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Regression analysis Relating two data matrices/tables to each other Purpose: prediction and interpretation Y-data X-data

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Typical examples Spectroscopy: Predict chemistry from spectral measurements Product development: Relating sensory to chemistry data Marketing: Relating sensory data to consumer preferences

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Topics covered Simple linear regression The selectivity problem: a reason why multivariate methods are needed The collinearity problem: a reason why data compression is needed The outlier problem: why and how to detect

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Simple linear regression One y and one x. Use x to predict y. Use a linear model/equation and fit it by least squares

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Data structure Y-variable X-variable Objects, same number in x and y-column 241...241... 768...768...

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b0b0 b1b1 y=b 0 +b 1 x+e x y Least squares (LS) used for estimation of regression coefficients Simple linear regression

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Model Data (X,Y) Regression analysis Future XPrediction Regression analysis Outliers? Pre-processing Interpretation

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The selectivity problem A reason why multivariate methods are needed

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Can be used for several Y’s also

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Multiple linear regression Provides –predicted values –regression coefficients –diagnostics If there are many highly collinear variables –unstable regression equations –difficult to interpret coefficients: many and unstable

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y=b 0 +b 1 x 1 +b 2 x 2 +e The two x’s have high correlation Leads to unstable equation/plane (in the direction with little variability) Collinearity, the problem of correlated X-variable Regression in this case is fitting a plane to the data (open circles)

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Possible solutions Select the most important wavelengths/variables (stepwise methods) Compress the variables to the most dominating dimensions (PCR, PLS) We will concentrate on the latter (can be combined)

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Data compression We will first discuss the situation with one y-variable Focus on ideas and principles Provides regression equation (as above) and plots for interpretation

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Model for data compression methods X=TP T +E y=Tq+f T-scores, carrier of information from X to y P,q –loadings E,f – residuals (noise) Centred X and y

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Regression by data compression Regression on scores PC1 t-score y q titi PCA to compress data x1x1 x2x2 x3x3

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x4 x1 x2 x3 x4 x2 x3 x1 x2 x4 x3 y y y t1 t2 MLR PCR PLS x1 t1 t2

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PCR and PLS For each factor/component PCR –Maximize variance of linear combinations of X PLS –Maximize covariance between linear combinations of X and y Each factor is subtracted before the next is computed

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Principal component regression (PCR) Uses principal components Solves the collinearity problem, stable solutions Provides plots for interpretation (scores and loadings) Well understood Outlier diagnostics Easy to modify But uses only X to determine components

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PLS-regression Easy to compute Stable solutions Provides scores and loadings Often less number of components than PCR Sometimes better predictions

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PCR and PLS for several Y- variables PCR is computed for each Y. Each Y is regressed onto the principal components PLS: The algorithm is easily modified. Maximises linear combinations of X and Y. For both methods: Regression equations and plots

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Validation is important Measure quality of the predictor Determine A – number of components Compare methods

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Prediction testing Calibration Estimate coefficients Testing/validation Predict y, use the coefficients

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Cross-validation Predict y, use the coefficients Calibrate, find y=f(x) estimate coefficients

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Validation Compute Plot RMSEP versus component Choose the number of components with best RMSEP properties Compare for different methods

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RMSEP NIR calibration of protein in wheat. 6 NIR wavelengths 12 calibration samples, 26 test samples MLR

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Conceptual illustration of important phenomena Estimation error Model error

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Prediction vs. cross-validation Prediction testing: Prediction ability of the predictor at hand. Requires much data. Cross-validation: Property of the method. Better for smaller data set.

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Validation One should also plot measured versus predicted y-value Correlation can be computed, but can sometimes be misleading

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Plot of measured and predicted protein NIR calibration Example, plot of y versus predicted y

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Outlier detection Instrument error or noise Drift of signal (over time) Misprints Samples outside normal range (different population)

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Outlier detection Outliers can be detected because –Model for spectral data (X=TP T +E) –Model for relationship between X and y (y=Tq+f)

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Outlier detection tools Residuals –X and y-residuals –X-residuals as before, y-residual is difference between measured and predicted y Leverage –h i

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