text corrections (#147)

Author: cwehmeyer

notebooks/08-common-problems.ipynb

@@ -18,7 +18,10 @@
     "- you can find the full documentation at [PyEMMA.org](http://www.pyemma.org).\n",
     "---\n",
     "\n",
-    "Most problems arise from bad sampling combined with a poor discretization. For estimating a Markov model, it is required to have a connected data set, i.e. we must have observed each process we want to describe in both directions. PyEMMA checks if this requirement is fulfilled, however in certain situations this might be less obvious."
+    "Most problems in Markov modeling of MD data arise from bad sampling combined with a poor discretization.\n",
+    "For estimating a Markov model, it is required to have a connected data set,\n",
+    "i.e., we must have observed each process we want to describe in both directions.\n",
+    "PyEMMA checks if this requirement is fulfilled but, however, in certain situations this might be less obvious."
    ]
   },
   {
@@ -40,7 +43,8 @@
    "source": [
     "## Case 1: preprocessed, two-dimensional data (toy model)\n",
     "### well-sampled double-well potential\n",
-    "Let's again have a look at the double-well potential. Since we are only interested in the problematic situations here, we will simplify our data a bit and work with a 1D projection."
+    "Let's again have a look at the double-well potential.\n",
+    "Since we are only interested in the problematic situations here, we will simplify our data a bit and work with a 1D projection."
    ]
   },
   {
@@ -97,7 +101,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We visualize the histogram along with the first part of the trajectory (left panel) and the MSM implied timescales (right panel):"
+    "As a reference, we visualize the histogram of this well-sampled trajectory along with the first $200$ steps (left panel) and the MSM implied timescales (right panel):"
    ]
   },
   {
@@ -122,13 +126,19 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We see a nice, reversibly connected trajectory. That means we have sampled transitions between the basins in both directions that are correctly resolved by the discretization. As we see from the almost perfect overlay of discrete and continuous trajectory, nearly no discretization error is made. \n",
+    "We see a nice, reversibly connected trajectory.\n",
+    "That means we have sampled transitions between the basins in both directions that are correctly resolved by the discretization.\n",
+    "As we see from the almost perfect overlay of discrete and continuous trajectory, nearly no discretization error is made. \n",
     "\n",
     "###  irreversibly connected double-well trajectories\n",
     "\n",
-    "In MD simulations, we often face the problem that a process is sampled only in one direction. For example, consider protein-protein binding. The unbinding might take on the order of seconds to minutes and is thus difficult to sample. We will have a look what happens with the MSM in this case. \n",
+    "In MD simulations, we often face the problem that a process is sampled only in one direction.\n",
+    "For example, consider protein-protein binding.\n",
+    "The unbinding might take on the order of seconds to minutes and is thus difficult to sample.\n",
+    "We will have a look what happens with the MSM in this case. \n",
     "\n",
-    "Our example are two trajectories sampled from a double-well potential, each started in a different basin. They will be color coded."
+    "Our example are two trajectories sampled from a double-well potential, each started in a different basin.\n",
+    "They will be color coded."
    ]
   },
   {