||
<span class="n">np</span><span class="o">.</span><span class="n">set_printoptions</span><span class="p">(</span><span class="n">precision</span> <span class="o">=</span> <span class="mi">6</span><span class="p">,</span> <span class="n">suppress</span> <span class="o">=</span> <span class="kc">True</span><span class="p">)</span>
|
||||
|
||||
<span class="c1">### CREATE X,Y ####</span>
|
||||
<span class="k">def</span> <span class="nf">create_xy</span><span class="p">(</span><span class="n">series</span><span class="p">,</span> <span class="n">window_size</span><span class="p">,</span> <span class="n">prediction_horizon</span><span class="p">,</span> <span class="n">shuffle</span> <span class="o">=</span> <span class="kc">False</span><span class="p">):</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">series</span><span class="p">)):</span>
|
||||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">series</span><span class="p">[(</span><span class="n">i</span> <span class="o">+</span> <span class="n">window_size</span><span class="p">):(</span><span class="n">i</span> <span class="o">+</span> <span class="n">window_size</span> <span class="o">+</span> <span class="n">prediction_horizon</span><span class="p">)])</span> <span class="o"><</span> <span class="n">prediction_horizon</span><span class="p">:</span>
|
||||
<span class="k">break</span>
|
||||
<span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">series</span><span class="p">[</span><span class="n">i</span><span class="p">:(</span><span class="n">i</span> <span class="o">+</span> <span class="n">window_size</span><span class="p">)])</span>
|
||||
<span class="n">y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">series</span><span class="p">[(</span><span class="n">i</span> <span class="o">+</span> <span class="n">window_size</span><span class="p">):(</span><span class="n">i</span> <span class="o">+</span> <span class="n">window_size</span> <span class="o">+</span> <span class="n">prediction_horizon</span><span class="p">)])</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span>
|
||||
|
||||
<span class="n">date_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span><span class="o">.</span><span class="n">strftime</span><span class="p">(</span><span class="s1">'%Y-%m-</span><span class="si">%d</span><span class="s1">'</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="n">start</span> <span class="o">=</span> <span class="s1">'2011-01-29'</span><span class="p">,</span> <span class="n">end</span> <span class="o">=</span> <span class="s1">'2016-04-24'</span><span class="p">)]</span>
|
||||
|
||||
<span class="n">df_sales</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">'../data/m5/sales_train_validation.csv'</span><span class="p">)</span>
|
||||
<span class="n">df_sales</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="n">columns</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">df_sales</span><span class="o">.</span><span class="n">columns</span><span class="p">[</span><span class="mi">6</span><span class="p">:],</span> <span class="n">date_list</span><span class="p">)),</span> <span class="n">inplace</span> <span class="o">=</span> <span class="kc">True</span><span class="p">)</span>
|
||||
|
||||
<span class="n">df_sales_total_by_store</span> <span class="o">=</span> <span class="n">df_sales</span><span class="o">.</span><span class="n">groupby</span><span class="p">([</span><span class="s1">'store_id'</span><span class="p">])</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
|
||||
|
||||
<span class="n">series</span> <span class="o">=</span> <span class="n">df_sales_total_by_store</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||||
|
||||
<span class="c1">### HYPERPARAMETERS ###</span>
|
||||
<span class="n">window_size</span> <span class="o">=</span> <span class="mi">365</span>
|
||||
<span class="n">prediction_horizon</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
|
||||
<span class="c1">### TRAIN VAL SPLIT ### (include shuffling later)</span>
|
||||
<span class="n">test_size</span> <span class="o">=</span> <span class="mi">28</span>
|
||||
<span class="n">split_time</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">series</span><span class="p">)</span> <span class="o">-</span> <span class="n">test_size</span>
|
||||
|
||||
<span class="n">train_series</span> <span class="o">=</span> <span class="n">series</span><span class="p">[:</span><span class="n">split_time</span><span class="p">]</span>
|
||||
<span class="n">test_series</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="n">split_time</span> <span class="o">-</span> <span class="n">window_size</span><span class="p">:]</span>
|
||||
|
||||
<span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span> <span class="o">=</span> <span class="n">create_xy</span><span class="p">(</span><span class="n">train_series</span><span class="p">,</span> <span class="n">window_size</span><span class="p">,</span> <span class="n">prediction_horizon</span><span class="p">)</span>
|
||||
<span class="n">test_x</span><span class="p">,</span> <span class="n">test_y</span> <span class="o">=</span> <span class="n">create_xy</span><span class="p">(</span><span class="n">test_series</span><span class="p">,</span> <span class="n">window_size</span><span class="p">,</span> <span class="n">prediction_horizon</span><span class="p">)</span>
|
||||
|
||||
<span class="n">train_y</span> <span class="o">=</span> <span class="n">train_y</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
|
||||
<span class="n">test_y</span> <span class="o">=</span> <span class="n">test_y</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
|
||||
<span class="ne">OSError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||||
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">f8af59ced7b0</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">2</span> <span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">3</span> <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="ne">----> </span><span class="mi">4</span> <span class="kn">import</span> <span class="nn">lightgbm</span> <span class="k">as</span> <span class="nn">lgb</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">5</span> <span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">6</span> <span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">RepeatedKFold</span>
|
||||
|
||||
<span class="o">~/</span><span class="n">opt</span><span class="o">/</span><span class="n">anaconda3</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">atsa</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="o">.</span><span class="mi">7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">lightgbm</span><span class="o">/</span><span class="fm">__init__</span><span class="o">.</span><span class="n">py</span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">6</span> <span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">absolute_import</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">7</span>
|
||||
<span class="ne">----> </span><span class="mi">8</span> <span class="kn">from</span> <span class="nn">.basic</span> <span class="kn">import</span> <span class="n">Booster</span><span class="p">,</span> <span class="n">Dataset</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">9</span> <span class="kn">from</span> <span class="nn">.callback</span> <span class="kn">import</span> <span class="p">(</span><span class="n">early_stopping</span><span class="p">,</span> <span class="n">print_evaluation</span><span class="p">,</span> <span class="n">record_evaluation</span><span class="p">,</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">10</span> <span class="n">reset_parameter</span><span class="p">)</span>
|
||||
|
||||
<span class="o">~/</span><span class="n">opt</span><span class="o">/</span><span class="n">anaconda3</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">atsa</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="o">.</span><span class="mi">7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">lightgbm</span><span class="o">/</span><span class="n">basic</span><span class="o">.</span><span class="n">py</span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">41</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">42</span>
|
||||
<span class="ne">---> </span><span class="mi">43</span> <span class="n">_LIB</span> <span class="o">=</span> <span class="n">_load_lib</span><span class="p">()</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">44</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">45</span>
|
||||
|
||||
<span class="nn">~/opt/anaconda3/envs/atsa/lib/python3.7/site-packages/lightgbm/basic.py</span> in <span class="ni">_load_lib</span><span class="nt">()</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">32</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lib_path</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">33</span> <span class="k">return</span> <span class="kc">None</span>
|
||||
<span class="ne">---> </span><span class="mi">34</span> <span class="n">lib</span> <span class="o">=</span> <span class="n">ctypes</span><span class="o">.</span><span class="n">cdll</span><span class="o">.</span><span class="n">LoadLibrary</span><span class="p">(</span><span class="n">lib_path</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">35</span> <span class="n">lib</span><span class="o">.</span><span class="n">LGBM_GetLastError</span><span class="o">.</span><span class="n">restype</span> <span class="o">=</span> <span class="n">ctypes</span><span class="o">.</span><span class="n">c_char_p</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">36</span> <span class="n">callback</span> <span class="o">=</span> <span class="n">ctypes</span><span class="o">.</span><span class="n">CFUNCTYPE</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="n">ctypes</span><span class="o">.</span><span class="n">c_char_p</span><span class="p">)</span>
|
||||
|
||||
<span class="nn">~/opt/anaconda3/envs/atsa/lib/python3.7/ctypes/__init__.py</span> in <span class="ni">LoadLibrary</span><span class="nt">(self, name)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">440</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">441</span> <span class="k">def</span> <span class="nf">LoadLibrary</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
|
||||
<span class="ne">--> </span><span class="mi">442</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dlltype</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">443</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">444</span> <span class="n">cdll</span> <span class="o">=</span> <span class="n">LibraryLoader</span><span class="p">(</span><span class="n">CDLL</span><span class="p">)</span>
|
||||
|
||||
<span class="nn">~/opt/anaconda3/envs/atsa/lib/python3.7/ctypes/__init__.py</span> in <span class="ni">__init__</span><span class="nt">(self, name, mode, handle, use_errno, use_last_error)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">362</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">363</span> <span class="k">if</span> <span class="n">handle</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
||||
<span class="ne">--> </span><span class="mi">364</span> <span class="bp">self</span><span class="o">.</span><span class="n">_handle</span> <span class="o">=</span> <span class="n">_dlopen</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_name</span><span class="p">,</span> <span class="n">mode</span><span class="p">)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">365</span> <span class="k">else</span><span class="p">:</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">366</span> <span class="bp">self</span><span class="o">.</span><span class="n">_handle</span> <span class="o">=</span> <span class="n">handle</span>
|
||||
|
||||
<span class="ne">OSError</span>: dlopen(/Users/prince.javier/opt/anaconda3/envs/atsa/lib/python3.7/site-packages/lightgbm/lib_lightgbm.so, 6): Library not loaded: /usr/local/opt/libomp/lib/libomp.dylib
|
||||
<span class="n">Referenced</span> <span class="n">from</span><span class="p">:</span> <span class="o">/</span><span class="n">Users</span><span class="o">/</span><span class="n">prince</span><span class="o">.</span><span class="n">javier</span><span class="o">/</span><span class="n">opt</span><span class="o">/</span><span class="n">anaconda3</span><span class="o">/</span><span class="n">envs</span><span class="o">/</span><span class="n">atsa</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">python3</span><span class="o">.</span><span class="mi">7</span><span class="o">/</span><span class="n">site</span><span class="o">-</span><span class="n">packages</span><span class="o">/</span><span class="n">lightgbm</span><span class="o">/</span><span class="n">lib_lightgbm</span><span class="o">.</span><span class="n">so</span>
|
||||
<span class="n">Reason</span><span class="p">:</span> <span class="n">image</span> <span class="ow">not</span> <span class="n">found</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">model</span> <span class="o">=</span> <span class="n">lgb</span><span class="o">.</span><span class="n">LGBMRegressor</span><span class="p">(</span><span class="n">first_metric_only</span> <span class="o">=</span> <span class="kc">True</span><span class="p">)</span>
|
||||
|
||||
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span><span class="p">,</span>
|
||||
<span class="n">eval_metric</span> <span class="o">=</span> <span class="s1">'l1'</span><span class="p">,</span>
|
||||
<span class="n">eval_set</span> <span class="o">=</span> <span class="p">[(</span><span class="n">test_x</span><span class="p">,</span> <span class="n">test_y</span><span class="p">)],</span>
|
||||
<span class="n">early_stopping_rounds</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span>
|
||||
<span class="n">verbose</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||||
|
||||
<span class="n">forecast</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">test_x</span><span class="p">)</span>
|
||||
<span class="n">s1_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">29</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
<span class="n">s7_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">35</span><span class="p">:</span><span class="o">-</span><span class="mi">7</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
<span class="n">s30_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">56</span><span class="p">:</span><span class="o">-</span><span class="mi">28</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
<span class="n">s365_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">364</span><span class="p">:</span><span class="o">-</span><span class="mi">336</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s1_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' s7-Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s7_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' s30-Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s30_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'s365-Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s365_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' LightGBM MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">forecast</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
|
||||
<span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="n">test_size</span><span class="p">:]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">marker</span> <span class="o">=</span> <span class="s1">'o'</span><span class="p">,</span> <span class="n">linestyle</span> <span class="o">=</span> <span class="s1">'--'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">forecast</span><span class="p">,</span> <span class="n">marker</span> <span class="o">=</span> <span class="s1">'o'</span><span class="p">,</span> <span class="n">linestyle</span> <span class="o">=</span> <span class="s1">'--'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> Naive MAE: 698.0000
|
||||
s7-Naive MAE: 372.2857
|
||||
s30-Naive MAE: 330.8214
|
||||
s365-Naive MAE: 247.9286
|
||||
LightGBM MAE: 208.9156
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="../_images/lightgbm_m5_tuning_2_1.png" src="../_images/lightgbm_m5_tuning_2_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="hyperparameter-tuning">
|
||||
<h2>Hyperparameter Tuning<a class="headerlink" href="#hyperparameter-tuning" title="Permalink to this headline">¶</a></h2>
|
||||
<div class="section" id="number-of-trees">
|
||||
<h3>Number of Trees<a class="headerlink" href="#number-of-trees" title="Permalink to this headline">¶</a></h3>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">trees</span> <span class="o">=</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">500</span><span class="p">,</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">2000</span><span class="p">]</span>
|
||||
|
||||
<span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">names</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">trees</span><span class="p">:</span>
|
||||
|
||||
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span>
|
||||
<span class="s1">'n_estimators'</span><span class="p">:</span> <span class="n">i</span><span class="p">,</span>
|
||||
<span class="p">}</span>
|
||||
|
||||
<span class="n">cv</span> <span class="o">=</span> <span class="n">RepeatedKFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">n_repeats</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">random_state</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">scores</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">lgb</span><span class="o">.</span><span class="n">LGBMRegressor</span><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">),</span> <span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span><span class="p">,</span> <span class="n">scoring</span> <span class="o">=</span> <span class="s1">'neg_mean_absolute_error'</span><span class="p">,</span> <span class="n">cv</span> <span class="o">=</span> <span class="n">cv</span><span class="p">,</span> <span class="n">n_jobs</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span>
|
||||
<span class="n">names</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%3d</span><span class="s1"> --- MAE: </span><span class="si">%.3f</span><span class="s1"> (</span><span class="si">%.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">scores</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">scores</span><span class="p">)))</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">boxplot</span><span class="p">(</span><span class="n">results</span><span class="p">,</span> <span class="n">labels</span> <span class="o">=</span> <span class="n">names</span><span class="p">,</span> <span class="n">showmeans</span> <span class="o">=</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 10 --- MAE: -385.639 (31.537)
|
||||
50 --- MAE: -283.540 (21.324)
|
||||
100 --- MAE: -285.091 (21.551)
|
||||
500 --- MAE: -288.237 (21.803)
|
||||
1000 --- MAE: -288.672 (21.822)
|
||||
2000 --- MAE: -288.734 (21.844)
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="../_images/lightgbm_m5_tuning_5_1.png" src="../_images/lightgbm_m5_tuning_5_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="tree-depth">
|
||||
<h3>Tree Depth<a class="headerlink" href="#tree-depth" title="Permalink to this headline">¶</a></h3>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">names</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">):</span>
|
||||
|
||||
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span>
|
||||
<span class="s1">'max_depth'</span><span class="p">:</span> <span class="n">i</span><span class="p">,</span>
|
||||
<span class="s1">'num_leaves'</span><span class="p">:</span> <span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span>
|
||||
<span class="p">}</span>
|
||||
|
||||
<span class="n">cv</span> <span class="o">=</span> <span class="n">RepeatedKFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">n_repeats</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">random_state</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">scores</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">lgb</span><span class="o">.</span><span class="n">LGBMRegressor</span><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">),</span> <span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span><span class="p">,</span> <span class="n">scoring</span> <span class="o">=</span> <span class="s1">'neg_mean_absolute_error'</span><span class="p">,</span> <span class="n">cv</span> <span class="o">=</span> <span class="n">cv</span><span class="p">,</span> <span class="n">n_jobs</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span>
|
||||
<span class="n">names</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%3d</span><span class="s1"> --- MAE: </span><span class="si">%.3f</span><span class="s1"> (</span><span class="si">%.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">scores</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">scores</span><span class="p">)))</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">boxplot</span><span class="p">(</span><span class="n">results</span><span class="p">,</span> <span class="n">labels</span> <span class="o">=</span> <span class="n">names</span><span class="p">,</span> <span class="n">showmeans</span> <span class="o">=</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 1 --- MAE: -292.235 (24.008)
|
||||
2 --- MAE: -284.534 (20.113)
|
||||
3 --- MAE: -278.984 (20.519)
|
||||
4 --- MAE: -276.622 (21.169)
|
||||
5 --- MAE: -280.497 (21.999)
|
||||
6 --- MAE: -283.877 (21.382)
|
||||
7 --- MAE: -284.667 (21.436)
|
||||
8 --- MAE: -288.216 (21.230)
|
||||
9 --- MAE: -287.373 (21.043)
|
||||
10 --- MAE: -288.503 (20.929)
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="../_images/lightgbm_m5_tuning_7_1.png" src="../_images/lightgbm_m5_tuning_7_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="learning-rate">
|
||||
<h3>Learning Rate<a class="headerlink" href="#learning-rate" title="Permalink to this headline">¶</a></h3>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rates</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0001</span><span class="p">,</span> <span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">]</span>
|
||||
|
||||
<span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">names</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rates</span><span class="p">:</span>
|
||||
|
||||
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span>
|
||||
<span class="s1">'learning_rate'</span><span class="p">:</span> <span class="n">i</span><span class="p">,</span>
|
||||
<span class="p">}</span>
|
||||
|
||||
<span class="n">cv</span> <span class="o">=</span> <span class="n">RepeatedKFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">n_repeats</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">random_state</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">scores</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">lgb</span><span class="o">.</span><span class="n">LGBMRegressor</span><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">),</span> <span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span><span class="p">,</span> <span class="n">scoring</span> <span class="o">=</span> <span class="s1">'neg_mean_absolute_error'</span><span class="p">,</span> <span class="n">cv</span> <span class="o">=</span> <span class="n">cv</span><span class="p">,</span> <span class="n">n_jobs</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span>
|
||||
<span class="n">names</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%.3f</span><span class="s1"> --- MAE: </span><span class="si">%.3f</span><span class="s1"> (</span><span class="si">%.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">scores</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">scores</span><span class="p">)))</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">boxplot</span><span class="p">(</span><span class="n">results</span><span class="p">,</span> <span class="n">labels</span> <span class="o">=</span> <span class="n">names</span><span class="p">,</span> <span class="n">showmeans</span> <span class="o">=</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.000 --- MAE: -748.947 (39.458)
|
||||
0.001 --- MAE: -695.973 (38.578)
|
||||
0.010 --- MAE: -391.718 (30.680)
|
||||
0.100 --- MAE: -285.091 (21.551)
|
||||
1.000 --- MAE: -437.648 (23.715)
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="../_images/lightgbm_m5_tuning_9_1.png" src="../_images/lightgbm_m5_tuning_9_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="boosting-type">
|
||||
<h3>Boosting Type<a class="headerlink" href="#boosting-type" title="Permalink to this headline">¶</a></h3>
|
||||
<ul class="simple">
|
||||
<li><p><code class="docutils literal notranslate"><span class="pre">gbdt</span></code>: Gradient Boosting Decision Tree (GDBT).</p></li>
|
||||
<li><p><code class="docutils literal notranslate"><span class="pre">dart</span></code>: Dropouts meet Multiple Additive Regression Trees (DART).</p></li>
|
||||
<li><p><code class="docutils literal notranslate"><span class="pre">goss</span></code>: Gradient-based One-Side Sampling (GOSS).</p></li>
|
||||
</ul>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">types</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'gbdt'</span><span class="p">,</span> <span class="s1">'dart'</span><span class="p">,</span> <span class="s1">'goss'</span><span class="p">]</span>
|
||||
|
||||
<span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">names</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">types</span><span class="p">:</span>
|
||||
|
||||
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span>
|
||||
<span class="s1">'boosting_type'</span><span class="p">:</span> <span class="n">i</span><span class="p">,</span>
|
||||
<span class="p">}</span>
|
||||
|
||||
<span class="n">cv</span> <span class="o">=</span> <span class="n">RepeatedKFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">n_repeats</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">random_state</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">scores</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">lgb</span><span class="o">.</span><span class="n">LGBMRegressor</span><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">),</span> <span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span><span class="p">,</span> <span class="n">scoring</span> <span class="o">=</span> <span class="s1">'neg_mean_absolute_error'</span><span class="p">,</span> <span class="n">cv</span> <span class="o">=</span> <span class="n">cv</span><span class="p">,</span> <span class="n">n_jobs</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span>
|
||||
<span class="n">names</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%s</span><span class="s1"> --- MAE: </span><span class="si">%.3f</span><span class="s1"> (</span><span class="si">%.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">scores</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">scores</span><span class="p">)))</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">boxplot</span><span class="p">(</span><span class="n">results</span><span class="p">,</span> <span class="n">labels</span> <span class="o">=</span> <span class="n">names</span><span class="p">,</span> <span class="n">showmeans</span> <span class="o">=</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>gbdt --- MAE: -285.091 (21.551)
|
||||
dart --- MAE: -481.225 (33.006)
|
||||
goss --- MAE: -291.263 (21.742)
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="../_images/lightgbm_m5_tuning_11_1.png" src="../_images/lightgbm_m5_tuning_11_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="more-than-one-variable">
|
||||
<h3>More than one variable…<a class="headerlink" href="#more-than-one-variable" title="Permalink to this headline">¶</a></h3>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">types</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'gbdt'</span><span class="p">,</span> <span class="s1">'dart'</span><span class="p">,</span> <span class="s1">'goss'</span><span class="p">]</span>
|
||||
|
||||
<span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">types</span><span class="p">:</span>
|
||||
|
||||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">):</span>
|
||||
|
||||
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span>
|
||||
<span class="s1">'n_estimators'</span><span class="p">:</span> <span class="mi">2000</span><span class="p">,</span>
|
||||
<span class="s1">'learning_rate'</span><span class="p">:</span> <span class="mf">0.1</span><span class="p">,</span>
|
||||
<span class="s1">'max_depth'</span><span class="p">:</span> <span class="n">j</span><span class="p">,</span>
|
||||
<span class="s1">'num_leaves'</span><span class="p">:</span> <span class="mi">2</span><span class="o">**</span><span class="n">j</span><span class="p">,</span>
|
||||
<span class="s1">'boosting_type'</span><span class="p">:</span> <span class="n">i</span><span class="p">,</span>
|
||||
<span class="p">}</span>
|
||||
|
||||
<span class="n">cv</span> <span class="o">=</span> <span class="n">RepeatedKFold</span><span class="p">(</span><span class="n">n_splits</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">n_repeats</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">random_state</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">scores</span> <span class="o">=</span> <span class="n">cross_val_score</span><span class="p">(</span><span class="n">lgb</span><span class="o">.</span><span class="n">LGBMRegressor</span><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">),</span> <span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span><span class="p">,</span> <span class="n">scoring</span> <span class="o">=</span> <span class="s1">'neg_mean_absolute_error'</span><span class="p">,</span> <span class="n">cv</span> <span class="o">=</span> <span class="n">cv</span><span class="p">,</span> <span class="n">n_jobs</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||||
<span class="n">results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">scores</span><span class="p">)</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">%s</span><span class="s1"> </span><span class="si">%d</span><span class="s1"> --- MAE: </span><span class="si">%.3f</span><span class="s1"> (</span><span class="si">%.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">scores</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">scores</span><span class="p">)))</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>gbdt 1 --- MAE: -287.860 (21.319)
|
||||
gbdt 2 --- MAE: -298.625 (20.007)
|
||||
gbdt 3 --- MAE: -288.742 (21.788)
|
||||
gbdt 4 --- MAE: -284.235 (20.843)
|
||||
gbdt 5 --- MAE: -285.302 (21.250)
|
||||
gbdt 6 --- MAE: -288.334 (21.073)
|
||||
gbdt 7 --- MAE: -288.306 (21.382)
|
||||
gbdt 8 --- MAE: -290.893 (21.207)
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/conda/lib/python3.8/site-packages/joblib/externals/loky/process_executor.py:688: UserWarning: A worker stopped while some jobs were given to the executor. This can be caused by a too short worker timeout or by a memory leak.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>gbdt 9 --- MAE: -291.163 (22.049)
|
||||
gbdt 10 --- MAE: -291.799 (21.347)
|
||||
dart 1 --- MAE: -277.039 (21.229)
|
||||
dart 2 --- MAE: -278.634 (19.236)
|
||||
dart 3 --- MAE: -274.650 (21.226)
|
||||
dart 4 --- MAE: -274.975 (22.260)
|
||||
dart 5 --- MAE: -276.816 (22.355)
|
||||
dart 6 --- MAE: -278.424 (22.700)
|
||||
dart 7 --- MAE: -280.250 (22.238)
|
||||
dart 8 --- MAE: -281.548 (23.202)
|
||||
dart 9 --- MAE: -282.776 (23.881)
|
||||
dart 10 --- MAE: -283.613 (23.785)
|
||||
goss 1 --- MAE: -301.392 (22.057)
|
||||
goss 2 --- MAE: -305.135 (17.874)
|
||||
goss 3 --- MAE: -297.664 (18.621)
|
||||
goss 4 --- MAE: -297.698 (19.858)
|
||||
goss 5 --- MAE: -296.022 (18.470)
|
||||
goss 6 --- MAE: -295.540 (20.386)
|
||||
goss 7 --- MAE: -290.380 (19.665)
|
||||
goss 8 --- MAE: -295.496 (20.014)
|
||||
goss 9 --- MAE: -292.827 (20.048)
|
||||
goss 10 --- MAE: -293.961 (22.087)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="final-model">
|
||||
<h2>Final Model<a class="headerlink" href="#final-model" title="Permalink to this headline">¶</a></h2>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">params</span> <span class="o">=</span> <span class="p">{</span>
|
||||
<span class="s1">'n_estimators'</span><span class="p">:</span> <span class="mi">2000</span><span class="p">,</span>
|
||||
<span class="s1">'max_depth'</span><span class="p">:</span> <span class="mi">4</span><span class="p">,</span>
|
||||
<span class="s1">'num_leaves'</span><span class="p">:</span> <span class="mi">2</span><span class="o">**</span><span class="mi">4</span><span class="p">,</span>
|
||||
<span class="s1">'learning_rate'</span><span class="p">:</span> <span class="mf">0.1</span><span class="p">,</span>
|
||||
<span class="s1">'boosting_type'</span><span class="p">:</span> <span class="s1">'dart'</span>
|
||||
<span class="p">}</span>
|
||||
|
||||
<span class="n">model</span> <span class="o">=</span> <span class="n">lgb</span><span class="o">.</span><span class="n">LGBMRegressor</span><span class="p">(</span><span class="n">first_metric_only</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span> <span class="o">**</span><span class="n">params</span><span class="p">)</span>
|
||||
|
||||
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">train_x</span><span class="p">,</span> <span class="n">train_y</span><span class="p">,</span>
|
||||
<span class="n">eval_metric</span> <span class="o">=</span> <span class="s1">'l1'</span><span class="p">,</span>
|
||||
<span class="n">eval_set</span> <span class="o">=</span> <span class="p">[(</span><span class="n">test_x</span><span class="p">,</span> <span class="n">test_y</span><span class="p">)],</span>
|
||||
<span class="c1">#early_stopping_rounds = 10,</span>
|
||||
<span class="n">verbose</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||||
|
||||
<span class="n">forecast</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">test_x</span><span class="p">)</span>
|
||||
<span class="n">s1_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">29</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
<span class="n">s7_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">35</span><span class="p">:</span><span class="o">-</span><span class="mi">7</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
<span class="n">s30_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">56</span><span class="p">:</span><span class="o">-</span><span class="mi">28</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
<span class="n">s365_naive</span> <span class="o">=</span> <span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="mi">364</span><span class="p">:</span><span class="o">-</span><span class="mi">336</span><span class="p">]</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">()</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s1_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' s7-Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s7_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' s30-Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s30_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'s365-Naive MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s365_naive</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">' LightGBM MAE: </span><span class="si">%.4f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">forecast</span> <span class="o">-</span> <span class="n">test_y</span><span class="p">))))</span>
|
||||
|
||||
<span class="n">series</span><span class="p">[</span><span class="o">-</span><span class="n">test_size</span><span class="p">:]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">marker</span> <span class="o">=</span> <span class="s1">'o'</span><span class="p">,</span> <span class="n">linestyle</span> <span class="o">=</span> <span class="s1">'--'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">forecast</span><span class="p">,</span> <span class="n">marker</span> <span class="o">=</span> <span class="s1">'o'</span><span class="p">,</span> <span class="n">linestyle</span> <span class="o">=</span> <span class="s1">'--'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> Naive MAE: 698.0000
|
||||
s7-Naive MAE: 372.2857
|
||||
s30-Naive MAE: 330.8214
|
||||
s365-Naive MAE: 247.9286
|
||||
LightGBM MAE: 200.5037
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="../_images/lightgbm_m5_tuning_15_1.png" src="../_images/lightgbm_m5_tuning_15_1.png" />
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<h1>Preface: Introduction to Time Series Analysis<a class="headerlink" href="#preface-introduction-to-time-series-analysis" title="Permalink to this headline">¶</a></h1>
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<p>This handbook extensively covers time series analysis and forecasting, delving from the most fundamental methods to the state-of-the-art. The handbook was made in Python and is designed such that readers can both learn the theory and apply them to real-world problems. Although chapters were made to be stand alone, it is recommended that readers go through the first few chapters to be able to fully appreciate the latter chapters. Moreover, the
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<strong><a class="reference external" href="https://www.kaggle.com/stytch16/jena-climate-2009-2016">Jena climate dataset</a></strong> is used across several chapters, with a summary of the performance of the models used at the end.</p>
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<li class="toc-h1 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#">
|
||||
Time Series Handbook
|
||||
</a>
|
||||
<ul class="nav section-nav flex-column">
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#outline">
|
||||
Outline
|
||||
</a>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h1 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#how-to-use-this-reference">
|
||||
How to use this reference
|
||||
</a>
|
||||
<ul class="nav section-nav flex-column">
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#setting-up-your-virtual-environment">
|
||||
Setting up your virtual environment
|
||||
</a>
|
||||
<ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#anaconda">
|
||||
Anaconda
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h3 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#python-venv">
|
||||
Python venv
|
||||
</a>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#rendering-the-notebooks">
|
||||
Rendering the notebooks
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#jupyterbook">
|
||||
Jupyterbook
|
||||
</a>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-h1 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#contributors">
|
||||
Contributors
|
||||
</a>
|
||||
</li>
|
||||
</ul>
|
||||
|
||||
</nav>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div id="main-content" class="row">
|
||||
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
|
||||
|
||||
<div>
|
||||
|
||||
<div class="section" id="time-series-handbook">
|
||||
<h1>Time Series Handbook<a class="headerlink" href="#time-series-handbook" title="Permalink to this headline">¶</a></h1>
|
||||
<p>This handbook extensively covers time series analysis and forecasting, delving from the most fundamental methods to the state-of-the-art. The handbook was made in Python and is designed such that readers can both learn the theory and apply them to real-world problems. Although chapters were made to be stand alone, it is recommended that readers go through the first few chapters to be able to fully appreciate the latter chapters. Moreover, the
|
||||
<a class="reference external" href="https://www.kaggle.com/stytch16/jena-climate-2009-2016">Jena climate dataset</a> is used across several chapters, with a summary of the performance of the models used at the end.</p>
|
||||
<p>The handbook is structured as follows: in the first part, classical forecasting methods are discussed in detail. The middle part is then dedicated to dynamical forecasting methods and as well as causality and correlations, topics that are particularly essential in understanding the intricacies of time series forecasting. Finally, the last part shows a glimpse into the current trends and open problems in time series forecasting and modeling.</p>
|
||||
<p>The aim of this handbook is to serve as a practitioner’s guide to forecasting, enabling them to better understand relationships in signals. It is made for an audience with a solid background in Statistics and Mathematics, as well as a basic knowledge of Python. Familiarity with Machine Learning methods is a plus, especially for the later chapters.</p>
|
||||
<div class="section" id="outline">
|
||||
<h2>Outline<a class="headerlink" href="#outline" title="Permalink to this headline">¶</a></h2>
|
||||
<p>This handbook contains a variety of techniques that you can use for time series analysis – from simple statistical models to some of the state-of-the-art algorithms as of writing. Here are the items that are covered in this material:</p>
|
||||
<ul class="simple">
|
||||
<li><p>Chapter 0: <span class="xref myst">Introduction to Time Series Analysis</span></p></li>
|
||||
<li><p>Chapter 1: <span class="xref myst">Autoregressive integrated moving average</span></p></li>
|
||||
<li><p>Chapter 2: <span class="xref myst">Linear Trend and Momentum Forecasting</span></p></li>
|
||||
<li><p>Chapter 3: <span class="xref myst">Vector Autoregressive Methods</span></p></li>
|
||||
<li><p>Chapter 4: <span class="xref myst">Granger Causality</span></p></li>
|
||||
<li><p>Chapter 5: <span class="xref myst">Simplex and S-map Projections</span></p></li>
|
||||
<li><p>Chapter 6: <span class="xref myst">Convergent Cross Mapping and Sugihara Causality</span></p></li>
|
||||
<li><p>Chapter 7: <span class="xref myst">Cross-Correlations, Fourier Transform and Wavelet Transform</span></p></li>
|
||||
<li><p>Chapter 8: <span class="xref myst">Winningest Methods</span></p></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="how-to-use-this-reference">
|
||||
<h1>How to use this reference<a class="headerlink" href="#how-to-use-this-reference" title="Permalink to this headline">¶</a></h1>
|
||||
<p>Each of the chapters mentioned above includes Jupyter notebook/s that contain/s the discussion of each topic (background, limitations, applications). Most of the datasets used in the handbook are included in this repository, and the details of each are described in the <span class="xref myst">data folder</span>.</p>
|
||||
<div class="section" id="setting-up-your-virtual-environment">
|
||||
<h2>Setting up your virtual environment<a class="headerlink" href="#setting-up-your-virtual-environment" title="Permalink to this headline">¶</a></h2>
|
||||
<p>To be able to run the contents of this repository, it is advised that you setup a virtual environment. You can install one via Anaconda or via Python’s native <code class="docutils literal notranslate"><span class="pre">venv</span></code> module.</p>
|
||||
<div class="section" id="anaconda">
|
||||
<h3>Anaconda<a class="headerlink" href="#anaconda" title="Permalink to this headline">¶</a></h3>
|
||||
<p>To set up a virtual environment called <code class="docutils literal notranslate"><span class="pre">atsa</span></code>, run the following in your terminal:</p>
|
||||
<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span><span class="c1"># this will create an anaconda environment</span>
|
||||
<span class="c1"># called atsa in 'path/to/anaconda3/envs/'</span>
|
||||
conda create -n atsa
|
||||
</pre></div>
|
||||
</div>
|
||||
<p>To activate and enter the environment, run <code class="docutils literal notranslate"><span class="pre">conda</span> <span class="pre">activate</span> <span class="pre">atsa</span></code>. To deactivate the environment, either run <code class="docutils literal notranslate"><span class="pre">conda</span> <span class="pre">deactivate</span> <span class="pre">atsa</span></code> or exit the terminal. For more information on setting up your virtual evironment using Anaconda, please visit <a class="reference external" href="https://docs.conda.io/projects/conda/en/latest/user-guide/tasks/manage-environments.html">this page</a>.</p>
|
||||
</div>
|
||||
<div class="section" id="python-venv">
|
||||
<h3>Python venv<a class="headerlink" href="#python-venv" title="Permalink to this headline">¶</a></h3>
|
||||
<p>To set up a virtual environment called <code class="docutils literal notranslate"><span class="pre">atsa</span></code>, run the following in your terminal:</p>
|
||||
<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span><span class="c1"># this will create a virtual environment</span>
|
||||
<span class="c1"># called atsa in your home directory</span>
|
||||
python3 -m venv ~/atsa
|
||||
</pre></div>
|
||||
</div>
|
||||
<p>To activate and enter the environment, run <code class="docutils literal notranslate"><span class="pre">source</span> <span class="pre">~/atsa/bin/activate</span></code>. To deactivate the environment, either run <code class="docutils literal notranslate"><span class="pre">deactivate</span></code> or exit the terminal. Note that every time you want to work on the assignment, you should rerun <code class="docutils literal notranslate"><span class="pre">source</span> <span class="pre">~/atsa/bin/activate</span></code>.</p>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="rendering-the-notebooks">
|
||||
<h2>Rendering the notebooks<a class="headerlink" href="#rendering-the-notebooks" title="Permalink to this headline">¶</a></h2>
|
||||
<p>To view the individual notebooks outside of Github without setting up a repository or installing any software, you may use <a class="reference external" href="https://nbviewer.jupyter.org/">The Jupyter Notebook Viewer</a>.</p>
|
||||
<ul class="simple">
|
||||
<li><p>Open <code class="docutils literal notranslate"><span class="pre">https://nbviewer.jupyter.org/</span></code></p></li>
|
||||
<li><p>Paste the link to the notebook.</p></li>
|
||||
</ul>
|
||||
<p>When a notebook rendered in nbviewer appears differently from the one rendered github, just append <code class="docutils literal notranslate"><span class="pre">?flush_cache=true</span></code> to the end of the nbviewer URL to force it to rerender.</p>
|
||||
</div>
|
||||
<div class="section" id="jupyterbook">
|
||||
<h2>Jupyterbook<a class="headerlink" href="#jupyterbook" title="Permalink to this headline">¶</a></h2>
|
||||
<p>To view all the chapters of this handbook, please visit this link: <a class="reference external" href="https://phdinds-aim.github.io/time_series_handbook/">Time Series Analysis Handbook</a></p>
|
||||
<p><img alt="Screenshot" src="_images/jupyterbook_handbook.png" /></p>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="contributors">
|
||||
<h1>Contributors<a class="headerlink" href="#contributors" title="Permalink to this headline">¶</a></h1>
|
||||
<ul class="simple">
|
||||
<li><p>Benjur Emmanuel Borja</p></li>
|
||||
<li><p>Gilbert Michael G. Chua</p></li>
|
||||
<li><p>Francis James Corpuz</p></li>
|
||||
<li><p>Carlo Vincienzo Dajac</p></li>
|
||||
<li><p>Sebastian C. Ibañez</p></li>
|
||||
<li><p>Prince Joseph Erneszer Javier</p></li>
|
||||
<li><p>Marissa P. Liponhay</p></li>
|
||||
<li><p>Maria Eloisa M. Ventura</p></li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<script type="text/x-thebe-config">
|
||||
{
|
||||
requestKernel: true,
|
||||
binderOptions: {
|
||||
repo: "binder-examples/jupyter-stacks-datascience",
|
||||
ref: "master",
|
||||
},
|
||||
codeMirrorConfig: {
|
||||
theme: "abcdef",
|
||||
mode: "python"
|
||||
},
|
||||
kernelOptions: {
|
||||
kernelName: "python3",
|
||||
path: "./."
|
||||
},
|
||||
predefinedOutput: true
|
||||
}
|
||||
</script>
|
||||
<script>kernelName = 'python3'</script>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
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|
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<div class='prev-next-bottom'>
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<footer class="footer mt-5 mt-md-0">
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<div class="container">
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<p>
|
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|
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By students of PhD in Data Science Batch 2023 at the Asian Institute of Management<br/>
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© Copyright 2020.<br/>
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Reference in New Issue