主成分分析のソースを表示

{{複数の問題
|出典の明記=2015年5月9日 (土) 01:24 (UTC)
|専門的=2015年5月9日 (土) 03:14 (UTC)
}}
[[File:GaussianScatterPCA.svg|thumb|right|{{math|(1, 3)}} を中心とし{{math|(0.866, 0.5)}} 方向の[[標準偏差]]が {{math|3}}、それに[[直交]]する方向の標準偏差が {{math|1}} の[[多変量正規分布]]に従うデータセットに対する主成分分析の結果。矢印で示される 2 つ[[ベクトル空間|ベクトル]]は[[共分散行列]]の[[固有ベクトル]]であり、ベクトルの長さはそれぞれの固有ベクトルの固有値の[[平方根]]に等しくなるようにスケールされている。また 2 つの固有ベクトルは分布の中心（平均）が原点となるように配置してある。]]

'''主成分分析'''（しゅせいぶんぶんせき、{{Lang-en-short|principal component analysis; PCA}}）は、[[相関]]のある多数の変数から相関のない少数で全体のばらつきを最もよく表す'''主成分'''と呼ばれる変数を合成する[[多変量解析]]の一手法{{sfn|Jolliffe|2002|p={{google books quote|id=TtVF-ao4fI8C|page=1|1}}}}。データの次元を削減するために用いられる。

主成分を与える変換は、第一主成分の[[分散 (確率論)|分散]]を[[最適化問題|最大化]]し、続く主成分はそれまでに決定した主成分と[[直交]]するという[[拘束条件]]の下で分散を最大化するようにして選ばれる。主成分の分散を最大化することは、観測値の変化に対する説明能力を可能な限り主成分に持たせる目的で行われる。選ばれた主成分は互いに直交し、与えられた観測値のセットを[[線型結合]]として表すことができる。言い換えると、主成分は観測値のセットの直交[[基底 (線型代数学)|基底]]となっている。主成分ベクトルの直交性は、主成分ベクトルが[[共分散行列]]（あるいは相関行列）の[[固有ベクトル]]になっており、共分散行列が実[[対称行列]]であることから導かれる。

主成分分析は純粋に固有ベクトルに基づく多変量解析の中で最も単純なものである。主成分分析は、データの分散をより良く説明するという観点から、そのデータの内部構造を明らかにするものだと考えられる。多くの場合、多変量データは次元が大きく、各変数を軸にとって視覚化することは難しいが、主成分分析によって情報をより少ない次元に集約することでデータを視覚化できる。集約によって得られる情報は、データセットを元のデータ変数の空間から主成分ベクトルのなす空間へ[[射影]]したものであり、元のデータから有用な情報を抜き出したものになっている<!--（[[#情報理論|情報理論]]の項も参照）-->。主成分分析によるデータ構造の可視化は、可視化に必要なだけ先頭から少数の主成分を選択することで実現される。

主成分分析は{{仮リンク|探索的データ解析|en|exploratory data analysis}}における主要な道具であり、{{仮リンク|予測モデル構築|en|predictive modeling}}にも使われる。主成分分析は観測値の[[共分散行列]]や[[相関行列]]に対する[[固有値分解]]、あるいは（大抵は[[正規化]]された）データ行列の[[特異値分解]]によって行われる{{sfn|Abdi|Williams|2010}}。主成分分析の結果は'''主成分得点'''（因子得点、{{lang-en-short|score}}）と'''主成分負荷量'''（因子負荷量、{{lang-en-short|loadings}}）によって評価される{{sfn|Shaw|2003|pp={{Page needed|date=2016-4-23}}}}。主成分得点とは、あるデータ点を主成分ベクトルで表現した場合の基底ベクトルにかかる係数であり、ある主成分ベクトルのデータ点に対する寄与の大きさを示す。主成分負荷量はある主成分得点に対する個々の（正規化された）観測値の重みであり、観測値と主成分の相関係数として与えられる。主成分分析は観測値の間の相対的なスケールに対して敏感である。

主成分分析による評価は主成分得点と主成分負荷量をそれぞれ可視化した主成分プロット、あるいは両者を重ね合わせたバイプロットを通して解釈される。主成分分析を実行するためのソフトウェアや関数によって、観測値の基準化の方法や数値計算のアルゴリズムに細かな差異が存在し、個々の方法は必ずしも互いに等価であるとは限らない（例えば、[[R言語]]における <code>prcomp</code> 関数と [[FactoMineR]] の <code>PCA</code> 関数の結果は異なる）。

== 直感的な説明 ==
主成分分析は与えられたデータを {{mvar|n}} 次元の[[楕円体]]にフィッティングするものであると考えることができる。このとき、それぞれの主成分は楕円体の軸に対応している。楕円体の軸が短いほどデータの[[分散 (確率論)|分散]]は小さく、短い軸に対応する主成分を無視することで、データの分散と同程度に小さな情報の損失だけで、データをより少ない[[変数 (数学)|変数]]で表現することができる。

楕円体の軸を見つけるには、データの[[平均]]を[[座標軸]]の[[原点]]に合わせる必要がある。そのため、データの[[共分散行列]]を計算し、共分散行列に対する[[固有値]]と[[固有ベクトル]]を計算する。また、それぞれの固有ベクトルを[[直交化]]し、[[正規化]]する必要がある。固有ベクトルの組として互いに[[直交]]する[[単位ベクトル]]が得られたなら、それらに対応する軸を持つ楕円体によってデータをフィッティングすることができる。それぞれの軸に対する'''寄与率'''（{{en|proportion of the variance}}: 分散の[[比]]）は、その軸に対応する固有ベクトルに対する固有値を、すべての固有値の和で割ったものとして得ることができる。

注意すべき点として、分散はデータのスケールに依存するため、主成分分析の結果はデータをスケール変換することで変わり得るということが挙げられる。

== 歴史と名称 ==
主成分分析は[[1901年]]に[[カール・ピアソン]]によって導入された{{sfn|Pearson|1901}}。ピアソンは[[力学]]における{{仮リンク|主軸定理|en|principal axis theorem}}からの類推によって主成分分析の方法を得た。主成分分析は、ピアソンとは独立に1930年代に[[ハロルド・ホテリング]]よっても導入され、ホテリングによって'''主成分分析''' {{en|(principal component analysis)}} と呼ばれるようになった{{sfn|Hotelling|1933}}{{sfn|Hotelling|1936}}。（{{harvtxt|Jolliffe|2002|loc=1.2 A Brief History of Principal Component Analysis}} 参照。）

主成分分析は応用分野によって様々な呼び名がある。

{|class=wikitable
!分野
!呼び名
|-
|[[信号処理]]
|
*'''離散（コサンビ・）カルフネン・ロエヴェ変換'''<ref>{{lang-en-short|(Kosambi–) Karhunen–Loève transform}}、KLT</ref>
*'''KL展開'''<ref>{{lang-en-short|Karhunen–Loève expansion}}</ref>
|-
|[[品質管理]]
|
*'''ホテリング変換'''<ref>{{lang-en-short|Hotelling transform}}</ref>
|-
|[[機械工学]]
|
*'''固有直交分解'''<ref>{{lang-en-short|proper orthogonal decomposition}}、POD</ref>
|-
|[[線型代数学]]
|
*行列 {{math|'''X'''}} の'''[[特異値分解]]'''
*{{math|'''X'''<sup>T</sup>'''X'''}} の'''固有値分解'''
|-
|計量心理学<ref>心理測定、心理統計学などとも呼ばれる。</ref>
|
*'''[[因子分析]]'''<ref>数学的な共通点は多いものの、厳密には主成分分析と因子分析は異なる手法である。両者の違いに関する議論は例えば {{harvnb|Jolliffe|2002|loc={{google books quote|id=TtVF-ao4fI8C|page=150|Chapter 7}}}} を参照。</ref>
*'''[[エッカート・ヤング定理]]'''
*'''[[シュミット・ミルスキー定理]]'''
|-
|[[気象学]]
|
*'''[[経験的直交関数]]'''
|-
|雑音・振動
|
*'''経験固有関数分解'''<ref>{{lang-en-short|empirical eigenfunction decomposition}}</ref>
*'''経験的成分分析'''<ref>{{lang-en-short|empirical component analysis}}</ref>
*'''[[準調和モード]]'''
*'''[[スペクトル分解]]'''
|-
|[[構造力学]]
|
*'''[[モーダル解析]]'''
|}


== 関連する手法 ==
主成分分析は[[因子分析]]によく似ている。因子分析は、データの背後にある構造に関する分野固有の[[仮設 (数学)|仮設]]と、主成分分析の場合とはわずかに異なった行列に対する固有ベクトルを求める手法である、と要約できる。

主成分分析は'''正準相関分析''' {{en|(canonical correlation analysis; CCA)}} とも関わりがある。正準相関分析は二つのデータセット間の相互共分散に基いて座標系を定める手続きだが、主成分分析は単一のデータセットの分散に基いて座標系を選択する手法である{{sfn|Barnett|Preisendorfer|1987}}{{sfn|Hsu|Kakade|Zhang|2012}}。

== 詳細 ==
数学的には主成分分析はデータの[[基底 (線型代数学)|基底]]に対し[[直交変換]]を行い、新たな[[座標系]]を得ることであり{{sfn|Jolliffe|2002}}{{page needed|date=2016-11}}、新しい座標系はその第一成分（第一主成分と呼ばれる）から順に、データの各成分に対する[[分散 (確率論)|分散]]が最大になるように選ばれる。

以下では、データ行列 {{math|'''X'''}} として、各列の[[標本平均]]が 0 になるものを考える<ref group="注">つまり事前処理として、生のデータの各成分から成分ごとの標本平均を引く。</ref>。データ行列の各列 {{mvar|p}} はそれぞれデータが持つ特定の指標に対応し、データ行列の各行 {{mvar|n}} はそれぞれ異なる事例に対する指標の組を表す<ref group="注">たとえば列のラベルには <code>"年齢", "性別", "身長", "体重"</code> など一般的な属性が入り、行のラベルには <code>"藤原", "木曽", "北条", "徳川"</code> など事例を特定する識別子が与えられる。行と列のどちらにラベルを与えるかは本質的ではなく、列と指標を対応させることは単に慣習による。</ref>。

主成分分析は {{mvar|p}} 次元ベクトル {{math|1='''w'''<sub>''k''</sub>}} によってデータ行列 {{math|'''X'''}} の各行 {{math|'''x'''<sub>''i''</sub>}} を主成分得点のベクトル {{math|1='''t'''<sub>(''i'')</sub> = (''t''<sub>1</sub>, ..., ''t<sub>k</sub>'')<sub>(''i'')</sub>}} に変換することであり、主成分得点{{math|''t<sub>k</sub>''<sub>(''i'')</sub>}} はデータ点 {{math|'''x'''<sub>''i''</sub>}} と負荷量ベクトル {{math|'''w'''<sub>''k''</sub>}} の[[内積]]によって与えられる。
:<math>{t_{k}}_{(i)} = \mathbf{x}_i \cdot \mathbf{w}_k</math>
負荷量ベクトル {{math|'''w'''}} は[[単位ベクトル]]であり、各主成分得点の分散を第一主成分から順に最大化するように選ばれる。負荷量ベクトルの個数（つまり主成分の数）{{mvar|k}} は、元の指標の数 {{mvar|p}} に等しいか、より小さい数が選ばれる ({{math|''k'' &le; ''p''}})。負荷量ベクトルの個数、つまり新しいデータ空間の次元を元の空間の次元より少なくとることで、{{仮リンク|次元削減|en|dimensionality reduction}}をすることができる（[[#次元削減]]を参照）。主成分分析による次元削減は、データの分散に関する情報を残すように行われる。

=== 第一主成分 ===
第一主成分に対応する負荷量ベクトル {{math|'''w'''<sub>1</sub>}} は以下の条件を満たす<ref group="注">{{math|{{underset|''x''|arg max}} ''f''(''x'')}} は {{math|''f''(''x'')}} が[[最大値]]をとるときの引数 {{mvar|x}} またはその[[集合]]を与える（[[arg max]] を参照）。[[作用素]] {{math|arg max}} によって与えられる集合の[[元 (数学)|元]]は最大値点と呼ばれることが多い。</ref>。
:<math>\mathbf{w}_1
  = \underset{\Vert \mathbf{w} \Vert = 1}\operatorname{arg\,max}
    \Vert \mathbf{Xw} \Vert^2.
</math>

さらに変数 {{math|'''w'''}} が[[単位ベクトル]]という制約を除けば、上述の条件は次の等価な条件に簡約化することができる<ref group="注">ゼロでない任意の[[ノルム]]のベクトルが方程式を満たすため、実際には以下の方程式の解から単位ベクトルとなるものを選ぶ。</ref>。
:<math>
\mathbf{w}_1 = \underset{\mathbf{w} \neq \mathbf{0}}{\operatorname{arg\,max}}
  \frac{\Vert \mathbf{X w} \Vert^2}{\Vert \mathbf{w} \Vert^2}.
</math>

右辺の最大化される量は {{math|'''X'''<sup>T</sup>'''X'''}} に対する[[レイリー商]]と見ることができる。{{math|'''X'''<sup>T</sup>'''X'''}} は[[対称行列]]だから、レイリー商の最大値は行列の最大[[固有値]]となり、それに伴い負荷量ベクトルは対応する[[固有ベクトル]]となる。

第一負荷量ベクトル {{math|'''w'''<sub>1</sub>}} が得られれば、データ点 {{math|'''x'''<sub>''i''</sub>}} に対応する主成分得点 {{math|1=''t''<sub>1(''i'')</sub> = '''x'''<sub>''i''</sub> &middot; '''w'''<sub>1</sub>}}、あるいは対応するベクトル {{math|('''x'''<sub>''i''</sub> &middot; '''w'''<sub>1</sub>)'''w'''<sub>1</sub>}} が得られる。

=== 他の主成分 ===
{{mvar|k}} 番目の主成分は {{math|''k'' &minus; 1}} 番目までの主成分をデータ行列 {{math|'''X'''}} から取り除くことで得られる：

:<math>\mathbf{\hat{X}}_k = \mathbf{X} - \sum_{s = 1}^{k - 1} \mathbf{X} \mathbf{w}_s \mathbf{w}_s^{\rm T} .</math>

負荷量ベクトルは新たなデータ行列に対して主成分得点の分散が最大となるようなベクトルとして与えられる。
:<math>
\mathbf{w}_k 
= \underset{\Vert \mathbf{w} \Vert = 1}{\operatorname{arg\,max}}
  \Vert \mathbf{\hat{X}}_{k} \mathbf{w} \Vert^2 
= \underset{\mathbf{w} \neq \mathbf{0}}{\operatorname{arg\,max}}
  \tfrac{\Vert \mathbf{\hat{X}}_{k} \mathbf{w} \Vert^2}{\Vert \mathbf{w} \Vert^2}.
</math>

このことから、新たな負荷量ベクトルは対称行列 {{math|'''X'''<sup>T</sup>'''X'''}} の固有ベクトルであり、右辺の括弧内の量の最大値は対応する固有値を与えることが分かる。したがって、すべての負荷量ベクトルは {{math|'''X'''<sup>T</sup>'''X'''}} の固有ベクトルである。

データ点 {{math|'''x'''<sub>''i''</sub>}} の第 {{mvar|k}} 主成分は主成分得点 {{math|1=''t''<sub>''k''(''i'')</sub> = '''x'''<sub>''i''</sub> &middot; '''w'''<sub>''k''</sub>}} として負荷量ベクトルを基底とする表示が与えられ、また対応するベクトルは主成分得点に対応する基底ベクトルをかけた {{math|('''x'''<sub>''i''</sub> &middot; '''w'''<sub>''k''</sub>) '''w'''<sub>''k''</sub>}} となる。ここで {{math|'''w'''<sub>''k''</sub>}} は行列 {{math|'''X'''<sup>T</sup>'''X'''}} の第 {{mvar|k}} 固有ベクトルである。

{{math|'''X'''}} の完全な主成分分解は以下のように表わすことができる。
:<math>\mathbf{T} = \mathbf{X} \mathbf{W}</math>
ここで {{math|'''W'''}} は {{math|''p'' &times; ''p''}} の[[正方行列]]であり、各列ベクトルは行列の {{math|'''X'''<sup>T</sup>'''X'''}} の[[固有ベクトル]]であり[[単位ベクトル]]である。

=== 共分散 ===

{{math|'''X'''<sup>T</sup>'''X'''}} はデータセット {{math|'''X'''}}から与えられる経験的な標本[[共分散行列]]に[[比例]]する。

データセット {{math|'''X'''}} に対する、2つの異なる主成分の間の標本共分散 {{mvar|Q}} は以下のようにして得られる：

:<math>\begin{align}
Q(\mathrm{PC}_j, \mathrm{PC}_k) 
& \propto (\mathbf{X}\mathbf{w}_j)^\mathrm{T} \cdot (\mathbf{X}\mathbf{w}_k) \\
& = \mathbf{w}_j^\mathrm{T} \mathbf{X}^\mathrm{T} \mathbf{X} \mathbf{w}_k \\
& ~\overset{(\ast)}{=} \mathbf{w}_j^\mathrm{T} \lambda_k \mathbf{w}_k 
    \qquad (\mathbf{X}^\mathrm{T}\mathbf{X}\mathbf{w}_k = \lambda_k\mathbf{w}_k) \\
& = \lambda_k \Vert \mathbf{w}_k \Vert^2.
\end{align}</math>

(&lowast;) の変形において、{{math|'''w'''<sub>''k''</sub>}} が[[行列]] {{math|'''X'''<sup>T</sup>'''X'''}} の[[固有値]] {{math|''&lambda;''<sub>''k''</sub>}} に対応する[[固有ベクトル]]であることを利用した。{{math|'''X'''<sup>T</sup>'''X'''}} は対称行列であり、対称行列の異なる固有値に対応する固有ベクトル達は互いに[[直交]]するから、結局データセット {{math|'''X'''}} に対する異なる主成分間の標本共分散 {{math|''Q''(PC<sub>''j''</sub>, PC<sub>''k''</sub>)}} はゼロとなる。

上述の結果を言い換えると、主成分変換は経験的な標本共分散行列を[[対角化]]する座標変換であると特徴づけられる。

元々の基底に対する経験共分散行列 {{math|'''Q'''}} は行列記法によって以下のように表わすことができる。
:<math>
\mathbf{Q} \propto \mathbf{X}^\mathrm{T} \mathbf{X} 
= \mathbf{W} \mathbf{\Lambda} \mathbf{W}^\mathrm{T}.
</math>
ここで {{math|'''&Lambda;'''}} は {{math|'''X'''<sup>T</sup>'''X'''}} の固有値 {{math|''λ''<sub>''k''</sub>}} からなる[[対角行列]]である。固有値 {{math|''λ''<sub>''k''</sub>}} は対応する添え字の主成分得点の二乗和に等しい。
:<math>
\lambda_k = \|\mathbf{X}\mathbf{w}_k\|^2 
= \sum_{i=1}^n (\mathbf{x}_i \cdot \mathbf{w}_k)^2 
= \sum_{i=1}^n t_{k(i)}^2.
</math>

行列 {{math|'''W'''}} が得られれば、行列 {{math|'''W'''}} の[[直交行列|直交性]]を利用して、主成分ベクトルを基底とする経験共分散行列として次の表示が得られる。
:<math>
\mathbf{W}^\mathrm{T} \mathbf{Q} \mathbf{W} 
\propto \mathbf{W}^\mathrm{T} \mathbf{W} \, \mathbf{\Lambda} \, \mathbf{W}^\mathrm{T} \mathbf{W}
=  \mathbf{\Lambda}.
</math>

=== 次元削減 ===
[[線型写像|線型変換]] {{math|1='''T''' = '''XW'''}} はデータ点 {{math|'''x'''<sub>''i''</sub>}} を元の {{mvar|p}} [[ハメル次元|次元]]の[[ベクトル空間|空間]]から、与えられたデータセットに対して各成分が互いに無[[相関]]になるような {{mvar|p}} 次元の空間へ[[写像|写す]]が、一部の主成分だけを残すような変換も考えることができる。第一主成分から順に、各主成分に関するデータの[[分散 (確率論)|分散]]が[[単調減少]]するように負荷量ベクトルが得られるため、最初の {{mvar|L}} 個の負荷量ベクトルだけを残し、残りの説明能力の低い負荷量ベクトルを無視すると、次のような変換が得られる。
:<math>\mathbf{T}_L = \mathbf{X} \mathbf{W}_L</math>
{{math|'''W'''<sub>''L''</sub>}} は {{math|''p'' &times; ''L''}} の行列であり、{{math|'''T'''<sub>''L''</sub>}} は {{math|''n'' &times; ''L''}} の行列である。上記の変換はデータ点 {{math|'''x''' &isin; R<sup>''p''</sup>}} に対する変換として<ref group="注">{{math|R<sup>''p''</sup>}} は {{mvar|p}} 次元の[[実数空間]]を表わす。</ref>、{{math|1=''t'' = '''W'''<sup>T</sup>'''x''' ('''t''' &isin; R<sup>''L''</sup>)}} と書くこともできる。つまり、主成分分析は {{mvar|p}} 個の特徴量を持つデータ点 {{math|'''x'''}} を {{mvar|L}} 個の互いに無相関な特徴量を持つ主成分得点 {{math|'''t'''}} へ写す線型変換 {{math|'''W''' : R<sup>''p''</sup> &rarr; R<sup>''L''</sup>}} を学習する手法であるといえる{{sfn|Bengio|Courville|Vincent|2013}}。
データ行列を変換することで得られる主成分得点行列は、元のデータセットの分散を保存し、二乗再構成誤差 {{en|(reconstruction error)}} の総和、
:<math>
\|\mathbf{T}\mathbf{W}^\mathrm{T} - \mathbf{T}_L\mathbf{W}_L^\mathrm{T}\|_2^2 
\qquad (\|\mathbf{X} - \mathbf{X}_L\|_2^2)
</math>
を最小化するように与えられる。

[[File:PCA of Haplogroup J using 37 STRs.png|thumb|right|354の個体について、37の[[Y染色体]]STRマーカーの反復回数から計算された {{仮リンク|Y-STR|en|Y-STR}} [[ハプロタイプ]]に対する主成分分析の結果。主成分分析により、個体のY染色体の遺伝的な系統について[[データ・クラスタリング|クラスタリング]]するようなマーカーの[[線型結合]]を得ることに成功している。]]
元のデータセットの分散をできる限り残すように次元削減することは、高次元のデータセットを可視化する上で重要である。例えば、主成分の数を {{math|1=''L'' = 2}} に選び、2つの主成分がなす平面にデータセットを射影すると、射影されたデータ点は主成分のなす平面に対して最もよく分散し、データに含まれる[[データ・クラスタリング|クラスタ]]はそれぞれ分離される。したがって、2つの主成分がなす平面はデータを平面上にプロットする上で都合がよい。射影平面として別の平面を選んだ場合、クラスタ間のばらつきは小さくなり互いに重なり合うようになるため、実質上はそれぞれのクラスタを[[分類 (統計学)|分類]]することが困難になってしまう。

[[回帰分析]]でも次元削減は有効である。回帰分析において、説明変数の数を増やすほど特定のデータに対して[[過剰適合]]したモデル、すなわち他のデータセットに対して誤った結果を与えるモデルを得がちである。モデル生成に使ったデータに対してモデルが過剰適合しないためには、説明変数の個数を適当に制限する必要があり、一つのアプローチとして、互いに強い[[相関]]を持つ説明変数を削減し、より少数の主成分によって回帰分析を行う方法がある。この方法を{{仮リンク|主成分回帰|en|principal component regression}}と呼ぶ。

次元削減は[[ノイズ]]の大きなデータを分析する上でも適切であることが多い。データ行列の各列、つまりそれぞれの特徴量に対して[[独立同分布]]な[[ガウシアンノイズ]]が含まれる場合、変換されたデータ行列 {{math|'''T'''}} の列にも同様に独立同分布なガウシアンノイズが含まれる（座標軸の回転操作 {{math|'''W'''}} に対して独立同分布な[[正規分布|ガウス分布]]は不変であるため）。しかしながら、最初の少数の主成分に関しては、全体の分散に比べてノイズに由来する分散が小さくなるため、[[SN比|シグナル・ノイズ比]]を高めることができる。主成分分析は主要な情報を少数の主成分に集中させるため、次元削減によってノイズが支配的な成分だけを捨て、データ構造を反映した有用な成分を取り出すことができる。

=== 特異値分解 ===
主成分変換は[[行列]]の[[特異値分解]]とも結び付けられる。行列 {{math|'''X'''}} の特異値分解は以下の形式で与えられる。
:<math>\mathbf{X} = \mathbf{U}\mathbf{\Sigma}\mathbf{W}^\mathrm{T}.</math>
ここで、{{math|'''Σ'''}} は {{math|''n'' &times; ''p''}} の矩形[[対角行列]]であり、対角成分 {{math|''σ''<sub>''k''</sub>}} が正の行列である。{{math|'''Σ'''}} の対角成分を行列 {{math|'''X'''}} の特異値という。{{math|'''U'''}} は {{math|''n'' &times; ''n''}} の[[正方行列]]であり、各列が互いに[[直交]]する {{mvar|n}} 次元の[[単位ベクトル]]<ref group="注">これらのベクトルは[[正規直交系]]をなす。</ref>となる行列（つまり[[直交行列]]）である。各々の単位ベクトルは行列 {{math|'''X'''}} の左特異ベクトルと呼ばれる。同様に {{math|'''W'''}} は、各列が互いに[[直交]]する {{mvar|p}} 次元の[[単位ベクトル]]となる {{math|''p'' &times; ''p''}} の正方行列である。こちらの単位ベクトルは行列 {{math|'''X'''}} の右特異ベクトルと呼ばれる。

{{math|'''X'''}} の特異値分解に基づいて {{math|'''X'''<sup>T</sup>'''X'''}} を表わせば、以下のようになる。
:<math>\begin{align}
\mathbf{X}^\mathrm{T}\mathbf{X} 
&= \mathbf{W}\mathbf{\Sigma}\mathbf{U}^\mathrm{T} \mathbf{U}\mathbf{\Sigma}\mathbf{W}^\mathrm{T} \\
&= \mathbf{W}\mathbf{\Sigma}^2\mathbf{W}^\mathrm{T}
\end{align}</math>

[[#共分散|前節で示した]] {{math|'''X'''<sup>T</sup>'''X'''}} の固有値分解と見比べると、{{math|'''X'''}} の右特異ベクトルの組 {{math|'''W'''}} はまた {{math|'''X'''<sup>T</sup>'''X'''}} の[[固有ベクトル]]の組でもあり、{{math|'''X'''}} の特異値 {{math|''σ''<sub>''k''</sub>}} は {{math|'''X'''<sup>T</sup>'''X'''}} の固有値 {{math|''λ''<sub>''k''</sub>}} の[[平方根]]に等しいことが分かる。

特異値分解を主成分得点行列 {{math|'''T'''}} に対して行うと、以下のような分解が得られる。
:<math>\begin{align}
\mathbf{T} & = \mathbf{X} \mathbf{W} \\
           & = \mathbf{U}\mathbf{\Sigma}\mathbf{W}^\mathrm{T} \mathbf{W} \\
           & = \mathbf{U}\mathbf{\Sigma}.
\end{align}</math>
{{math|'''T'''}} の各列は {{math|'''X'''}} の左特異ベクトルに対応する特異値をかけたものとして表わされることが分かる。この結果は {{math|'''T'''}} の{{仮リンク|極分解|en|polar decomposition}}によっても得られる。

主成分分析の実装として、{{math|'''X'''}} の特異値分解の[[アルゴリズム]]がしばしば利用される。

{{math|''n'' × ''L''}} に次元削減された主成分得点行列 {{math|'''T'''<sub>L</sub>}} は、[[#次元削減|固有値分解の場合]]と同様に、寄与の大きい最初の {{mvar|L}} 個の特異値とそれに対応する左特異ベクトルだけを残すことによっても得られる：
:<math>\mathbf{T}_L = \mathbf{U}_L\mathbf{\Sigma}_L = \mathbf{X} \mathbf{W}_L. </math>
特異値分解から寄与の小さな特異値を除いて {{math|'''T'''<sub>''L''</sub>}} を作るということは、元の行列との[[フロベニウスノルム]]で測った差を最小化するような[[行列の階数|階数]] {{mvar|L}} の行列を選ぶことに相当する。この結果はエッカート・ヤング定理として知られる。
<!--
== Further considerations ==

Given a set of points in [[Euclidean space]], the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line. The second principal component corresponds to the same concept after all correlation with the first principal component has been subtracted from the points. The singular values (in '''Σ''') are the square roots of the [[eigenvalue]]s of the matrix '''X'''<sup>T</sup>'''X'''. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is correlated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see [[Principle Component Analysis#PCA and Information Theory|below]]). PCA is often used in this manner for [[dimensionality reduction]]. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example and when applicable, to the [[discrete cosine transform]], and in particular to the DCT-II which is simply known as the "DCT". [[Nonlinear dimensionality reduction]] techniques tend to be more computationally demanding than PCA.

PCA is sensitive to the scaling of the variables. If we have just two variables and they have the same [[sample variance]] and are positively correlated, then the PCA will entail a rotation by 45° and the "loadings" for the two variables with respect to the principal component will be equal. But if we multiply all values of the first variable by 100, then the first principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Note that Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.

Mean subtraction (a.k.a. "mean centering") is necessary for performing PCA to ensure that the first principal component describes the direction of maximum variance. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data. A mean of zero is needed for finding a basis that minimizes the [[Minimum mean square error|mean square error]] of the approximation of the data.<ref>A. A. Miranda, Y. A. Le Borgne, and G. Bontempi. [http://www.ulb.ac.be/di/map/yleborgn/pub/NPL_PCA_07.pdf New Routes from Minimal Approximation Error to Principal Components], Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer</ref>

Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: ''Pearson Product-Moment Correlation''). Also see the article by Kromrey & Foster-Johnson (1998) on ''"Mean-centering in Moderated Regression: Much Ado About Nothing".''

An [[autoencoder]] [[Artificial neural network|neural network]] with a linear hidden layer is similar to PCA. Upon convergence, the weight vectors of the ''K'' neurons in the hidden layer will form a basis for the space spanned by the first ''K'' principal components. Unlike PCA, this technique will not necessarily produce [[orthogonal]] vectors.

PCA is a popular primary technique in [[pattern recognition]]. It is not, however, optimized for class separability.<ref>{{Cite book| author=Fukunaga, Keinosuke | title = Introduction to Statistical Pattern Recognition |publisher=Elsevier | year = 1990 | url=https://books.google.com/books?visbn=0-12-269851-7| isbn=0-12-269851-7}}</ref> An alternative is the [[linear discriminant analysis]], which does take this into account.
-->
<!--
== Table of symbols and abbreviations ==

{| class="wikitable"
|-
! Symbol
! Meaning
! Dimensions
! Indices
|-
| <math>\mathbf{X} = \{ X[i,j] \}</math>
| data matrix, consisting of the set of all data vectors, one vector per row
| <math> n \times p</math>
| <math> i = 1 \ldots n </math> <br /> <math> j = 1 \ldots p </math>
|-
| <math>n \,</math>
| the number of row vectors in the data set
| <math>1 \times 1</math>
| ''scalar''
|-
| <math>p \,</math>
| the number of elements in each row vector (dimension)
| <math>1 \times 1</math>
| ''scalar''
|-
| <math>L \,</math>
| the number of dimensions in the dimensionally reduced subspace, <math> 1 \le L \le p </math>
| <math>1 \times 1</math>
| ''scalar''
|-
| <math>\mathbf{u} = \{ u[j] \}</math>
| vector of empirical [[mean]]s, one mean for each column ''j'' of the data matrix
| <math> p \times 1</math>
| <math> j = 1 \ldots p </math>
|-
| <math>\mathbf{s} = \{ s[j] \}</math>
| vector of empirical [[standard deviation]]s, one standard deviation for each column ''j'' of the data matrix
| <math> p \times 1</math>
| <math> j = 1 \ldots p </math>
|-
| <math>\mathbf{h} = \{ h[i] \}</math>
| vector of all 1's
| <math> 1 \times n</math>
| <math> i = 1 \ldots n </math>
|-
| <math>\mathbf{B} = \{ B[i,j] \}</math>
| [[Standard deviation|deviations]] from the mean of each column ''j'' of the data matrix
| <math> n \times p</math>
| <math> i = 1 \ldots n </math> <br /> <math> j = 1 \ldots p </math>
|-
| <math>\mathbf{Z} = \{ Z[m,n] \} </math>
| [[z-score]]s, computed using the mean and standard deviation for each row ''m'' of the data matrix
| <math> n \times p</math>
| <math> i = 1 \ldots n </math> <br /> <math> j = 1 \ldots p </math>
|-
| <math>\mathbf{C} = \{ C[k, l] \} </math>
| [[covariance matrix]]
| <math> p \times p </math>
| <math> k = 1 \ldots p </math> <br /> <math> l = 1 \ldots p </math>
|-
| <math>\mathbf{R} = \{ R[k, l] \} </math>
| [[correlation matrix]]
| <math> p \times p </math>
| <math> k = 1 \ldots p </math> <br /> <math> l = 1 \ldots p </math>
|-
| <math> \mathbf{V} = \{ V[j, k] \} </math>
| matrix consisting of the set of all [[eigenvectors]] of '''C''', one eigenvector per column
| <math> p \times p </math>
| <math> j = 1 \ldots p </math> <br /> <math> k = 1 \ldots p </math>
|-
| <math>\mathbf{D} = \{ D[k, l] \} </math>
| [[diagonal matrix]] consisting of the set of all [[eigenvalues]] of '''C''' along its [[principal diagonal]], and 0 for all other elements
| <math> p \times p </math>
| <math> k = 1 \ldots p </math> <br /> <math> l = 1 \ldots p </math>
|-
| <math>\mathbf{W} = \{ W[j, k] \} </math>
| matrix of basis vectors, one vector per column, where each basis vector is one of the eigenvectors of '''C''', and where the vectors in '''W''' are a sub-set of those in '''V'''
| <math> p \times L</math>
| <math> j = 1 \ldots p </math> <br /> <math> k = 1 \ldots L</math>
|-
| <math>\mathbf{T} = \{ T[i, k] \} </math>
| matrix consisting of ''n'' row vectors, where each vector is the projection of the corresponding data vector from matrix '''X''' onto the basis vectors contained in the columns of matrix '''W'''.
| <math> n \times L</math>
| <math> i = 1 \ldots n </math> <br /> <math> k = 1 \ldots L</math>
|}
-->
<!--
== Properties and limitations of PCA ==

=== Properties{{harvnb|Jolliffe|2002}} ===

:<big>'''''Property 1'':'''</big> For any integer ''q, 1 ≤ q ≤ p,'' consider the orthogonal [[linear transformation]]
::<math>y =\mathbf{B'}x</math>
:where <math>y</math> is a ''q-element'' vector and <math>\mathbf{B'}</math> is a ''(q × p)'' matrix, and let <math>\mathbf{{\Sigma}}_y = \mathbf{B'}\mathbf{\Sigma}\mathbf{B}</math> be the [[variance]]-[[covariance]] matrix for <math>y</math>. Then the trace of <math>\mathbf{\Sigma}_y</math>, denoted <math>\text{tr} (\mathbf{\Sigma}_y)</math>, is maximized by taking <math>\mathbf{B} = \mathbf{A}_q</math>, where <math>\mathbf{A}_q</math> consists of the first ''q'' columns of <math>\mathbf{A}</math> <math>(\mathbf{B'}</math> is the transposition of <math>\mathbf{B})</math>.

:<big>'''''Property 2'':'''</big> Consider again the [[orthonormal transformation]]
::<math>y = \mathbf{B'}x</math>
:with <math>x, \mathbf{B}, \mathbf{A}</math> and <math>\mathbf{\Sigma}_y</math> defined as before. Then <math>\text{tr}(\mathbf{\Sigma}_y)</math> is minimized by taking <math>\mathbf{B} = \mathbf{A}_q^*,</math> where <math>\mathbf{A}_q^*</math> consists of the last ''q'' columns of <math>\mathbf{A}</math>.

The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear relationships between the elements of {{mvar|x}}, and they may also be useful in [[regression analysis|regression]], in selecting a subset of variables from {{mvar|x}}, and in outlier detection.

:<big>'''''Property 3'':'''</big> (Spectral Decomposition of {{math|'''Σ'''}})
::<math>\mathbf{{\Sigma}} = \lambda_{1}\alpha_{1}\alpha_{1}' + \cdots + \lambda_{p}\alpha_{p}\alpha_{p}'</math>

Before we look at its usage, we first look at [[diagonal]] elements,
:<math>\text{Var}(x_j) = \sum_{k=1}^P \lambda_{k}\alpha_{kj}^2</math>
Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of {{mvar|x}} into decreasing contributions due to each PC, but we can also decompose the whole [[covariance matrix]] into contributions <math>\lambda_{k}\alpha_{k}\alpha_{k}'</math> from each PC. Although not strictly decreasing, the elements of <math>\lambda_{k}\alpha_{k}\alpha_{k}'</math> will tend to become smaller as <math>k</math> increases, as <math>\lambda_{k}\alpha_{k}\alpha_{k}'</math> decreases for increasing <math>k</math>, whereas the elements of <math>\alpha_{k}</math> tend to stay 'about the same size'because of the normalization constraints: <math>\alpha_{k}'\alpha_{k}=1, k=1,\cdots, p</math>

=== Limitations ===
As noted above, the results of PCA depend on the scaling of the variables. A scale-invariant form of PCA has been developed.<ref name=Leznik>Leznik, M; Tofallis, C. 2005 [uhra.herts.ac.uk/bitstream/handle/2299/715/S56.pdf Estimating Invariant Principal Components Using Diagonal Regression.]</ref>

The applicability of PCA is limited by certain assumptions<ref>Jonathon Shlens, [http://arxiv.org/abs/1404.1100 A Tutorial on Principal Component Analysis.]</ref> made in its derivation.

=== PCA and information theory ===
The claim that the PCA used for dimensionality reduction preserves most of the "information" of the data is misleading. Indeed, without any assumption on the signal model, PCA cannot help but reduce the amount of information lost during dimensionality reduction, where information is measured using [[Entropy (information theory)|Shannon entropy]].<ref>{{cite journal|last=Geiger|first=Bernhard|author2=Kubin, Gernot|title=Relative Information Loss in the PCA|journal=Proc. IEEE Information Theory Workshop|date=Sep 2012|pages=562–566|url=http://arxiv.org/abs/1204.0429}}</ref>

Under the assumption that

:<math>\mathbf{x}=\mathbf{s}+\mathbf{n}</math>

i.e., that the data vector <math>\mathbf{x}</math> is the sum of the desired information-bearing signal <math>\mathbf{s}</math> and a noise signal <math>\mathbf{n}</math> one can show that PCA can be optimal for dimensionality reduction also from an information-theoretic point-of-view.

In particular, Linsker showed that if <math>\mathbf{s}</math> is Gaussian and <math>\mathbf{n}</math> is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the [[mutual information]] <math>I(\mathbf{y};\mathbf{s})</math> between the desired information <math>\mathbf{s}</math> and the dimensionality-reduced output <math>\mathbf{y}=\mathbf{W}^\mathrm{T}_L\mathbf{x}</math>.<ref>{{cite journal|last=Linsker|first=Ralph|title=Self-organization in a perceptual network|journal=IEEE Computer|date=March 1988|volume=21|issue=3|pages=105–117|doi=10.1109/2.36}}</ref>

If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (i.e., the components of the vector <math>\mathbf{n}</math> are [[iid]]), but the information-bearing signal <math>\mathbf{s}</math> is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the ''information loss'', which is defined as<ref>{{cite book|last=Deco & Obradovic|title=An Information-Theoretic Approach to Neural Computing|year=1996|publisher=Springer|location=New York, NY}}</ref><ref>{{cite journal|last=Plumbley|first=Mark|title=Information theory and unsupervised neural networks|year=1991}}Tech Note</ref>

:<math>I(\mathbf{x};\mathbf{s})-I(\mathbf{y};\mathbf{s}).</math>

The optimality of PCA is also preserved if the noise <math>\mathbf{n}</math> is iid and at least more Gaussian (in terms of the [[Kullback–Leibler divergence]]) than the information-bearing signal <math>\mathbf{s}</math>.<ref>{{cite journal|last=Geiger|first=Bernhard|author2=Kubin, Gernot|title=Signal Enhancement as Minimization of Relevant Information Loss|journal=Proc. ITG Conf. on Systems, Communication and Coding|date=January 2013|url=http://arxiv.org/abs/1205.6935}}</ref> In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise <math>\mathbf{n}</math> becomes dependent.
-->
<!--
== Computing PCA using the covariance method ==

The following is a detailed description of PCA using the covariance method (see also [http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf here]) as opposed to the correlation method.<ref>{{cite web|title=Engineering Statistics Handbook Section 6.5.5.2|url=http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc552.htm|accessdate=19 January 2015}}</ref> But note that it is better to use the singular value decomposition (using standard software). {{Citation needed|date = December 2013}}

The goal is to transform a given data set '''X''' of dimension ''p'' to an alternative data set '''Y''' of smaller dimension ''L''. Equivalently, we are seeking to find the matrix '''Y''', where '''Y''' is the  [[Karhunen–Loève theorem|Kosambi-Karhunen–Loève]] transform (KLT) of matrix '''X''':

:<math> \mathbf{Y} = \mathbb{KLT} \{ \mathbf{X} \} </math>

=== Organize the data set ===

'''Suppose''' you have data comprising a set of observations of ''p'' variables, and you want to reduce the data so that each observation can be described with only ''L'' variables, ''L'' < ''p''.  Suppose further, that the data are arranged as a set of ''n'' data vectors <math>\mathbf{x}_1 \ldots \mathbf{x}_n</math> with each <math>\mathbf{x}_i </math> representing a single grouped observation of the ''p'' variables.

* Write <math>\mathbf{x}_1 \ldots \mathbf{x}_n</math> as row vectors, each of which has ''p'' columns.
* Place the row vectors into a single matrix '''X''' of dimensions ''n'' × ''p''.

=== Calculate the empirical mean ===

* Find the empirical mean along each dimension ''j'' = 1,&nbsp;...,&nbsp;''p''.
* Place the calculated mean values into an empirical mean vector '''u''' of dimensions ''p'' × 1.
::<math>u[j] = {1 \over n} \sum_{i=1}^n X[i,j] </math>

=== Calculate the deviations from the mean ===
Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data.<ref>A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. [http://www.ulb.ac.be/di/map/yleborgn/pub/NPL_PCA_07.pdf New Routes from Minimal Approximation Error to Principal Components], Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer</ref> Hence we proceed by centering the data as follows:
* Subtract the empirical mean vector '''u''' from each row of the data matrix '''X'''.
* Store mean-subtracted data in the ''n'' × ''p'' matrix '''B'''.

::<math>\mathbf{B} = \mathbf{X} - \mathbf{h}\mathbf{u}^{T} </math>
:: where '''h''' is an {{math|''n'' × 1}} column vector of all&nbsp;1s:

:::<math>h[i] = 1 \, \qquad \qquad \text{for } i = 1, \ldots, n </math>

=== Find the covariance matrix ===

* Find the ''p'' × ''p'' empirical [[covariance matrix]] '''C''' from the [[outer product]] of matrix '''B''' with itself:
::<math>\mathbf{C} = { 1 \over {n-1} } \mathbf{B}^{*} \cdot \mathbf{B}</math>
:where <math> *</math> is the [[conjugate transpose]] operator. Note that if '''B''' consists entirely of real numbers, which is the case in many applications, the "conjugate transpose" is the same as the regular [[transpose]].

* Please note that outer products apply to vectors. For tensor cases we should apply tensor products, but the covariance matrix in PCA is a sum of outer products between its sample vectors; indeed, it could be represented as B*.B. See the covariance matrix sections on the discussion page for more information.
* The reasoning behind using {{math|''N'' − 1}} instead of ''N'' to calculate the covariance is [[Bessel's correction]]

=== Find the eigenvectors and eigenvalues of the covariance matrix ===

* Compute the matrix '''V''' of [[eigenvector]]s which [[diagonalizable matrix|diagonalizes]] the covariance matrix '''C''':

::<math>\mathbf{V}^{-1} \mathbf{C} \mathbf{V} = \mathbf{D} </math>

: where '''D''' is the [[diagonal matrix]] of [[eigenvalue]]s of '''C'''. This step will typically involve the use of a computer-based algorithm for computing eigenvectors and eigenvalues. These algorithms are readily available as sub-components of most [[matrix algebra]] systems, such as [[R (programming language)|R]], [[MATLAB]],<ref>[http://www.mathworks.com/access/helpdesk/help/techdoc/ref/eig.html#998306 eig function] Matlab documentation</ref><ref>[http://www.mathworks.com/matlabcentral/fileexchange/24634 MATLAB PCA-based Face recognition software]</ref> [[Mathematica]],<ref>[http://reference.wolfram.com/mathematica/ref/Eigenvalues.html Eigenvalues function] Mathematica documentation</ref> [[SciPy]], [[IDL (programming language)|IDL]] ([[Interactive Data Language]]), or [[GNU Octave]] as well as [[OpenCV]].
* Matrix '''D''' will take the form of an ''p'' × ''p'' diagonal matrix, where
::<math>D[k,l] = \lambda_k \qquad \text{for } k = l</math>
:is the ''j''th eigenvalue of the covariance matrix '''C''', and
::<math>D[k,l] = 0 \qquad \text{for } k \ne l.</math>
* Matrix '''V''', also of dimension ''p'' × ''p'', contains ''p'' column vectors, each of length ''p'', which represent the ''p'' eigenvectors of the covariance matrix '''C'''.
* The eigenvalues and eigenvectors are ordered and paired. The ''j''th eigenvalue corresponds to the ''j''th eigenvector.

=== Rearrange the eigenvectors and eigenvalues ===

* Sort the columns of the eigenvector matrix '''V''' and eigenvalue matrix '''D''' in order of ''decreasing'' eigenvalue.
* Make sure to maintain the correct pairings between the columns in each matrix.

=== Compute the cumulative energy content for each eigenvector ===

* The eigenvalues represent the distribution of the source data's energy{{Clarify|date=March 2011}} among each of the eigenvectors, where the eigenvectors form a [[basis (linear algebra)|basis]] for the data. The cumulative energy content ''g'' for the ''j''th eigenvector is the sum of the energy content across all of the eigenvalues from 1 through ''j'':

::<math>g[j] = \sum_{k=1}^j D[k,k] \qquad \mathrm{for} \qquad j = 1,\dots,p </math>{{Citation needed|date=March 2011}}

=== Select a subset of the eigenvectors as basis vectors ===

* Save the first ''L'' columns of '''V''' as the ''p'' × ''L'' matrix '''W''':

::<math> W[k,l] = V[k,l] \qquad \mathrm{for} \qquad k = 1,\dots,p \qquad l = 1,\dots,L </math>

: where

::<math>1 \leq L \leq p.</math>
* Use the vector '''g''' as a guide in choosing an appropriate value for ''L''. The goal is to choose a value of ''L'' as small as possible while achieving a reasonably high value of ''g'' on a percentage basis. For example, you may want to choose ''L'' so that the cumulative energy ''g'' is above a certain threshold, like 90 percent. In this case, choose the smallest value of ''L'' such that

::<math> \frac{g[L]}{g[p]} \ge 0.9\, </math>

=== Convert the source data to z-scores (optional) ===

* Create an ''p'' × 1 empirical standard deviation vector '''s''' from the square root of each element along the main diagonal of the diagonalized covariance matrix '''C'''. (Note, that scaling operations do not commute with the KLT thus we must scale by the variances of the already-decorrelated vector, which is the diagonal of '''C''') :
::<math> \mathbf{s} = \{ s[j] \} = \{ \sqrt{C[j,j]} \} \qquad \text{for } j = 1, \ldots, p </math>
* Calculate the ''n'' × ''p'' [[standard score|z-score]] matrix:
::<math> \mathbf{Z} = { \mathbf{B} \over \mathbf{h} \cdot \mathbf{s}^{T} } </math> (divide element-by-element)
* Note: While this step is useful for various applications as it normalizes the data set with respect to its variance, it is not integral part of PCA/KLT

=== Project the z-scores of the data onto the new basis ===

* The projected vectors are the columns of the matrix
::<math> \mathbf{T} =  \mathbf{Z} \cdot \mathbf{W} = \mathbb{KLT} \{ \mathbf{X} \}.</math>
* The rows of matrix '''T''' represent the  [[Karhunen–Loève theorem|Kosambi-Karhunen–Loève]] transforms (KLT) of the data vectors in the rows of matrix&nbsp;'''X'''.
-->
<!--
== Derivation of PCA using the covariance method ==

Let '''X''' be a ''d''-dimensional random vector expressed as column vector. Without loss of generality, assume '''X''' has zero mean.

We want to find <math>(\ast)\,</math> a {{math|''d'' × ''d''}} [[orthonormal basis|orthonormal transformation matrix]] '''P''' so that '''PX''' has a diagonal covariant matrix (''i.e.'' '''PX''' is a random vector with all its distinct components pairwise uncorrelated).

A quick computation assuming <math>P</math> were unitary yields:

:<math>\begin{align}
\operatorname{var}(PX) &= \mathbb{E}[PX~(PX)^{\dagger}]\\
	&= \mathbb{E}[PX~X^{\dagger}P^{\dagger}]\\
	&= P~\mathbb{E}[XX^{\dagger}]P^{\dagger}\\
	&= P~\operatorname{var}(X)P^{-1}\\
\end{align}</math>

Hence <math>(\ast)\,</math> holds if and only if <math>\operatorname{var}(X)</math> were diagonalisable by <math>P</math>.

This is very constructive, as var('''X''') is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.

=== Iterative computation ===

In practical implementations especially with high dimensional data (large {{mvar|p}}), the covariance method is rarely used because it is not efficient. One way to compute the first principal component efficiently<ref name="roweis">Roweis, Sam. "EM Algorithms for PCA and SPCA." Advances in Neural Information Processing Systems. Ed. Michael I. Jordan, Michael J. Kearns, and Sara A. Solla The MIT Press, 1998.</ref> is shown in the following pseudo-code, for a data matrix {{math|'''X'''}} with zero mean, without ever computing its covariance matrix.

 {{math|'''r'''}} = a random vector of length {{mvar|p}}
 do {{mvar|c}} times:
       {{math|1='''s''' = 0}} (a vector of length {{mvar|p}})
       {{nowrap|for each row <math>\mathbf{x} \in \mathbf{X}</math>}}
             {{nowrap|<math>\mathbf{s} = \mathbf{s} + (\mathbf{x} \cdot \mathbf{r})\mathbf{x}</math>}}
       {{nowrap|<math>\mathbf{r} = \frac{\mathbf{s}}{|\mathbf{s}|}</math>}}
 return {{math|'''r'''}}

This algorithm is simply an efficient way of calculating {{math|'''X<sup>T</sup>X r'''}}, normalizing, and placing the result back in {{math|'''r'''}} ([[power iteration]]). It avoids the {{math|''np''<sup>2</sup>}} operations of calculating the covariance matrix.
{{math|'''r'''}} will typically get close to the first principal component of {{math|'''X'''}} within a small number of iterations, {{mvar|c}}. (The magnitude of {{math|'''s'''}} will be larger after each iteration. Convergence can be detected when it increases by an amount too small for the precision of the machine.)

Subsequent principal components can be computed by subtracting component {{math|'''r'''}} from {{math|'''X'''}} (see [[Gram–Schmidt]]) and then repeating this algorithm to find the next principal component. However this simple approach is not numerically stable if more than a small number of principal components are required, because imprecisions in the calculations will additively affect the estimates of subsequent principal components. More advanced methods build on this basic idea, as with the closely related [[Lanczos algorithm]].

One way to compute the eigenvalue that corresponds with each principal component is to measure the difference in mean-squared-distance between the rows and the centroid, before and after subtracting out the principal component. The eigenvalue that corresponds with the component that was removed is equal to this difference.

=== The NIPALS method ===

{{Main|Non-linear iterative partial least squares}}

For very-high-dimensional datasets, such as those generated in the *omics sciences (e.g., [[genomics]], [[metabolomics]]) it is usually only necessary to compute the first few PCs. The [[non-linear iterative partial least squares]] (NIPALS) algorithm calculates '''t<sub>1</sub>''' and '''w<sub>1</sub>'''<sup>T</sup> from '''X'''. The outer product, '''t<sub>1</sub>w<sub>1</sub>'''<sup>T</sup> can then be subtracted from '''X''' leaving the residual matrix '''E<sub>1</sub>'''. This can be then used to calculate subsequent PCs.<ref>{{Cite journal
  | last = Geladi
  | first = Paul
  | author-link =
  | last2 = Kowalski
  | first2 = Bruce
  | author2-link =
  | title = Partial Least Squares Regression:A Tutorial
  | journal = Analytica Chimica Acta
  | volume = 185
  | issue =
  | pages = 1–17
  | year = 1986
  | url =
  | doi = 10.1016/0003-2670(86)80028-9
  | id = }}
</ref> This results in a dramatic reduction in computational time since calculation of the covariance matrix is avoided.

However, for large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality due to machine precision limitations accumulated in each iteration step.<ref>{{cite book |last=Kramer |first=R. |year=1998 |title=Chemometric Techniques for Quantitative Analysis |publisher=CRC Press |location=New York |isbn= }}</ref> A Gram–Schmidt (GS) re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.<ref>{{cite journal |first=M. |last=Andrecut |title=Parallel GPU Implementation of Iterative PCA Algorithms |journal=Journal of Computational Biology |volume=16 |issue=11 |year=2009 |pages=1593–1599 |doi=10.1089/cmb.2008.0221 }}</ref>

=== Online/sequential estimation ===
In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.<ref>{{Cite journal
  | last = Warmuth
  | first = M. K.
  | last2 = Kuzmin
  | first2 = D.
  | title = Randomized online PCA algorithms with regret bounds that are logarithmic in the dimension
  | journal = Journal of Machine Learning Research
  | volume = 9
  | issue =
  | pages = 2287–2320
  | year = 2008
  | url =
  | doi = 
  | id = }}
</ref>
-->
<!--
== PCA and qualitative variables ==
In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These data were subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable ''species''.
For this, the following results are produced.
* Identification, on the factorial planes, of the different species e.g. using different colors.
* Representation, on the factorial planes, of the centers of gravity of plants belonging to the same species.
* For each center of gravity and each axis,  p-value to judge the significance of the difference between the center of gravity and origin.

These results are what is called ''introducing a qualitative variable as supplementary element''. This procedure is detailed in and Husson, Lê & Pagès 2009 and Pagès 2013.
Few software offer this option in an "automatic" way. This is the case of [http://www.coheris.com/produits/analytics/logiciel-data-mining/ SPAD] that historically, following the work of [[Ludovic Lebart]], was the first to propose this option, and the R package [http://factominer.free.fr/ FactoMineR].
-->
<!--
== Applications ==

=== Neuroscience ===
A variant of principal components analysis is used in [[neuroscience]] to identify the specific properties of a stimulus that increase a [[neuron]]'s probability of generating an [[action potential]].<ref name="brenner00">Brenner, N., Bialek, W., & de Ruyter van Steveninck, R.R. (2000).</ref> This technique is known as [[Spike-triggered covariance|spike-triggered covariance analysis]]. In a typical application an experimenter presents a [[white noise]] process as a stimulus (usually either as a sensory input to a test subject, or as a [[Electric current|current]] injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result. Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the [[covariance matrix]] of the ''spike-triggered ensemble'', the set of all stimuli (defined and discretized over a finite time window, typically on the order of 100 ms) that immediately preceded a spike. The [[Eigenvectors and eigenvalues|eigenvectors]] of the difference between the spike-triggered covariance matrix and the covariance matrix of the ''prior stimulus ensemble'' (the set of all stimuli, defined over the same length time window) then indicate the directions in the [[Vector space|space]] of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble. Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spike-triggered ensemble showed the largest positive change compared to the variance of the prior. Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features.

In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential. [[Spike sorting]] is an important procedure because [[Electrophysiology#Extracellular recording|extracellular]] recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs [[Cluster analysis|clustering analysis]] to associate specific action potentials with individual neurons.

PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, i.e. [[order parameters]], during [[phase transitions]] in the brain.<ref>{{cite journal|last1=Jirsa|first1=Victor|last2=Friedrich|first2=R|last3=Haken|first3=Herman|last4=Kelso|first4=Scott|title=A theoretical model of phase transitions in the human brain.|journal=Biological Cybernetics|date=1994|volume=71|issue=1|pages=27–35|doi=10.1007/bf00198909}}</ref>
-->
<!--
== Relation between PCA and ''K''-means clustering ==
It was asserted in <ref>{{cite journal|authors=H. Zha, C. Ding, M. Gu, X. He and H.D. Simon|title=Spectral Relaxation for K-means Clustering|journal=Neural Information Processing Systems vol.14 (NIPS 2001)|pages=1057–1064|location=Vancouver, Canada|date=Dec 2001|url=http://ranger.uta.edu/~chqding/papers/Zha-Kmeans.pdf}}</ref><ref>{{cite journal|authors=Chris Ding and Xiaofeng He|title=K-means Clustering via Principal Component Analysis|work=Proc. of Int'l Conf. Machine Learning (ICML 2004)|pages=225–232|date=July 2004|url=http://ranger.uta.edu/~chqding/papers/KmeansPCA1.pdf}}</ref> that the relaxed solution of {{math|<var>k</var>}}-means clustering, specified by the cluster indicators, is given by the PCA (principal component analysis) principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of k-means clustering was not a new result (see, for example,<ref>{{cite journal | title = Clustering large graphs via the singular value decomposition | journal = Machine learning | year = 2004 | first = P. | last = Drineas |author2=A. Frieze |author3=R. Kannan |author4=S. Vempala |author5=V. Vinay  | volume = 56 | pages = 9–33| id = | url = http://www.cc.gatech.edu/~vempala/papers/dfkvv.pdf | accessdate = 2012-08-02 | doi=10.1023/b:mach.0000033113.59016.96}}</ref>), and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.<ref>{{cite journal | title = Dimensionality reduction for k-means clustering and low rank approximation (Appendix B) | journal =  | arxiv = | year = 2014 | first = M. | last = Cohen |author2=S. Elder |author3=C. Musco |author4=C. Musco |author5=M. Persu | url = http://arxiv.org/abs/1410.6801 | accessdate = 2014-11-29}}</ref>
-->
<!--
== Relation between PCA and factor analysis<ref>http://www.linkedin.com/groups/What-is-difference-between-factor-107833.S.162765950</ref> ==
Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The principal components can be used to find clusters in a set of data. PCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (i.e., translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors.

[[Factor analysis]] is similar to principal component analysis, in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlation-focused approach seeking to reproduce the inter-correlations among variables, in which the factors "represent the common variance of variables, excluding unique variance<ref>Timothy A. Brown. Confirmatory Factor Analysis for Applied Research Methodology in the social sciences. Guilford Press, 2006</ref>" . In terms of the correlation matrix, this corresponds with focusing on explaining the off-diagonal terms (i.e. shared co-variance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations.<ref>I.T. Jolliffe. Principal Component Analysis, Second Edition. Chapter 7. 2002</ref> Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (i.e., latent constructs or factors) or causal modeling.
-->
<!--
== Correspondence analysis ==

'''[[Correspondence analysis]]''' (CA)
was developed by [[Jean-Paul Benzécri]]<ref>{{Cite book
 | author = Benzécri, J.-P.
 | publisher=Dunod |location= Paris, France
 | year = 1973
 | title = L'Analyse des Données. Volume II. L'Analyse des Correspondances
 }}</ref>
and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to [[contingency tables]].
CA decomposes the [[chi-squared statistic]] associated to this table into orthogonal factors.<ref>{{Cite book
 | author = Greenacre, Michael
 | publisher=Academic Press |location= London
 | year = 1983
 | title = Theory and Applications of Correspondence Analysis
 | isbn = 0-12-299050-1
 }}</ref>
Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not.
Several variants of CA are available including [[detrended correspondence analysis]] and [[canonical correspondence analysis]]. One special extension is [[multiple correspondence analysis]], which may be seen as the counterpart of principal component analysis for categorical data.<ref>{{Cite book
 | author = Le Roux, Brigitte and Henry Rouanet
 | publisher=Kluwer|location= Dordrecht
 | year = 2004
 | title = Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis
 | isbn =
 }}</ref>
-->
<!--
== Generalizations ==

=== Nonlinear generalizations ===
[[File:Elmap breastcancer wiki.png|thumb|300px| Linear PCA versus nonlinear Principal Manifolds<ref>[[Alexander Nikolaevich Gorban|A. N. Gorban]], A. Y. Zinovyev, [http://arxiv.org/abs/0809.0490 Principal Graphs and Manifolds], In: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Olivas E.S. et al Eds. Information Science Reference, IGI Global: Hershey, PA, USA, 2009. 28–59.</ref> for [[Scientific visualization|visualization]] of [[breast cancer]] [[microarray]] data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and cannot be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The "basal" breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the [[elastic map]]s algorithm. Data are available for public competition.<ref>{{cite journal |last=Wang |first=Y. |last2=Klijn |first2=J. G. |last3=Zhang |first3=Y. |last4=Sieuwerts |first4=A. M. |last5=Look |first5=M. P. |last6=Yang |first6=F. |last7=Talantov |first7=D. |last8=Timmermans |first8=M. |last9=Meijer-van Gelder |first9=M. E. |last10=Yu |first10=J. |title=Gene expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer |journal=[[The Lancet]] |volume=365 |issue=9460 |pages=671–679 |year=2005 |doi=10.1016/S0140-6736(05)17947-1 |display-authors=etal}} [http://www.ihes.fr/~zinovyev/princmanif2006/ Data online]</ref> Software is available for free non-commercial use.<ref>{{cite web |first=A. |last=Zinovyev |url=http://bioinfo-out.curie.fr/projects/vidaexpert/ |title=ViDaExpert – Multidimensional Data Visualization Tool |work=[[Curie Institute (Paris)|Institut Curie]] |location=Paris }} (free for non-commercial use)</ref>]]

Most of the modern methods for [[nonlinear dimensionality reduction]] find their theoretical and algorithmic roots in PCA or K-means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of data points. '''Principal [[curve]]s and [[manifold]]s'''<ref>A.N. Gorban, B. Kegl, D.C. Wunsch, A. Zinovyev  (Eds.),  [http://pca.narod.ru/contentsgkwz.htm Principal Manifolds for Data Visualisation and Dimension Reduction,]
LNCSE 58, Springer, Berlin – Heidelberg – New York, 2007. ISBN 978-3-540-73749-0</ref> give the natural geometric framework for PCA generalization and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold for data [[approximation]], and by encoding using standard geometric [[Projection (mathematics)|projection]] onto the manifold, as it is illustrated by Fig.
See also the [[elastic map]] algorithm and [[principal geodesic analysis]]. Another popular generalization is [[kernel PCA]], which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel.

=== Multilinear generalizations ===
In [[multilinear subspace learning]],<ref>{{cite journal
 |first=Haiping |last=Lu
 |first2=K.N. |last2=Plataniotis
 |first3=A.N. |last3=Venetsanopoulos
 |url=http://www.dsp.utoronto.ca/~haiping/Publication/SurveyMSL_PR2011.pdf
 |title=A Survey of Multilinear Subspace Learning for Tensor Data
 |journal=Pattern Recognition
 |volume=44 |number=7 |pages=1540–1551 |year=2011
 |doi=10.1016/j.patcog.2011.01.004
}}</ref> PCA is generalized to [[multilinear principal component analysis|multilinear PCA]] (MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA.

=== Higher order ===
''N''-way principal component analysis may be performed with models such as [[Tucker decomposition]],  [[PARAFAC]], multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.

=== Robustness – weighted PCA ===
While PCA finds the mathematically optimal method (as in minimizing the squared error), it is sensitive to [[outlier]]s in the data that produce large errors PCA tries to avoid. It therefore is common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify. For example, in [[data mining]] algorithms like [[correlation clustering]], the assignment of points to clusters and outliers is not known beforehand. A recently proposed generalization of PCA<ref>{{Cite journal | doi = 10.1007/978-3-540-69497-7_27 | title = A General Framework for Increasing the Robustness of PCA-Based Correlation Clustering Algorithms | isbn = 978-3-540-69476-2 | series = Lecture Notes in Computer Science | journal = Scientific and Statistical Database Management| year = 2008 | last1 = Kriegel | first1 = H. P. | last2 = Kröger | first2 = P. | last3 = Schubert | first3 = E. | last4 = Zimek | first4 = A. | volume = 5069 | pages = 418–435 }}</ref> based on a '''weighted PCA''' increases robustness by assigning different weights to data objects based on their estimated relevancy.

=== Robust PCA via decomposition in low-rank and sparse matrices ===
[[Robust principal component analysis]] (RPCA) is a modification of the widely used statistical procedure principal component analysis (PCA) which works well with respect to grossly corrupted observations.

=== Sparse PCA ===
A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. [[Sparse PCA]] overcomes this disadvantage by finding linear combinations that contain just a few input variables.

== Similar techniques ==

=== Network component analysis ===
Given a matrix <math>E</math>, it tries to decompose it into two matrices such that <math>E=AP
</math>. A key difference from techniques such as PCA and ICA is that some of the entries of <math>A</math> are constrained to be 0. Here <math>P</math> is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :-
# <math>A</math> has full column rank
# Each column of <math>A</math> must have at least <math>L-1</math>  zeroes where <math>L</math> is the number of columns of <math>A</math> (or alternatively the number of rows of <math>P</math>). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank.
# <math>P</math> must have full row rank.
then the decomposition is unique up to multiplication by a scalar.<ref>{{Cite web|url = http://www.pnas.org/content/100/26/15522.full.pdf|title = Network component analysis: Reconstruction of regulatory signals in biological systems|date = |accessdate = February 10, 2015|website = |publisher = }}</ref>
-->

==ソフトウェア==
*[[Origin (グラフ作成ソフト)|Origin]] 「Pro」バージョンに主成分分析を含む[[多変量解析]]機能が含まれる。
*[http://www.okada.jp.org/RWiki/index.php?R%A4%CE%B4%F0%CB%DC%A5%D1%A5%C3%A5%B1%A1%BC%A5%B8%C3%E6%A4%CE%C2%BF%CA%D1%CE%CC%B2%F2%C0%CF%B4%D8%BF%F4%B0%EC%CD%F7 Rの基本パッケージ中の多変量解析関数一覧] 統計解析ツール「[[R言語]]」は主成分分析を始め[[多変量解析]]を標準で行える[[フリーウェア]]。他統計ソフトやExcelのファイル取込やODBC接続も可能。[[アメリカ食品医薬品局|FDA]]の申請にも使用を認められ、CRANという仕組で世界の膨大なアプリケーションを無償で使える。[[可視化]]機能に優れる。[[マルチプラットフォーム]]。
*SAS [http://www.ie.reitaku-u.ac.jp/~tak/datB/datB_prin00.pdf 主成分分析 (PCA: Principal Component Analysis)]
*SPSS [http://www.kisc.meiji.ac.jp/~nino/2005/nissi/0111.html 多変量解析の選び方・SPSSによる主成分分析] [http://www-06.ibm.com/software/jp/analytics/spss/analysis/pca.html IBM 主成分分析]

==脚注==
=== 注釈 ===
{{reflist|group="注"}}

=== 出典 ===
{{Reflist|30em}}

== 参考文献 ==
<!-- 本文中で引用されていないものをコメントアウト -->
* {{Cite journal|last = Pearson |first = K. |authorlink=カール・ピアソン |year = 1901 |title = On Lines and Planes of Closest Fit to Systems of Points in Space |journal = Philosophical Magazine |volume = 2 |issue = 11 |pages = 559–572 |url = http://stat.smmu.edu.cn/history/pearson1901.pdf |format=PDF |doi=10.1080/14786440109462720|ref = harv}}
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==関連項目==

* {{仮リンク|カルフネン・ロエヴェ定理|en|Karhunen–Loève theorem}}
* {{仮リンク|カーネル主成分分析|en|Kernel principal component analysis}}
* {{仮リンク|スパース主成分分析|en|Sparse PCA}}
* {{仮リンク|汎関数主成分分析|en|Functional principal component analysis}}
* {{仮リンク|点群統計モデル|en|Point distribution model}}
* [[最小二乗法]]
* {{仮リンク|主成分回帰|en|Principal component regression}}
* [[固有値]]
* [[特異値分解]]
* {{仮リンク|低ランク近似|en|Low-rank approximation}}
* {{仮リンク|CUR分解|en|CUR matrix approximation}}
* {{仮リンク|非負値行列因子分解|en|Non-negative matrix factorization}}
* {{仮リンク|特異スペクトル解析|en|Singular spectrum analysis}}
* [[固有顔]]
* [[独立成分分析]]
* {{仮リンク|対応分析|en|Correspondence analysis}}
* {{仮リンク|多重対応分析|en|Multiple correspondence analysis}}
* {{仮リンク|除歪対応分析|en|Detrended correspondence analysis}}
* [[因子分析]]
* {{仮リンク|混合データ因子分析<!--定訳なし-->|en|Factor analysis of mixed data}}
* [[正準相関]]
* {{仮リンク|動的モード分解|en|Dynamic mode decomposition}}
* {{仮リンク|幾何学的データ分析<!--定訳なし-->|en|Geometric data analysis}}
* {{仮リンク|非線形次元削減|en|Nonlinear dimensionality reduction}}
* {{仮リンク|オヤの法則|en|Oja's rule}}

==外部リンク==
{{commons category}}
* [http://www.cis.doshisha.ac.jp/mjin/R/24/24.html Rと主成分分析]
* {{YouTube|UUxIXU_Ob6E|Rasmus Bro のコペンハーゲン大学での動画}}
* {{YouTube|id=ey2PE5xi9-A#t=2385|title=Stanford University video by Andrew Ng}}
* {{YouTube|BfTMmoDFXyE|A layman's introduction to principal component analysis}}

{{統計学}}
{{Statistics-stub}}

{{DEFAULTSORT:しゆせいふんふんせき}}
[[Category:統計学]]
[[Category:機械学習]]
[[Category:データ分析]]
[[Category:行列の分解]]
[[Category:数学に関する記事]]