Fixed typos in tutorial

Author: seanlaw

docs/Tutorial_Multidimensional_Motif_Discovery.ipynb

@@ -8,9 +8,9 @@
     "\n",
     "## Finding a Motif in Multidimensional Time Series Data with MSTUMP\n",
     "\n",
-    "This tutorial utilizes the main takeways from the [Matrix Profile VI](https://www.cs.ucr.edu/~eamonn/Motif_Discovery_ICDM.pdf) research paper and requires STUMPY v1.6.1 or newer. Also, the word “dimensionality” is overloaded for multi-dimensional time series since it is often used to refer to both the number of time series and to the number of data points in a subsequence. For clarity, we restrict our use of \"dimensions\" to refer only to the number of time series and not to the number of data points.\n",
+    "This tutorial utilizes the main takeaways from the [Matrix Profile VI](https://www.cs.ucr.edu/~eamonn/Motif_Discovery_ICDM.pdf) research paper and requires STUMPY v1.6.1 or newer. Also, the word “dimensionality” is overloaded for multi-dimensional time series since it is often used to refer to both the number of time series and to the number of data points in a subsequence. For clarity, we restrict our use of \"dimensions\" to refer only to the number of time series and not to the number of data points.\n",
     "\n",
-    "Previously, we had introduced a concept called [time series motifs](https://stumpy.readthedocs.io/en/latest/Tutorial_STUMPY_Basics.html), which are conserved patterns found within a 1-dimensional time series, $T$, that can be discovered by computing its [matrix profile](https://stumpy.readthedocs.io/en/latest/Tutorial_The_Matrix_Profile.html) using STUMPY. This process of computing a matrix profile with one time series is commonly known as a \"self-join\" since the subsequences within time series $T$ are only being compared with itself. Since the first 1-dimensional motif discovery algorithm was introduced in 2002, a lot of effort has been made to generalize motif-finding to the multi-dimensional case but producing multi-dimensional matrix profiles are computationally expensive and so extra care must be taken to minimize the added time complexity. Also, while it may be tempting to find motifs in all avaialble dimensions (i.e., a motif must exist in all dimensions and occur simultaneously), it has been shown that this rarely produces meaningful motifs except in the most contrived situations. Instead, given a set of time series dimensions, we should filter them down to a subset of \"useful\" dimensions before assigning a subsequence as a motif. For example, take a look at this motion capture of a boxer throwing some punches:"
+    "Previously, we had introduced a concept called [time series motifs](https://stumpy.readthedocs.io/en/latest/Tutorial_STUMPY_Basics.html), which are conserved patterns found within a 1-dimensional time series, $T$, that can be discovered by computing its [matrix profile](https://stumpy.readthedocs.io/en/latest/Tutorial_The_Matrix_Profile.html) using STUMPY. This process of computing a matrix profile with one time series is commonly known as a \"self-join\" since the subsequences within time series $T$ are only being compared with itself. Since the first 1-dimensional motif discovery algorithm was introduced in 2002, a lot of effort has been made to generalize motif-finding to the multi-dimensional case but producing multi-dimensional matrix profiles are computationally expensive and so extra care must be taken to minimize the added time complexity. Also, while it may be tempting to find motifs in all available dimensions (i.e., a motif must exist in all dimensions and occur simultaneously), it has been shown that this rarely produces meaningful motifs except in the most contrived situations. Instead, given a set of time series dimensions, we should filter them down to a subset of \"useful\" dimensions before assigning a subsequence as a motif. For example, take a look at this motion capture of a boxer throwing some punches:"
    ]
   },
   {
@@ -239,7 +239,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "And when we plot the raw times series again (below) along with their independently discovered motifs (thick red lines), we can correctly match the visually obvious motif pairs in `T1` and `T2` starting near locations `150` and `350` (dotted vertical lines). Notice that these two motifs aren't perfectly aligned in time (i.e., they aren't occuring simultaneously) but they are reasonably close to each and their motif pair values are `1.1` and `1.0`, respectively. This is a great start!"
+    "And when we plot the raw times series again (below) along with their independently discovered motifs (thick red lines), we can correctly match the visually obvious motif pairs in `T1` and `T2` starting near locations `150` and `350` (dotted vertical lines). Notice that these two motifs aren't perfectly aligned in time (i.e., they aren't occurring simultaneously) but they are reasonably close to each and their motif pair values are `1.1` and `1.0`, respectively. This is a great start!"
    ]
   },
   {
@@ -425,7 +425,7 @@
     "\n",
     "Additionally, it may seem counterintuitive, but as demonstrated above, the lower dimensional motif(s) may or may not necessarily be a subset of the higher dimensional motif, since the lower dimensional motif pair could be closer than any subset of dimensions in the higher dimensional motif pair. In general, this is a subtle but important point to keep in mind.\n",
     "\n",
-    "So then how do choose the \"right\" $k$? One straightforward approach is to turn this into a classic elbow/knee finding problem by plotting the minimum matrix profile value in each dimension against $k$ and then you look for the \"turning point\" (i.e., the point of maximum curvature):"
+    "So then how do we choose the \"right\" $k$? One straightforward approach is to turn this into a classic elbow/knee finding problem by plotting the minimum matrix profile value in each dimension against $k$ and then you look for the \"turning point\" (i.e., the point of maximum curvature):"
    ]
   },
   {