k-means クラスタリング
このトピックでは k-means クラスタリングを紹介し、Statistics and Machine Learning Toolbox™ の関数 kmeans を使用してデータ セットに最適なクラスタリング ソリューションを見つける例について説明します。
k-means クラスタリングの紹介
k-means クラスタリングは、分割を行うための方法です。関数 kmeans は、データを互いに排他的な k 個のクラスターに分割し、各観測値が割り当てられたクラスターのインデックスを返します。kmeans は、データ内の各観測値を、空間内のある位置をもつオブジェクトとして処理します。この関数は、各クラスター内のオブジェクトができるだけ互いに近くにあり、他のクラスターのオブジェクトからはできるだけ遠くにあるような分割を探します。距離計量を選択して、データの属性に基づいて kmeans と一緒に使用することができます。多くのクラスタリング手法と同じように、k-means クラスタリングでもクラスタリングを行う前にクラスター数 k を指定しなければなりません。
階層クラスタリングと異なり、k-means クラスタリングは、データ内の観測値の各ペア間の非類似度ではなく、実際の観測値で動作します。また、k-means クラスタリングでは、クラスターの多重レベルの階層ではなく、単一レベルのクラスターを作成します。そのため、k-means クラスタリングは、大量のデータを処理する場合に階層クラスタリングより適していることが多くあります。
k-means の分割内の各クラスターは、メンバー オブジェクトと重心 (または中心) で構成されます。各クラスター内で、kmeans は、重心とクラスターのすべてのメンバー オブジェクトの間の合計距離を最小化します。kmeans は、サポートされている距離計量の重心クラスターを異なる方法で計算します。詳細については、'Distance'を参照してください。
kmeans で使用可能な名前と値のペアの引数を使用して、最小化の詳細を制御できます。たとえば、クラスターの重心の初期値や、アルゴリズムの最大反復回数を指定できます。既定では、kmeans はクラスターの重心を初期化するために k-means++ アルゴリズムを使用し、距離を決定するために 2 乗ユークリッド距離計量を使用します。
k-means クラスタリングを実行するときは、次のベストプラクティスに従ってください。
k のさまざまな値について k-means クラスタリングの解を比較して、自分のデータにとって最適なクラスター数を決定する。
シルエット プロットとシルエット値を調べることでクラスタリングの解を評価する。また、関数
evalclustersを使用して、ギャップ値、シルエット値、Davies-Bouldin インデックス値、Calinski-Harabasz インデックス値などの基準に基づいてクラスタリングの解を評価することもできます。無作為に選択したさまざまな重心からのクラスタリングを複製し、すべての複製の中で距離の総和が最小である解を返す。
k-means クラスタリングを対話的に実行するには、[データのクラスタリング] ライブ エディター タスクを使用します。
k-means クラスタリングの解の比較
この例では、4 次元のデータ セットに対する k-means クラスタリングについて確認します。この例では、シルエット プロットやシルエット値を使用することでデータ セットに適切なクラスター数を決定して、さまざまな k-means クラスタリングの解の結果を分析する方法を示します。また、名前と値のペアの引数 'Replicates' を使用して、可能性のある解を指定された数だけテストして、距離の総和が最小である解を返す方法も示します。
データ セットの読み込み
kmeansdata データ セットを読み込みます。
rng('default') % For reproducibility load('kmeansdata.mat') size(X)
ans = 1×2
560 4
データ セットは 4 次元であり、可視化は容易ではありません。ただし、kmeans を使用してデータ中にグループ構造があるかどうかを調べることができます。
クラスターの作成と間隔の確認
k-means クラスタリングを使用して、データ セットを 3 つのクラスターに分割します。尺度として市街地距離を指定し、既定の k-means++ アルゴリズムをクラスター中心の初期化に使用します。'Display' 名前と値のペアの引数を使用して、その解に最終的な距離の合計を表示します。
[idx3,C,sumdist3] = kmeans(X,3,'Distance','cityblock', ... 'Display','final');
Replicate 1, 7 iterations, total sum of distances = 12.33359.9739211.091610.116811.700510.187913.532612.580711.219112.853510.704410.77138.1630912.171811.45579.8538112.795914.45599.6384511.238811.47769.7562110.123212.32649.889067.9706610.418212.44569.5151813.42537.2736915.41889.27369.8963410.768712.716310.249114.86810.06114.32529.8808910.224410.9798.194249.6843810.12610.37459.6912610.432918.862112.424610.39059.7947712.0569.237947.192279.8355711.795813.549418.02813.513310.538115.215712.89.2484611.52888.301189.2878714.401612.558510.48410.101410.28399.737857.5376310.087512.188111.936816.910110.36532.424443.605043.009376.141930.7777841.275414.437837.683983.299182.974482.164524.141572.512513.691782.439292.260422.138720.3402413.059593.563791.504873.504481.640413.24173.071245.230583.621591.815782.63742.231127.019662.391782.75715.896420.6779552.677142.014591.678420.5832696.274682.599293.145995.006793.620742.74974.378482.172092.514132.438561.897813.279913.415875.648724.072633.611244.724717.301022.696322.746784.426383.89472.534974.733063.241314.133842.646882.442330.889671.519181.458941.612132.854781.571712.325032.15933.76761.06813.353431.085264.871233.935892.697852.955311.566992.421531.331792.126571.121152.591592.476291.079673.813861.357062.856981.336592.617471.469792.846445.400326.266552.010324.869181.624674.274263.272334.068951.366133.987062.402827.719572.692591.583231.749414.244762.257793.045331.817942.349951.317742.891512.062391.841353.024056.257353.894233.212662.465583.087333.196456.468895.055881.578383.829412.847735.021947.242854.409553.820693.416131.554374.686762.595072.783435.921083.788213.724522.426563.828561.592593.265851.944974.448221.95850.9445461.232322.978152.227741.720342.340593.055992.181499.579881.994862.509331.870951.987523.21696.594761.659552.606422.473987.124972.289892.748422.446481.12951.429993.671496.682094.405384.846433.293162.569161.408522.538991.540382.621443.226881.920264.184716.529713.003374.678543.481483.064462.715841.659710.9387373.506092.938462.829382.99012.378763.13991.729052.329211.716221.534693.631311.4264.504781.347640.9666442.715343.752922.991482.456372.344163.321973.001276.299943.210953.82422.618236.897514.000036.634695.061853.944296.593628.401773.720713.118053.489322.658992.872313.632082.225950.8828163.319422.922281.022671.868071.303553.539033.794393.6481.768263.422975.897644.255962.67984.48661.991481.409152.165113.123423.336836.073052.2654.391941.1681.542493.598338.56184.953921.530273.065412.230056.847555.549352.671972.185062.2427911.4684.366762.229123.181826.636053.901067.207545.533744.548713.004341.989721.254471.496712.088621.881631.692231.305738.704622.331792.570672.527942.617058.464034.935452.805255.110151.898143.375360.9778493.264722.864252.684771.436129.013262.44463.983841.704511.420123.131071.901315.424463.892031.229182.825833.444072.051535.60644.720963.602781.873483.863843.974692.436382.067874.07954.13224.609053.285962.745673.44613.607193.481222.576364.811530.891972.507010.4076293.235464.14985.64743.400672.942361.605714.800314.289742.029511.808715.022014.119913.653584.235262.408924.747940.9184210.9380626.022292.149263.952862.65231.473832.904876.019271.437680.9252253.11951.367062.306645.604272.205082.644647.931324.286935.577791.3393.511392.408346.755854.839241.1444.056041.837293.318472.374675.648681.755874.691081.605943.165321.989192.937344.754833.094567.388235.82343.942322.582552.291162.416673.35712.277261.76032.701683.688493.019361.526613.023412.175051.635656.544232.603592.085315.107912.402074.905294.190414.307886.729182.112541.983072.47596.588732.781192.230742.537285.03830.7022944.95294.080833.748713.645825.11992.633530.7536962.921783.544072.691234.159253.544612.066853.36262.018013.104162.199141.084125.693791.636896.090332.29823.84711.348190.6705253.620344.00251.719842.640425.939321.99731.783964.767265.337361.984899.134861.292222.713231.880253.772266.164281.925753.527832.422754.910284.647242.91052.572332.644713.50167. Best total sum of distances = 2459.98
idx3 には、X の各行に対するクラスター割り当てを示すクラスター インデックスが含まれています。生成されたクラスターが十分に分離されているかどうかを確認するには、シルエット プロットを作成できます。
シルエット プロットは、1 個のクラスター中の各点が近隣のクラスター中の点にどれくらい接近しているかについて基準を表示します。この尺度の範囲は、(隣接するクラスターから非常に遠い点を示す) 1 から、(いずれかのクラスター内に含まれているかどうかがあいまいな点を示す) 0 を通り、(適切でないクラスターに割り当てられた確率が高い点を示す) –1 までです。silhouetteは、その 1 番目の出力でこれらの値を返します。
idx3 からシルエット プロットを作成します。k-means クラスタリングが差の絶対値の総和に基づくことを指示するために、距離計量に対して 'cityblock' を指定します。
[silh3,h] = silhouette(X,idx3,'cityblock'); xlabel('Silhouette Value') ylabel('Cluster')
シルエット プロットは、2 番目のクラスターの大部分の点が (0.6 よりも大きい) 大きなシルエット値をもっており、そのクラスターが近隣のクラスターからある程度分離されていることを示しています。一方、3 番目のクラスターにはシルエット値の低い点が多く含まれており、また 1 番目と 3 番目のクラスターには負の値をもつ点がいくつか含まれていることから、これら 2 つのクラスターが十分には分離されていないことがわかります。
kmeans がデータのより適切なグループ化を見つけることができるかどうか確認するには、クラスターの数を 4 に増やします。'Display' 名前と値のペアの引数を使用して、各反復に関する情報を表示します。
idx4 = kmeans(X,4,'Distance','cityblock','Display','iter');
iter phase num sum
1 1 560 2.50747
8.304961e-01 2.110459e+00 2.735457e+00 2.46836
2.036070e+00 3.770269e+00 2.866086e+00 1.73312
3.063747e+00 1.697008e+00 1.218750e+00 5.63006
2.345771e+00 2.756564e+00 2.958386e+00 2.96989
4.629916e+00 2.612240e+00 1.476521e+00 2.3564
3.561821e+00 1.910498e+00 3.722177e+00 7.8805
5.601491e+00 3.110286e+00 2.683279e+00 2.05194
3.662951e+00 6.267081e+00 6.011975e+00 3.36686
5.494719e+00 3.108410e+00 3.386063e+00 2.76565
5.105645e+00 2.498318e+00 4.562881e+00 1.6913
5.627059e+00 1.306599e+00 2.944986e+00 4.14043
2.716523e+00 2.287602e+00 8.619236e-01 3.16994
9.099832e+00 2.662265e+00 4.798414e+00 1.07115
2.230025e+00 2.151256e+00 3.884083e+00 3.10084
1.614047e+00 3.367665e+00 8.265723e+00 3.75101
9.444974e-01 5.453364e+00 2.618340e+00 1.2037
2.351116e+00 3.856289e+00 2.856166e+00 4.63925
3.351290e+00 1.401212e+00 1.637554e+00 0.794629
1.079287e+00 4.871440e+00 2.579814e+00 2.42574
1.955302e+00 7.147768e+00 1.649958e+00 2.42486
3.605452e+00 2.964771e+00 6.141513e+00 0.869295
1.264563e+00 4.527010e+00 7.683559e+00 3.22126
3.052403e+00 2.164106e+00 4.063642e+00 2.59043
3.692193e+00 2.494324e+00 2.210258e+00 2.13913
2.852117e-01 3.059171e+00 3.564207e+00 1.47113
3.504066e+00 1.695442e+00 3.241280e+00 3.12799
5.230998e+00 3.621177e+00 1.826625e+00 2.54821
2.186516e+00 7.019246e+00 2.380939e+00 2.75668
5.896008e+00 8.113252e-01 2.699909e+00 1.96998
1.600499e+00 6.078903e-01 6.275097e+00 2.73266
3.223911e+00 5.007212e+00 3.621158e+00 2.74928
4.378065e+00 2.250014e+00 2.513717e+00 2.39396
1.975736e+00 3.279492e+00 3.415449e+00 5.6483
4.072211e+00 3.610826e+00 4.769311e+00 7.30144
2.730053e+00 2.824707e+00 4.371346e+00 3.89428
2.445791e+00 4.732644e+00 3.240896e+00 4.13426
2.568956e+00 2.308963e+00 8.346409e-01 1.55292
1.592312e+00 1.645866e+00 2.721409e+00 1.70508
2.414212e+00 2.237220e+00 3.801339e+00 1.03436
3.353847e+00 9.960760e-01 4.782047e+00 3.84671
2.698268e+00 2.877385e+00 1.566577e+00 2.37692
1.386233e+00 2.048648e+00 1.194570e+00 2.5024
2.475873e+00 1.045929e+00 3.735935e+00 1.30203
2.856561e+00 1.425771e+00 2.695392e+00 1.3806
2.846854e+00 5.399904e+00 6.266968e+00 1.93239
4.869597e+00 1.569636e+00 4.274673e+00 3.27191
4.146875e+00 1.365716e+00 4.031667e+00 2.34779
7.719155e+00 2.658849e+00 1.494046e+00 1.71567
4.299784e+00 2.213185e+00 3.045752e+00 1.90713
2.294917e+00 1.262713e+00 2.936118e+00 2.06197
1.875087e+00 3.101973e+00 6.257766e+00 3.89381
3.213079e+00 2.554766e+00 3.176509e+00 3.10727
6.468477e+00 5.056296e+00 1.633412e+00 3.82899
2.792700e+00 5.022362e+00 7.243271e+00 4.40914
3.821107e+00 3.416549e+00 1.554790e+00 4.60884
2.517145e+00 2.794271e+00 5.920664e+00 3.83281
3.591145e+00 2.460295e+00 3.828974e+00 1.54799
3.221248e+00 1.945387e+00 4.447802e+00 2.0031
9.891492e-01 1.098953e+00 3.033176e+00 2.22816
1.853709e+00 2.429771e+00 2.978063e+00 2.23651
9.579459e+00 2.028599e+00 2.509751e+00 1.86011
1.909599e+00 3.183159e+00 6.595179e+00 1.71458
2.572681e+00 2.551908e+00 7.125384e+00 2.24529
2.881790e+00 2.501508e+00 1.129920e+00 1.46746
3.671077e+00 6.682502e+00 4.405794e+00 4.84685
3.293577e+00 2.514132e+00 1.408936e+00 2.5394
1.451196e+00 2.632279e+00 3.271479e+00 2.05364
4.184291e+00 6.530124e+00 3.058399e+00 4.67896
3.481900e+00 3.019852e+00 2.770868e+00 1.79308
9.049999e-01 3.451058e+00 2.938039e+00 3.05574
4.055653e+00 2.688796e+00 1.887670e+00 1.1077
1.076977e+00 1.678056e+00 1.761055e+00 2.37755
2.678227e+00 4.732665e+00 1.993789e+00 1.61402
2.755025e+00 2.634154e+00 2.227330e+00 1.71136
3.088988e+00 4.575729e+00 1.935710e+00 6.33811
2.000878e+00 3.597843e+00 2.450322e+00 8.15126
2.746277e+00 5.380933e+00 3.814996e+00 2.69053
7.847375e+00 7.148015e+00 2.466952e+00 3.156
3.261437e+00 3.724547e+00 1.796477e+00 2.7035
1.330813e+00 6.707674e-01 2.695014e+00 4.12267
4.493791e-01 1.288885e+00 2.367579e+00 2.2868
4.020752e+00 2.583969e+00 1.167962e+00 3.38481
4.643887e+00 3.003726e+00 1.427571e+00 3.23285
2.695786e+00 2.662904e+00 1.937228e+00 3.84411
4.590588e+00 4.819293e+00 1.391349e+00 5.6457
2.271765e+00 9.566677e-01 2.532769e+00 7.30805
4.347703e+00 1.380990e+00 1.865818e+00 1.61294
8.101306e+00 4.297118e+00 1.574103e+00 2.13363
2.080578e+00 1.181923e+01 4.908753e+00 3.29467
4.435576e+00 5.382291e+00 2.658990e+00 5.95379
4.279986e+00 5.802465e+00 1.750587e+00 3.24348
7.583504e-01 1.534875e+00 8.648290e-01 0.800769
2.060692e+00 2.369757e+00 7.450861e+00 2.48238
1.506635e+00 1.462386e+00 3.138898e+00 7.21028
3.681698e+00 1.739696e+00 3.856392e+00 1.85845
2.123130e+00 7.247386e-01 2.247476e+00 1.79869
1.935554e+00 1.915771e+00 7.759504e+00 2.97069
2.730087e+00 1.317219e+00 1.380430e+00 1.87731
2.966867e+00 6.678214e+00 5.145785e+00 0.976481
3.135821e+00 3.405912e+00 2.055097e+00 4.35264
5.974719e+00 4.856537e+00 2.238908e+00 2.61161
2.720935e+00 1.672036e+00 2.295760e+00 2.82574
5.385957e+00 3.356817e+00 3.058072e+00 2.00609
2.192349e+00 2.354963e+00 2.460035e+00 1.51233
3.745978e+00 1.118331e+00 1.254780e+00 1.02624
3.463346e+00 4.111640e+00 6.901155e+00 4.65443
2.306591e+00 9.567764e-01 4.227085e+00 5.41763
1.334880e+00 1.821775e+00 4.901148e+00 3.99905
3.717171e+00 4.171673e+00 2.229404e+00 4.6099
7.470399e-01 1.058921e+00 9.926443e+00 2.21285
4.073716e+00 2.773153e+00 1.594685e+00 3.17683
6.140131e+00 1.748451e+00 9.315060e-01 2.84753
1.303468e+00 2.034670e+00 5.483409e+00 2.14149
2.708225e+00 8.052183e+00 4.407786e+00 5.30582
1.610961e+00 3.390528e+00 2.344753e+00 6.635
4.718381e+00 1.282042e+00 3.935182e+00 1.5148
3.590432e+00 2.460955e+00 5.527822e+00 1.87673
4.570219e+00 1.726796e+00 3.286179e+00 1.86833
2.665379e+00 4.633968e+00 3.215418e+00 11.8457
5.702546e+00 4.063183e+00 2.518959e+00 2.30423
2.295809e+00 3.236241e+00 2.264191e+00 1.84782
2.765270e+00 3.550448e+00 3.140222e+00 1.40575
3.295379e+00 1.903086e+00 1.548134e+00 6.66509
2.774972e+00 2.172827e+00 4.987054e+00 2.26403
4.784434e+00 4.069548e+00 4.428735e+00 11.1005
1.840577e+00 1.711109e+00 2.153413e+00 10.6973
2.868710e+00 2.109883e+00 2.741946e+00 5.15916
8.854788e-01 4.832042e+00 4.201692e+00 3.62786
3.766674e+00 4.999042e+00 2.569938e+00 0.921282
2.700322e+00 3.557141e+00 2.419263e+00 4.28011
3.272646e+00 2.204892e+00 3.634561e+00 1.87997
2.983304e+00 2.292084e+00 1.147709e+00 5.81465
1.700480e+00 6.211184e+00 2.026233e+00 3.67572
1.519567e+00 9.424900e-01 3.561244e+00 4.12336
1.840696e+00 2.317934e+00 6.060179e+00 1.89123
1.720373e+00 4.646405e+00 5.475396e+00 2.12293
1.193030e+01 1.463597e+00 2.649639e+00 2.00111
3.859778e+00 6.043424e+00 2.248234e+00 3.40697
2.335237e+00 5.031142e+00 4.526379e+00 2.92357
2.451467e+00 2.506669e+00 3.622532e+00 2 1 6 2.50747
8.304961e-01 2.110459e+00 2.735457e+00 2.46836
2.036070e+00 3.770269e+00 2.866086e+00 1.73312
3.063747e+00 1.697008e+00 1.218750e+00 5.63006
2.345771e+00 2.756564e+00 2.958386e+00 2.96989
4.629916e+00 2.612240e+00 1.476521e+00 2.3564
3.561821e+00 1.910498e+00 3.722177e+00 7.8805
5.601491e+00 3.110286e+00 2.683279e+00 2.05194
3.662951e+00 6.267081e+00 6.011975e+00 3.36686
5.494719e+00 3.108410e+00 3.386063e+00 2.76565
5.105645e+00 2.498318e+00 4.562881e+00 1.6913
5.627059e+00 1.306599e+00 2.944986e+00 4.14043
2.716523e+00 2.287602e+00 8.619236e-01 3.16994
9.099832e+00 2.662265e+00 4.798414e+00 1.07115
2.230025e+00 2.151256e+00 3.884083e+00 3.10084
1.614047e+00 3.367665e+00 8.265723e+00 3.75101
9.444974e-01 5.453364e+00 2.618340e+00 1.2037
2.351116e+00 3.856289e+00 2.856166e+00 4.63925
3.351290e+00 1.401212e+00 1.637554e+00 0.794629
1.079287e+00 4.871440e+00 2.579814e+00 2.42574
1.955302e+00 7.147768e+00 1.649958e+00 2.42444
3.605035e+00 3.009374e+00 6.141930e+00 0.777784
1.275406e+00 4.437826e+00 7.683976e+00 3.29918
2.974479e+00 2.164523e+00 4.141565e+00 2.51251
3.691776e+00 2.439294e+00 2.260420e+00 2.13872
3.402411e-01 3.059588e+00 3.563790e+00 1.50487
3.504483e+00 1.640412e+00 3.241697e+00 3.07124
5.230581e+00 3.621594e+00 1.815782e+00 2.6374
2.231120e+00 7.019663e+00 2.391782e+00 2.7571
5.896425e+00 6.779550e-01 2.677140e+00 2.01459
1.678423e+00 5.832689e-01 6.274680e+00 2.59929
3.145988e+00 5.006794e+00 3.620740e+00 2.7497
4.378482e+00 2.172090e+00 2.514135e+00 2.43856
1.897812e+00 3.279909e+00 3.415866e+00 5.64872
4.072629e+00 3.611243e+00 4.724708e+00 7.30102
2.696316e+00 2.746783e+00 4.426376e+00 3.8947
2.534974e+00 4.733061e+00 3.241313e+00 4.13384
2.646879e+00 2.442333e+00 8.896703e-01 1.51918
1.458942e+00 1.612129e+00 2.854779e+00 1.57171
2.325028e+00 2.159296e+00 3.767602e+00 1.0681
3.353429e+00 1.085260e+00 4.871230e+00 3.93589
2.697851e+00 2.955308e+00 1.566995e+00 2.42153
1.331785e+00 2.126572e+00 1.121146e+00 2.59159
2.476290e+00 1.079666e+00 3.813858e+00 1.35706
2.856978e+00 1.336587e+00 2.617469e+00 1.46979
2.846437e+00 5.400321e+00 6.266551e+00 2.01032
4.869179e+00 1.624665e+00 4.274256e+00 3.27233
4.068951e+00 1.366133e+00 3.987064e+00 2.40282
7.719572e+00 2.692586e+00 1.583229e+00 1.74941
4.244755e+00 2.257789e+00 3.045335e+00 1.81794
2.349946e+00 1.317742e+00 2.891514e+00 2.06239
1.841350e+00 3.024049e+00 6.257349e+00 3.89423
3.212662e+00 2.465583e+00 3.087326e+00 3.19645
6.468894e+00 5.055879e+00 1.578383e+00 3.82941
2.847729e+00 5.021945e+00 7.242854e+00 4.40955
3.820690e+00 3.416132e+00 1.554373e+00 4.68676
2.595068e+00 2.783428e+00 5.921081e+00 3.78821
3.724515e+00 2.426557e+00 3.828557e+00 1.59259
3.265852e+00 1.944970e+00 4.448219e+00 1.9585
9.445455e-01 1.232323e+00 2.978147e+00 2.22774
1.720339e+00 2.340588e+00 3.055987e+00 2.18149
9.579876e+00 1.994862e+00 2.509334e+00 1.87095
1.987523e+00 3.216896e+00 6.594762e+00 1.65955
2.606418e+00 2.473984e+00 7.124966e+00 2.28989
2.748420e+00 2.446479e+00 1.129503e+00 1.42999
3.671495e+00 6.682085e+00 4.405377e+00 4.84643
3.293160e+00 2.569162e+00 1.408519e+00 2.53899
1.540380e+00 2.621436e+00 3.226875e+00 1.92026
4.184708e+00 6.529707e+00 3.003369e+00 4.67854
3.481483e+00 3.064456e+00 2.715839e+00 1.65971
9.387371e-01 3.506088e+00 2.938456e+00 3.03276
4.078405e+00 2.657386e+00 1.917555e+00 1.08471
1.106861e+00 1.646879e+00 1.738070e+00 2.40896
2.648342e+00 4.708156e+00 1.962380e+00 1.58413
2.784676e+00 2.658663e+00 2.196153e+00 1.74254
3.057578e+00 4.544319e+00 1.912959e+00 6.36928
2.025388e+00 3.620828e+00 2.473074e+00 8.11985
2.777687e+00 5.412343e+00 3.839506e+00 2.72194
7.815965e+00 7.179425e+00 2.498361e+00 3.13302
3.285946e+00 3.747298e+00 1.820987e+00 2.72801
1.361989e+00 6.393576e-01 2.672029e+00 4.09126
4.179693e-01 1.313394e+00 2.391855e+00 2.31668
3.997767e+00 2.559692e+00 1.197613e+00 3.35363
4.675296e+00 3.033610e+00 1.457455e+00 3.26426
2.664377e+00 2.631494e+00 1.961737e+00 3.81422
4.559179e+00 4.850702e+00 1.360172e+00 5.61429
2.240355e+00 9.252579e-01 2.510018e+00 7.33946
4.378880e+00 1.410641e+00 1.843067e+00 1.58176
8.069896e+00 4.327003e+00 1.549827e+00 2.15639
2.056068e+00 1.146796e+01 4.877343e+00 3.31742
4.404166e+00 5.413701e+00 2.683500e+00 5.9852
4.311396e+00 5.771055e+00 1.781996e+00 3.21207
7.895268e-01 1.566052e+00 8.960054e-01 0.823753
2.029282e+00 2.394033e+00 7.482271e+00 2.45249
1.482358e+00 1.439634e+00 3.161650e+00 7.24169
3.713107e+00 1.716944e+00 3.887802e+00 1.8288
2.153015e+00 7.559150e-01 2.278652e+00 1.77594
1.904378e+00 1.940047e+00 7.790913e+00 2.93928
2.761497e+00 1.287567e+00 1.350779e+00 1.90872
2.989619e+00 6.646804e+00 5.114376e+00 1.00613
3.104411e+00 3.374735e+00 2.077849e+00 4.38405
5.943310e+00 4.825128e+00 2.209023e+00 2.64149
2.752345e+00 1.695020e+00 2.271250e+00 2.85715
5.354547e+00 3.386702e+00 3.082582e+00 1.97644
2.223759e+00 2.384848e+00 2.428859e+00 1.48805
3.723227e+00 1.095346e+00 1.284664e+00 0.996356
3.438836e+00 4.080463e+00 6.869745e+00 4.62302
2.283606e+00 9.271250e-01 4.202575e+00 5.38622
1.366056e+00 2.007330e+00 4.605721e+00 3.70362
3.590079e+00 4.298765e+00 2.212184e+00 4.71374
5.726686e-01 1.354348e+00 5.606007e+00 2.08576
4.369144e+00 3.068581e+00 1.890112e+00 3.36239
6.435558e+00 1.812950e+00 9.887276e-01 2.66198
1.430561e+00 1.849116e+00 5.187982e+00 2.26859
2.581133e+00 8.347610e+00 4.703214e+00 5.12027
1.796515e+00 3.095101e+00 2.539749e+00 6.33957
4.422954e+00 1.209312e+00 3.639755e+00 1.4503
3.775987e+00 2.478175e+00 5.232395e+00 2.17216
4.274792e+00 2.022223e+00 3.581606e+00 1.5795
2.479825e+00 4.338540e+00 3.510846e+00 6.97194
5.407119e+00 4.358610e+00 2.646051e+00 2.29819
2.000382e+00 2.940814e+00 2.270227e+00 1.86504
2.644887e+00 3.654284e+00 3.435649e+00 1.11032
3.480934e+00 1.717532e+00 1.720605e+00 6.96052
2.949344e+00 2.190047e+00 4.691627e+00 2.36786
4.489007e+00 3.774121e+00 4.724163e+00 6.31289
1.655022e+00 1.525554e+00 2.088915e+00 6.17244
3.054264e+00 1.814455e+00 2.927500e+00 5.45459
1.048046e+00 4.536615e+00 4.497119e+00 3.33243
4.062101e+00 4.703615e+00 2.697030e+00 0.98578
2.683102e+00 3.551104e+00 2.233709e+00 4.57553
3.087091e+00 2.101056e+00 3.820115e+00 1.98381
2.687877e+00 2.587511e+00 1.020617e+00 6.11008
1.573388e+00 6.506611e+00 1.840679e+00 3.50135
1.693938e+00 1.128045e+00 3.457408e+00 4.41879
2.136123e+00 2.253435e+00 6.355607e+00 1.87401
1.847466e+00 4.350977e+00 5.371560e+00 2.01909
8.718572e+00 1.637969e+00 2.776731e+00 2.29653
3.876997e+00 5.747996e+00 2.312732e+00 3.11154
2.318018e+00 5.326570e+00 4.230952e+00 2.91753
2.176051e+00 2.610506e+00 3.917959e+00 Best total sum of distances = 1771.1
4 つのクラスターのシルエット プロットを作成します。
[silh4,h] = silhouette(X,idx4,'cityblock'); xlabel('Silhouette Value') ylabel('Cluster')
シルエット プロットは、これら 4 つのクラスターが、前の解の 3 つのクラスターよりも、より適切に分離されていることを示します。2 つの場合については、平均のシルエット値を計算することで、2 つの解をより定量的に比較できます。
平均のシルエット値を計算します。
cluster3 = mean(silh3)
cluster3 = 0.5352
cluster4 = mean(silh4)
cluster4 = 0.6400
4 つのクラスターの平均シルエット値は、3 つのクラスターの平均値よりも高くなります。こうした値は、シルエット プロットに示された結果を立証しています。
最後に、データ内で 5 つのクラスターを見つけます。シルエット プロットを作成し、5 つのクラスターの平均シルエット値を計算します。
idx5 = kmeans(X,5,'Distance','cityblock','Display','final');
Replicate 1, 7 iterations, total sum of distances = 2.507470.8304962.110462.735462.468362.036073.770272.866091.733123.063751.697011.218755.630062.345772.756562.958392.969894.629922.612241.476522.35643.561821.91053.722187.88055.601493.110292.683282.051943.662956.267086.011973.366865.494723.108413.386062.765655.105642.498324.562881.69135.627061.30662.944994.140432.716522.28760.8619243.169949.099832.662264.798411.071152.230032.151263.884083.100841.614053.367668.265723.751010.9444975.453362.618341.20372.351123.856292.856174.639253.351291.401211.637550.7946291.079294.871442.579812.425741.95537.147771.649962.424443.605043.009376.141930.7777841.275414.437837.683983.299182.974482.164524.141572.512513.691782.439292.260422.138720.3402413.059593.563791.504873.504481.640413.24173.071245.230583.621591.815782.63742.231127.019662.391782.75715.896420.6779552.677142.014591.678420.5832696.274682.599293.145995.006793.620742.74974.378482.172092.514132.438561.897813.279913.415875.648724.072633.611244.724717.301022.696322.746784.426383.89472.534974.733063.241314.133842.646882.442330.889671.519181.458941.612132.854781.571712.325032.15933.76761.06813.353431.085264.871233.935892.697852.955311.566992.421531.331792.126571.121152.591592.476291.079673.813861.357062.856981.336592.617471.469792.846445.400326.266552.010324.869181.624674.274263.272334.068951.366133.987062.402827.719572.692591.583231.749414.244762.257793.045331.817942.349951.317742.891512.062391.841353.024056.257353.894233.212662.465583.087333.196456.468895.055881.578383.829412.847735.021947.242854.409553.820693.416131.554374.686762.595072.783435.921083.788213.724522.426563.828561.592593.265851.944974.448221.95850.9445461.232322.978152.227741.720342.340593.055992.181499.579881.994862.509331.870951.987523.21696.594761.659552.606422.473987.124972.289892.748422.446481.12951.429993.671496.682094.405384.846433.293162.569161.408522.538991.540382.621443.226881.920264.184716.529713.003374.678543.481483.064462.715841.659710.9387373.506092.938463.032764.07842.657391.917551.084711.106861.646881.738072.408962.648344.708161.962381.584132.784682.658662.196151.742543.057584.544321.912966.369282.025393.620832.473078.119852.777695.412343.839512.721947.815977.179422.498363.133023.285953.74731.820992.728011.361990.6393582.672034.091260.4179691.313392.391852.316683.997772.559691.197613.353634.67533.033611.457463.264262.664382.631491.961743.814224.559184.85071.360175.614292.240360.9252582.510027.339464.378881.410641.843071.581768.06994.3271.549832.156392.0560711.4684.877343.317424.404175.41372.68355.98524.31145.771061.7823.212070.7895271.566050.8960050.8237532.029282.394037.482272.452491.482361.439633.161657.241693.713111.716943.88781.82882.153020.7559152.278651.775941.904381.940057.790912.939282.76151.287571.350781.908722.989626.64685.114381.006133.104413.374742.077854.384055.943314.825132.209022.641492.752341.695022.271252.857155.354553.38673.082581.976442.223762.384852.428861.488053.723231.095351.284660.9963563.438844.080466.869754.623022.283610.9271254.202585.386221.366061.416912.002551.048373.202663.209162.189823.510192.407591.283672.672333.175362.010030.9807961.285861.807994.076452.22081.401492.096131.753161.968262.254311.701282.178785.98852.34414.011612.227452.221181.382243.405892.549572.381752.225781.893572.221592.650542.7251.050031.341120.9917621.977671.360772.458681.564041.611624.038272.727351.99951.556441.899361.843232.179222.980962.377042.535683.058452.271191.823351.926543.087941.815814.60142.387962.480351.882441.164322.775911.876632.365053.379222.355162.825632.246963.238772.601041.337372.357123.095481.783881.680042.23070.8111371.702991.769941.672442.557753.01253.995942.144082.720221.88283.159272.265721.548461.388491.636432.110223.750972.333974.14752.164662.509511.093361.412262.118772.059681.436743.037834.00541.851642.508552.605254.032922.142125.78491.385293.218570.8601453.652472.814322.246780.3586292.750123.660821.904853.064961.920912.456592.3188. Best total sum of distances = 1647.26
[silh5,h] = silhouette(X,idx5,'cityblock'); xlabel('Silhouette Value') ylabel('Cluster')
mean(silh5)
ans = 0.5721
このシルエット プロットは、2 つのクラスターにはほとんど低いシルエット値をもつ点が含まれ、5 番目のクラスターには負の値をもつ点がいくつか含まれているため、5 がおそらくクラスターの正しい数ではないことを示しています。また、5 つのクラスターの平均シルエット値は 4 つのクラスターの値よりも低くなります。どれだけ多くのクラスターがデータ内にあるかについて分からない場合、ある範囲内の複数の k の値 (クラスターの数) を使用して実験することをお勧めします。
クラスターの数が増加すると、距離の合計が減少することに注意してください。たとえば、クラスターの数が 3 から 4、そして 5 に増加すると、距離の合計は 2459.98 から 1771.1、そして 1647.26 に減少します。そのため、距離の合計は最適なクラスター数を決定するのに役立ちません。
ローカルな最小値の回避
既定の設定では、関数 kmeans は無作為に選択された初期の重心位置の集合を使用して処理を開始します。kmeans アルゴリズムは、ローカルな (グローバルではない) 最小値である解に収束する可能性があります。つまり、kmeans は、任意の 1 点を異なるクラスターに移動すると距離の総和が大きくなるようにデータを分割する可能性があります。ただし、他の多くのタイプの数値的最小化と同様に、kmeans によって到達する解は、開始点に依存する場合があります。したがって、そのデータについて、距離の総和がより小さい他の解 (ローカルな最小値) が存在する可能性があります。'Replicates' 名前と値のペアの引数を使用して異なる解をテストできます。複数の複製を指定すると、kmeans は、無作為に選択した重心から始まるクラスタリング プロセスを各複製について繰り返し、すべての複製の中で距離の総和が最小である解を返します。
データ内の 4 つのクラスターを見つけて、クラスタリングを 5 回複製します。また、尺度として市街地距離を指定し、'Display' 名前と値のペアの引数を使用して、それぞれの解に最終的な距離の合計を表示します。
[idx4,cent4,sumdist] = kmeans(X,4,'Distance','cityblock', ... 'Display','final','Replicates',5);
Replicate 1, 2 iterations, total sum of distances = 2.507470.8304962.110462.735462.468362.036073.770272.866091.733123.063751.697011.218755.630062.345772.756562.958392.969894.629922.612241.476522.35643.561821.91053.722187.88055.601493.110292.683282.051943.662956.267086.011973.366865.494723.108413.386062.765655.105642.498324.562881.69135.627061.30662.944994.140432.716522.28760.8619243.169949.099832.662264.798411.071152.230032.151263.884083.100841.614053.367668.265723.751010.9444975.453362.618341.20372.351123.856292.856174.639253.351291.401211.637550.7946291.079294.871442.579812.425741.95537.147771.649962.424443.605043.009376.141930.7777841.275414.437837.683983.299182.974482.164524.141572.512513.691782.439292.260422.138720.3402413.059593.563791.504873.504481.640413.24173.071245.230583.621591.815782.63742.231127.019662.391782.75715.896420.6779552.677142.014591.678420.5832696.274682.599293.145995.006793.620742.74974.378482.172092.514132.438561.897813.279913.415875.648724.072633.611244.724717.301022.696322.746784.426383.89472.534974.733063.241314.133842.646882.442330.889671.519181.458941.612132.854781.571712.325032.15933.76761.06813.353431.085264.871233.935892.697852.955311.566992.421531.331792.126571.121152.591592.476291.079673.813861.357062.856981.336592.617471.469792.846445.400326.266552.010324.869181.624674.274263.272334.068951.366133.987062.402827.719572.692591.583231.749414.244762.257793.045331.817942.349951.317742.891512.062391.841353.024056.257353.894233.212662.465583.087333.196456.468895.055881.578383.829412.847735.021947.242854.409553.820693.416131.554374.686762.595072.783435.921083.788213.724522.426563.828561.592593.265851.944974.448221.95850.9445461.232322.978152.227741.720342.340593.055992.181499.579881.994862.509331.870951.987523.21696.594761.659552.606422.473987.124972.289892.748422.446481.12951.429993.671496.682094.405384.846433.293162.569161.408522.538991.540382.621443.226881.920264.184716.529713.003374.678543.481483.064462.715841.659710.9387373.506092.938463.032764.07842.657391.917551.084711.106861.646881.738072.408962.648344.708161.962381.584132.784682.658662.196151.742543.057584.544321.912966.369282.025393.620832.473078.119852.777695.412343.839512.721947.815977.179422.498363.133023.285953.74731.820992.728011.361990.6393582.672034.091260.4179691.313392.391852.316683.997772.559691.197613.353634.67533.033611.457463.264262.664382.631491.961743.814224.559184.85071.360175.614292.240360.9252582.510027.339464.378881.410641.843071.581768.06994.3271.549832.156392.0560711.4684.877343.317424.404175.41372.68355.98524.31145.771061.7823.212070.7895271.566050.8960050.8237532.029282.394037.482272.452491.482361.439633.161657.241693.713111.716943.88781.82882.153020.7559152.278651.775941.904381.940057.790912.939282.76151.287571.350781.908722.989626.64685.114381.006133.104413.374742.077854.384055.943314.825132.209022.641492.752341.695022.271252.857155.354553.38673.082581.976442.223762.384852.428861.488053.723231.095351.284660.9963563.438844.080466.869754.623022.283610.9271254.202585.386221.366062.007334.605723.703623.590084.298772.212184.713740.5726691.354355.606012.085764.369143.068581.890113.362396.435561.812950.9887282.661981.430561.849125.187982.268592.581138.347614.703215.120271.796523.09512.539756.339574.422951.209313.639751.45033.775992.478185.232392.172164.274792.022223.581611.57952.479824.338543.510856.971945.407124.358612.646052.298192.000382.940812.270231.865042.644893.654283.435651.110323.480931.717531.720616.960522.949342.190054.691632.367864.489013.774124.724166.312891.655021.525552.088916.172443.054261.814462.92755.454591.048054.536614.497123.332434.06214.703622.697030.985782.68313.55112.233714.575533.087092.101063.820121.983812.687882.587511.020626.110081.573396.506611.840683.501351.693941.128043.457414.418792.136122.253446.355611.874011.847474.350985.371562.019098.718571.637972.776732.296533.8775.7482.312733.111542.318025.326574.230952.917532.176052.610513.91796. Replicate 2, 3 iterations, total sum of distances = 2.507470.8304962.110462.735462.468362.036073.770272.866091.733123.063751.697011.218755.630062.345772.756562.958392.969894.629922.612241.476522.35643.561821.91053.722187.88055.601493.110292.683282.051943.662956.267086.011973.366865.494723.108413.386062.765655.105642.498324.562881.69135.627061.30662.944994.140432.716522.28760.8619243.169949.099832.662264.798411.071152.230032.151263.884083.100841.614053.367668.265723.751010.9444975.453362.618341.20372.351123.856292.856174.639253.351291.401211.637550.7946291.079294.871442.579812.425741.95537.147771.649962.424443.605043.009376.141930.7777841.275414.437837.683983.299182.974482.164524.141572.512513.691782.439292.260422.138720.3402413.059593.563791.504873.504481.640413.24173.071245.230583.621591.815782.63742.231127.019662.391782.75715.896420.6779552.677142.014591.678420.5832696.274682.599293.145995.006793.620742.74974.378482.172092.514132.438561.897813.279913.415875.648724.072633.611244.724717.301022.696322.746784.426383.89472.534974.733063.241314.133842.646882.442330.889671.519181.458941.612132.854781.571712.325032.15933.76761.06813.353431.085264.871233.935892.697852.955311.566992.421531.331792.126571.121152.591592.476291.079673.813861.357062.856981.336592.617471.469792.846445.400326.266552.010324.869181.624674.274263.272334.068951.366133.987062.402827.719572.692591.583231.749414.244762.257793.045331.817942.349951.317742.891512.062391.841353.024056.257353.894233.212662.465583.087333.196456.468895.055881.578383.829412.847735.021947.242854.409553.820693.416131.554374.686762.595072.783435.921083.788213.724522.426563.828561.592593.265851.944974.448221.95850.9445461.232322.978152.227741.720342.340593.055992.181499.579881.994862.509331.870951.987523.21696.594761.659552.606422.473987.124972.289892.748422.446481.12951.429993.671496.682094.405384.846433.293162.569161.408522.538991.540382.621443.226881.920264.184716.529713.003374.678543.481483.064462.715841.659710.9387373.506092.938463.032764.07842.657391.917551.084711.106861.646881.738072.408962.648344.708161.962381.584132.784682.658662.196151.742543.057584.544321.912966.369282.025393.620832.473078.119852.777695.412343.839512.721947.815977.179422.498363.133023.285953.74731.820992.728011.361990.6393582.672034.091260.4179691.313392.391852.316683.997772.559691.197613.353634.67533.033611.457463.264262.664382.631491.961743.814224.559184.85071.360175.614292.240360.9252582.510027.339464.378881.410641.843071.581768.06994.3271.549832.156392.0560711.4684.877343.317424.404175.41372.68355.98524.31145.771061.7823.212070.7895271.566050.8960050.8237532.029282.394037.482272.452491.482361.439633.161657.241693.713111.716943.88781.82882.153020.7559152.278651.775941.904381.940057.790912.939282.76151.287571.350781.908722.989626.64685.114381.006133.104413.374742.077854.384055.943314.825132.209022.641492.752341.695022.271252.857155.354553.38673.082581.976442.223762.384852.428861.488053.723231.095351.284660.9963563.438844.080466.869754.623022.283610.9271254.202585.386221.366062.007334.605723.703623.590084.298772.212184.713740.5726691.354355.606012.085764.369143.068581.890113.362396.435561.812950.9887282.661981.430561.849125.187982.268592.581138.347614.703215.120271.796523.09512.539756.339574.422951.209313.639751.45033.775992.478185.232392.172164.274792.022223.581611.57952.479824.338543.510856.971945.407124.358612.646052.298192.000382.940812.270231.865042.644893.654283.435651.110323.480931.717531.720616.960522.949342.190054.691632.367864.489013.774124.724166.312891.655021.525552.088916.172443.054261.814462.92755.454591.048054.536614.497123.332434.06214.703622.697030.985782.68313.55112.233714.575533.087092.101063.820121.983812.687882.587511.020626.110081.573396.506611.840683.501351.693941.128043.457414.418792.136122.253446.355611.874011.847474.350985.371562.019098.718571.637972.776732.296533.8775.7482.312733.111542.318025.326574.230952.917532.176052.610513.91796. Replicate 3, 3 iterations, total sum of distances = 2.507470.8304962.110462.735462.468362.036073.770272.866091.733123.063751.697011.218755.630062.345772.756562.958392.969894.629922.612241.476522.35643.561821.91053.722187.88055.601493.110292.683282.051943.662956.267086.011973.366865.494723.108413.386062.765655.105642.498324.562881.69135.627061.30662.944994.140432.716522.28760.8619243.169949.099832.662264.798411.071152.230032.151263.884083.100841.614053.367668.265723.751010.9444975.453362.618341.20372.351123.856292.856174.639253.351291.401211.637550.7946291.079294.871442.579812.425741.95537.147771.649962.424443.605043.009376.141930.7777841.275414.437837.683983.299182.974482.164524.141572.512513.691782.439292.260422.138720.3402413.059593.563791.504873.504481.640413.24173.071245.230583.621591.815782.63742.231127.019662.391782.75715.896420.6779552.677142.014591.678420.5832696.274682.599293.145995.006793.620742.74974.378482.172092.514132.438561.897813.279913.415875.648724.072633.611244.724717.301022.696322.746784.426383.89472.534974.733063.241314.133842.646882.442330.889671.519181.458941.612132.854781.571712.325032.15933.76761.06813.353431.085264.871233.935892.697852.955311.566992.421531.331792.126571.121152.591592.476291.079673.813861.357062.856981.336592.617471.469792.846445.400326.266552.010324.869181.624674.274263.272334.068951.366133.987062.402827.719572.692591.583231.749414.244762.257793.045331.817942.349951.317742.891512.062391.841353.024056.257353.894233.212662.465583.087333.196456.468895.055881.578383.829412.847735.021947.242854.409553.820693.416131.554374.686762.595072.783435.921083.788213.724522.426563.828561.592593.265851.944974.448221.95850.9445461.232322.978152.227741.720342.340593.055992.181499.579881.994862.509331.870951.987523.21696.594761.659552.606422.473987.124972.289892.748422.446481.12951.429993.671496.682094.405384.846433.293162.569161.408522.538991.540382.621443.226881.920264.184716.529713.003374.678543.481483.064462.715841.659710.9387373.506092.938463.032764.07842.657391.917551.084711.106861.646881.738072.408962.648344.708161.962381.584132.784682.658662.196151.742543.057584.544321.912966.369282.025393.620832.473078.119852.777695.412343.839512.721947.815977.179422.498363.133023.285953.74731.820992.728011.361990.6393582.672034.091260.4179691.313392.391852.316683.997772.559691.197613.353634.67533.033611.457463.264262.664382.631491.961743.814224.559184.85071.360175.614292.240360.9252582.510027.339464.378881.410641.843071.581768.06994.3271.549832.156392.0560711.4684.877343.317424.404175.41372.68355.98524.31145.771061.7823.212070.7895271.566050.8960050.8237532.029282.394037.482272.452491.482361.439633.161657.241693.713111.716943.88781.82882.153020.7559152.278651.775941.904381.940057.790912.939282.76151.287571.350781.908722.989626.64685.114381.006133.104413.374742.077854.384055.943314.825132.209022.641492.752341.695022.271252.857155.354553.38673.082581.976442.223762.384852.428861.488053.723231.095351.284660.9963563.438844.080466.869754.623022.283610.9271254.202585.386221.366062.007334.605723.703623.590084.298772.212184.713740.5726691.354355.606012.085764.369143.068581.890113.362396.435561.812950.9887282.661981.430561.849125.187982.268592.581138.347614.703215.120271.796523.09512.539756.339574.422951.209313.639751.45033.775992.478185.232392.172164.274792.022223.581611.57952.479824.338543.510856.971945.407124.358612.646052.298192.000382.940812.270231.865042.644893.654283.435651.110323.480931.717531.720616.960522.949342.190054.691632.367864.489013.774124.724166.312891.655021.525552.088916.172443.054261.814462.92755.454591.048054.536614.497123.332434.06214.703622.697030.985782.68313.55112.233714.575533.087092.101063.820121.983812.687882.587511.020626.110081.573396.506611.840683.501351.693941.128043.457414.418792.136122.253446.355611.874011.847474.350985.371562.019098.718571.637972.776732.296533.8775.7482.312733.111542.318025.326574.230952.917532.176052.610513.91796. Replicate 4, 6 iterations, total sum of distances = 12.33359.9739211.091610.116811.700510.187913.532612.580711.219112.853510.704410.77138.1630912.171811.45579.8538112.795914.45599.6384511.238811.47769.7562110.123212.32649.889067.9706610.418212.44569.5151813.42537.2736915.41889.27369.8963410.768712.716310.249114.86810.06114.32529.8808910.224410.9798.194249.6843810.12610.37459.6912610.432918.862112.424610.39059.7947712.0569.237947.192279.8355711.795813.549418.02813.513310.538115.215712.89.2484611.52888.301189.2878714.401612.558510.48410.101410.28399.737857.5376310.087512.188111.936816.910110.36531.28121.444461.831823.83261.925682.380063.395055.132271.988752.181191.207142.831132.089161.423251.591542.444650.7232572.150871.090921.695891.466191.581991.559561.409812.256783.603791.069882.374381.208032.467874.467952.604762.102963.644632.244252.350491.438651.624111.685264.006152.598813.160072.738262.371721.146511.826772.26220.7026322.132622.337491.779910.9564373.097012.218471.30473.498485.032492.964442.544744.027841.913471.852372.181351.181191.865312.250652.583972.095372.131282.277832.187932.641352.349061.458941.70354.192652.360892.416141.555223.439462.504121.483641.958011.015611.369441.403892.055161.507512.609891.946592.01872.57961.996211.985351.562442.169661.534322.224952.848614.125321.730762.600651.012742.204720.7206213.398822.455232.760842.275185.167862.851322.191711.866622.699762.19061.917081.423951.67311.319272.311811.072962.26642.464313.988821.727231.762212.279623.296512.622174.450763.416632.272631.784932.075882.753415.01392.131751.822361.185990.8665873.376332.113692.954213.369373.228424.064962.851611.5732.433752.020590.9577251.896511.055891.641262.236582.422952.041432.435652.112442.402141.371627.028172.388311.248162.211381.645112.871284.326231.761212.338291.931564.856442.261392.961852.292921.139032.275561.980744.413562.789162.57791.024631.879920.8612971.664762.583162.600082.000652.473681.6334.649411.458382.566771.343312.610781.428492.244142.156612.548462.455032.829382.99012.378763.13991.729052.329211.716221.534693.631311.4264.504781.347640.9666442.715343.752922.991482.456372.344163.321973.001276.299943.210953.82422.618236.897514.000036.634695.061853.944296.593628.401773.720713.118053.489322.658992.872313.632082.225950.8828163.319422.922281.022671.868071.303553.539033.794393.6481.768263.422975.897644.255962.67984.48661.991481.409152.165113.123423.336836.073052.2654.391941.1681.542493.598338.56184.953921.530273.065412.230056.847555.549352.671972.185062.2427911.46754.366762.229123.181826.636053.901067.207545.533744.548713.004341.989721.254471.496712.088621.881631.692231.305738.704622.331792.570672.527942.617058.464034.935452.805255.110151.898143.375360.9778493.264722.864252.684771.436129.013262.44463.983841.704511.420123.131071.901315.424463.892031.229182.825833.444072.051535.60644.720963.602781.873483.863843.974692.436382.067874.07954.13224.609053.285962.745673.44613.607193.481222.576364.811530.891972.507010.4076293.235464.14985.64743.400672.942361.605714.800314.289742.029511.808715.022014.119913.653584.235262.408924.747940.9184210.9380626.022292.149263.952862.65231.473832.904876.019271.437680.9252253.11951.367062.306645.604272.205082.644647.931324.286935.577791.3393.511392.408346.755854.839241.1444.056041.837293.318472.374675.648681.755874.691081.605943.165321.989192.937344.754833.094567.388235.82343.942322.582552.291162.416673.35712.277261.76032.701683.688493.019361.526613.023412.175051.635656.544232.603592.085315.107912.402074.905294.190414.307886.729182.112541.983072.47596.588732.781192.230742.537285.03830.7022944.95294.080833.748713.645825.11992.633530.7536962.921783.544072.691234.159253.544612.066853.36262.018013.104162.199141.084125.693791.636896.090332.29823.84711.348190.6705253.620344.00251.719842.640425.939321.99731.783964.767265.337361.984899.134861.292222.713231.880253.772266.164281.925753.527832.422754.910284.647242.91052.572332.644713.50167. Replicate 5, 2 iterations, total sum of distances = 2.507470.8304962.110462.735462.468362.036073.770272.866091.733123.063751.697011.218755.630062.345772.756562.958392.969894.629922.612241.476522.35643.561821.91053.722187.88055.601493.110292.683282.051943.662956.267086.011973.366865.494723.108413.386062.765655.105642.498324.562881.69135.627061.30662.944994.140432.716522.28760.8619243.169949.099832.662264.798411.071152.230032.151263.884083.100841.614053.367668.265723.751010.9444975.453362.618341.20372.351123.856292.856174.639253.351291.401211.637550.7946291.079294.871442.579812.425741.95537.147771.649962.424443.605043.009376.141930.7777841.275414.437837.683983.299182.974482.164524.141572.512513.691782.439292.260422.138720.3402413.059593.563791.504873.504481.640413.24173.071245.230583.621591.815782.63742.231127.019662.391782.75715.896420.6779552.677142.014591.678420.5832696.274682.599293.145995.006793.620742.74974.378482.172092.514132.438561.897813.279913.415875.648724.072633.611244.724717.301022.696322.746784.426383.89472.534974.733063.241314.133842.646882.442330.889671.519181.458941.612132.854781.571712.325032.15933.76761.06813.353431.085264.871233.935892.697852.955311.566992.421531.331792.126571.121152.591592.476291.079673.813861.357062.856981.336592.617471.469792.846445.400326.266552.010324.869181.624674.274263.272334.068951.366133.987062.402827.719572.692591.583231.749414.244762.257793.045331.817942.349951.317742.891512.062391.841353.024056.257353.894233.212662.465583.087333.196456.468895.055881.578383.829412.847735.021947.242854.409553.820693.416131.554374.686762.595072.783435.921083.788213.724522.426563.828561.592593.265851.944974.448221.95850.9445461.232322.978152.227741.720342.340593.055992.181499.579881.994862.509331.870951.987523.21696.594761.659552.606422.473987.124972.289892.748422.446481.12951.429993.671496.682094.405384.846433.293162.569161.408522.538991.540382.621443.226881.920264.184716.529713.003374.678543.481483.064462.715841.659710.9387373.506092.938463.032764.07842.657391.917551.084711.106861.646881.738072.408962.648344.708161.962381.584132.784682.658662.196151.742543.057584.544321.912966.369282.025393.620832.473078.119852.777695.412343.839512.721947.815977.179422.498363.133023.285953.74731.820992.728011.361990.6393582.672034.091260.4179691.313392.391852.316683.997772.559691.197613.353634.67533.033611.457463.264262.664382.631491.961743.814224.559184.85071.360175.614292.240360.9252582.510027.339464.378881.410641.843071.581768.06994.3271.549832.156392.0560711.4684.877343.317424.404175.41372.68355.98524.31145.771061.7823.212070.7895271.566050.8960050.8237532.029282.394037.482272.452491.482361.439633.161657.241693.713111.716943.88781.82882.153020.7559152.278651.775941.904381.940057.790912.939282.76151.287571.350781.908722.989626.64685.114381.006133.104413.374742.077854.384055.943314.825132.209022.641492.752341.695022.271252.857155.354553.38673.082581.976442.223762.384852.428861.488053.723231.095351.284660.9963563.438844.080466.869754.623022.283610.9271254.202585.386221.366062.007334.605723.703623.590084.298772.212184.713740.5726691.354355.606012.085764.369143.068581.890113.362396.435561.812950.9887282.661981.430561.849125.187982.268592.581138.347614.703215.120271.796523.09512.539756.339574.422951.209313.639751.45033.775992.478185.232392.172164.274792.022223.581611.57952.479824.338543.510856.971945.407124.358612.646052.298192.000382.940812.270231.865042.644893.654283.435651.110323.480931.717531.720616.960522.949342.190054.691632.367864.489013.774124.724166.312891.655021.525552.088916.172443.054261.814462.92755.454591.048054.536614.497123.332434.06214.703622.697030.985782.68313.55112.233714.575533.087092.101063.820121.983812.687882.587511.020626.110081.573396.506611.840683.501351.693941.128043.457414.418792.136122.253446.355611.874011.847474.350985.371562.019098.718571.637972.776732.296533.8775.7482.312733.111542.318025.326574.230952.917532.176052.610513.91796. Best total sum of distances = 1771.1
複製 4 で、kmeans はローカルな最小値を見つけます。複製はそれぞれ無作為に選択した異なる重心の初期値から始まっているため、kmeans は 2 つ以上のローカルな最小値を見つける場合があります。しかし、kmeans によって返される最終的な解は、すべての複製のうちで距離の総和が最小値となる解になります。
kmeans によって返される最終的な解に対して、点と重心間の距離のクラスター内合計の総和を求めます。
sum(sumdist)
ans = 1.7711e+03
参考
[データのクラスタリング] | kmeans | silhouette
