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Update Chapter 7

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eloiventura 2021-02-04 22:42:46 +08:00
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07_CrosscorrelationsFourierTransformandWaveletTransform/07_CrosscorrelationsFourierTransformandWaveletTransform.ipynb
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@ -25,7 +25,20 @@
"\n",
" 3) Wavelet Transform\n",
"\n",
"and test their use on the Jena Climate Dataset (2009-2016) along with a handful of other datasets. To understand"
"and test their use on the Jena Climate Dataset (2009-2016) along with a handful of other datasets.\n",
"\n",
"The contents of this notebook are complied from the following references:\n",
"\n",
" - Chapters 1-3 of Priestley, M.B. (1981). Spectral Analysis and Time Series, Vols. 1 and 2, Academic Press, New York.\n",
"\n",
" - <a href='https://towardsdatascience.com/multiple-time-series-classification-by-using-continuous-wavelet-transformation-d29df97c0442'>Multiple Time Series Classification by Using Continuous Wavelet Transformation</a>\n",
" \n",
" - <a href='https://pywavelets.readthedocs.io/en/latest/'>PyWavelets Documentation</a>\n",
"\n",
"\n",
"Organized by:\n",
"\n",
"Gilbert Michael Go Chua"
]
},
{
@ -106,10 +119,15 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Cross-correlation was discussed previously in the chapter on `Vector Autoregressive Methods` in a more strict statistical light. This time, we will be using a slightly modified definition more apt for signals. \n",
"\n",
"The cross-correlation between two signals $X_{1}$ and $X_{2}$ is given by:\n",
"\n",
"\\begin{align} R_{2,1}(s) = E[(X_{1,t}-\\mu_{1})*(X_{2,t+s}-\\mu_{2})] \\end{align}\n",
"\n",
"\n",
"which is essentially the expected value of the <a href='https://en.wikipedia.org/wiki/Convolution'>convolution</a> between the covariance of two signals $X_{1}$ and $X_{2}$.\n",
"\n",
"The normalized version is given by:\n",
"\n",
"\\begin{align} \\rho_{21}(s) = \\frac{R_{2,1}(s)}{(R_{1,1}(0)R_{2,2}(0))^{\\frac{1}{2}}}\\end{align}\n",

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07_CrosscorrelationsFourierTransformandWaveletTransform/README.md
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@ -15,3 +15,5 @@ We introduce three techniques:
3. Wavelet Transform
and test their use on the Jena Climate Dataset (2009-2016) along with a handful of other datasets.