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Principal Components Regression: multivariate inverse least squares regression.


model = pcr(x,y,ncomp,options)  %identifies model (calibration step)
pred = pcr(x,model,options)  %applies model to a new X-block
valid = pcr(x,y,model,options)  %applies model to a new X-block, with corresponding new Y values
pcr % Launches an Analysis window with PCR as the selected method.


PCR calculates a single principal components regression model using the given number of components ncomp to predict y from measurements x, OR applies an existing PCR model to a new set of data x

To make predictions, the inputs are x the new predictor x-block (2-way array class "double" or "dataset"), and model the PCR model. The output pred is a structure, similar to model, that contains scores, predictions, etc. for the new data.

If new y-block measurements are also available for the new data, then the inputs are x the new x-block (2-way array class "double" or "dataset"), y the new y-block (2-way array class "double" or "dataset"), and model the PCR model to apply. The output valid is a structure, similar to model, that contains scores, predictions, and additional y-block statistics etc. for the new data.

In prediction and validation modes, the same model structure is used but predictions are provided in the model.detail.pred field.

Note: Calling pcr with no inputs starts the graphical user interface (GUI) for this analysis method.


  • x = X-block data (2-way array or DataSet Object)
  • y = Y-block data (2-way array or DataSet Object)
  • ncomp = number of components to to be calculated (positive integer scalar).

Optional Inputs

  • options discussed below


The output is a standard model structure with the following fields (see Standard Model Structure):

  • modeltype: 'PCR',
  • datasource: structure array with information about input data,
  • date: date of creation,
  • time: time of creation,
  • info: additional model information,
  • reg: regression vector,
  • loads: cell array with model loadings for each mode/dimension,
  • pred: 2 element cell array containing model predictions for each input block (when options.blockdetail='normal' x-block predictions are not saved and this will be an empty array), and the y-block predictions.
  • tsqs: cell array with T2 values for each mode,
  • ssqresiduals: cell array with sum of squares residuals for each mode,
  • description: cell array with text description of model, and
  • detail: sub-structure with additional model details and results.


options = a structure array with the following fields:

  • display: [ 'off' | {'on'} ], governs level of display to command window,
  • plots: [ 'none' | {'final'} ], governs level of plotting,
  • outputversion: [ 2 | {3} ], governs output format (discussed below),
  • preprocessing: {[] []}, two element cell array containing preprocessing structures (see PREPROCESS) defining preprocessing to use on the x- and y-blocks (first and second elements respectively),
  • algorithm: [ {'svd'} | ' robustpcr' | ' correlationpcr' | 'frpcr' ], governs which algorithm to use.
    • 'svd' = standard singular value decomposition algorithm.
    • 'robustpcr' = robust algorithm with automatic outlier detection.
    • 'correlationpcr' = standard PCR with re-ordering of factors in order of y-variance captured.
    • 'frpcr' = full-ratio PCR (a.k.a. optimized scaling) with automatic sample scale correction. Note that with FRPCR, models generally perform better without mean-centering on the x-block.
  • blockdetails: [ 'compact' | {'standard'} | 'all' ] level of detail (predictions, raw residuals, and calibration data) included in the model.
  • ‘Standard’ = the predictions and raw residuals for the X-block as well as the X-block itself are not stored in the model to reduce its size in memory. Specifically, these fields in the model object are left empty: 'model.pred{1}', 'model.detail.res{1}', 'model.detail.data{1}'.
  • ‘Compact’ = for this function, 'compact' is identical to 'standard'.
  • 'All' = keep predictions, raw residuals for both X- & Y-blocks as well as the X- & Y-blocks themselves.
  • confidencelimit: [ {'0.95'} ], confidence level for Q and T2 limits. A value of zero (0) disables calculation of confidence limits,
  • roptions: structure of options to pass to rpcr (robust PCR engine from the Libra Toolbox). Only used when algorithm is 'robustpcr',
  • alpha : [ {0.75} ], (1-alpha) measures the number of outliers the algorithm should resist. Any value between 0.5 and 1 may be specified. These options are only used when algorithm is 'robustpcr'.
  • intadjust : [ {0} ], if equal to one, the intercept adjustment for the LTS-regression will be calculated. See ltsregres for details (Libra Toolbox).

The default options can be retreived using: options = pcr('options');.


By default (options.outputversion = 3) the output of the function is a standard model structure model. If options.outputversion = 2, the output format is:

[b,ssq,t,p] = pcr(x,y,ncomp,options)

where the outputs are

  • b = matrix of regression vectors or matrices for each number of principal components up to ncomp,
  • ssq = the sum of squares information,
  • t = x-block scores, and
  • p = x-block loadings.

Note: The regression matrices are ordered in b such that each Ny (number of y-block variables) rows correspond to the regression matrix for that particular number of principal components.

See Also

analysis, crossval, frpcr, mlr, modelstruct, pca, pls, preprocess, ridge