method : one of the names eigenvector or svd; controls if the principal component analysis uses the eigenvector method (on either the covariance or correlation matrix) or the singular values method. By default this is set to svd.

columns : non-negative integer; controls the number of dimensions of data to retain. The default is set to the same number of dimensions as the original dataset. Columns of data are discarded based on their effect on the variability in the data with the columns accounting for the most variability being discarded last.

tolerance : float; controls the number of dimensions of data to discard. The default is set to 0.0. Columns are discarded if their standard deviations are less than or equal to the value for tolerance multiplied by the standard deviation of the first component.

correlation : truefalse; controls the choice of the transformation matrix in the computation. By default, correlation is set to false, and the covariance matrix is used.

center : truefalse, numeric, procedure; controls if the returned list of values is centered or not. The default is set to true, which uses the Statistics[Mean] command to compute the center. If this is set to false, the list is not centered and the list of values will have the same central mean value as before. If a numeric value is entered, the list is centered using that value as its center. If a procedure is entered, the list is centered using the value returned from the procedure.  If a procedure is entered as the first value of a list, subsequent arguments in the list are passed to the procedure.

scale : truefalse, numeric, procedure, or identical to the name auto; controls if the returned list of values is scaled or not. The default is set to auto. When set to auto, if the correlation matrix is used in the computation, scale is automatically set to true. If the covariance matrix is used, scale is set to false. If scale is set to false, each column is not scaled and therefore has the same standard deviation as before. When set to true, the Statistics[StandardDeviation] command is used to compute the standard deviation. If a numeric value is entered, the list is scaled using that value as its standard deviation. If a procedure is entered, the list is scaled using the value returned from the procedure.  If a procedure is entered as the first value of a list, subsequent arguments in the list are passed to the procedure.

output : list; controls the form of the returned solution. The default is record. Output options include:

record : a record which contains all of the following output options. Each of the outputs can be queried using :-outputname.

values : the singular values or eigenvalues

varianceproportion : the proportion of the total variance for each value

stdev : the square root of the singular values or eigenvalues

rotation : a matrix whose columns correspond to the eigenvectors

principalcomponents : a matrix whose columns correspond to the resulting principal components

ignore : truefalse; controls how missing data is handled. Missing items are represented by undefined or Float(undefined). If ignore=false and the dataset contains missing data, the PCA command will return undefined. If ignore = true, all rows with any missing items in dataset will be removed. The default value is false. The ignore option is not passed to any procedures entered for either of the center or scale options.

summarize : truefalse; controls the display of a printed summary of the singular values or eigenvalues. The default is false

The PCA command is used to perform a principal component analysis on a set of data. Principal Component Analysis transforms a multi-dimensional data set to a new set of perpendicular axes (or components) that describe decreasing amounts of variance in the data.   

The PCA command returns a record containing values for values, varianceproportion, stdev, rotation and principalcomponents by default. The output option can also be used to return specific elements from the principal component analysis.

The default method is svd, which is generally the recommended method for numerical accuracy.

The PrincipalComponentAnalysis command is provided as an alias.

To print out the results of the principal component analysis, set infolevel[Statistics] to 1, or use the summarize option.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$\mathrm{infolevel}\left[\mathrm{Statistics}\right]≔1\:$

$\mathrm{data}≔⟨⟨2\,4\,5\,6\,8\,9⟩|⟨-2\,-1\,0\,1\,2\,4⟩⟩$

$\mathrm{data}≔\left[\right]$

$\mathrm{PCAnalysis}≔\mathrm{PCA}\left(\mathrm{data}\right)\:$

summary:

Values   proportion of variance  St. Deviation
11.2240     0.9904                 3.3502
0.1093      0.0096                 0.3306

$\mathrm{PCAnalysis}:-\mathrm{rotation}$

$\mathrm{PCAnalysis}:-\mathrm{principalcomponents}$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{dataplot}\left(\mathrm{data}\,'\mathrm{points}'\,\mathrm{color}=''Red''\right)\,\mathrm{dataplot}\left(\mathrm{PCAnalysis}:-\mathrm{principalcomponents}\,'\mathrm{points}'\right)\right)$

$\mathrm{plotopts}≔\mathrm{size}=\left[600\,100\right],\mathrm{view}=\left[-10..10\,\mathrm{default}\right],\mathrm{symbol}=\mathrm{solidcircle},\mathrm{symbolsize}=20\:$

$\mathrm{splots}≔\left[\left[\mathrm{Statistics}:-\mathrm{ScatterPlot}\left(\mathrm{data}\left[..,1\right]\,\mathrm{color}=''Red''\,\mathrm{plotopts}\,\mathrm{title}=''X''\right)\,\mathrm{Statistics}:-\mathrm{ScatterPlot}\left(\mathrm{PCAnalysis}:-\mathrm{principalcomponents}\left[..,1\right]\,\mathrm{color}=''Orange''\,\mathrm{plotopts}\,\mathrm{title}=''1st\; Component''\right)\right]\,\left[\mathrm{Statistics}:-\mathrm{ScatterPlot}\left(\mathrm{data}\left[..,2\right]\,\mathrm{color}=''DarkBlue''\,\mathrm{plotopts}\,\mathrm{title}=''Y''\right)\,\mathrm{Statistics}:-\mathrm{ScatterPlot}\left(\mathrm{PCAnalysis}:-\mathrm{principalcomponents}\left[..,2\right]\,\mathrm{color}=''RoyalBlue''\,\mathrm{plotopts}\,\mathrm{title}=''2nd\; Component''\right)\right]\right]\:$

$\mathrm{DocumentTools}:-\mathrm{Tabulate}\left(\mathrm{splots}\right)$

$''Tabulate''$

$\mathrm{infolevel}\left[\mathrm{Statistics}\right]≔0\:$

$\mathrm{data2}≔⟨⟨2.5\left|2.4\right|10.5⟩\,⟨0.5\left|0.7\right|0.785⟩\,⟨2.2\left|2.9\right|1.286⟩\,⟨1.9\left|2.2\right|2.35⟩\,⟨3.1\left|3.0\right|2.202⟩\,⟨2.3\left|2.7\right|1.351⟩\,⟨2.0\left|1.6\right|2.021⟩\,⟨1.0\left|1.1\right|1.247⟩\,⟨1.5\left|1.6\right|2.503⟩\,⟨1.1\left|0.9\right|1.214⟩⟩$

$\mathrm{data2}≔\left[\right]$

$\mathrm{PCAnalysis2}≔\mathrm{PCA}\left(\mathrm{data2}\,\mathrm{columns}=2\,\mathrm{summarize}=\mathrm{true}\right)\:$

summary:

Values   proportion of variance  St. Deviation
8.3208     0.8782                 2.8846
1.1119     0.1173                 1.0545
0.0425     0.0045                 0.2061

$\mathrm{PCAnalysis2}:-\mathrm{principalcomponents}$

$\mathrm{EVectors}≔\mathrm{PCAnalysis2}:-\mathrm{rotation}\left[1..3,1..2\right]$

$\mathrm{EVectors}≔\left[\right]$

$\mathrm{TempMatrix}≔\mathrm{EVectors}·{\mathrm{PCAnalysis2}:-\mathrm{principalcomponents}}^{\mathrm{\%T}}\:$

$\mathrm{RecData}≔\mathrm{Matrix}\left(\mathrm{upperbound}\left(\mathrm{data2}\right)\,\left(i\,j\right)\mapsto \mathrm{TempMatrix}\left[j,i\right]+\mathrm{Mean}\left(\mathrm{data2}\right)\left[j\right]\right)$

$\mathrm{RecData}≔\left[\right]$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{dataplot}\left(\mathrm{data2}\,'\mathrm{points}'\,\mathrm{color}=''Red''\right)\,\mathrm{dataplot}\left(\mathrm{convert}\left(\mathrm{RecData}\,\mathrm{Matrix}\right)\,'\mathrm{points}'\right)\right)$

$\mathrm{PCA}\left(\mathrm{data2}\,\mathrm{method}=\mathrm{eigenvector}\,\mathrm{correlation}=\mathrm{true}\,\mathrm{output}=\left[\mathrm{stdev}\,\mathrm{principalcomponents}\right]\right)$

$\left[?\right],\left[?\right]$

$\mathrm{data3}≔\mathrm{DataFrame}\left(⟨⟨2.5\left|2.4\right|10.5\left|0.1\right|0.5⟩\,⟨0.5\left|0.7\right|0.785\left|4.3\right|2.0⟩\,⟨2.2\left|2.9\right|1.286\left|5.4\right|7.0⟩\,⟨1.9\left|2.2\right|2.35\left|6.7\right|3.1⟩\,⟨3.1\left|3.0\right|2.202\left|8.1\right|12⟩\,⟨0.1\left|0.4\right|0.5\left|0.6\right|0.9⟩⟩\,\mathrm{columns}=\left[a\,b\,c\,d\,e\right]\right)$

$\mathrm{data3}≔\left[\right]$

$\mathrm{PCAnalysis3}≔\mathrm{PCA}\left(\mathrm{data3}\,\mathrm{summarize}=\mathrm{true}\right)\:$

summary:

Values   proportion of variance  St. Deviation
31.4897     0.6658                 5.6116
13.2270     0.2797                 3.6369
2.3575      0.0498                 1.5354
0.2180      0.0046                 0.4669
0.0031      0.0001                 0.0558

$\mathrm{ScreePlot}\left(\mathrm{PCAnalysis3}\right)$

$\mathrm{PCA}\left(\mathrm{data3}\,\mathrm{tolerance}=0.01\right):-\mathrm{principalcomponents}$

$\mathrm{Biplot}\left(\mathrm{PCAnalysis3}\,\mathrm{pointlabels}=\mathrm{true}\,\mathrm{points}=\mathrm{false}\right)$

The Statistics[PCA] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.