An Introduction to
Partial Least Squares Regression
Randall D. Tobias, SAS Institute Inc., Cary, NC
Abstract
Partial least squares is a popular method for soft
modelling in industrial applications. This paper intro-
duces the basic concepts and illustrates them with
a chemometric example. An appendix describes the
experimental PLS procedure of SAS/STAT software.
Introduction
Research in science and engineering often involves
using controllable and/or easy-to-measure variables
(factors) to explain, regulate, or predict the behavior of
other variables (responses). When the factors are few
in number, are not significantly redundant (collinear),
and have a well-understood relationship to the re-
sponses, then multiple linear regression (MLR) can
be a good way to turn data into information. However,
if any of these three conditions breaks down, MLR
can be inefficient or inappropriate. In such so-called
soft science applications, the researcher is faced with
many variables and ill-understood relationships, and
the object is merely to construct a good predictive
model. For example, spectrographs are often used
C o m p o n e n t 1 = 0 . 3 7 0
C o m p o n e n t 2 = 0 . 1 5 2
C o m p o n e n t 3 = 0 . 3 3 7
C o m p o n e n t 4 = 0 . 4 9 4
C o m p o n e n t 5 = 0 . 5 9 3
Figure 2: Spectrograph for a mixture
to estimate the amount of different compounds in a
chemical sample. (See Figure 2.) In this case, the
factors are the measurements that comprise the spec-
trum; they can number in the hundreds but are likely
to be highly collinear. The responses are component
amounts that the researcher wants to predict in future
samples.
Partial least squares (PLS) is a method for construct-
ing predictive models when the factors are many and
highly collinear. Note that the emphasis is on pre-
dicting the responses and not necessarily on trying
to understand the underlying relationship between the
variables. For example, PLS is not usually appropriate
for screening out factors that have a negligible effect
on the response. However, when prediction is the
goal and there is no practical need to limit the number
of measured factors, PLS can be a useful tool.
PLS was developed in the 1960’s by Herman Wold
as an econometric technique, but some of its most
avid proponents (including Wold’s son Svante) are
chemical engineers and chemometricians. In addi-
tion to spectrometric calibration as discussed above,
PLS has been applied to monitoring and controlling
industrial processes; a large process can easily have
hundreds of controllable variables and dozens of out-
puts.
The next section gives a brief overview of how PLS
works, relating it to other multivariate techniques such
as principal components regression and maximum re-
dundancy analysis. An extended chemometric exam-
ple is presented that demonstrates how PLS models
are evaluated and how their components are inter-
preted. A final section discusses alternatives and
extensions of PLS. The appendices introduce the ex-
perimental PLS procedure for performing partial least
squares and related modeling techniques.
How Does PLS Work?
In principle, MLR can be used with very many factors.
However, if the number of factors gets too large (for
example, greater than the number of observations),
you are likely to get a model that fits the sampled
data perfectly but that will fail to predict new data well.
This phenomenon is called over-fitting. In such cases,
although there are many manifest factors, there may
be only a few underlying or latent factors that account
for most of the variation in the response. The general
idea of PLS is to try to extract these latent factors,
accounting for as much of the manifest factor variation
1

as possible while modeling the responses well. For
this reason, the acronym PLS has also been taken
to mean ‘‘projection to latent structure.’’ It should be
noted, however, that the term ‘‘latent’’ does not have
the same technical meaning in the context of PLS as
it does for other multivariate techniques. In particular,
PLS does not yield consistent estimates of what are
called ‘‘latent variables’’ in formal structural equation
modelling (Dykstra 1983, 1985).
Figure 3 gives a schematic outline of the method.
The overall goal (shown in the lower box) is to use
Factors Responses
Population
Sample
Factors Responses
T U
Figure 3: Indirect modeling
the factors to predict the responses in the population.
This is achieved indirectly by extracting latent vari-
ables T and U from sampled factors and responses,
respectively. The extracted factors T (also referred
to as X-scores) are used to predict the Y-scores U ,
and then the predicted Y-scores are used to construct
predictions for the responses. This procedure actu-
ally covers various techniques, depending on which
source of variation is considered most crucial.
 Principal Components Regression (PCR):
The X-scores are chosen to explain as much
of the factor variation as possible. This ap-
proach yields informative directions in the factor
space, but they may not be associated with the
shape of the predicted surface.
 Maximum Redundancy Analysis (MRA) (van
den Wollenberg 1977): The Y-scores are cho-
sen to explain as much of the predicted Y varia-
tion as possible. This approach seeks directions
in the factor space that are associated with the
most variation in the responses, but the predic-
tions may not be very accurate.
 Partial Least Squares: The X- and Y-scores
are chosen so that the relationship between
successive pairs of scores is as strong as pos-
sible. In principle, this is like a robust form of
redundancy analysis, seeking directions in the
factor space that are associated with high vari-
ation in the responses but biasing them toward
directions that are accurately predicted.
Another way to relate the three techniques is to note
that PCR is based on the spectral decomposition of
X
0
X, where X is the matrix of factor values; MRA is
based on the spectral decomposition of ^Y 0 ^Y , where
^
Y is the matrix of (predicted) response values; and
PLS is based on the singular value decomposition of
X
0
Y . In SAS software, both the REG procedure and
SAS/INSIGHT software implement forms of principal
components regression; redundancy analysis can be
performed using the TRANSREG procedure.
If the number of extracted factors is greater than or
equal to the rank of the sample factor space, then
PLS is equivalent to MLR. An important feature of the
method is that usually a great deal fewer factors are
required. The precise number of extracted factors is
usually chosen by some heuristic technique based on
the amount of residual variation. Another approach
is to construct the PLS model for a given number of
factors on one set of data and then to test it on another,
choosing the number of extracted factors for which
the total prediction error is minimized. Alternatively,
van der Voet (1994) suggests choosing the least
number of extracted factors whose residuals are not
significantly greater than those of the model with
minimum error. If no convenient test set is available,
then each observation can be used in turn as a test
set; this is known as cross-validation.
Example: Spectrometric Calibra-
tion
Suppose you have a chemical process whose yield
has five different components. You use an instrument
to predict the amounts of these components based
on a spectrum. In order to calibrate the instrument,
you run 20 different known combinations of the five
components through it and observe the spectra. The
results are twenty spectra with their associated com-
ponent amounts, as in Figure 2.
PLS can be used to construct a linear predictive
model for the component amounts based on the spec-
trum. Each spectrum is comprised of measurements
at 1,000 different frequencies; these are the factor
levels, and the responses are the five component
amounts. The left-hand side of Table shows the
individual and cumulative variation accounted for by
2

Table 2: PLS analysis of spectral calibration, with cross-validation
Number of Percent Variation Accounted For Cross-validation
PLS Factors Responses Comparison
Factors Current Total Current Total PRESS P
0 1.067 0
1 39.35 39.35 28.70 28.70 0.929 0
2 29.93 69.28 25.57 54.27 0.851 0
3 7.94 77.22 21.87 76.14 0.728 0
4 6.40 83.62 6.45 82.59 0.600 0.002
5 2.07 85.69 16.95 99.54 0.312 0.261
6 1.20 86.89 0.38 99.92 0.305 0.428
7 1.15 88.04 0.04 99.96 0.305 0.478
8 1.12 89.16 0.02 99.98 0.306 0.023
9 1.06 90.22 0.01 99.99 0.304 *
10 1.02 91.24 0.01 100.00 0.306 0.091
the first ten PLS factors, for both the factors and the
responses. Notice that the first five PLS factors ac-
count for almost all of the variation in the responses,
with the fifth factor accounting for a sizable proportion.
This gives a strong indication that five PLS factors are
appropriate for modeling the five component amounts.
The cross-validation analysis confirms this: although
the model with nine PLS factors achieves the absolute
minimum predicted residual sum of squares (PRESS),
it is insignificantly better than the model with only five
factors.
The PLS factors are computed as certain linear combi-
nations of the spectral amplitudes, and the responses
are predicted linearly based on these extracted fac-
tors. Thus, the final predictive function for each
response is also a linear combination of the spectral
amplitudes. The trace for the resulting predictor of
the first response is plotted in Figure 4. Notice that
Figure 4: PLS predictor coefficients for one response
a PLS prediction is not associated with a single fre-
quency or even just a few, as would be the case if
we tried to choose optimal frequencies for predicting
each response (stepwise regression). Instead, PLS
prediction is a function of all of the input factors. In
this case, the PLS predictions can be interpreted as
contrasts between broad bands of frequencies.
Discussion
As discussed in the introductory section, soft science
applications involve so many variables that it is not
practical to seek a ‘‘hard’’ model explicitly relating
them all. Partial least squares is one solution for such
problems, but there are others, including
 other factor extraction techniques, like principal
components regression and maximum redun-
dancy analysis
 ridge regression, a technique that originated
within the field of statistics (Hoerl and Kennard
1970) as a method for handling collinearity in
regression
 neural networks, which originated with attempts
in computer science and biology to simulate the
way animal brains recognize patterns (Haykin
1994, Sarle 1994)
Ridge regression and neural nets are probably the
strongest competitors for PLS in terms of flexibility
and robustness of the predictive models, but neither
of them explicitly incorporates dimension reduction---
that is, linearly extracting a relatively few latent factors
that are most useful in modeling the response. For
more discussion of the pros and cons of soft modeling
alternatives, see Frank and Friedman (1993).
There are also modifications and extensions of partial
least squares. The SIMPLS algorithm of de Jong
3

(1993) is a closely related technique. It is exactly
the same as PLS when there is only one response
and invariably gives very similar results, but it can
be dramatically more efficient to compute when there
are many factors. Continuum regression (Stone and
Brooks 1990) adds a continuous parameter , where
0    1, allowing the modeling method to vary
continuously between MLR ( = 0), PLS ( = 0:5),
and PCR ( = 1). De Jong and Kiers (1992) de-
scribe a related technique called principal covariates
regression.
In any case, PLS has become an established tool in
chemometric modeling, primarily because it is often
possible to interpret the extracted factors in terms
of the underlying physical system---that is, to derive
‘‘hard’’ modeling information from the soft model. More
work is needed on applying statistical methods to the
selection of the model. The idea of van der Voet
(1994) for randomization-based model comparison is
a promising advance in this direction.
For Further Reading
PLS is still evolving as a statistical modeling tech-
nique, and thus there is no standard text yet that gives
it in-depth coverage. Geladi and Kowalski (1986) is
a standard reference introducing PLS in chemomet-
ric applications. For technical details, see Naes and
Martens (1985) and de Jong (1993), as well as the
references in the latter.
References
Dijkstra, T. (1983), ‘‘Some comments on maximum
likelihood and partial least squares methods,’’
Journal of Econometrics, 22, 67-90.
Dijkstra, T. (1985). Latent variables in linear stochas-
tic models: Reflections on maximum likelihood
and partial least squares methods. 2nd ed. Ams-
terdam, The Netherlands: Sociometric Research
Foundation.
Geladi, P, and Kowalski, B. (1986), ‘‘Partial least-
squares regression: A tutorial,’’ Analytica Chim-
ica Acta, 185, 1-17.
Frank, I. and Friedman, J. (1993), ‘‘A statistical view
of some chemometrics regression tools,’’ Tech-
nometrics, 35, 109-135.
Haykin, S. (1994). Neural Networks, a Comprehen-
sive Foundation. New York: Macmillan.
Helland, I. (1988), ‘‘On the structure of partial least
squares regression,’’ Communications in Statis-
tics, Simulation and Computation, 17(2), 581-
607.
Hoerl, A. and Kennard, R. (1970), ‘‘Ridge regression:
biased estimation for non-orthogonal problems,’’
Technometrics, 12, 55-67.
de Jong, S. and Kiers, H. (1992), ‘‘Principal covari-
ates regression,’’ Chemometrics and Intelligent
Laboratory Systems, 14, 155-164.
de Jong, S. (1993), ‘‘SIMPLS: An alternative approach
to partial least squares regression,’’ Chemomet-
rics and Intelligent Laboratory Systems, 18, 251-
263.
Naes, T. and Martens, H. (1985), ‘‘Comparison of pre-
diction methods for multicollinear data,’’ Com-
munications in Statistics, Simulation and Com-
putation, 14(3), 545-576.
Ranner, Lindgren, Geladi, and Wold, ‘‘A PLS kernel
algorithm for data sets with many variables and
fewer objects,’’ Journal of Chemometrics, 8, 111-
125.
Sarle, W.S. (1994), ‘‘Neural Networks and Statis-
tical Models,’’ Proceedings of the Nineteenth
Annual SAS Users Group International Confer-
ence, Cary, NC: SAS Institute, 1538-1550.
Stone, M. and Brooks, R. (1990), ‘‘Continuum regres-
sion: Cross-validated sequentially constructed
prediction embracing ordinary least squares,
partial least squares, and principal components
regression,’’ Journal of the Royal Statistical So-
ciety, Series B, 52(2), 237-269.
van den Wollenberg, A.L. (1977), ‘‘Redundancy
Analysis--An Alternative to Canonical Correla-
tion Analysis,’’ Psychometrika, 42, 207-219.
van der Voet, H. (1994), ‘‘Comparing the predictive ac-
curacy of models using a simple randomization
test,’’ Chemometrics and Intelligent Laboratory
Systems, 25, 313-323.
SAS, SAS/INSIGHT, and SAS/STAT are registered
trademarks of SAS Institute Inc. in the USA and other
countries.  indicates USA registration.
4

Appendix 1: PROC PLS: An Exper-
imental SAS Procedure for Partial
Least Squares
An experimental SAS/STAT software procedure,
PROC PLS, is available with Release 6.11 of the
SAS System for performing various factor-extraction
methods of modeling, including partial least squares.
Other methods currently supported include alternative
algorithms for PLS, such as the SIMPLS method of de
Jong (1993) and the RLGW method of Rannar et al.
(1994), as well as principal components regression.
Maximum redundancy analysis will also be included in
a future release. Factors can be specified using GLM-
type modeling, allowing for polynomial, cross-product,
and classification effects. The procedure offers a wide
variety of methods for performing cross-validation on
the number of factors, with an optional test for the
appropriate number of factors. There are output data
sets for cross-validation and model information as well
as for predicted values and estimated factor scores.
You can specify the following statements with the PLS
procedure. Items within the brackets <> are optional.
PROC PLS <options>;
CLASS class-variables;
MODEL responses = effects < / option >;
OUTPUT OUT=SAS-data-set <options>;
PROC PLS Statement
PROC PLS <options>;
You use the PROC PLS statement to invoke the PLS
procedure and optionally to indicate the analysis data
and method. The following options are available:
DATA = SAS-data-set
specifies the input SAS data set that con-
tains the factor and response values.
METHOD = factor-extraction-method
specifies the general factor extraction
method to be used. You can specify any
one of the following:
METHOD=PLS < (PLS-options) >
specifies partial least squares. This is
the default factor extraction method.
METHOD=SIMPLS
specifies the SIMPLS method of de
Jong (1993). This is a more effi-
cient algorithm than standard PLS; it
is equivalent to standard PLS when
there is only one response, and it
invariably gives very similar results.
METHOD=PCR
specifies principal components re-
gression.
You can specify the following PLS-options
in parentheses after METHOD=PLS:
ALGORITHM=PLS-algorithm
gives the specific algorithm used to
compute PLS factors. Available algo-
rithms are
ITER the usual iterative NIPALS al-
gorithm
SVD singular value decomposi-
tion of X0Y , the most exact
but least efficient approach
EIG eigenvalue decomposition of
Y
0
XX
0
Y
RLGW an iterative approach that
is efficient when there are
many factors
MAXITER=number
gives the maximum number of itera-
tions for the ITER and RLGW algo-
rithms. The default is 200.
EPSILON=number
gives the convergence criterion for
the ITER and RLGW algorithms. The
default is 1012.
CV = cross-validation-method
specifies the cross-validation method to
be used. If you do not specify a cross-
validation method, the default action is
not to perform cross-validation. You can
specify any one of the following:
CV = ONE
specifies one-at-a-time cross- valida-
tion
CV = SPLIT < ( n ) >
specifies that every nth observation
be excluded. You may optionally
specify n; the default is 7.
CV = BLOCK < ( n ) >
specifies that blocks of n observa-
tions be excluded. You may option-
ally specify n; the default is 7.
CV = RANDOM < ( cv-random-opts ) >
5

specifies that random observations
be excluded.
CV = TESTSET(SAS-data-set)
specifies a test set of observations to
be used for cross-validation.
You also can specify the following cv-
random-opts in parentheses after CV =
RANDOM:
NITER = number
specifies the number of random sub-
sets to exclude.
NTEST = number
specifies the number of observations
in each random subset chosen for
exclusion.
SEED = number
specifies the seed value for random
number generation.
CVTEST < ( cv-test-options ) >
specifies that van der Voet’s (1994)
randomization-based model comparison
test be performed on each cross-validated
model. You also can specify the follow-
ing cv-test-options in parentheses after
CVTEST:
PVAL = number
specifies the cut-off probability for
declaring a significant difference. The
default is 0.10.
STAT = test-statistic
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. T2 is the default.
NSAMP = number
specifies the number of randomiza-
tions to perform. The default is 1000.
LV = number
specifies the number of factors to extract.
The default number of factors to extract is
the number of input factors, in which case
the analysis is equivalent to a regular least
squares regression of the responses on
the input factors.
OUTMODEL = SAS-data-set
specifies a name for a data set to contain
information about the fit model.
OUTCV = SAS-data-set
specifies a name for a data set to contain
information about the cross-validation.
CLASS Statement
CLASS class-variables;
You use the CLASS statement to identify classifica-
tion variables, which are factors that separate the
observations into groups.
Class-variables can be either numeric or character.
The PLS procedure uses the formatted values of
class-variables in forming model effects. Any variable
in the model that is not listed in the CLASS statement
is assumed to be continuous. Continuous variables
must be numeric.
MODEL Statement
MODEL responses = effects < / INTERCEPT >;
You use the MODEL statement to specify the re-
sponse variables and the independent effects used
to model them. Usually you will just list the names
of the independent variables as the model effects,
but you can also use the effects notation of PROC
GLM to specify polynomial effects and interactions.
By default the factors are centered and thus no inter-
cept is required in the model, but you can specify the
INTERCEPT option to override this behavior.
OUTPUT Statement
OUTPUT OUT=SAS-data-set keyword = names
< : : :keyword = names >;
You use the OUTPUT statement to specify a data
set to receive quantities that can be computed for
every input observation, such as extracted factors
and predicted values. The following keywords are
available:
PREDICTED predicted values for responses
YRESIDUAL residuals for responses
XRESIDUAL residuals for factors
XSCORE extracted factors (X-scores, latent
vectors, T )
YSCORE extracted responses (Y-scores, U )
STDY standardized Y variables
STDX standardized X variables
H approximate measure of influence
PRESS predicted residual sum of squares
T2 scaled sum of squares of scores
6

XQRES sum of squares of scaled residuals
for factors
YQRES sum of squares of scaled residuals
for responses
Appendix 2: Example Code
The data for the spectrometric calibration example is
in the form of a SAS data set called SPECTRA with
20 observations, one for each test combination of the
five components. The variables are
X1 : : :X1000 - the spectrum for this combination
Y1 : : :Y5 - the component amounts
There is also a test data set of 20 more observations
available for cross-validation. The following state-
ments use PROC PLS to analyze the data, using the
SIMPLS algorithm and selecting the number of factors
with cross-validation.
proc pls data = spectra
method = simpls
lv = 9
cv = testset(test5)
cvtest(stat=press);
model y1-y5 = x1-x1000;
run;
The listing has two parts (Figure 5), the first part
summarizing the cross-validation and the second part
showing how much variation is explained by each ex-
tracted factor for both the factors and the responses.
Note that the extracted factors are labeled ‘‘latent
variables’’ in the listing.
7

The PLS Procedure
Cross Validation for the Number of Latent Variables
Test for larger
residuals than
minimum
Number of Root
Latent Mean Prob >
Variables PRESS PRESS
-----------------------------------
0 1.0670 0
1 0.9286 0
2 0.8510 0
3 0.7282 0
4 0.6001 0.00500
5 0.3123 0.6140
6 0.3051 0.6140
7 0.3047 0.3530
8 0.3055 0.4270
9 0.3045 1.0000
10 0.3061 0.0700
Minimum Root Mean PRESS = 0.304457 for 9 latent variables
Smallest model with p-value > 0.1: 5 latent variables
The PLS Procedure
Percent Variation Accounted For
Number of
Latent Model Effects Dependent Variables
Variables Current Total Current Total
----------------------------------------------------------
1 39.3526 39.3526 28.7022 28.7022
2 29.9369 69.2895 25.5759 54.2780
3 7.9333 77.2228 21.8631 76.1411
4 6.4014 83.6242 6.4502 82.5913
5 2.0679 85.6920 16.9573 99.5486
Figure 5: PROC PLS output for spectrometric calibration example
8