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On the Interpretability of O-PLS Filtered Models
Abstract
Orthogonal PLS, introduced originally by Trygg and Wold in 2002 [1], is a patented algorithm that has received much attention for its perceived ability to simplify model interpretation. Since its introduction, it has been shown by Ergon [2]
and Kemsley and Tapp [3] that results identical to the original O-PLS formulation can be obtained by post-processing conventional PLS models in a nonpatented way. This demonstrated, unequivocally, that O-PLS models have predictive properties identical to their non-rotated versions. The authors did not,
however, consider the interpretability of the models at length. In this poster we
explore the issue of interpretability of O-PLS models by applying the method to
carefully constructed simple systems and well characterized data.
Jeremy M. Shaver and Barry M. Wise
Eigenvector Research, Inc.
Binary Expression Simulation
•• 10 Expressed Proteins (variables), 500 Subjects
•• 1 "primary" effect with loading:
1 2 3 4 5 6 7 8 9 10
+ + x x 0 0 0 0 0
0
(must have +, cannot have x, 0 have no effect)
•• 7 "background" effects (rank 1 patterns with positive and negative
correlations as for primary)
O-PLS Simple System Example
Pseudo Gasoline Example
•• Synthetic example with three constituents
50
100
•• Solve for pure components via CLS
•• Vary correlation between analyte and interferents
150
•• Use pure spectra to create synthetic scenarios
- Start with orthogonal concentrations, then
1) All analytes positively correlated
2) One interferent positive, three negative
0.045
Interferent 1
Analyte
Interferent 2
0.04
Positive
•• Five component mixture measured by NIR
•• Evenly spaced Gaussian peaks, analyte in middle
Orthogonalize Model
•• Only samples with primary loading expression for 1-4 (after mixing
all effects) will exhibit property of interest (e.g. disease)
0.035
200
250
Negative
300
350
400
0.03
Example Set 1
450
0.025
500
1
2
3
4
5
6
7
8
9
10
0.02
20
40
60
80
100
120
140
160
180
200
1)
Gaussian Peaks Scenarios
- 1) Go from orthogonal (r=0) to positively correlated
concentrations (r=1)
- 2) One interferent positively correlated, one negatively
correlated
•• What kinds of sensitivities does O-PLS have to
noise, rotational ambiguity, and correlated signals?
First O-PLS Component Loading
•• What do we need to be aware of when interpreting
O-PLS recovered components?
2)
First O-PLS Component Loading
1)
Co
rre
la
tio
nr
2)
Co
rre
la
ble
Varia
tio
nr
b
Varia
le
Interferent Concentration
2.3
Samples/Scores Plot of spec1
30
0.015
28
26
24
Y CV Predicted 1
0.02
5 Chemical Components:
•• Heptane
•• Toluene
•• Xylene
•• Decane
•• Iso-Octane
Mixtures at known concentrations
Pure Component Spectra from CLS shown
0.01
Regression Coefficient
First O-PLS Component reflects pure component response when
r = 0 but changes rapidly when analyte and interferent are correlated,
even for r = 0.5, R2 = 0.25!
NIR of Pseudo-Gasoline Samples
R2 = 0.988
5 Latent Variables
RMSEC = 0.46945
RMSECV = 0.68306
Calibration Bias = ï1.0658eï14
CV Bias = 0.0375
2.2
2.1
2
1.9
1.6
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
12
100
Cor
re
50
latio
nr
ngth
Wavele
(nm)
0
ï50
ï150
800
Regression vector is the same for both
scenarios, and all values of r except 1.
900
1000
1100
1200
1300
1400
1500
1600
Wavelength (nm)
tio
nr
ble
Varia
Regression vector is the same for both
scenarios, and all values of r except 1.
Percent Variance Captured by Regression Model
-----X-Block----- -----Y-Block----Comp This
Total
This
Total
---- ------------- ------------1
91.17
91.17
8.36
8.36
2
7.40
98.57
7.19
15.55
3
0.93
99.50
32.81
48.36
4
0.46
99.96
26.18
74.54
5
0.02
99.98
24.90
99.44
14
150
ï100
Co
rre
la
2.4
Analyte Concentration
20
16
nr
)
h (nm
gt
elen
v
a
W
1.8
Note: this is what a correlation
of r = 0.5 looks like!
0.005
tio
1.7
22
18
First O-PLS Loading varies
strongly with analyte
correlations!
Regression Coefficient
Questions
Co
rre
la
Example Set 3
0
First O-PLS Component Loading
0
Example Set 1
0.005
Example Set 2
0.01
First O-PLS Component Loading
0.015
Conclusions
0
900
1000
1100
1200
1300
Wavelength (nm)
1400
1500
10
10
1600
12
14
16
18
20
22
Y Measured 1
24
26
28
30
Compare to Selectivity Ratio
Regular and O-PLS Filtered
Regression Vectors
Variables/Loadings Plot for spec1
Variables/Loadings Plot for spec1
25
1.5
Kvalheim’’s Selectivity Ratio
not a strong function of
correlation in analyte
concentrations, correctly
picks variables that are
most selective for analyte
of interest
20
15
1
Regular PLS
Reg Vector for Y 1
Reg Vector for Y 1
5
0
ï5
ï10
0
ï0.5
ï1
ï15
ï1.5
ï20
ï25
800
O-PLS Filtered
0.5
10
900
1000
1100
1200
Variable
1300
1400
1500
1600
ï2
800
Pure Component
Spectrum
900
1000
1100
1200
Variable
1300
•• O-PLS does simplify regression vectors. It is
CLOSER to underlying bilinear response……
•• …… HOWEVER, result generally NOT the same
as a first principles model.
Selectivity Ratio (normalized)
800
•• O-PLS results are a strong function of the correlation in the concentrations
Co
rre
la
1400
1500
1600
tio
nr
ble
Varia
•• O-PLS recovered component is more sensitive
to chance correlation than regression vector
(Problem seen even with 2000 samples!)
References
[1]
J. Trygg and S. Wold, ““Orthogonal Projections to Latent Structures
(O-PLS),”” J. Chemo, 16, 119-128, 2002.
[2]
R. Ergon, ““PLS post-processing by similarity transformation (PLS+ST): a
simple alternative to OPLS,”” J. Chemo, 19, 1-4, 2005.
[3]
E.K. Kemsley and H.S. Tapp, ““OPLS filtered data can be obtained directly
from non-orthogonalized PLS1,”” J. Chemo, 23, 263-264, 2009