Comparison of principal components regression, partial least squares regression, multi-block partial least squares regression, and serial partial least squares regression algorithms

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Comparison of principal components regression, partial least squaresregression, multi-block partial least squares regression, and serial partial leastsquares regression algorithms for the analysis of Fe in iron ore using LIBS

P. Yaroshchyk,* D. L. Death and S. J. Spencer

Received 30th May 2011, Accepted 10th October 2011

DOI: 10.1039/c1ja10164a

The objective of the current research was to compare different data-driven multivariate statistical

predictive algorithms for the quantitative analysis of Fe content in iron ore measured using Laser-

Induced Breakdown Spectroscopy (LIBS). The algorithms investigated were Principal Components

Regression (PCR), Partial Least Squares Regression (PLS), Multi-Block Partial Least Squares (MB-

PLS), and Serial Partial Least Squares Regression (S-PLS). Particular emphasis was placed on the

issues of the selection and combination of atomic spectral data available from two separate

spectrometers covering 208–222 nm and 300–855 nm ranges, which include many of the spectral

features of interest. Standard PLS and PCR models produced similar prediction accuracy, although in

the case of PLS there were notably less latent variables in use by the model. It was further shown that

MB-PLS and S-PLS algorithms which both treated available UV and VIS data blocks separately,

demonstrated inferior performance in comparison with both PCR and PLS.

1. Introduction

Laser-Induced Breakdown Spectroscopy (LIBS) is a well-estab-

lished analytical technique used for the quantitative analysis of

a wide range of materials in many industrial and scientific

applications. Examples may be found in some recent review

articles.1–3

The use of multivariate data analysis algorithms to interpret

LIBS data sets has recently received a great deal of attention. In

particular, such techniques as Principal Components Analysis

(PCA) have been applied to material classification and identifi-

cation,4–6 while Principal Components Regression (PCR)7,8 and

Partial Least Squares Regression (PLS)5,6,9 have proven

successful in quantitative analysis. As LIBS data normally spans

a spectral range of tens or hundreds of nanometres recorded with

the use of a Czerny–Turner or an Echelle spectrometer, it is

logical to seek a data-driven analysis algorithm that optimally

utilises all the available spectral bandwidth to establish the

strongest possible correlations between the spectral data and

elemental composition. This is generally done by formulating

a predictive model based on reducing the dimensionality of

a data set of input raw spectral variables to a limited number of

latent variables (or principal components in the case of PCR) in

a spectral matrix X in such a way that they robustly correlate

with the known concentrations y of the species analysed. It has

CSIRO Process Science and Engineering, Lucas Heights Science andTechnology Centre, Locked Bag 2005, Kirawee NSW 2232, Australia.E-mail: [email protected]

92 | J. Anal. At. Spectrom., 2012, 27, 92–98

been demonstrated that the use of data-driven multivariate

statistical analysis techniques makes possible robust quantitative

elemental analysis of species which LIBS normally struggles

with, such as phosphorous,8 and even the prediction of material

physical parameters which are not directly related to single

species’ elemental composition, for example, loss on ignition.10

The question of selecting the most suitable data-driven

multivariate statistical analysis algorithm from a number of

available options remains. Generally PLS is expected to perform

slightly better than PCR, as the former is able to better cope with

the problem which arises when significant changes in the analyte

concentration give rise to only small variations in X, and also if

there is a high degree of spectral interferences in the data. In an

extreme case some important information can be interpreted by

PCR as noise11 and discarded. Further it is not necessarily clear

which method should be chosen if there is data available from

multiple spectrometers, especially if the spectra differ by

dimensions, resolution and recorded intensity levels. It is

expected that if two datasets (based onmeasurements of the same

material with different spectrometers) are available for analysis,

it is best to add them together to form an extended spectral

matrix X, than, for example, use a multi-block regression anal-

ysis.12 However, there is no guarantee that a combined block will

provide better results compared to one of the single original

blocks. For instance, the improvement can be compromised by

the increased noise coming from the additional Xmatrix (block),

or there simply may not be much useful information contained in

it relative to the analyte. Also, it has been recently demonstrated

that S-PLS, a multi-block analysis technique, is nevertheless

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capable of achieving the same or better results as PLS when

analysing a combination of MIR and NIR spectra.13,14

The LIBS data used in this study consisted of two blocks,

which are referred to as UV (covering the range from 208 nm to

222 nm) and VIS (covering the range from 300nm to 855nm),

which were collected using a Czerny–Turner spectrometer and an

Echelle spectrometer respectively. Both spectrometers were fitted

with intensified multichannel detectors. As PCR and PLS both

use a single X block, three different analysis options were

investigated. In the first case the data available from two spec-

trometers were put together to form a single extended X matrix.

In other two cases the X matrix contained the data originating

from only one of the spectrometers, taking either the UV or the

VIS data as the X matrix.

MB-PLS is an extension of the standard PLS method and it is

similar to PLS except that MB-PLS contains all measured vari-

ables into several blocks according to some predetermined

criterion or a priori knowledge of the measurement or

process.15–17Unlike the standard PLS, theMB-PLS operates with

super scores and super weights obtained from multiple blocks

and uses those for modelling and subsequent prediction. With

S-PLS the data is treated as multiple X blocks separately in

a serial mode, allowing the use of a different number of latent

variables for modelling each data block.18

The aim of this present study was to compare the accuracy of

analysis for Fe content in an iron ore matrix by applying the

PCR, PLS, MB-PLS, and S-PLS algorithms to LIBS data sets.

The emphasis is made on finding the optimal analytical strategy

for the case when the spectral data is available is numerous

blocks, which may or may not come from separate spectrome-

ters. While the first two algorithms have been extensively used in

LIBS for elemental analysis of a range of species, to our

knowledge there are no reports of the use of the S-PLS or MB-

PLS to analyse atomic spectral data. As the PCR, PLS, MB-PLS

and S-PLS multivariate statistical algorithms have already been

successfully applied to the analysis of NIR and MIR spectra, it

seemed useful to investigate whether MB-PLS and S-PLS are

useful for analysis of LIBS data.

2. Theory

2.1 Principal components regression

PCR decomposes the predictor block X (n � p), using a selected

number of principal components, so that important features of X

are retained in a truncated matrix T:

X ¼ TPT + E (1)

In this case n is the number of samples measured and p is the

number of pixels corresponding to the LIBS spectra. E is the

matrix of residuals, the columns of matrix T are the scores and

the columns of P are the loadings that linearly relate the principal

components to the ‘raw’ spectral data. The scores and loadings

are subsequently used to compute regression coefficients b for

prediction of the unknown response vector y (n � 1), in this case

being the predicted concentration of the species of interest in

each sample:

y ¼ Xb (2)


A training data set is used to generate the model in a calibra-

tion phase and a separate validation data set is used to evaluate

the performance of the predictive model. Details of the imple-

mentation of PCR including the SVD factorisation and NIPALS

algorithm are given by Jørgensen and Goegebeur.11

2.2 Partial least squares regression

In contrast to PCR, PLS uses a selected number of latent vari-

ables to decompose both the predictor block X (n � p) and the

response vector y (n � 1). This leads to two equations:

X ¼ TPT + E (3)

y ¼ TqT + f (4)

Here q is the loading vector for y, and f is a vector of residuals for

y. Predicted response vector y is computed from the following

linear equation, where b is an array of regression coefficients:

y ¼ Xb (5)

Separate training and validation data sets are used for cali-

bration and evaluation of model predictive performance. Details

on the implementation of the SIMPLS PLS algorithm used in

this study are given by deJong.19

2.3 Multi-block partial least squares regression

MB-PLS is an extension to the standard PLS algorithm with the

main difference being that (in our case) two predictor blocks X1

(n � p1) and X2 (n � p2) are used to model one response vector y

(n � 1).

X1¼TPT1 + E1 (6)

X2¼TPT2 + E2 (7)

y ¼ TqT + f (8)

Here p1 and p2 stand for the number of data points (pixels)

in spectral blocks X1 (UV) and X2 (VIS), and T is a matrix of

super scores, which contains the scores of predictor blocks X1

and X2.

Two variations of MB-PLS algorithms have been reported in

literature. One version uses the block scores to calculate the

loadings and residuals,15 while the second version uses the super-

scores.17 The algorithm used in the present study was based on

the latter version. In the calibration phase, block loadings and

weights as well as super weights were computed from X1, X2, and

y, and stored for prediction. The predicted response y was esti-

mated from the predictor data X1 and X2 corresponding to the

unknown samples as well as the calibration loadings and weights,

see Westerhuis et al.17 and Chong and Lee20 for implementation

details.

2.4 Serial partial least squares regression

In S-PLS, predictor blocks X1, X2, and response vector y are

formed using the following relationships:

J. Anal. At. Spectrom., 2012, 27, 92–98 | 93


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X1¼T1PT1 + E1 (9)

X2¼T2PT2 + E2 (10)

y ¼ T1qT1+T2q

T2 + f (11)

Here Ti and Pi are correspondingly the scores and the loadings

for the predictor block Xi, and qi is the loading vector for y. Ei

and f are the residuals for Xi and y. Prediction was performed

using the following linear model, where b1 and b2 are the

regression coefficients for predictor blocks X1 and X2:

y ¼ X1b1 + X2b2 (12)

The major difference between the S-PLS and the MB-PLS

algorithms is that in the former method decomposition of the

predictor blocks (reduction of data dimension) is performed in

a serial mode, and it is thus possible to use a different number of

latent variables for blocks X1 and X2. Details on the imple-

mentation of the S-PLS algorithm used in this study are given by

Felicio et al.13 and Berglund and Wold.18

3. Experimental

3.1 Laboratory setup

Fig. 1 shows a schematic diagram of the experimental setup used

in this study. The fundamental output of a Q-switched Nd:YAG

laser (Quanta-Ray INDI, Spectra Physics), pulse duration 5–8

ns, operating at 20 Hz was directed using steering mirrors and

focused onto the sample surface with a 200 mm FL plano-convex

lens. The beam was attenuated using the combination of

a Brewster plate polariser and a half-wave plate. A constant

laser-pulse energy of 40 mJ was used for this study.

The UV and VIS components of the LIBS plasma emission

were collected by two sets of imaging optics and delivered to two

separate spectrometers. A combination of two fused silica 100

mmFL, 25 mm diameter plano-convex lenses was used to collect,

collimate and deliver UV emission to the triple Czerny–Turner

spectrometer (Spex 1877C). This spectrometer was tuned to

cover the 208 nm to 222 nm spectral range. The Spex

Fig. 1 Schematic diagram o


spectrometer was equipped with a gated and intensified linear

photodiode array detector (PDA, 1024 pixels long) which was

controlled using an optical multi-channel analyser (OMA,

Princeton Instruments ST-120 controller) and a PG-200 gate

pulse generator.

A 100 mm FL plano-convex lens of 25 mm diameter sampled

and collimated the VIS LIBS plasma emission and a second

similar lens was used to focus the light into a 200 mm core optical

fibre. The fibre was connected to the Echelle spectrometer

(Spectra Pro HRE Echelle, Princeton Instruments) equipped

with gated intensified CCD detector (1024 � 1024 pixel array,

PI-MAX Princeton Instruments). The spectrometer nominally

covers the 200–1000 nm range; however, due to the limitations of

the optical components used in the experiment reported, the

interval from 300 nm to 855 nm was used for the analysis. Both

the PDA and ICCD detectors were externally triggered by the

laser and set to 0.8 ms gate width and 2 ms gate delay between the

laser Q-switch and acquisition start. The data has been collected

simultaneously with the two detector systems. The detectors were

triggered independently and not synchronised with each other.

3.2 Samples

The sample set used in this study has been previously described

by Yaroshchyk et al.10 The sample set consisted of ring-milled

and mechanically pressed iron ore pellets. The raw material was

obtained from five different iron ore bodies. A total of 63 original

samples with Fe ranging from 12.6 to 67.9% were used for model

calibration, while another 60 original samples with Fe ranging

from 14.2 to 67.9% were available for model validation. Their

sub-samples were sent for XRF elemental analysis, with the

results used as reference concentration values. All of the original

samples were split in two resulting in a calibration set of 126 and

validation set of 120 sub-samples. The splitting has been done

mainly in order to double the size of the datasets, and it also

provided easy means of monitoring the precision of measure-

ments. A motorised sample-stage accommodating up to 105

pressed pellets was used to position the samples under the laser

beam. A pair of stepper motors moved the stage in the orbital

f the experimental setup.



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pattern under the control of a National Instruments MID-7604

controller operated using LabView software.

3.3 Data pre-treatment

Our previous work had shown the initial data pre-treatment to

have a significant effect on the performance of the multivariate

statistical analysis algorithms when applied to LIBS data.10 For

every sub-sample used in this study a total of 50 spectra were

collected. Each of those spectra was an average spectrum

recorded with 40 and 50 laser shots for the OMA and PI-MAX

detectors respectively. The UV and VIS data blocks were treated

separately. Firstly a dark-noise background spectrum was sub-

tracted from each of the laser-shot averaged data. Then an

automatic filtering routine was employed to remove outlying

spectra based on the integrated total energy of the spectral data

(TEi). Spectra with TE greater that 2 standard deviations from

the mean ðTEÞ for those 50 spectra were excluded from the

spectral data set to be analysed. Here TEi is defined as:

TEi ¼Xp

j¼1

I ji (13)

Where Iji was the counts in energy channel j for spectrum i. Each

of the retained spectra were subsequently normalised to its total

integrated energy TEi. Depending on the analysis algorithm

employed, the spectral data was collated into a single predictor

block X or used as separate predictor blocks, X1 and X2. The X

Table 1 Summary of results obtained with PCR and PLS-based algorithms

Algorithm Data setOpt. Num. ofLV (PC)

PCR UV 8VIS 18UV and VIS 13

PLS UV 6VIS 11UV and VIS 6

MB-PLS UV(X1) and VIS(X2) 3S-PLS UV (X1) and VIS(X2) 3 and 1

VIS(X1) and UV (X2) 5 and 5

Fig. 2 Results of the PCR analysis: MSECV corresponding to the 10-fold cr


blocks have not been separately scaled prior to the concatenation.

TheMATLAB software equipped with the Statistics Toolbox has

been used for all computations reported in this study.

4. Results and discussion

4.1 Calibration

For all methods reported in this study, a global sample dataset,

y, comprising of 126 sub-samples originating from the five

different ore bodies was used for model calibration. Separate

global dataset, yval, consisted of 120 sub-samples. The number of

latent variables (or principal components in case of PCR) was

chosen by performing the 10-fold cross-validation based on the

metric of minimal mean squared error of cross-validation

(MSECV):

MSECV ¼ 1

n

Xn

i¼1

ðyi � yiÞ2 (14)

The remaining subset yval was used to provide an independent

validation of the model with the metric of validation being the

estimated root mean squared error of prediction (RMSEP):

RMSEP ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1

ðyvali � yvali Þ2

n

vuuut(15)

CalibrationR2

ValidationR2

RMSEP/wt(%)

0.96 0.94 3.10.99 0.90 3.80.99 0.97 2.20.96 0.94 3.00.99 0.90 3.90.99 0.97 2.20.96 0.94 3.10.89 0.83 5.10.95 0.87 4.5

oss-validation (left), calibration plot (centre), and validation plot (right).



Fig. 3 Results of the PLS analysis: MSECV corresponding to the 10-fold cross-validation (left), calibration plot (centre), and validation plot (right).

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4.2 Analysis

The PCR, PLS, S-PLS, and MB-PLS algorithms were used to

predict the Fe content in the iron ore matrix from the measured

LIBS data.

4.2.1 PCR Analysis. Three data combinations of the spectral

data were used to construct the X dataset as used for PCR

analysis:

1) UV data / X

2) VIS data / X

3) Concatenated UV and VIS data / X

For each of the three cases, the optimal number of principal

components was found using the procedure described in section

4.1 and was equal to 8 (UV), 18 (VIS), and 13 (UV and VIS

combined) respectively, see Table 1 for details. It was demon-

strated that all three combinations of spectral data are capable

of providing adequate calibration with goodness-of-fit R2 in the

range of 0.96–0.99. However, the use of concatenated UV and

VIS data for the X block resulted in the best prediction. Fig. 2

Fig. 4 Results of the MB-PLS analysis: MSECV corresponding to the 10-

(right).


exhibits the MSECV plot for the UV and VIS data/ X case

as a function of number of principal components. Generally

there is a danger of over-fitting the data by taking a large

number of principal components however, as may be seen in

the figure, MSECV comes to a minimum with the choice of 13

PCs. Fig. 2 also shows the corresponding calibration and

validation plots.

4.2.2 PLS analysis. The same three data combinations were

used in the PLS analysis. Again, the optimal number of latent

variables differed significantly between the three cases and was

equal to 6 (UV), 11 (VIS), and 6 (UV and VIS combined). As

generally expected, PLS provided similar prediction accuracy

with a lesser number of latent variables as compared to PCR

(see Table 1). The best validation results were R2 ¼ 0.97 and

RMSEP ¼ 2.2%. In common with the PCR analysis this was

achieved using the UV and VIS data combined as the X block.

Fig. 3 shows the correspondingMSECV plot for the UV and VIS

data as a function of a number of latent variables, as well as

calibration and validation plots.

fold cross-validation (left), calibration plot (centre), and validation plot



Fig. 5 Results of the S-PLS analysis: MSECV corresponding to the 10-fold cross-validation (left), calibration plot (centre), and validation plot (right).

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4.2.3 MB-PLS analysis. A total of 3 latent variables were

found to be optimal for the MB-PLS algorithm applied to this

dataset. The associated goodness-of-fit parameters for calibra-

tion and validation were R2Cal ¼ 0.96 and R2

Val ¼ 0.94 respec-

tively with RMSEP ¼ 3.1%. The UV and VIS LIBS data were

applied as theX1 andX2 blocks. SwappingX1 forX2 did not have

an effect on the outcome of the analysis. This reflects the

robustness of the MB-PLS approach. See Fig. 4 for theMSECV,

calibration, and prediction plots.

4.2.4 S-PLS analysis. In contrast to MB-PLS, the S-PLS

algorithm is sensitive to the choice of the order of the X blocks

used for analysis.14 Both combinations were considered in the

application of S-PLS to the available LIBS data:

1) UV / X1; VIS / X2

2) VIS / X1; UV / X2

Fig. 6 Regression coefficients corresponding to the PLS


The calibration procedures described in section 4.1 were

applied for both cases above acknowledging that the optimal

number of latent variables, LV1, used to decompose X1 and y is

not necessarily equal to the optimal number of latent variables,

LV2, used to decompose X2 and y.

Indeed the number of latent variables found to be optimal for

combination one differed quite significantly for those required

for combination two, see Table 1. Taking into account that the

UV data block resulted in better performance when used by the

standard PLS method, it was somewhat surprising that the X1 ¼VIS and X2 ¼ UV combination demonstrated better prediction

results compared to the alternative as the serial nature of the

analysis would see the UV X2 block fine tuning the VIS X1 PLS

outcome. We also note that S-PLS demonstrated much stronger

dependency on the number of LVs used for cross-validation

computations, which resulted in significantly higher errors

analysis (top) and to the S-PLS analysis (bottom).



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achieved for models that used a far from optimal number of LVs.

Fig. 5 and Table 1 show that the S-PLS with VIS data in block 1

provides better Fe abundance prediction results with calibration

R2 ¼ 0.95, validation R2 ¼ 0.87, and RMSEP ¼ 4.5%.

The regression coefficients corresponding to the best per-

forming S-PLS (bottom) and PLS (top) models are demonstrated

in Fig. 6. When comparing the two, it is seen that the VIS

components have a range of common features, however the

values corresponding to the S-PLS case are generally up to

a factor of 5 larger. The most noticeable difference is perhaps

seen in the UV channel. In contrast to the PLS, for which some of

the stronger correlating peaks are located in the UV region, the

values of the same coefficients calculated using the S-PLS are

relatively small. This difference seems fair considering that these

coefficients correspond to the calibration case when the UV

spectra were used in a secondary data block. The regression

coefficients as such are not used for prediction in the case of the

MB-PLS calibration, and therefore not shown in the manuscript.

The regression coefficients obtained for the PCR model are very

similar to the ones demonstrated in Fig. 6 (top).

5. Conclusion

PCR, PLS, MB-PLS, and S-PLS algorithms have been applied to

and compared for the analysis of Fe in iron ore matrices using

laser induced breakdown spectroscopy. To our knowledge,

neither MB-PLS nor S-PLS have been used in this way for the

quantitative estimation of elemental concentrations from atomic

spectroscopy even though they have been used successfully for

the analysis of molecular spectra previously.

Two data blocks recorded using the Czerny–Turner and

Echelle spectrometers fitted with gated and intensified detectors

and covering spectral ranges from 208 nm to 222 nm and 300 nm

to 855 nm respectively were analysed. When the combination of

UV and VIS data was used for analysis, PCR and PLS provided

superior performance for Fe determination with validation R2 of

0.97 and RMSEP of 2.2%.

The sole use of the VIS or UV data was found less precise in

predicting iron content with validation in the range of R2 ¼ 0.90–

0.94 and RMSEP in the range of 3.0–3.9% for PCR and PLS. As

expected, PLS required significantly less latent variables for

calibration compared to PCR.

Results produced by the MB-PLS algorithm which treats the

two spectral data blocks separately in a parallel mode were third


best, with validation R2 ¼ 0.94, and RMSEP ¼ 3.1%. S-PLS

algorithm which treats the two data blocks in a serial mode,

demonstrated relatively weak performance with validation R2 ¼0.87 and RMSEP ¼ 4.5%. In both cases these results did not

compare favourably with the outcome of the standard PLS and

PCR analysis on combined data sets.

The present study shows that in general multivariate data

analysis techniques are very well suited for quantitative elemental

analysis in iron ore using LIBS spectra. With a range of

commercial instruments available for LIBS comprising several

compact spectrometers, it seems reasonable to put all available

spectral data into a single X block and consider PLS and PCR as

first candidates to be used for calibration.

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Comparison of principal components regression, partial least squares regression, multi-block partial least squares regression, and serial partial least squares regression algorithms