Linear Regression 1Daniel Baur / Numerical Methods for Chemical Engineers / Linear Regression Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften

1Daniel Baur / Numerical Methods for Chemical Engineers / Linear Regression

Linear Regression

0 1i i iY x

Daniel Baur

ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften

ETH Hönggerberg / HCI F128 – Zürich

E-Mail: [email protected]

http://www.morbidelli-group.ethz.ch/education/index

2

Linear regression model

As inputs for our model we use two vectors x and Y, where xi is the i-th observation Yi is the i-th response

The model reads:

At this point, we make a fundamental assumption:

As outputs from our regression we get estimated values for the regression parameters:

Daniel Baur / Numerical Methods for Chemical Engineers / Linear Regression

0 1 0 1or i i iY x Y x

The errors are mutually independent and normally distributed with mean zero and variance σ2:

20,i N

0 1ˆ ˆ, A regression is called linear if

it is linear in the parameters!


The errors ε Since the errors are assumed to be normally distributed,

the following is true for the expectation values and variance of the model responses

0 1 0 12

0 1

( ) ( )0,

var( ) var( ) var( )i i i i i

ii i i i

E Y E x xN

Y x

0 1 iE Y x

2,i iY


Example: Boiling Temperature and Pressure


Parameter estimation

1 11

,

1obs obsN N

x Y

X Y

x Y

a = confidence interval


Residuals

1

1

0

0

obs

obs

N

ii

N

i ii

x

Outlier


Removing the Outlier


Goodness of fit measures Coefficient of determination

Total sum of squares

Sum of squares due to regression

Sum of squares due to error

2

1

obsN

ii

SSTO Y Y

2

1

ˆobsN

ii

SSR Y Y

22

1 1

ˆobs obsN N

i i ii i

SSE Y Y

R2 = 1 i = 0

R2 = 0 regression does not explain variation of Y

2 1SSR SSE

RSSTO SSTO


The LinearModel and dataset classes

Matlab 2012 features two classes that are designed specifically for statistical analysis and linear regression

dataset creates an object that holds data and meta-data like variable names,

options for inclusion / exclusion of data points, etc.

LinearModel is constructed from datasets or X, Y pairs (as with the regress

function) and a model description automatically does linear regression and holds all important

regression outputs like parameter estimates, residuals, confidence intervals etc.

includes several useful functions like plots, residual analysis, exclusion of parameters etc.


Classes in Matlab

Classes define a set of properties (variables) and methods (functions) which operate on those properties

This is useful for bundling information together with ways of treating and modifying this information

When a class is instantiated, an object of this class is created which can be used with the methods of the class, e.g. mdl = LinearModel.fit(X,Y);

Properties can be accessed with the dot operator, like with structs (e.g. mdl.Coefficients)

Methods can be called either with the dot operator, or by having an object of the class as first input argument (e.g. plot(mdl) or mdl.plot())


Working with LinearModel and dataset

First, we define our observed and measured variables, giving them appropriate names, since these names will be used by the dataset and the LinearModel as meta-data



Next, we construct the dataset from our variables



After defining the relationship between our data (a model), we can use the dataset and the model to construct a LinearModel object This will automatically fit the data, perform residual analysis and

much more


LinearModel: Plot

Now that we have the model, we have many analysis and plotting tools at our disposal

90 92 94 96 98 100 102-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02LogP vs. Temp

Temp

LogP

Data

Fit

Confidence bounds


Linear Model: Tukey-Anscombe Plot

Plot residuals vs. fitted values; These should be randomly distributed around 0

-0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02-4

-2

0

2

4

6

8

10

12

14x 10

-3

Fitted values

Res

idua

ls

Plot of residuals vs. fitted values

Outlier?


LinearModel: Cook’s Distance

The Cook’s distance measures the effect of removing one measurement from the data

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Row number

Coo

k's

dist

ance

Case order plot of Cook's distance


90 92 94 96 98 100 102-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02LogP vs. Temp

Temp

LogP

Data

Fit

Confidence bounds

Linear Model: Removing the Outlier

After identifying an outlier, it can be easily removed

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Row number

Coo

k's

dist

ance

Case order plot of Cook's distance

-0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

Fitted values

Res

idua

ls

Plot of residuals vs. fitted values


Multiple linear regression

Approximate model

Residuals

Least squares

ˆ Y Xβ ε1 1,1 1, 1 0 0

,1 , 1 1

ˆ1

ˆ1

p

n n n p np

Y x x

Y x x

ˆ r Y Y

22 ˆmin min r Y Y ˆT TX Xβ X Y


Exercise

The data file asphalt.dat (online), contains data from a degradation experiment for different concrete mixtures[1]

The rutting (erosion) in inches per million cars (RUT) is measured as a function of viscosity (VISC) percentage of asphalt in the surface course (ASPH) percentage of asphalt in the base course (BASE) an operating mode 0 or 1 (RUN) percentage (*10) of fines in the surface course (FINES) percentage of voids in the surface course (VOIDS)

[1] R.V. Hogg and J. Ledolter, Applied Statistics for Engineers and Physical

Scientists, Maxwell Macmillan International Editions, 1992, p.393.


Assignment

The LinearModel class only exists in Matlab 2012 or newer There are two versions of the assignment, one for Matlab

2012 and one for older versions, do one of the two


Assignment (Matlab 2012 and newer only)

1. Find online the file readVars.m that will read the data file and assign the variables RUT, VISC, ASPH, BASE, RUN, FINES and VOIDS; You can copy and paste this script into your own file.

2. Create a dataset using the variables from 1.

3. Set the RUN variable to be a discrete variable Assuming your dataset is called ds, useds.RUN = nominal(ds.RUN);

4. Create a modelspec string To include multiple variables in the modelspec, use the plus sign

5. Fit your model using LinearModel.fit, display the model output and plot the model.


Assignment (Continued)

6. Which variables most likely have the largest influence?

7. Generate the Tukey-Anscombe plot. Is there any indication of nonlinearity, non-constant variance or of a skewed distribution of residuals?

8. Plot the adjusted responses for each variable, using the plotAllResponses function you can find online

9. The variables seem to show a rather random response, except for VISC which seems to mostly lie on one of the axes. Try and transform the system by defining logRUT = log10(RUT); logVISC = log10(VISC);

10.Define a new dataset and modelspec using the transformed variables.


Assignment (Continued)

11. Fit a new model with the transformed variables and repeat the analysis from before.

12.With the new model, try to remove variables that have a small influence. To do this systematically, use the function step, which will remove and/or add variables one at a time: reduced_model = step(mdl2, 'nsteps', 20); Which variables have been removed and which of the remaining

ones most likely have the largest influence?


Assignment (older versions than Matlab 2012)

1. Find online the file readVars.m that will read the data file and assign the variables RUT, VISC, ASPH, BASE, RUN, FINES and VOIDS; You can copy and paste this script into your own file.

2. Create the matrix X using the variables from 1 except RUT and a column of ones.

3. Create the vector Y using RUT

4. Fit your model using regress and and alpha = 0.05

5. Display the estimated values of beta and the confidence intervals

6. Are any of the values not significantly different from 0, i.e. does 0 lie inside the confidence interval?


Assignment (Continued)7. Generate the Tukey-Anscombe plot. Is there any

indication of nonlinearity, non-constant variance or of a skewed distribution of residuals?

8. Plot the response of all the variables using plotmatrix(aspData(:,1:6), RUT). The variables seem to show a rather random response, except for VISC. Try and transform the system by defining logRUT = log10(RUT); logVISC = log10(VISC);

9. Define a new X matrix and a new Y vector and regress again

10.Comment again on the estimates and their significance

11. Reproduce the Tukey-Anscombe plot. Did anything change?

Documents

Linear Regression 1Daniel Baur / Numerical Methods for Chemical Engineers / Linear Regression Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften