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CONSENSUS MULTIVARIATE CALIBRATION OR MAINTENANCE WITHOUT REFERENCE SAMPLES USING TIKHONOV TYPE REGULARIZATION APPROACHES . John Kalivas, Josh Ottaway , Jeremy Farrell, Parviz Shahbazikah Department of Chemistry Idaho State University Pocatello, Idaho 83209 USA. Outline.
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CONSENSUS MULTIVARIATE CALIBRATION OR MAINTENANCE WITHOUT REFERENCE SAMPLES USING TIKHONOV TYPE REGULARIZATION APPROACHES John Kalivas, Josh Ottaway, Jeremy Farrell,ParvizShahbazikah Department of Chemistry Idaho State University Pocatello, Idaho 83209 USA
Outline • Multivariate calibration • Tikhonov regularization (TR) • TR calibration maintenance with reference samples to form full wavelength or sparse models • Selecting “a” model • Selecting a collection of models • Comparison to PLS • TR calibration or maintenance without reference samples • Examples with comparison to PLS • Summary TR variant equations
Spectral Multivariate Calibration • y = Xb y = m x 1 vector of analyte reference values for m calibration samples X = m x n matrix of spectra for n wavelengths b = n x 1 regression (model) vector MLR solution; requires m≥ p (wavelength selection) Biased regression solutions such as TR, RR (a TR variant), PLS, and PCR • Requires meta-parameter (tuning parameter) selection
Quantitation by Tikhonov Regularization (TR) = Euclidian vector 2-norm (vector magnitude or length) • General TR in 2-norm • Ridge regression (RR) when L = I Depending on the calibration goal, L can have different forms • RR is regularized by using I and selecting ηto minimize prediction errors (low bias) simultaneously shrinking the model vector (low variance)
overfitting best model underfitting Selecting η • Cross-validation • L-curve graphic (can use with RMSEC) • Bias/Variance can be assessed • Useful for putting RR, PLS, etc. on one plot for objective comparison • C.L. Lawson, et.al., Solving Least-Squares Problems. Prentice-Hall, (1974) • P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Press (1998)
Calibration Maintenance • Need primary model to function over time and/or under new secondary conditions • Prepare calibration samples to span all potential spectral variances • Not possible with a seasonal or geographical effects in some data sets • Preprocess primary and secondary data to be robust to new conditions • Adjust spectra measured under new conditions to fit the primary model • Update the primary model to predict in the new conditions
Calibration Maintenance with TR2 • Model updating a RR model requires a new penalty term • Minimize prediction errors for a few samples from new secondary conditions M = spectra from secondaryconditions yM= analyte reference values • Avoid measuring many samples by tuning with λ • Local centering • Respectively mean center X, y, M, and yM • Validation spectra centered to M
Pharmaceutical Example • M. Dyrby, et. al., Appl. Spectrosc. 56 (2002) 579-585 • http://www.models.life.ku.dk/datasets; Dept. of Food Sciences, Univ. of Copenhagen • 310 Escitolopram tablets measured in NIR from 7,400-10,507 cm-1 at resolution 6 cm-1for 404 wavelengths • Four tablet types based on nominal weight: type 1, type 2, type 3, and type 4 • Three tablet batches (production scale): laboratory, pilot, and full • 30 tablets for each batch tablet type combination Lab, type1 Lab, type 2 Lab, type 3 Lab, type 4 Full, type 1 Full, type 2 Full, type 3 Full, type 4
Objective • Using laboratory produced tablets as the primary calibration set • Determine active pharmaceutical ingredient (API) concentration in new tablets produced in full production (secondary condition) Primary Calibration Space: 30 random lab batch samples with 15 from types 1 and 2 each Secondary Calibration Space: 30 random full batch samples with 15 from types 1 and 2 each Standardization Set M: 4 random full batch samples with 2 from types 1 and 2 each Validation Space: Remaining 30 full batch types 1 and 2 • Other batch type combinations studied
Example Model Merit Landscapes RMSEC η η λ λ RMSEM η η λ λ
Model Merit Landscapes RMSEC η η RMSEM λ λ • Convergence at small λ • Secondary conditions are not included in new model • Amounts to using primary RR with local centering where secondary validation samples are centered to the mean of M Best local centered models A tradeoff region Prediction of primary degrades while the prediction of secondary improves
Model Merit Landscapes RMSEC η η RMSEM λ λ too large η A tradeoff region Prediction of primary degrades while the prediction of secondary improves • Further tradeoffs • Tradeoff region between and RMSEC and RMSEM • Can use an L-curve at a fixed λ value λ
Model Merit Evaluations • Multiple merits can be used to assess tradeoff • Respective RMSEC and RMSEM landscapes for R2, slope, and intercept • L-curves at selected η and λ values λ = 54.29 RMSEV η H η λ λ = 54.29
Model Updating Results Updating Primary Lab Batch Types 1 and 2 to Predict Secondary Full Batch Types 1 and 2 • Updated primary models predicts equivalently to the secondary model predicting the secondary validation samples Lab and Full Batches Types 1 and 2 Self Predicting Using RR
Model Vectors RR Lab Batch TR2 Wavelength, cm-1 RR Full Batch Wavelength, cm-1 Wavelength, cm-1
Using PLS • PLS (and other methods) can also be used • With PLS, the PLS latent vectors (PLS factors) replace the η values TR2 PLS
PLS Model Merit Landscapes RMSEM RMSEC Factors • Similar landscape trends • The discrete factor aspect of PLS can make it difficult to capture the underlying continuity of the landscapes λ λ RMSEV Factors λ λ
PLS and TR2 Model Updating Results Updating Primary Lab Batch Types 1 and 2 to Predict Secondary Full Batch Types 1 and 2 • PLS prediction equivalent to TR2 • The discrete factor aspect of PLS can make it difficult to capture the underlying continuity of the landscapes
Sparse TR Calibration Maintenance • TR2: • TR2b (sparse model): • L = diagonal matrix with lii = 1/│bi│ • Gorodnitsky IF, RaoBD. IEEE Transactions on Signal Processing 1997; 45: 600-616. • TR2-1 (sparse model):
TR2-1 Sparse Model Merit Landscapes RMSEC RMSEM • Similar landscape trends • For small λ values, the η values are the same across λ • At greater λvalues, the η values vary across λ Models with increasing η RMSEV λ λ
TR2 and TR2-1 Model Updating Results Updating Primary Lab Batch Types 1 and 2 to Predict Secondary Full Batch Types 1 and 2 TR2-1 prediction results improve over TR2 and PLS TR2 TR2-1 PLS cm-1 cm-1 cm-1
Other TR Sparse Maintenance Methods • TR2-1b (sparse models when L = I or L≠I): • TR1-2b (full wavelength when L = I): • TR1 (sparse models when L = I or L≠I):
Other TR Applications • Updating a primary model: • for extra virgin olive oil adulterant quantitation to a new geographical region (applicable to new seasons) • to a new temperature • formed on one instrument to work on another
Summary • Only a few samples needed for M with appropriate weighting • Same samples measured in primary and secondary conditions are not needed • Avoids long term stability issue • PLS and other methods can be used • Discrete nature (PLS factors) can limit landscapes • Need to select a pair of tuning parameters for “a” model • Requires reference values foryM
Consensus (Ensemble) Modeling • Samples predicted with a collection of models • Composite (fused) prediction is formed • Simple mean prediction used here • Typically form models by random sampling across calibration samples and/or variables • From collection, filter for model quality • Ideal models: • High degree of prediction accuracy • Small but noteworthy difference between selected models (model diversity)
Consensus TR and PLS Modeling • Models formed from varying tuning parameter values • Plot predicted values against reference values for X,yand M,yM • Use respective R2, slope, and intercept model merit values • Natural target values: • R2 → 1 • Slope → 1 • Intercept → 0
TR and PLS Consensus Models (RMSEV) 1 PLS Model 348 TR2 Models η Factors λ 628 TR2-1 Models • Fewer PLS models selected due to sharpness of landscapes from the discrete factor nature of PLS • Number of “good” models can be made to increase by reducing the increment sizes of η and λ Model with increasi8ng η λ
Consensus Mean Model Updating Results Updating Primary Lab Batch Types 1 and 2 to Predict Secondary Full Batch Types 1 and 2 • The one PLS model predicts best • PLS limited to discrete factors where TR allows 0 ≤ η < ∞ to more fully resolve the landscape
Consensus Models and Correlations TR2 348 models TR2-1 628 models cm-1
Summary • Only a few samples needed for M with appropriate weighting • Same samples measured in primary and secondary conditions are not needed • Avoids long term stability issue • Can select “a” model or a collection of models • Natural target values (thresholds) with model merits R2, slope, and intercept for primary and secondary standardization sets • Work in progress • Requires reference values foryM
Without Reference Samples Beer’s law: x =yaka+yiki+ m + + n ka= pure component (PC) analyte spectrum ki= PC spectrum of ith interferent (drift, background, etc.) m = rest of the sample matrix n = spectral noise • Ideal situation: WHEN: THEN: • Cannot simultaneously satisfy 1, 2, and 3 to obtain 4
Compromise PCTR2 Model N= spectra without analyte, e.g., ki • Minimizing the sum requires a tradeoff between the three conditions • The closer the three conditions are met, the more likely • Updating the non-matrix effected PC ka to predict in current conditions (spanned by N)
Sources of N • PC interferent spectra • Reference values are 0 • Matrix effected samples without the analyte • Reference values are 0 • Constant analyte samples • Reference values are 0 after spectra are mean centered • Estimate using samples with reference values • Samples for N need to be measured at current conditions Goicoechea et al. Chemom. Intell. Lab. Syst. 56 (2001) 73-81
Extra Virgin Olive Oil Adulteration • EVOO samples: Crete, Peloponnese, and Zakynthos • RR calibration: 56 samples spiked 5, 10, and 15% (wt/wt) sunflower oil • Primary: PC sunflower oil, 1 sample • Secondary: EVOO, 25 samples • Validation: 22 spiked samples • Synchronous fluorescence spectra 270 to 340 nm at Δλ=20 nm Zakynthos
Model Merit Landscapes RMSEN RMSEPC λ λ η η RMSEV η η λ λ
H Values RR with PCTR2 cal samples PCTR2 at η= 9.1e3 η λ RR full cal η
Model Updating From PC Sunflower • Updated PC predicts better than a full calibration yi = 0.422xi + 0.048 yi = 0.807xi - 0.0074 yi
Temperature Data Set • Wülfert, et al., Anal. Chem. 70 (1998) 1761-1767 • hhttp://www.models.life.ku.dk/datasets; Dept. of Food Sciences, Univ. of Copenhagen • Water, 2-propanol, ethanol (analyte) • 850 to 1049 nm at 1 nm intervals at 30, 40, 50, 60, and 70°C • Calibration: 13 mixtures from 0% to 67% at 30°C • Validation: 6 mixtures from 16% to 66% at 70°C • Primary: PC ethanol at 30°C • Non-analyte matrix (standardization set) N at 70°C • PC interferents water and 2-propoanol (2 samples) • Blanks (3 samples) • Constant analyte (CA, 5 samples)
PCTR2 Model Merit Landscapes RMSEPC RMSEN η RMSEV η λ λ
Model Updating From PC 30°C to 70°C Updated PC predicts as well as secondary model predicting the secondary validation samples
PCTR2 and PLS Modeling Temperature Updating analyte PC at 30°C to 70°C using interferent PC and blanks PLS and PCTR2 predict similarly PCTR2 RMSEV PLS RMSEV η Factors λ λ
PCTR2 Consensus Modeling Temperature • On-going work • Cannot use R2, slope, and intercept for respective predicted values of the PC and N • Set thresholds for RMSEN, RMSPC, and based on preliminary inspection of landscapes • Tradeoff needed between , RMSEN and RMSEPC • Can further filter based on predicted values • Majority vote • Remove outliers • Combine predicted value of analyte pure component sample with predicated non-analyte samples to obtain R2, slope, and intercept
PCTR Variants (Calibration or Maintenance) • No reference values • With current condition sample reference values • A combination of N and M • Replace with or to obtain sparse models
Summary • PCTR2 calibrates (updates) to current conditions without reference samples • Only a few new samples needed • Can predict better than a full calibration • More focused toorthogonalize to the sample matrix • Requires PC analyte spectrum • Does not have to be matrix effected • Requires non-analyte samples • Can be estimated with reference samples bias variance
Other On-Going Consensus Modeling • In addition to combining a set of models, can combine TR2, PLS, PCTR2, … sets of model predictions Consensus TR2 models Consensus PLS models Consensus PCTR2 models Final prediction Consensus TR2-1 models