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Regression analysis

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Relating two data matrices/tables to each other

Purpose: prediction and interpretation

Y-data

X-data

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

- 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

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

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

- 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

Leads to unstable equation/plane

(in the direction with little variability)

- 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)

- 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

P,q –loadings

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

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

- Maximize covariance between linear combinations of X and y

- 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

- Easy to compute
- Stable solutions
- Provides scores and loadings
- Often less number of componentsthan PCR
- Sometimes better predictions

- 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

- 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

- Compute
- Plot RMSEP versus component
- Choose the number of components with best RMSEP properties

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

- 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

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

- Outliers can be detected because
- Model for spectral data (X=TPT+E)
- Model for relationship between X and y (y=Tq+f)

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

- Leverage
- hi