Regression analysis - PowerPoint PPT Presentation

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Regression analysis. Relating two data matrices/tables to each other. Purpose: prediction and interpretation. Y-data. X-data. Typical examples. Spectroscopy: Predict chemistry from spectral measurements Product development: Relating sensory to chemistry data

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Regression analysis

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Regression analysis

Relating two data matrices/tables to each other

Purpose: prediction and interpretation

Y-data

X-data

Typical examples

• Spectroscopy: Predict chemistry from spectral measurements

• Product development: Relating sensory to chemistry data

• Marketing: Relating sensory data to consumer preferences

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

Simple linear regression

• One y and one x. Use x to predict y.

• Use a linear model/equation and fit it by least squares

Data structure

X-variable

Y-variable

2

4

1

.

.

.

7

6

8

.

.

.

Objects, same number

in x and y-column

Least squares (LS) used

for estimation of regression coefficients

y

y=b0+b1x+e

b1

b0

x

Simple linear regression

Model

Regression analysis

Data (X,Y)

Future X

Prediction

Regression analysis

Interpretation

Outliers?

Pre-processing

The selectivity problem

A reason why multivariate methods are needed

Can be used for several Y’s also

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

Collinearity, the problem of correlated X-variable

y=b0+b1x1+b2x2+e

Regression in this case is fitting a

plane to the data (open circles)

The two x’s have high correlation

(in the direction with little variability)

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)

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

Model for data compression methods

X=TPT+E

Centred X and y

y=Tq+f

T-scores, carrier of information from X to y

E,f – residuals (noise)

x3

PCA

to compress data

x2

ti

x1

y

q

t-score

Regression by data compression

PC1

Regression on scores

x1

x2

MLR

y

x3

x4

x1

t1

x2

PCR

y

t2

x3

x4

x1

t1

y

x2

PLS

x3

t2

x4

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

Principal component regression (PCR)

• Uses principal components

• Solves the collinearity problem, stable solutions

• Well understood

• Outlier diagnostics

• Easy to modify

• But uses only X to determine components

PLS-regression

• Easy to compute

• Stable solutions

• Often less number of componentsthan PCR

• Sometimes better predictions

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

Validation is important

• Measure quality of the predictor

• Determine A – number of components

• Compare methods

Prediction testing

Calibration

Estimate coefficients

Testing/validation

Predict y, use the

coefficients

Calibrate, find y=f(x)

estimate coefficients

Predict y, use the coefficients

Validation

• Compute

• Plot RMSEP versus component

• Choose the number of components with best RMSEP properties

• Compare for different methods

• RMSEP

MLR

NIR calibration of protein in wheat. 6 NIR wavelengths

12 calibration samples, 26 test samples

Estimation error

Model error

Conceptual illustration of important phenomena

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.

Validation

• One should also plot measured versus predicted y-value

• Correlation can be computed, but can sometimes be misleading

Example, plot of y versus predicted y

Plot of measured and predicted protein

NIR calibration

Outlier detection

• Instrument error or noise

• Drift of signal (over time)

• Misprints

• Samples outside normal range (different population)

Outlier detection

• Outliers can be detected because

• Model for spectral data (X=TPT+E)

• Model for relationship between X and y (y=Tq+f)

Outlier detectiontools

• Residuals

• X and y-residuals

• X-residuals as before, y-residual is difference between measured and predicted y

• Leverage

• hi