関数
>
ベクトルと行列
>
配列の特徴
>
例: 共分散と主成分分析
例: 共分散と主成分分析
データセットの共分散を計算し、
eigenvals
関数と
eigenvec
関数を使用して主成分分析を行います。
1.
次の行列
D
を定義します。
<region id="ID0EGKFMB" top="364.8" left="19.2" xml:lang="ja" height="490.6" width="136.696666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="0" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="Variable">D</id> <matrix rows="26" cols="3" xml:lang="ja"> <real xml:lang="ja">7.5</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">4.5</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">7.0</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">4.5</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">4.5</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">6.5</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">6.5</real> <real xml:lang="ja">12.0</real> <real xml:lang="ja">8.5</real> <real xml:lang="ja">12.0</real> <real xml:lang="ja">8.5</real> <real xml:lang="ja">9.0</real> <real xml:lang="ja">8.5</real> <real xml:lang="ja">8.5</real> <real xml:lang="ja">12.0</real> <real xml:lang="ja">12.0</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">8.0</real> <real xml:lang="ja">11.0</real> <real xml:lang="ja">8.5</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">8.0</real> <real xml:lang="ja">9.0</real> <real xml:lang="ja">7.5</real> <real xml:lang="ja">8.0</real> <real xml:lang="ja">8.0</real> <real xml:lang="ja">11.5</real> <real xml:lang="ja">8.5</real> <real xml:lang="ja">10.5</real> <real xml:lang="ja">12.0</real> <real xml:lang="ja">12.0</real> <real xml:lang="ja">8.0</real> <real xml:lang="ja">10.0</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">4.5</real> <real xml:lang="ja">3.5</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">4.5</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">3.5</real> <real xml:lang="ja">3.5</real> <real xml:lang="ja">3.5</real> <real xml:lang="ja">4.5</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">4.5</real> <real xml:lang="ja">5.5</real> <real xml:lang="ja">5.0</real> <real xml:lang="ja">4.0</real> <real xml:lang="ja">4.0</real> </matrix> </define> </math> </region>
D
の各行は 1 つの観測値を表し、各列は測定された特徴を表します。
2.
サンプルの共分散行列を計算します。
<region id="ID0EFMFMB" top="710.4" left="38.4" xml:lang="ja" height="25.6" width="132.716666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="1" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="None">i</id> <range xml:lang="ja"> <real xml:lang="ja">0</real> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">rows</id> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D</id> </apply> <real xml:lang="ja">1</real> </apply> </range> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0EXMFMB" top="748.8" left="38.4" xml:lang="ja" height="25.6" width="127.45" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="2" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="None">j</id> <range xml:lang="ja"> <real xml:lang="ja">0</real> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">cols</id> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D</id> </apply> <real xml:lang="ja">1</real> </apply> </range> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0EJNFMB" top="768" left="38.4" xml:lang="ja" height="38.952" width="166.983333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="3" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label="None">X</id> <sequence xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label="None">i</id> <id xml:space="preserve" xml:lang="ja" label="None">j</id> </sequence> </apply> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D</id> <sequence xml:lang="ja"> <id xml:space="preserve" xml:lang="ja">i</id> <id xml:space="preserve" xml:lang="ja">j</id> </sequence> </apply> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">mean</id> <apply xml:lang="ja"> <matcol xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D</id> <id xml:space="preserve" xml:lang="ja">j</id> </apply> </apply> </apply> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0E2NFMB" top="825.6" left="38.4" xml:lang="ja" height="62.8" width="292.253333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="4" xml:lang="ja"> <eval xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <define xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label="None">S1</id> <apply xml:lang="ja"> <div xml:lang="ja" /> <apply xml:lang="ja"> <mult xml:lang="ja" /> <apply xml:lang="ja"> <transpose xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja">X</id> </apply> <id xml:space="preserve" xml:lang="ja">X</id> </apply> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja">rows</id> <id xml:space="preserve" xml:lang="ja">D</id> </apply> <real xml:lang="ja">1</real> </apply> </apply> </define> <unitOverride xml:lang="ja"> <placeholder xml:lang="ja" /> </unitOverride> </eval> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
3.
固有値を計算して並べ替えます。
<region id="ID0EQPFMB" top="921.6" left="38.4" xml:lang="ja" height="25.6" width="233.816666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="5" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="None">V</id> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">reverse</id> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">sort</id> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">eigenvals</id> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">S1</id> </apply> </apply> </apply> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0ECQFMB" top="979.2" left="38.4" xml:lang="ja" height="62.8" width="94.7566666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="6" xml:lang="ja"> <eval xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">V</id> <unitOverride xml:lang="ja"> <placeholder xml:lang="ja" /> </unitOverride> </eval> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
4.
変換行列を計算します。
<region id="ID0EXRFMB" top="1075.2" left="38.4" xml:lang="ja" height="36.8" width="150.513333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="7" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label="Variable">x</id> <id xml:space="preserve" xml:lang="ja" label="None">j</id> </apply> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">eigenvec</id> <sequence xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">S1</id> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">V</id> <id xml:space="preserve" xml:lang="ja">j</id> </apply> </sequence> </apply> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0EJSFMB" top="1132.8" left="38.4" xml:lang="ja" height="36.8" width="166.673333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="8" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="None">T</id> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">augment</id> <sequence xml:lang="ja"> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">x</id> <real xml:lang="ja">0</real> </apply> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">x</id> <real xml:lang="ja">1</real> </apply> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">x</id> <real xml:lang="ja">2</real> </apply> </sequence> </apply> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0E2SFMB" top="1209.6" left="38.4" xml:lang="ja" height="62.8" width="204.193333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="9" xml:lang="ja"> <eval xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">T</id> <unitOverride xml:lang="ja"> <placeholder xml:lang="ja" /> </unitOverride> </eval> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
5.
元のデータを変換します。
<region id="ID0EQUFMB" top="1305.6" left="38.4" xml:lang="ja" height="25.6" width="75.6933333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="10" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="None">D2</id> <apply xml:lang="ja"> <mult xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D</id> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">T</id> </apply> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
6.
変換後のデータの共分散行列を計算します。
<region id="ID0EFWFMB" top="1382.4" left="38.4" xml:lang="ja" height="25.6" width="140.696666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="11" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="None">i</id> <range xml:lang="ja"> <real xml:lang="ja">0</real> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">rows</id> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D2</id> </apply> <real xml:lang="ja">1</real> </apply> </range> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0EXWFMB" top="1420.8" left="38.4" xml:lang="ja" height="25.6" width="135.43" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="12" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" xml:lang="ja" label="None">j</id> <range xml:lang="ja"> <real xml:lang="ja">0</real> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">cols</id> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D2</id> </apply> <real xml:lang="ja">1</real> </apply> </range> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0EJXFMB" top="1459.2" left="38.4" xml:lang="ja" height="38.952" width="182.943333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="13" xml:lang="ja"> <define xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label="None">X</id> <sequence xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label="None">i</id> <id xml:space="preserve" xml:lang="ja" label="None">j</id> </sequence> </apply> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <indexer xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D2</id> <sequence xml:lang="ja"> <id xml:space="preserve" xml:lang="ja">i</id> <id xml:space="preserve" xml:lang="ja">j</id> </sequence> </apply> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Function">mean</id> <apply xml:lang="ja"> <matcol xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja" label-is-contextual="true" label="Variable">D2</id> <id xml:space="preserve" xml:lang="ja">j</id> </apply> </apply> </apply> </define> <resultFormat xml:lang="ja"> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
<region id="ID0E2XFMB" top="1516.8" left="38.4" xml:lang="ja" height="62.8" width="300.233333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="14" xml:lang="ja"> <eval xml:lang="ja" xmlns="http://schemas.mathsoft.com/math50"> <define xml:lang="ja"> <id xml:space="preserve" xml:lang="ja" label="None">S2</id> <apply xml:lang="ja"> <div xml:lang="ja" /> <apply xml:lang="ja"> <mult xml:lang="ja" /> <apply xml:lang="ja"> <transpose xml:lang="ja" /> <id xml:space="preserve" xml:lang="ja">X</id> </apply> <id xml:space="preserve" xml:lang="ja">X</id> </apply> <apply xml:lang="ja"> <minus xml:lang="ja" /> <apply xml:lang="ja"> <id xml:space="preserve" xml:lang="ja">rows</id> <id xml:space="preserve" xml:lang="ja">D2</id> </apply> <real xml:lang="ja">1</real> </apply> </apply> </define> <unitOverride xml:lang="ja"> <placeholder xml:lang="ja" /> </unitOverride> </eval> <resultFormat xml:lang="ja"> <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="ja" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="ja" /> </resultFormat> </math> </region>
データの主成分は行列
S2
の対角要素です。
式をコピー
