updated discussion in example notebook

06_ConvergentCrossMappingandSugiharaCausality/ccm_sugihara.ipynb

@@ -25,7 +25,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Importance of understanding causality\n",
+    "### Importance of Understanding Causality\n",
     "We define causality as the influence of one variable on another. Understanding causality is important because it allows us to:\n",
     "* Explain phenomena that they are not necessarily supernatural/paranormal\n",
     "* Anticipate the future and make the necessary adjustments\n",
@@ -48,7 +48,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### CCM addresses limits of Granger's causality\n",
+    "### CCM addresses limits of Granger causality\n",
     "Convergent Cross Mapping is designed to identify and quantify causalities in systems whose variables are not separable. Beyond this, it is designed to identify and quantify weak to moderate causalities, which Granger Causality may miss. Finally, CCM is independent of a predictive model so we're not at the mercy of model performance."
    ]
   },
@@ -81,13 +81,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## How and why does CCM work?\n",
+    "## Overview of CCM\n",
     "\n",
     "<img src=https://media.springernature.com/lw785/springer-static/image/chp%3A10.1007%2F978-3-319-58895-7_27/MediaObjects/429072_1_En_27_Fig1_HTML.gif width=400>\n",
     "\n",
     "Tsonis A.A., Deyle E.R., Ye H., Sugihara G. (2018) Convergent Cross Mapping: Theory and an Example. In: Tsonis A. (eds) Advances in Nonlinear Geosciences. Springer, Cham. https://doi.org/10.1007/978-3-319-58895-7_27\n",
     "\n",
-    "### Why Cross Mapping?\n",
+    "### Cross Mapping\n",
     "In a system $Y=f(X,Y)$, cross Mapping means given the points on the manifold of one variable $M_y$, we look for the corresponding points on $M_x$, i.e. points at the same time $t$.\n",
     "\n",
     "If $X$ causes $Y$, then we say some information about $X$ gets stored in $Y$. If this is the case, then we can predict the values of $X$ given values from $Y$. Note that this sounds counter intuitive. We contrast this with Granger causality which says that $X$ causes $Y$ if we can predict $Y$ better given $X$.\n",
@@ -96,7 +96,7 @@
     "\n",
     "If we use the analogy that manifolds are \"summaries\" of system, we can think of $M_x$ and $M_y$ as summaries of $X$ and $Y$. If $X$ causes $Y$, information from $X$ gets embedded in $Y$. It follows that we can then use $M_x$ to predict $Y$, i.e. $\\hat{Y}|M_x$. Finally, we can use the \"accuracy\" (MAE, MSE, or correlation) of this prediction as our metric for causality. \n",
     "\n",
-    "### Why Convergence?\n",
+    "### Convergence\n",
     "Convergence in CCM means that for variables with causalities, the longer our observation period (or more data we gather), the better we can predict one variable using the other. In the attractor example above, we can imagine the attractor will get denser as time goes on since our system will eventually fill in some of the gaps in the manifold. Stated another way, more defined manifolds imply that for variables with causalities, we can expect the accuracy of $\\hat{Y}|M_x$ to improve. If two variables have no causal link, then improving their manifolds will not translate to improvement in predictions.\n",
     "\n",
     "Cross mapping and convergence are two requirements in the CCM framework that we need to satisfy to consider the possiblity of causality between variables."