{"text_format":"1","workbook":"xlsx","n_sheets":2,"sheets":["Animation","Calculation"],"not_rendered":{"charts":14,"images":1,"macros":1,"external_links":0,"defined_names":0,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"Animation","index":0,"n_rows":75,"n_cols":18,"rows_kept":75,"cut":false,"hidden":false,"merged":[[39,0,40,7],[41,0,43,7],[13,0,15,2],[8,5,8,6],[0,0,1,10],[2,0,3,10],[13,3,15,5],[13,6,15,6],[13,7,15,14],[5,0,5,10],[4,0,4,10]]}
["EM or \"soft\" clustering.  Start by guessing 3 cluster centers.  For each data point, figure out the probability that it came from each of the cluster centers (the closest center is most likely).  Then recompute the centers based on the data points fractionally assigned to them."]
[]
["Note: The Viterbi approximation to this algorithm (easier to implement and converges in a few iterations) is called \"k-means\" or \"hard\" clustering.  One can get the same effect by using a small variance."]
[]
["To watch the progress of the algorithm, use the scrollbar between the graphs."]
["To see the underlying calculations, click the \"Calculation\" worksheet tab at the bottom of this worksheet."]
[]
[null,"Initial clusters (centers and variances)"]
[null,null,"x","y","Variance","Reestimate variance?"]
[null,"Red",2,1,1,false,"Warning: Setting this to TRUE is a nice idea but tends to trap us in local maxima"]
[null,"Green",2,2,{"$f":"E10","v":1}]
[null,"Blue",-1,5,{"$f":"E10","v":1}]
[]
["Switch to other window (Window menu) for other views.  Scrollbar may not work in all views.",null,null," ITERATION:",null,null,0,"If you need more than 20 iterations, use Edit / Copy and Edit / PasteSpecial / Values to copy the values from the \"Final clusters\" table (below) over the \"Initial clusters\" table (above).  Each repetition of the PasteSpecial command will get you another 6 iterations."]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"Data points",null,"Posterior probabilities (curr. iteration)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"x","y","p(red|point)","p(green|point)","p(blue|point)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,3,4,{"$f":"OFFSET(Calculation!I17,0,$G$14*11)","v":7.5684807429077489E-2},{"$f":"OFFSET(Calculation!J17,0,$G$14*11)","v":0.92202970942174223},{"$f":"OFFSET(Calculation!K17,0,$G$14*11)","v":2.2854831491801947E-3}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,5,2,{"$f":"OFFSET(Calculation!I18,0,$G$14*11)","v":0.3775406652190435},{"$f":"OFFSET(Calculation!J18,0,$G$14*11)","v":0.62245932530091364},{"$f":"OFFSET(Calculation!K18,0,$G$14*11)","v":9.4800429162403711E-9}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,1,1.5,{"$f":"OFFSET(Calculation!I19,0,$G$14*11)","v":0.49986176713468167},{"$f":"OFFSET(Calculation!J19,0,$G$14*11)","v":0.49986176713468167},{"$f":"OFFSET(Calculation!K19,0,$G$14*11)","v":2.7646573063666899E-4}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,1,0,{"$f":"OFFSET(Calculation!I20,0,$G$14*11)","v":0.81757355980719759},{"$f":"OFFSET(Calculation!J20,0,$G$14*11)","v":0.18242531933290182},{"$f":"OFFSET(Calculation!K20,0,$G$14*11)","v":1.1208599006050614E-6}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,{"$f":"OFFSET(Calculation!I21,0,$G$14*11)","v":0.81755607047358336},{"$f":"OFFSET(Calculation!J21,0,$G$14*11)","v":0.18242141693509159},{"$f":"OFFSET(Calculation!K21,0,$G$14*11)","v":2.2512591325174201E-5}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,2,3.5,{"$f":"OFFSET(Calculation!I22,0,$G$14*11)","v":0.11804785075397697},{"$f":"OFFSET(Calculation!J22,0,$G$14*11)","v":0.87226219157932872},{"$f":"OFFSET(Calculation!K22,0,$G$14*11)","v":9.6899576666944019E-3}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,5,6,{"$f":"OFFSET(Calculation!I23,0,$G$14*11)","v":1.0960073809423017E-2},{"$f":"OFFSET(Calculation!J23,0,$G$14*11)","v":0.98659440316624181},{"$f":"OFFSET(Calculation!K23,0,$G$14*11)","v":2.4455230243351773E-3}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,6,2,{"$f":"OFFSET(Calculation!I24,0,$G$14*11)","v":0.37754066861995245},{"$f":"OFFSET(Calculation!J24,0,$G$14*11)","v":0.62245933090806405},{"$f":"OFFSET(Calculation!K24,0,$G$14*11)","v":4.7198354905279915E-10}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,1,3,{"$f":"OFFSET(Calculation!I25,0,$G$14*11)","v":0.17529039214003667},{"$f":"OFFSET(Calculation!J25,0,$G$14*11)","v":0.78559703458927599},{"$f":"OFFSET(Calculation!K25,0,$G$14*11)","v":3.9112573270687429E-2}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,5,0,{"$f":"OFFSET(Calculation!I26,0,$G$14*11)","v":0.81757447618801293},{"$f":"OFFSET(Calculation!J26,0,$G$14*11)","v":0.18242552380510027},{"$f":"OFFSET(Calculation!K26,0,$G$14*11)","v":6.8868089667484473E-12}]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
["In these graphs, the three solid circles represent the cluster centroids.  (Their areas are the cluster variances, so their radii represent about 0.564 standard deviations.)  "]
[]
["The white circles of area 1 are the data points. Each data point carries hollow circles in red, green, and blue, whose areas are proportional to p(red | point), p(green | point), and p(blue | point)."]
[]
[]
[]
[null,"Current clusters",null,null,null,null,"Final clusters"]
[null,null,"x","y","Variance",null,null,"x","y","Variance"]
[null,"Red",{"$f":"OFFSET(C55,$G$14,0)","v":2},{"$f":"OFFSET(D55,$G$14,0)","v":1},{"$f":"OFFSET(E55,$G$14,0)","v":1},null,"Red",{"$f":"Calculation!HN7","v":0.76880911919303829},{"$f":"Calculation!HO7","v":1.1087223363005023},{"$f":"Calculation!HP7","v":1}]
[null,"Green",{"$f":"OFFSET(F55,$G$14,0)","v":2},{"$f":"OFFSET(G55,$G$14,0)","v":2},{"$f":"OFFSET(H55,$G$14,0)","v":1},null,"Green",{"$f":"Calculation!HN8","v":5.3311710873039564},{"$f":"Calculation!HO8","v":1.3329229818827504},{"$f":"Calculation!HP8","v":1}]
[null,"Blue",{"$f":"OFFSET(I55,$G$14,0)","v":-1},{"$f":"OFFSET(J55,$G$14,0)","v":5},{"$f":"OFFSET(K55,$G$14,0)","v":1},null,"Blue",{"$f":"Calculation!HN9","v":3.2707617309702215},{"$f":"Calculation!HO9","v":4.4437520092340739},{"$f":"Calculation!HP9","v":1}]
[]
[null,"Evolution of clusters over time"]
[null,null,"Red",null,null,"Green",null,null,"Blue"]
[null,"Iteration","x","y","Variance","x","y","Variance","x","y","Variance"]
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["The underlying calculations for EM or \"soft\" clustering.  To change the data or the initial clusters, or view the results, click the Animation worksheet tab at the bottom of this worksheet.",null,null,null,null,null,null,null,null,null,null,"In the next iteration, we recompute the cluster centers.  To recompute the (x,y) center of the red cluster, just take (average x-coordinate of red points, average y-coordinate of red points).  These are weighted averages; a point contributes to the average to the degree that it is red."]
[]
["This is the first iteration.  The initial parameter table, below, is copied from the Animation worksheet."]
[]
["Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances",null,null,null,null,null,null,null,null,null,null,"Cluster Centers and Variances"]
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[]
[]
[]
[]
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[]
[]
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[0," ITERATIONS",null,null,"In these graphs, the three solid circles represent the cluster centroids.  (Their areas are the cluster variances, so the diameter is about 1.1 standard deviations.)  The white circles of area 1 are the data points. Each data point carries hollow circles in red, green, and blue indicating how strongly it is associated with each of those clusters.  The area of the hollow red circle is proportional to p(point | red) or p(red | point).",null,null,null,null,null,null,{"$f":"A52+1","v":1}," ITERATION",null,null,"In these graphs, each cluster's center has been updated to be the centroid of  the data points (or fractions of data points) that were placed in that cluster in the previous iteration.",null,null,null,null,null,null,{"$f":"L52+1","v":2}," ITERATIONS",null,null,null,null,null,null,null,null,null,{"$f":"W52+1","v":3}," ITERATIONS",null,null,null,null,null,null,null,null,null,{"$f":"AH52+1","v":4}," ITERATIONS",null,null,null,null,null,null,null,null,null,{"$f":"AS52+1","v":5}," ITERATIONS"]