Simple View on Simple Interval Calculation (SIC) Alexey Pomerantsev, Oxana Rodionova Institute of Chemical Physics, Moscow and Kurt Varmuza Vienna Technical University © Kurt Varmuza 10.02.05 WSC-4 1 CAC, Lisbon, September 2004 10.02.05 WSC-4 2 Leisured Agenda 1. Why errors are limited? 2. Simple calculations, indeed! Univariate case 3. Complicated SIC. Bivariate case 4. Conclusions 10.02.05 WSC-4 3 Part I. Why errors are limited? 10.02.05 WSC-4 4 Water in wheat. NIR spectra by Lumex Co 2 1 0 -1 -2 9058. 10.02.05 9290. 9521. 9753. 9984. WSC-4 10216 10447 10679 5 Histogram for Y (water contents) 40 141 samples 30 20 10 0 8 10.02.05 9 10 11 WSC-4 12 13 14 6 Normal Probability Plot for Y 99.65 89.01 78.37 67.73 57.09 42.91 32.27 38% 21.63 10.99 21% 3% 0.35 8 10.02.05 9 10 11 WSC-4 12 13 14 7 PLS Regression. Whole data set 10.02.05 WSC-4 8 PLS Regression. Marked “outliers” 10.02.05 WSC-4 9 PLS Regression. Revised data set 10.02.05 WSC-4 10 Histogram for Y. Revised data set 40 124 samples 30 20 10 0 8 10.02.05 10 12 WSC-4 14 11 Normal Probability Plot. Revised data set 99.60 89.92 80.24 70.56 60.89 39.11 29.44 96% 19.76 10.08 81% 31% 0.40 9 10.02.05 10 11 WSC-4 12 13 14 12 Histogram for Y. Revised data set 40 30 20 10 0 10 m-3s 10.02.05 12 m-2s m-s m WSC-4 14 m+s m+2s m+3s 13 Error Distribution Normal distribution -b s s s s s s Truncated normal distribution 3.5s +b s s s s -b s +b s s s s s s s s s s s Both distributions -b +b s s s s s s 10.02.05 WSC-4 s s s s s 14 Main SIC postulate All errors are limited! There exists Maximum Error Deviation, b, such that for any error e Prob{| e | > b }= 0 Error distribution b 10.02.05 b WSC-4 15 Part 2. Simple calculations 10.02.05 WSC-4 16 Case study. Simple Univariate Model Data 7 Test y C1 1.0 1.28 C2 2.0 1.68 C3 4.0 4.25 C4 5.0 5.32 T1 3.0 3.35 T2 4.5 6.19 T3 5.5 5.40 6 C4 5 Response, y Training x T2 T3 4 T1 C3 3 2 C1 C2 1 Error distribution Variable, x 0 0 1 2 3 4 5 6 Model b 10.02.05 y=ax+e b WSC-4 17 OLS calibration OLS Calibration is minimizing the Sum of Least Squares 7 T2 6 C4 Response, y 5 Sum of Squares T3 4 T1 3 2 C1 C2 1 a 0.5 10.02.05 C3 Variable, x 0 0 0.7 0.8 0.9 1.044 1.1 1.2 1.3 1.4 1.5 WSC-4 1 2 3 4 5 6 18 Uncertainties in OLS 7 T2 6 C4 Response, y 5 t3(P) is quantile of Student's t-distribution for probability P with 3 degrees of freedom T3 4 T1 C3 3 2 C1 C2 1 Variable, x 0 0 10.02.05 WSC-4 1 2 3 4 5 6 19 SIC calibration |e|<b 7 Maximum Error Deviation is known: 6 b = 0.7 (=2.5s) C4 2b Response, y 5 C3 2b 4 3 2 C2 2b 2b C1 1 Variable, x 0 0 10.02.05 WSC-4 1 2 3 4 5 6 20 SIC calibration 7 6 C4 Response, y 5 C3 4 3 2 C2 Training 10.02.05 amin amax x y C1 1.0 1.28 0.58 1.98 C2 2.0 1.68 0.49 1.19 C3 4.0 4.25 0.89 1.24 C4 5.0 5.32 0.92 1.20 C1 1 Variable, x 0 0 WSC-4 1 2 3 4 5 6 21 Region of Possible Values amin amax Training x y C1 1.0 1.28 0.58 1.98 C2 2.0 1.68 0.49 1.19 C3 4.0 4.25 0.89 1.24 C4 5.0 5.32 0.92 1.20 C4 C3 C2 C1 RPV a min=0.92 10.02.05 a a max=1.19 WSC-4 22 SIC prediction 7 T2 6 T3 C4 Response, y 5 Test x y v- v+ T1 3.0 3.35 2.77 3.57 T2 4.5 6.19 4.16 5.36 T3 5.5 5.40 5.08 6.55 C3 4 T1 3 2 C2 C1 1 Variable, x 0 0 10.02.05 WSC-4 1 2 3 4 5 6 23 Object Status. Calibration Set amin amax y C1 1.0 1.28 0.58 1.98 C2 2.0 1.68 0.49 1.19 C3 4.0 4.25 0.89 1.24 C4 5.0 5.32 0.92 1.20 Samples C2 & C4 are the boundary objects. They form RPV. 7 6 C4 5 Response, y Training x C3 4 3 2 C2 Samples C1 & C3 are insiders. They could be removed from the calibration set and RPV doesn’t C1 1 Variable, xx Variable, 0 change. 10.02.05 0 WSC-4 1 2 3 4 5 6 24 Object Status. Test Set Let’s consider what happens when a 7 new sample is added to the calibration set. 6 C4 Response, y 5 4 3 2 C2 C4 1 C2 a a min =0.92 10.02.05 RPV Variable, x 0 a max=1.19 0 WSC-4 1 2 3 4 5 6 25 Object Status. Insider If we add sample T1, RPV doesn’t change. 7 This object is an insider. 6 Prediction interval lies 5 C4 Response, y inside error interval 4 T1 3 2 T1 C2 C4 1 C2 a a min=0.92 10.02.05 RPV a max=1.19 Variable, x 0 0 WSC-4 1 2 3 4 5 6 26 Object Status. Outlier If we add sample T2, RPV disappears. 7 This object is an outlier. T2 6 C4 Prediction Interval 5 Response, y lies out error interval 4 3 2 T2 C2 C4 1 C2 a Variable, x 0 a min=0.92 10.02.05 a max=1.19 0 WSC-4 1 2 3 4 5 6 27 Object Status. Outsider If we add sample T3, 7 RPV becomes smaller. 6 This object is an outsider. T3 C4 5 Response, y Prediction interval overlaps error interval 4 3 2 T3 C2 C4 1 C2 a a min =0.92 10.02.05 RPV a max=1.11 Variable, x 0 0 WSC-4 1 2 3 4 5 6 28 SIC-Residual and SIC-Leverage They characterize interactions between prediction and error intervals Definition 1. SIC-residual is defined as – v+ bh This is a characteristic of bias y–b Definition 2. SIC-leverage is defined as – br v– y y+b This is a normalized precision 10.02.05 WSC-4 29 Object Status Plot Using simple algebraic calculus one can prove the following statements T2 Statement 1 An object (x, y) is an insider, iff B 1 Statement 2 An object (x, y) is an outlier, iff SIC-residual, r | r (x, y) | 1 – h (x) Presented by triangle BCD C1 C4 C T1 C3 C2 -1 1 SIC-Leverage, h T3 D | r (x, y) | > 1 + h (x) Presented by lines AB and DE 10.02.05 A E WSC-4 30 Object Status Classification Insiders Absolute outsiders Outliers Outsiders 10.02.05 WSC-4 31 OLS Confidence versus SIC Prediction 7 True response value, y, is always located within the SIC prediction P=0.95 P=0.999 P=0.99 T2 6 C4 interval. This has been confirmed 5 times. Thus Prob{ v- < y < v+ } = 1.00 Confidence intervals tends to infinity when P is increased. Response, y by simulations repeated 100,000 T3 4 T1 C3 3 2 C1 C2 1 Confidence intervals are unreasonably wide! 10.02.05 Variable, x 0 0 WSC-4 1 2 3 4 5 6 32 Beta Estimation. Minimum b 0.6 0.5 0.4 0.3 bb==0.7 C4 C4 C4 C4 C4 C3 C3 C3 C3 C3 C2C2C2C2C2 C1 C1 C1 C1 C1 RPV RPV RPV RPV a b > bmin = 0.3 10.02.05 WSC-4 33 Beta Estimation from Regression Residuals e = ymeasured – ypredicted bOLS= max {|e1|, |e2|, ... , |en |} bOLS = 0.4 bSIC= bOLS C(n) Prob{b < bSIC}=0.90 10.02.05 WSC-4 bSIC = 0.8 34 1-2-3-4 Sigma Rule 10.02.05 1s RMSEC RMSEC = 0.2 = 1s 2s bmin bmin = 0.3 = 1.5s 3s bOLS bOLS = 0.4 = 2s 4s bSIC bSIC = 0.8 = 4s WSC-4 35 Part 3. Complicated SIC. Bivariate case 10.02.05 WSC-4 36 Octane Rating Example (by K. Esbensen) Test set =13 samples 0.6 0.4 0.2 Training set = 24 samples 0.6 0 0.4 1100 1200 1300 1400 1500 X-values are NIR-measurements over 226 wavelengths 0.2 Y-values are reference measurements of octane number. 0 1100 10.02.05 1200 1300 1400 1500 WSC-4 37 Calibration 0.4 PC2 Scores 4 0.2 2 0 0 -0.2 PC1 -0.4 -0.2 0 RMSE Elements: 37 Slope: 9.643866 Offset: 0.006391 Correlation: 0.991227 -2 0.2 T Scores -0.1 RESULT4, X-expl: 85%,12% Y-expl: 85%,13% 4 U Scores 0 0.1 0.2 0.3 0.4 RESULT4, PC(X-expl,Y-expl): 2(12%,13%) Root Mean Square Error 94 Predicted Y Slope 0.981975 0.919002 Offset 1.608816 7.082160 Corr. 0.990947 0.972058 92 2 90 88 0 PCs PC_01 PC_02 RESULT4, Variable: c.octane v.octane 10.02.05 PC_03 86 PC_04 Measured Y 86 88 90 92 RESULT4, (Y-var, PC): (octane,2) (octane,2) WSC-4 38 PLS Decomposition 1 p PLS = y n b p n X 1 2PC 1 1 10.02.05 WSC-4 = y – y0 1 n a n n T 2 1 39 1-2-3-4 Sigma Rule for Octane Example RMSEC = 0.27 = 1s bmin = 0.48 = 1.8s bOLS = 0.58 = 2.2s bSIC = 0.88 = 3.3s b = bSIC = 0.88 10.02.05 WSC-4 40 RPV in Two-Dimensional Case y1 – y0– b t11a1 + t12a2 y1 – y0 + b y2 – y0– b t21a1 + t22a2 y2 – y0 + b ... yn – y0– b tn1a1 + tn2a2 yn – y0 + b We have a system of 2n =48 inequalities regarding two parameters a1 and a2 10.02.05 WSC-4 41 Region of Possible Values 40 a2 35 30 25 RPV 20 15 10 5 5 a 11 0 0 0 0 10.02.05 5 5 10 10 15 15 20 20 WSC-4 25 25 30 30 35 35 40 40 42 Close view on RPV. Calibration Set RPV in parameter space a2 1 18+ 3 24 23 4 9 – 12 RPV 9 20 13 SIC-Residual 28 Object Status Plot – 2 + 2 0 3 23 1 18 24 7+ 5 12 16 18 20 Samples 24 7 13 a1 SIC-Leverage -1 22 Boundary Samples C7 10.02.05 10 1 6 14 6 22 4 20 11 14– 12 19 8 15 21 5 1617 0 1 16 14 C9 C13 C14 C18 C23 —— —— —— —— —— —— WSC-4 43 SIC Prediction with Linear Programming v+ v– Vertex # a1 a2 t ta y 1 13.91 16.36 -0.40 88.86 2 14.22 18.36 -0.35 88.90 3 16.79 26.66 -0.24 89.01 4 19.91 26.61 -0.46 88.79 5 20.41 13.16 -0.96 88.30 6 17.44 13.52 -0.74 88.52 Linear Programming Problem 10.02.05 WSC-4 44 Octane Prediction. Test Set Prediction intervals: SIC & PLS 94 Object Status Plot Reference values PLS 2RMSEP SIC prediction 2 13 SIC-Residual Octane Number 92 90 88 11 6 1 8 7 9 -1 1 5 10 4 3 0 12 2 1 2 3 SIC-Leverage 86 1 2 3 4 5 6 7 8 9 10 11 12 13 Test Samples 10.02.05 -2 WSC-4 45 Conclusions • Real errors are limited. The truncated normal distribution is a much more realistic model for the practical applications than unlimited error distribution. s s s s s s s s s s s • Postulating that all errors are limited we can draw out a new concept of data modeling that is the SIC method. It is based on this single assumption and nothing else. • SIC approach let us a new view on the old chemometrics problems, like outliers, influential samples, etc. I think that this is interesting and helpful view. 10.02.05 WSC-4 46 OLS versus SIC SIC-Residuals vs. OLS-Residuals SIC-Leverages vs. OLS-Leverages OLS-Leverage OLS-Residual T2 C1 0.0 0.0 C2 T3 C3 0.4 C3 -1.0 C4 0.6 1.0 T2 SIC-Residual T1 C4 1.0 2.0 0.2 C2 T1 C1 T3 SIC-Leverage 0.0 0.0 -1.0 0.5 SIC Object Status Plot SIC-residual 1.0 OLS/PLS Influence Plot T2 OLS-variance T2 2 1.0 C1 T1 0.0 0.5 C4 C3 SIC-Leverage 1.0 1 C2 -1.0 C2 T3 C1 T3 T1 C3 C4 OLS-Leverage 0 0.0 10.02.05 WSC-4 0.5 1.0 47 Statistical view on OLS & SIC Let’s have a sampling {x1,...xn} from a distribution with finite support [-1,+1]. The mean value a is unknown! OLS SIC Statistics -1 +1 Deviation a=? 0.3 2.5s truncated normal distribution, n=100 OLS SIC 0.2 0.1 0.0 -0.1 -0.2 -0.3 1 10.02.05 20 40 60 WSC-4 80 100 48