Pcr

Purpose

Principal components regression: multivariate inverse least squares regession.

Synopsis

model = pcr(x,y,ncomp,options) %calibration

pred = pcr(x,model,options) %prediction

valid = pcr(x,y,model,options) %validation

Description

PCR calculates a single principal components regression model using the given number of components ncomp to predict y from measurements x.

To construct a PCR model, the inputs are x the predictor x-block (2-way array class "double" or "dataset"), y the predicted y-block (2-way array class "double" or "dataset"), ncomp the number of components to to be calculated (positive integer scalar) and the optional structure, options. The output is a standard model structure model with the following fields (see MODELSTRUCT):

• 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 with 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.

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 then the inputs are x the new predictor x-block (2-way array class "double" or "dataset"), y the new predicted block (2-way array class "double" or "dataset"), and model the PCR model. 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.

Options

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' ], governs which algorithm to use. 'svd' is standard algorithm. 'robustpcr' is robust algorithm with automatic outlier detection. 'correlationpcr' is standard PCR with re-ordering of factors in order of y-variance captured.

• blockdetails: ['compact' | {'standard'} | 'all'], extent of predictions and raw residuals included in model. 'standard' = only y-block, 'all' x and y blocks.

• 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.m for details (Libra Toolbox).

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

OUTPUTVERSION

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, modelstruct, pca, pls, preprocess, analysis, ridge