added reference to manuscript regarding ITS convergence (#147)

Author: cwehmeyer

notebooks/03-msm-estimation-and-validation.ipynb

@@ -85,7 +85,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The first step after obtaining the discretized dynamics is finding a suitable lag time. The systematic approach is to estimate MSMs at various lag times and observe how the implied timescales (ITSs) of these models behave. To this aim, PyEMMA provides the `its()` function which we use to track the first three (`nits=3`) implied timescales:"
+    "The first step after obtaining the discretized dynamics is finding a suitable lag time.\n",
+    "The systematic approach is to estimate MSMs at various lag times and observe how the implied timescales (ITSs) of these models behave.\n",
+    "In particular, we are looking for lag time ranges in which the implied timescales are constant\n",
+    "(i.e., lag time independent as described in the manuscript in Section 2.1).\n",
+    "To this aim, PyEMMA provides the `its()` function which we use to track the first three (`nits=3`) implied timescales:"
    ]
   },
   {
@@ -117,11 +121,20 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The above plot tells us that there is one resolved process with an ITS of approximately $8.5$ steps (blue) which is largely invariant to the MSM lag time. The other two ITSs (green, red) are smaller than the lag time (black line, grey-shaded area); they correspond to processes which are faster than the lag time and, thus, are not resolved.\n",
+    "The above plot tells us that there is one resolved process with an ITS of approximately $8.5$ steps (blue) which is largely invariant to the MSM lag time.\n",
+    "The other two ITSs (green, red) are smaller than the lag time (black line, grey-shaded area);\n",
+    "they correspond to processes which are faster than the lag time and, thus, are not resolved.\n",
+    "Since the implied timescales are, like the corresponding eigenvalues, sorted in decreasing order, we know that all other remaining processes must be even faster.\n",
     "\n",
-    "As MSMs tend to underestimate the true ITSs, we are looking for a converged maximum in the ITS plot. In our case, any lag time before the slow process (blue line) crosses the lag time threshold (black line) would work. To maximize the kinetic resolution, we choose the lag time $1$ step.\n",
+    "As MSMs tend to underestimate the true ITSs, we are looking for a converged maximum in the ITS plot.\n",
+    "In our case, any lag time before the slow process (blue line) crosses the lag time threshold (black line) would work.\n",
+    "To maximize the kinetic resolution, we choose the lag time $1$ step.\n",
     "\n",
-    "To see whether our model satisfies Markovianity, we perform (and visualize) a Chapman-Kolmogorow (CK) test. Since we aim at modeling the dynamics between metastable states rather than between microstates, this will be conducted in the space of metastable states. The latter are identified automatically using PCCA++ (which is explained in a later notebook). We usually choose the number of metastable states according to the implied timescales plot by identifying a gap between the ITS. For a single process, we can assume that there are two metastable states between which the process occurs."
+    "To see whether our model satisfies Markovianity, we perform (and visualize) a Chapman-Kolmogorow (CK) test.\n",
+    "Since we aim at modeling the dynamics between metastable states rather than between microstates, this will be conducted in the space of metastable states.\n",
+    "The latter are identified automatically using PCCA++ (which is explained in [Notebook 05 📓](05-pcca-tpt.ipynb)).\n",
+    "We usually choose the number of metastable states according to the implied timescales plot by identifying a gap between the ITS.\n",
+    "For a single process, we can assume that there are two metastable states between which the process occurs."
    ]
   },
   {