SAS Visual Statistics 8.1 Programming Guide Partial Least Squares Action Set

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SAS ^® Visual Statistics 8.1 Programming Guide

Partial Least Squares

Action Set

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Chapter 4 The Partial Least Squares Action Set

Contents

Details. . . . 43

Results . . . . 44

Syntax. . . . 47

Examples . . . . 57

Example 4.1: Spectrometric Calibration . . . . 57

References . . . . 64

Details

The^plsaction fits models in SAS Viya by using any one of a number of linear predictive methods, including partial least squares (PLS). The^plsaction performs the underlying computations for the PLSMOD procedure in SAS Viya.

Ordinary least squares regression has the single goal of minimizing sample response prediction error, and it seeks linear functions of the predictors that explain as much variation in each response as possible. The plsaction implements techniques that have the additional goal of accounting for variation in the predictors, under the assumption that directions in the predictor space that are well sampled should provide better prediction for new observations when the predictors are highly correlated. All the techniques that thepls action implements work by extracting successive linear combinations of the predictors, called factors (also called components, latent vectors, or latent variables), which optimally address one or both of these two goals: explaining response variation and explaining predictor variation. In particular, the method of partial least squares balances the two objectives by seeking factors that explain both response and predictor variation.

The name “partial least squares” also applies to a more general statistical method that is not implemented in this action. The partial least squares method was originally developed in the 1960s by the econometrician Herman Wold (1966) for modeling “paths” of causal relation between any number of “blocks” of variables.

However, the^plsaction fits only predictive partial least squares models that have one “block” of predictors and one “block” of responses.

The^plsaction implements the following methods:

principal component regression, which extracts factors to explain as much predictor sample variation as possible

reduced rank regression, which extracts factors to explain as much response variation as possible. This technique, also known as (maximum) redundancy analysis, differs from multivariate linear regression only when there are multiple responses.

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44 F Chapter 4: The Partial Least Squares Action Set

partial least squares regression, which balances the two objectives of explaining response variation and explaining predictor variation. Two different formulations for partial least squares are available: the original predictive method ofWold(1966) and the straightforward implementation of a statistically inspired modification of the partial least squares (SIMPLS) method ofDe Jong(1993).

Theplsaction has the following features:

provides model-building syntax with classification variables, continuous variables, interactions, and nestings

provides effect-construction syntax for polynomial and spline effects

partitions your data into training and testing subsets

provides test set validation to choose the number of extracted factors, where the model is fit to only part of the available data (the training set) and the fit is evaluated over the other part of the data (the test set)

produces an output data table that contains predicted values and other observationwise statistics Because the^plsaction runs on CAS, it also does the following:

enables you to run on a cluster of machines that distribute the data and the computations

exploits all the available cores and concurrent threads

For more information about predictive partial least squares models and the capabilities of the^plsaction, see the section “Details: PLSMOD Procedure” (Chapter 8, SAS Visual Data Mining and Machine Learning:

Statistical Procedures).

Results

The results that the plsaction produces are stored in tables. You can access these tables by using their names, which are shown inTable 4.1. For more information about the contents of these tables, see the section

“Displayed Output” (Chapter 8, SAS Visual Data Mining and Machine Learning: Statistical Procedures).

The table names are the last level of the path of the output objects; paths depend on the grouping struc- ture of the output tables. For example, the ResidualSummary table can have a path like the following:

CrossValidation.ResidualSummary. If you use PROC CAS on the SAS client to submit the action, CAS.plsis added as a prefix to the pathname. For example, the ResidualSummary table pathname could be CAS.pls.ResidualSummary. You can obtain the full paths by submitting code as shown inExample 4.1.

Table 4.1 Results Tables Produced by the^plsAction

Table Name Description Parameter Subparameter

BSplineDetails B-spline basis details ^spline basis=bspline

and^details

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Results F ⁴⁵

Table 4.1 continued

Table Name Description Parameter Subparameter

CVResults Results of test set validation partitionByVaror partitionByFrac CenScaleParms Parameter estimates for centered and

scaled data

model solution

ClassInfo Level information for classification variables

classor^classvars

CodedCoef Coded regression coefficients ^details

CollectionLevelInfo Level information for collection effects ^collection ^details

Dimensions Model dimensions Default

MMLevelInfo Level information for multimember effects

multimember details

ModelInfo Information about the modeling environment

Default NObs Number of observations read and used Default OutputCasTables Library and name of the output data

table, and number of rows and columns in the table

outputor outputTables

ParameterEstimates Parameter estimates for raw data model solution PercentVariation Predictor and response variation that are

accounted for by each factor

Default PolyDetails Number of variables and columns,

polynomial degree, and standardization method

polynomial details

PolyScaling Centering and scaling details polynomial standardize

and^details ResidualSummary Residual summary from test set

validation

partitionByVaror partitionByFrac

SplineKnots Knot and boundary knot values ^spline ^details

Timing Absolute and relative times for tasks performed by the action

Default TPFSplineDetails Truncated power function spline basis

details

spline basis=tpfor naturalcubic and^details XEffectCenScale Centering and scaling information for

predictor effects

cenScale

XLoadings Loadings for predictor effects ^details XPercentVariation Variation that is accounted for by each

factor for predictor effects

varss

XWeights Weights for predictor effects ^details

YPercentVariation Variation that is accounted for by each factor for responses

varss

YVariableCenScale Centering and scaling information for responses

cenScale

YWeights Weights for responses ^details

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NOTE: You use these table names when you specify thedisplayandoutputTablesparameters.

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Partial Least Squares Action Set: Syntax pls Action

CASL Syntax

pls.pls <result =results> <status=rc> / attributes={{

format="string",

formattedLength=integer, label="string",

*name="variable-name", nfd=integer,

nfl=integer }, {...}}

cenScale=TRUE | FALSE class={{

countMissing=TRUE | FALSE, descending=TRUE | FALSE, ignoreMissing=TRUE | FALSE, levelizeRaw=TRUE | FALSE, maxLev=integer,

order="FORMATTED" | "FREQ" | "FREQFORMATTED" | "FREQINTERNAL" | "INTERNAL",

"REFERENCE",

ref="FIRST" | "LAST" | "string", split=TRUE | FALSE,

*vars={"variable-name-1" <, "variable-name-2", ...>}

}, {...}}

classGlobalOptions={

countMissing=TRUE | FALSE, descending=TRUE | FALSE, ignoreMissing=TRUE | FALSE, levelizeRaw=TRUE | FALSE, maxLev=integer,

order="FORMATTED" | "FREQ" | "FREQFORMATTED" | "FREQINTERNAL" | "INTERNAL",

"REFERENCE",

ref="FIRST" | "LAST" | "string", split=TRUE | FALSE }

classLevelsPrint=TRUE | FALSE collection={{

details=TRUE | FALSE,

*name="string",

}, {...}}

cvTest={

nSamp=integer, pValue=double, seed=integer, stat="PRESS" | "T2"

}

details=TRUE | FALSE display={

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caseSensitive=TRUE | FALSE, exclude=TRUE | FALSE, excludeAll=TRUE | FALSE, keyIsPath=TRUE | FALSE, names={"string-1" <, "string-2", ...>}, pathType="LABEL" | "NAME", traceNames=TRUE | FALSE }

inputs={{

nfl=integer }, {...}}

* method={

algorithm="EIG" | "NIPALS" | "SVD", epsilon=double,

maxIter=integer,

*name="PCR" | "PLS" | "RRR" | "SIMPLS"

} model={

depVars={{

name="variable-name"

}, {...}}, effects={{

interaction="BAR" | "CROSS" | "NONE", maxInteract=integer,

nest={"string-1" <, "string-2", ...>},

*vars={"string-1" <, "string-2", ...>}

}, {...}},

entry="variable-name", intercept=TRUE | FALSE, solution=TRUE | FALSE, }

multimember={{

details=TRUE | FALSE,

noEffect=TRUE | FALSE, stdize=TRUE | FALSE,

*vars={"variable-name-1" <, "variable-name-2", ...>}, weight={"variable-name-1" <, "variable-name-2", ...>}

}, {...}}

nClassLevelsPrint=integer nFactors=integer

noCVStdize=TRUE | FALSE noCenter=TRUE | FALSE noScale=TRUE | FALSE nominals={{

*name="variable-name",

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nfd=integer, nfl=integer }, {...}}

output={

*casOut={

caslib="string"

compress=TRUE | FALSE label="string"

maxMemSize=64-bit-integer name="table-name"

onDemand=TRUE | FALSE promote=TRUE | FALSE replace=TRUE | FALSE replication=integer timeStamp="string"

where={"string-1" <, "string-2", ...>}

},

copyVars="ALL" | "ALL_MODEL" | "ALL_NUMERIC" | {"variable-name-1" <, "variable-name-2", ...>}, h="string",

predicted="string", press="string", role="string", t2="string", xResidual="string", xScore="string", xStd="string", xStdsse="string", yResidual="string", yScore="string", yStd="string", yStdsse="string"

}

outputTables={

groupByVarsRaw=TRUE | FALSE,

names={"string-1" <, "string-2", ...>} | {key-1={casouttable-1} <, key-2={casouttable-2}, ...>}, repeated=TRUE | FALSE,

replace=TRUE | FALSE }

partitionByFrac={

seed=integer, test=double, }

partitionByVar={

*name="variable-name", test="string",

train="string", }

polynomial={{

degree=integer, details=TRUE | FALSE, labelStyle={

expand=TRUE | FALSE exponent="string"

includeName=TRUE | FALSE

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productSymbol="NONE" | "string"

},

mDegree=integer,

noSeparate=TRUE | FALSE, standardize={

method="MOMENTS" | "MRANGE" | "WMOMENTS"

options="CENTER" | "CENTERSCALE" | "NONE" | "SCALE"

prefix="NONE" | "string"

},

}, {...}}

spline={{

degree=integer, details=TRUE | FALSE, knotMax=double, knotMethod={

equal=integer

list={double-1 <, double-2, ...>}

listWithBoundary={double-1 <, double-2, ...>}

multiscale={

endScale=integer startScale=integer }

rangeFractions={double-1 <, double-2, ...>}

},

knotMin=double,

naturalCubic=TRUE | FALSE, separate=TRUE | FALSE, split=TRUE | FALSE,

}, {...}}

* table={

caslib="string",

computedOnDemand=TRUE | FALSE, computedVars={{

nfl=integer }, {...}},

computedVarsProgram="string", groupBy={{

nfl=integer

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}, {...}},

groupByMode="NOSORT" | "REDISTRIBUTE",

importOptions={fileType="AUTO" | "BASESAS" | "CSV" | "DTA" | "ESP" | "EXCEL" | "FMT" | "HDAT" | "JMP" | "LASR" | "SPSS" | "XLS", fileType- specific-parameters},

*name="table-name", onDemand=TRUE | FALSE, orderBy={{

nfl=integer }, {...}},

singlePass=TRUE | FALSE, vars={{

nfl=integer }, {...}},

where="where-expression"

}

target="string"

varss=TRUE | FALSE

;

Parameter Descriptions

attributes={{casinvardesc-1} <, {casinvardesc-2}, ...>}

alters attributes on variables used in this action.

Alias attribute

To simplify, you can specify only a value for the name parameter. In this case, the other parameters in the list use default values.

For more information about this common parameter, see casinvardesc.

cenScale=TRUE | FALSE

when set to True, displays the centering and scaling information.

Default FALSE

class={{classStatement-1} <, {classStatement-2}, ...>}

specifies the classification variables to be used as explanatory variables in the analysis.

Alias classVars

To simplify, you can specify only a value for the vars parameter. In this case, the other parameters in the list use default values.

For more information about this common parameter, see classStatement.

classGlobalOptions={classopts}

specifies options that apply to all classification variables.

Alias classGlobalOpts

To simplify, you can specify only a value for the param parameter. In this case, the other parameters in the list use default values.

For more information about this common parameter, see classopts.

classLevelsPrint=TRUE | FALSE

when set to False, suppresses the display of class levels.

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Default TRUE

collection={{collection-1} <, {collection-2}, ...>}

defines a set of variables that are treated as a single effect that has multiple degrees of freedom.

cvTest={cvTestOptions}

performs van der Voet's randomization-based model comparison test.

To simplify, you can specify only a value for the stat parameter. In this case, the other parameters in the list use default values.

The cvTestOptions value can be one or more of the following:

nSamp=integer

specifies the number of randomizations to perform.

Default 1000 Range 0–MACINT pValue=double

specifies the cutoff probability for declaring an insignificant difference.

Alias pVal Alias pVal Default 0.1 Range 0–1 seed=integer

specifies the seed value for the random number stream.

Default 0

Minimum value 0 stat="PRESS" | "T2"

specifies the test statistic for the model comparison. You can specify either T2, for Hotelling's T^2 statistic, or PRESS, for the predicted residual sum of squares.

Default T2

details=TRUE | FALSE

when set to True, displays the details of the fitted model.

Alias detail

display={displayTables}

specifies a list of result tables to send to the client for display.

To simplify, you can specify only a value for the names parameter. In this case, the other parameters in the list use default values.

For more information about this common parameter, see displayTables.

inputs={{casinvardesc-1} <, {casinvardesc-2}, ...>}

specifies variables to use for analysis.

Alias input

* method={methodOptions}

specifies the settings for the general factor extraction method.

The methodOptions value can be one or more of the following:

algorithm="EIG" | "NIPALS" | "SVD"

specifies the algorithm used to compute extracted PLS factors.

Alias alg Alias alg Default NIPALS

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epsilon=double

specifies the convergence criterion for the NIPALS algorithm.

Alias eps Alias eps Default 1e-12 Range 0–1

maxIter=integer

specifies the maximum number of iterations for the NIPALS algorithm.

Default 200 Range 0–MACINT

* name="PCR" | "PLS" | "RRR" | "SIMPLS"

specifies the name of the general factor extraction method to use.

model={modelOptions}

specifies the responses and the predictors, which determine the Y and X matrices of the model, respectively.

To simplify, you can specify only a value for the depVars parameter. In this case, the other parameters in the list use default values.

The modelOptions value can be one or more of the following:

depVars={{responsevar-1} <, {responsevar-2}, ...>}

specifies a list of variables to use as response variables in the model.

Alias depVar, target

The responsevar value can be one or more of the following:

name="variable-name"

defines a response variable.

effects={{effect-1} <, {effect-2}, ...>}

specifies a list of effects that define the model. Each term in this list is made up of variables specified in the vars parameter and their interaction (which can be NONE, CROSS, or BAR). When the interaction is BAR, it can be limited by the maxInteract parameter.

To simplify, you can specify only a value for the vars parameter. In this case, the other parameters in the list use default values.

The effect value can be one or more of the following:

interaction="BAR" | "CROSS" | "NONE"

specifies the type of interaction for the variables.

Alias interact Alias interact Default NONE maxInteract=integer

eliminates higher-order interaction effects when used in conjunction with the BAR interaction.

nest={"string-1" <, "string-2", ...>}

specifies the variables to be nested within the term defined by the vars parameter. For terms with a BAR or CROSS interaction, the nest corresponds to the last variable in the vars parameter. For terms with no interaction, the nest is distributed across all variables that are listed in the vars parameter.

* vars={"string-1" <, "string-2", ...>}

specifies the variables to use in defining a term of the effect. You must specify at least one variable.

entry="variable-name"

specifies the entry variable.

intercept=TRUE | FALSE

when set to True, includes the intercept term in the model.

Default FALSE solution=TRUE | FALSE

when set to True, displays the coefficients of the final predictive model for the responses.

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Default FALSE

multimember={{multimember-1} <, {multimember-2}, ...>}

uses one or more classification variables specified in the vars parameter in such a way that each observation can be associated with one or more levels of the union of the levels of the classification variables.

nClassLevelsPrint=integer

limits the display of class levels. The value 0 suppresses all levels.

Minimum value 0

nFactors=integer

specifies the number of factors to extract.

Alias nFac, lv

Default 0

Minimum value 0

noCVStdize=TRUE | FALSE

when set to True, suppresses re-centering and rescaling of the responses and predictors when cross validating.

Default FALSE

noCenter=TRUE | FALSE

when set to True, suppresses centering of the responses and predictors before fitting.

Default FALSE

noScale=TRUE | FALSE

when set to True, suppresses scaling of the responses and predictors before fitting.

Default FALSE

nominals={{casinvardesc-1} <, {casinvardesc-2}, ...>}

specifies nominal variables to use for analysis.

Alias nominal

output={outputOptions}

creates a data table on the server that contains observationwise statistics, which are computed after fitting the model.

To simplify, you can specify only a value for the casOut parameter. In this case, the other parameters in the list use default values.

The outputOptions value can be one or more of the following:

* casOut={casouttable}

specifies the settings for an output table.

For more information about this common parameter, see casouttable.

copyVars="ALL" | "ALL_MODEL" | "ALL_NUMERIC" | {"variable-name-1" <, "variable-name-2", ...>}

specifies a list of one or more variables to be copied from the input table to the output table. You can alternatively specify the value ALL, ALL_MODEL, or

ALL_NUMERIC, which respectively requests that all variables, all variables used in the modeling, or all numeric variables be copied from the input table to the output table.

h="string"

requests the approximate leverage. If set to an empty string, the prefix H is used for naming the output variable.

predicted="string"

requests predicted values for each response. If set to an empty string, the prefix Pred is used for naming the output variables.

Alias p, pred press="string"

requests approximate predicted residuals for each response. If set to an empty string, the prefix PRESS is used for naming the output variables.

role="string"

requests numeric values that indicate the role played by each observation in fitting the model. If set to an empty string, the prefix _ROLE_ is used for naming the output variable.

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t2="string"

requests scaled sum of squares of score values. If set to an empty string, the prefix TSquare is used for naming the output variable.

Alias tSquare xResidual="string"

requests residuals for each predictor. If set to an empty string, the prefix XResid is used for naming the output variables.

Alias xr, xResid xScore="string"

requests extracted factors (X-scores, latent vectors, latent variables, T) for each selected model factor. If set to an empty string, the prefix XScore is used for naming the output variables.

xStd="string"

requests standardized (centered and scaled) predictor values for each predictor. If set to an empty string, the prefix StdX is used for naming the output variables.

Alias stdX xStdsse="string"

requests the sum of squares of residuals for standardized predictors. If set to an empty string, the prefix StdXSSE is used for naming the output variable.

Alias xQres, stdXsse yResidual="string"

requests residuals for each response. If set to an empty string, the prefix YResid is used for naming the output variables.

Alias yr, yResid yScore="string"

requests extracted responses (Y-scores, U) for each selected model factor. If set to an empty string, the prefix YScore is used for naming the output variables.

yStd="string"

requests standardized (centered and scaled) response values for each response. If set to an empty string, the prefix StdY is used for naming the output variables.

Alias stdY yStdsse="string"

requests the sum of squares of residuals for standardized responses. If set to an empty string, the prefix StdYSSE is used for naming the output variable.

Alias yQres, stdYsse

outputTables={outputTables}

lists the names of result tables to save as CAS tables on the server.

To simplify, you can specify only a value for the names parameter. In this case, the other parameters in the list use default values.

For more information about this common parameter, see outputTables.

partitionByFrac={partByFracStatement}

specifies the fractions of the data to be used for training and testing.

Alias partByFrac

To simplify, you can specify only a value for the test parameter. In this case, the other parameters in the list use default values.

seed=integer

specifies the seed to use in the random number generator that is used for partitioning the data.

Default 0 test=double

randomly assigns the specified proportion of observations in the input table to the testing role. The sum of the fractions that are specified in the test and validate parameters must be less than 1.

Range 0–1

partitionByVar={partByVarStatement}

specifies the variable and its values used to partition the data into training and testing roles.

Alias partByVar

* name="variable-name"

names the variable in the input table whose values are used to assign rows to each observation.

test="string"

specifies the formatted value of the variable that is used to assign observations to the testing role.

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train="string"

specifies the formatted value of the variable that is used to assign observations to the training role. If you do not specify the train parameter, then all observations whose roles are not determined by the test and validate parameters are assigned to training.

polynomial={{polynomial-1} <, {polynomial-2}, ...>}

specifies a polynomial effect. All specified variables must be numeric. A design matrix column is generated for each term of the specified polynomial. By default, each of these terms is treated as a separate effect for the purpose of model building.

Alias poly

spline={{spline-1} <, {spline-2}, ...>}

expands variables into spline bases whose form depends on the specified parameters.

* table={castable}

specifies the settings for an input table.

For more information about this common parameter, see castable.

target="string"

specifies target variable to use for analysis.

varss=TRUE | FALSE

when set to True, displays the amount of variation accounted for in each response and predictor.

Default FALSE

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Examples F ⁵⁷

Examples

Example 4.1: Spectrometric Calibration

Spectrometric Calibration Using PROC CAS This section contains PROC CAS code.

NOTE: Input data must be in a CAS table that is accessible in your CAS session. This table has a two-level name; the first level is your CAS engine libref and the second level is the table name. You refer to this table in the CAS procedure by specifying only the second level. For more information about two-level names, see Chapter 2, “Shared Concepts” (SAS Visual Data Mining and Machine Learning: Statistical Procedures).

For more information about PROC CAS and programming in CASL, see Getting Started with the CASL Programming Language, SAS Cloud Analytic Services: CAS Procedure Programming Guide and Reference, and SAS Viya: System Programming Guide.

The example in this section illustrates basic features of theplsaction in theplsaction set. The data are reported in Umetrics (1995); the original source isLindberg, Persson, and Wold(1983). Suppose you are researching pollution in the Baltic Sea and you want to use the fluorescence spectra of seawater samples to determine the amounts of three compounds present in those samples: lignin sulfonate (ls: pulp industry pollution), humic acids (ha: natural forest products), and optical whitener from laundry detergent (dt).

Spectrometric calibration is a type of problem in which partial least squares can be very effective. The predictors are the spectra emission intensities at different frequencies in a sample spectrum, and the responses are the amounts of various chemicals in the sample.

For the purpose of calibrating the model, samples that have known compositions are used. The calibration data consist of 16 samples of known concentrations ofls,ha, anddt, with spectra based on 27 frequencies (or, equivalently, wavelengths). In order to demonstrate the use of test set validation, the data contain the variableRole, which is used to assign observations to the training and testing roles. In this case, the training role has nine samples and the testing role has seven samples.

The following DATA step creates themycas.Sampledata table, which provides the calibration data, in your CAS session. This DATA step assume that your CAS engine libref is namedmycas, but you can substitute any appropriately defined CAS engine libref.

data mycas.Sample;

input obsnam $ v1-v27 ls ha dt Role $5. @@;

datalines;

EM1 2766 2610 3306 3630 3600 3438 3213 3051 2907 2844 2796 2787 2760 2754 2670 2520 2310 2100 1917 1755 1602 1467

1353 1260 1167 1101 1017 3.0110 0.0000 0.00 TRAIN EM2 1492 1419 1369 1158 958 887 905 929 920 887 800

710 617 535 451 368 296 241 190 157 128 106 89 70 65 56 50 0.0000 0.4005 0.00 TEST EM3 2450 2379 2400 2055 1689 1355 1109 908 750 673 644

640 630 618 571 512 440 368 305 247 196 156

120 98 80 61 50 0.0000 0.0000 90.63 TRAIN EM4 2751 2883 3492 3570 3282 2937 2634 2370 2187 2070 2007

1974 1950 1890 1824 1680 1527 1350 1206 1080 984 888 810 732 669 630 582 1.4820 0.1580 40.00 TEST

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EM5 2652 2691 3225 3285 3033 2784 2520 2340 2235 2148 2094 2049 2007 1917 1800 1650 1464 1299 1140 1020 909 810

726 657 594 549 507 1.1160 0.4104 30.45 TEST EM6 3993 4722 6147 6720 6531 5970 5382 4842 4470 4200 4077

4008 3948 3864 3663 3390 3090 2787 2481 2241 2028 1830

1680 1533 1440 1314 1227 3.3970 0.3032 50.82 TRAIN EM7 4032 4350 5430 5763 5490 4974 4452 3990 3690 3474 3357

3300 3213 3147 3000 2772 2490 2220 1980 1779 1599 1440

1320 1200 1119 1032 957 2.4280 0.2981 70.59 TRAIN EM8 4530 5190 6910 7580 7510 6930 6150 5490 4990 4670 4490

4370 4300 4210 4000 3770 3420 3060 2760 2490 2230 2060

1860 1700 1590 1490 1380 4.0240 0.1153 89.39 TRAIN EM9 4077 4410 5460 5857 5607 5097 4605 4170 3864 3708 3588

3537 3480 3330 3192 2910 2610 2325 2064 1830 1638 1476 1350 1236 1122 1044 963 2.2750 0.5040 81.75 TEST EM10 3450 3432 3969 4020 3678 3237 2814 2487 2205 2061 2001

1965 1947 1890 1776 1635 1452 1278 1128 981 867 753 663 600 552 507 468 0.9588 0.1450 101.10 TRAIN EM11 4989 5301 6807 7425 7155 6525 5784 5166 4695 4380 4197

4131 4077 3972 3777 3531 3168 2835 2517 2244 2004 1809

1620 1470 1359 1266 1167 3.1900 0.2530 120.00 TRAIN EM12 5340 5790 7590 8390 8310 7670 6890 6190 5700 5380 5200

5110 5040 4900 4700 4390 3970 3540 3170 2810 2490 2240 2060 1870 1700 1590 1470 4.1320 0.5691 117.70 TEST EM13 3162 3477 4365 4650 4470 4107 3717 3432 3228 3093 3009

2964 2916 2838 2694 2490 2253 2013 1788 1599 1431 1305

1194 1077 990 927 855 2.1600 0.4360 27.59 TRAIN EM14 4380 4695 6018 6510 6342 5760 5151 4596 4200 3948 3807

3720 3672 3567 3438 3171 2880 2571 2280 2046 1857 1680

1548 1413 1314 1200 1119 3.0940 0.2471 61.71 TRAIN EM15 4587 4200 5040 5289 4965 4449 3939 3507 3174 2970 2850

2814 2748 2670 2529 2328 2088 1851 1641 1431 1284 1134 1020 918 840 756 714 1.6040 0.2856 108.80 TEST EM16 4017 4725 6090 6570 6354 5895 5346 4911 4611 4422 4314

4287 4224 4110 3915 3600 3240 2913 2598 2325 2088 1917 1734 1587 1452 1356 1257 3.1620 0.7012 60.00 TEST

;

To isolate a few underlying spectral factors that provide a good predictive model, you can fit a PLS model to the 16 samples by using theplsaction in theplsaction set in the following PROC CAS statements:

proc cas;

action pls.pls / table='SAMPLE', method='PLS',

model={depVars={'ls', 'ha', 'dt'},

effects={'v1', 'v2', 'v3', 'v4', 'v5', 'v6', 'v7', 'v8', 'v9', 'v10', 'v11', 'v12', 'v13', 'v14', 'v15', 'v16', 'v17', 'v18', 'v19', 'v20', 'v21', 'v22', 'v23', 'v24', 'v25', 'v26', 'v27'}};

run;

By default, the^plsaction extracts at most 15 factors. The default output from this analysis is presented in Output 4.1.1andOutput 4.1.2.

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Example 4.1: Spectrometric Calibration F ⁵⁹

Output 4.1.1displays the “Model Information,” “Dimensions,” and “Number of Observations” tables.

The “Model Information” table identifies the data source and shows that the factor extraction method is partial least squares regression and that the nonlinear iterative partial least squares (NIPALS) algorithm (which is the default) is used to compute extracted PLS factors.

The “Dimensions” table shows the number of response variables, the number of effects, the number of predictor parameters, and the number of factors to extract.

The “Number of Observations” table shows that all 16 of the sample observations in the input data are used in the analysis because all the samples contain complete data.

Output 4.1.1 Model Information, Dimensions, and Number of Observations R e s u l t s f r o m p l s . p l s

R e s u l t s f r o m p l s . p l s

M o d e l I n f o r m a t i o n

D a t a S o u r c e S A M P L E

F a c t o r E x t r a c t i o n M e t h o d P a r t i a l L e a s t S q u a r e s

P L S A l g o r i t h m N I P A L S

V a l i d a t i o n M e t h o d N o n e

D i m e n s i o n s

N u m b e r o f R e s p o n s e V a r i a b l e s 3

N u m b e r o f E f f e c t s 2 7

N u m b e r o f P r e d i c t o r P a r a m e t e r s 2 7

N u m b e r o f F a c t o r s 1 5

N u m b e r o f O b s e r v a t i o n s R e a d 1 6 N u m b e r o f O b s e r v a t i o n s U s e d 1 6

Output 4.1.2lists the amount of variation, both individual and cumulative, that is accounted for by each of the 15 factors. All the variation in both the predictors and the responses is accounted for by only 15 factors because there are only 16 sample observations. More important, almost all the variation is accounted for by even fewer factors—one or two for the predictors and three to eight for the responses.

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Output 4.1.2 PLS Variation Summary

P e r c e n t a g e V a r i a t i o n A c c o u n t e d f o r b y P a r t i a l L e a s t S q u a r e s F a c t o r s

M o d e l E f f e c t s

R e s p o n s e V a r i a b l e s N u m b e r o f

E x t r a c t e d

F a c t o r s C u r r e n t T o t a l C u r r e n t T o t a l

1 9 7 . 4 6 0 6 8 9 7 . 4 6 0 6 8 4 1 . 9 1 5 4 6 4 1 . 9 1 5 4 6 2 2 . 1 8 2 9 6 9 9 . 6 4 3 6 5 2 4 . 2 4 3 5 5 6 6 . 1 5 9 0 0 3 0 . 1 7 8 0 6 9 9 . 8 2 1 7 0 2 4 . 5 3 3 9 3 9 0 . 6 9 2 9 3 4 0 . 1 1 9 7 3 9 9 . 9 4 1 4 3 3 . 7 8 9 7 8 9 4 . 4 8 2 7 1 5 0 . 0 4 1 4 6 9 9 . 9 8 2 8 9 1 . 0 0 4 5 4 9 5 . 4 8 7 2 5 6 0 . 0 1 0 5 8 9 9 . 9 9 3 4 7 2 . 2 8 0 8 4 9 7 . 7 6 8 0 9 7 0 . 0 0 1 6 8 9 9 . 9 9 5 1 5 1 . 1 6 9 3 5 9 8 . 9 3 7 4 4 8 0 . 0 0 0 9 7 5 8 6 9 9 . 9 9 6 1 3 0 . 5 0 4 1 0 9 9 . 4 4 1 5 3 9 0 . 0 0 1 4 2 9 9 . 9 9 7 5 5 0 . 1 2 2 9 2 9 9 . 5 6 4 4 6 1 0 0 . 0 0 0 9 7 0 3 7 9 9 . 9 9 8 5 2 0 . 1 1 0 2 7 9 9 . 6 7 4 7 2 1 1 0 . 0 0 0 3 2 7 2 5 9 9 . 9 9 8 8 4 0 . 1 5 2 2 7 9 9 . 8 2 6 9 9 1 2 0 . 0 0 0 2 9 3 3 8 9 9 . 9 9 9 1 4 0 . 1 2 9 0 7 9 9 . 9 5 6 0 6 1 3 0 . 0 0 0 2 4 7 9 2 9 9 . 9 9 9 3 9 0 . 0 3 1 2 1 9 9 . 9 8 7 2 7 1 4 0 . 0 0 0 4 2 7 4 2 9 9 . 9 9 9 8 1 0 . 0 0 6 5 1 9 9 . 9 9 3 7 8 1 5 0 . 0 0 0 1 8 6 3 9 1 0 0 . 0 0 0 0 0 0 . 0 0 6 2 2 1 0 0 . 0 0 0 0 0

A PLS model is not complete until you choose the number of factors. You can choose the number of factors by using test set validation, in which the data table is divided into two groups called the training data and the test data. You fit the model to the training data, and then you check the capability of the model to predict responses for the test data. The predicted residual sum of squares (PRESS) statistic is based on the residuals that are generated by this process.

To select the number of extracted factors by test set validation, you use the partitionByVar or partitionByFracparameter to specify how to logically divide observations in the input data table into two subsets for model training and testing. For example, you can designate a variable in the input data table and a set of formatted values of that variable to determine the role of each observation, as in the following PROC CAS statements. The ODS TRACE command displays the full paths of the output tables in the SAS log.

ods trace on;

proc cas;

action pls.pls / table='SAMPLE', method='PLS',

model={depVars={'ls', 'ha', 'dt'},

effects={'v1', 'v2', 'v3', 'v4', 'v5', 'v6', 'v7', 'v8', 'v9', 'v10', 'v11', 'v12', 'v13', 'v14', 'v15', 'v16', 'v17', 'v18', 'v19', 'v20', 'v21', 'v22', 'v23', 'v24', 'v25', 'v26', 'v27'}},

partitionByVar={name='Role', train='TRAIN', test='TEST'};

run;

The resulting output is shown inOutput 4.1.3throughOutput 4.1.5.

SAS Visual Statistics 8.1 Programming Guide Partial Least Squares Action Set