mention eigenvector interpretation (#147)

Author: cwehmeyer

notebooks/04-msm-analysis.ipynb

@@ -243,7 +243,9 @@
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    "metadata": {},
    "source": [
-    "We will see further uses of the stationary distribution later. But for now, we continue the analysis of our model by visualizing its (right) eigenvectors. First, we notice that the first right eigenvector is a constant $1$."
+    "We will see further uses of the stationary distribution later.\n",
+    "But for now, we continue the analysis of our model by visualizing its (right) eigenvectors which encode the dynamical processes.\n",
+    "First, we notice that the first right eigenvector is a constant $1$."
    ]
   },
   {
@@ -288,11 +290,15 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The right eigenvectors can be used to visualize the processes governed by the corresponding implied timescales. The first right eigenvector (always) is $(1,\\dots,1)^\\top$ for an MSM transition matrix and it corresponds to the stationary process (infinite implied timescale).\n",
+    "The right eigenvectors can be used to visualize the processes governed by the corresponding implied timescales.\n",
+    "The first right eigenvector (always) is $(1,\\dots,1)^\\top$ for an MSM transition matrix and it corresponds to the stationary process (infinite implied timescale).\n",
     "\n",
-    "The second right eigenvector corresponds to the slowest process; its entries are negative for one group of discrete states and positive for the other group. This tells us that the slowest process happens between these two groups and that the process relaxes on the slowest ITS ($\\approx 8.5$ steps).\n",
+    "The second right eigenvector corresponds to the slowest process;\n",
+    "its entries are negative for one group of discrete states and positive for the other group.\n",
+    "This tells us that the slowest process happens between these two groups and that the process relaxes on the slowest ITS ($\\approx 8.5$ steps).\n",
     "\n",
-    "The third and fourth eigenvectors show a larger spread of values and no clear grouping. In combination with the ITS convergence plot, we can safely assume that these eigenvectors contain just noise and do not indicate any resolved processes.\n",
+    "The third and fourth eigenvectors show a larger spread of values and no clear grouping.\n",
+    "In combination with the ITS convergence plot, we can safely assume that these eigenvectors contain just noise and do not indicate any resolved processes.\n",
     "\n",
     "We then continue to validate our MSM with a CK test for $2$ metastable states which are already indicated by the second right eigenvector."
    ]