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Notebook[{

Cell[CellGroupData[{
Cell["Cluster Analysis", "Subtitle"],

Cell[BoxData[
    \(Needs["\<Statistics`MultiDescriptiveStatistics`\>"]\)], "Input"],

Cell["\<\
This package is needed for the data: in general, the principal \
components of the original data are used in this  clustering procedure rather \
than the data itself.\
\>", "Text"],

Cell["\<\
The main function is recursive, so special cases are included to \
avoid infinite recursion\
\>", "Text"],

Cell[BoxData[
    \(clusterData::usage = 
      "\<clusterData clusters data comprising points in multidimensional \
space into clusters of radius no greater than threshold.\>"\)], "Input"],

Cell[BoxData[
    \(clusterData[currentclusters : {__?MatrixQ}, data_?MatrixQ, 
          threshold_Real] /; 
        \((\((Length[data] > 1)\)\  && \ 
            \((And\ @@\ 
                \((\ Thread[
                    \((\(\(Dimensions[#]\)\[LeftDoubleBracket]2
                              \[RightDoubleBracket]\ &\)\ /@\ currentclusters)
                        \) == \(Dimensions[data]\)\[LeftDoubleBracket]2
                        \[RightDoubleBracket]\ ])\)\ )\)\ )\) := \n\t
      clusterData[updateclusters[currentclusters, First[data], threshold], 
        Rest[data], threshold]\)], "Input"],

Cell[BoxData[
    \(clusterData[currentclusters : {__?MatrixQ}, data : {_?VectorQ}, 
          threshold_Real]\  /; 
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            \((\ Thread[
                \((\(\(Dimensions[#]\)\[LeftDoubleBracket]2
                          \[RightDoubleBracket]\ &\)\ /@\ currentclusters)
                    \) == \(Dimensions[data]\)\[LeftDoubleBracket]2
                    \[RightDoubleBracket]\ ])\)\ )\)\  := \n\t
      updateclusters[currentclusters, First[data], threshold]\)], "Input"],

Cell[BoxData[
    \(clusterData[currentclusters : {__?MatrixQ}, {{}}, threshold_Real] /; 
        \((Equal\ @@
            \((\ \(\((
                    \(Dimensions[#]\)\[LeftDoubleBracket]2
                      \[RightDoubleBracket])\)\ &\)\ /@\ clusters)\))\) := \ 
      currentclusters\)], "Input"],

Cell["\<\
This is the meat of the program.  The two main versions of \
clusterData (other than the \"empty points\" one) call updateclusters as the \
means of allocating each new point.\
\>", "Text"],

Cell[BoxData[
    \(updateclusters::usage = 
      "\<updateclusters is an auxiliary function that determines which \
cluster to place a new point into.\>"\)], "Input"],

Cell[BoxData[
    \(\(updateclusters[currentclusters : {__?MatrixQ}, newpoint_?VectorQ, 
          threshold_Real] /; \n\t
        \((And\ @@\ 
            \((\ Thread[
                \((\(\((\(Dimensions[#]\)\[LeftDoubleBracket]2
                            \[RightDoubleBracket])\)\ &\)\ /@\ 
                      currentclusters)\) == Length[newpoint]\ ])\))\) := \n\t
      With[{centroids = 
            Sqrt[Apply[Plus, \ 
                \((\(\((\((First[#] - newpoint)\)^2)\)&\)\ /@currentclusters)
                  \), {1}]]}, \n\t
        With[{closest = 
              \(Flatten[Position[centroids, Min[centroids]]]
                  \)\[LeftDoubleBracket]1\[RightDoubleBracket]}, \n\t\t
          If[centroids\[LeftDoubleBracket]closest\[RightDoubleBracket] < 
              threshold, \n\t\t\t
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                  \[RightDoubleBracket] == 1, \n\t\t\t\t
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              Insert[currentclusters, 
                newpoint, {closest, 
                  Length[currentclusters\[LeftDoubleBracket]closest
                        \[RightDoubleBracket]] + 1}]], 
            Join[currentclusters, {{newpoint}}]]\ ]]\ \)\)], "Input"],

Cell["\<\
These are two information statistics that are intended to help \
determine the best number of clusters. 
clusterLambdaStatistic is the ratio of the determinant of the matrix of sums \
of sums of squares and crossproducts of each cluster, to the determinant of \
the matrix of the sums of squares and cross products for the whole dataset.

Think of this as the ratio of the sum of the variations within each cluster, \
to the total variation in the data set.  Smaller numbers are better.

MarriotClusterStatistic trades off parsimony for explanatory power, being the \
product of the square of the number of clusters, and the determinant of the \
matrix of sums of sums of squares and crossproducts of each cluster.

Think of this as the product of the total within-cluster variation and the \
square of the number of clusters.  The more clusters, the less variation \
within each of them, but this is traded off against the number of clusters.  \
Again, the smaller the value of the statistic, the better.\
\>", "Text"],

Cell[BoxData[
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        \((Equal\ @@\ 
            \((\(\((\(Dimensions[#]\)\[LeftDoubleBracket]2
                      \[RightDoubleBracket])\)\ &\)\ /@\ clusters)\))\) := \n
      \tWith[{joined = Join@@\ clusters}, \n\t\t
        Det[Plus\ @@
              \((\(Apply[Plus, 
                      Apply\ [Plus, Outer[Times, Transpose[#], #], {1}], {1}]&
                    \)\ /@\ clusters)\)]/\n\t\t\t\t
          Det[Apply[Plus, 
              Apply\ [Plus, 
                Outer[Times, Transpose[\((joined)\)], \((joined)\)], {1}], {
                1}]]\ ]\)], "Input"],

Cell[BoxData[
    \(MarriotClusterStatistic[clusters : {__?MatrixQ}] /; 
        \((Equal\ @@\ 
            \((\(\((\(Dimensions[#]\)\[LeftDoubleBracket]2
                      \[RightDoubleBracket])\)\ &\)\ /@\ clusters)\))\) := \n
      \t\t\tLength[clusters]^2*
        Det[Plus\ @@
            \((\(Apply[Plus, 
                    Apply\ [Plus, Outer[Times, Transpose[#], #], {1}], {1}]&
                  \)\ /@\ clusters)\)]\)], "Input"]
}, Open  ]]
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]


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