Transcript
Page 1: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Empirical Model Building Ib: Objectives:

By the end of this class you should be able to:

• Determine the coefficients for any of the basic two parameter models

• Plot the data and resulting fits• Calculate and describe residuals

Palm, Section 5.5

Download file FnDiscovery.mat and load into MATLAB

Page 2: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

1. Below are three graphs of the same dataset. What is the name and equation of the likely model

that would match this data?

0 10 20 300

20

40

60

80

100

x

y

Linear Graph

0 10 20 3010

-1

100

101

102

x

y

Semilog Graph

100

101

102

10-1

100

101

102

x

y

Log-Log Graph

Page 3: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

2. Here are the plots for another dataset. Name the model and write its equation for this case

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

x

y

Linear Graph

0 5 10 15 20 25 3010

-1

100

101

102

103

x

y

Semilog Graph

100

101

102

10-1

100

101

102

103

x

y

Log-Log Graph

Page 4: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

How would you

define the Best Fit

line?

0 0 . 5 1 1 . 5 20

2

4

6

8

1 0

F o r c e ( lb s . )

Leng

th In

crea

se (i

n.)

Page 5: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Fitting a Linear equation via matricese.g., Fitting the Spring data

• Model: y = mx + b • Setup: 1. Design Matrix: >> X = [ones(length(Force),1),

Force]2. Response Vector >> Y= Length

• Fit: find the fitted parameters >> B = X \ Y B will be

• Predict: calculate predicted y for each x>> Lhat = X*B

• Plot: plot the result >> plot(Force, Length, ‘p’, Force, Lhat) (plus

labels ...)

m

b

Page 6: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

A Linear Model & it’s Design Matrix

64.11

15.11

47.01

01

X

y = b(1) + mx Linear Model:

Design Matrix:

>> X = [ ones(4, 1), L’ ]Matlab Syntax: (to convert a row vector of x values to a design matrix)

Page 7: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Fitting a Linear Equation in Matrix Form

4

3

2

1

64.11

15.11

47.01

01

2.8

9.5

5.2

0

m

b

XY

>> B = X\Y

Matrix Equation:

The Full Matrices

MATAB Syntax for finding the parameter matrix

Page 8: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Linear Equation in Matrix Form

00.5

08.0

64.11

15.11

47.01

01

3.8

8.5

4.2

08.0

XY ˆ

fits: >> yhat = X*B

residuals: >> res = Y - X*B

Page 9: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Fitting Transformed models

• Same as linear model except set up design matrix (X) and response vector (Y) using the transformed variables

• e.g., the capacitor discharge from last time

• straight line on a semilog plot what model is implied?

Exponential y = b10mx

or in this example V = b10mt

what is its linearized (transformed) formlog(V) = log(b) + mt

Page 10: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

E.G., Fitting the capacitor discharge data

Model: Last lecture we found the data was straight on a semilog plot implying an exponential model. For the base-ten model the equations are: V = b10mt or log(V) = log(b) + mt

Setup: 1. Design Matrix: >> X = [ones(length(t),1), t]2. Response Vector >> Y= log10(V)

Fit: determine parameters >> B = X \ Y

Predict: Predict: >> logVhat = X*BUntransform: >> Vhat = 10.^logVhat

orUntransform >> b = 10^B(1), m = B(2)Predict >> Vhat = b*10.^(m.*t)

Plot: either on linear or semilog plot

Page 11: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Equation Fit Parameters

linear x vs. yb = B(1)m= B(2)

power log(x) vs. log(y)

b=10^B(1) m=B(2),

exponential

x vs. ln(y)b=e^B(1) m=B(2),

x vs. log(y)b=10^B(1) m=B(2)

Function Discovery (Review) 2. Fitting Parameters (m & b)

bmxy

mbxy

mxbey

mxby 10

Page 12: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Fitting a 2-parameter models

Model: Identify Functional Form• Plot data

•is it linear ? •is it monotonic?

• Log-Log (loglog(x,y)) semilog (semilogy(x,y))• look for straight graph

Setup:

Transform Data to Linearize

Create X & Y matrices Fit linear model to transformed data

Predict and Untransform Parameters to m & b

Plot:

Plot data and predicted equation.

“Normal Data” “Transformed Data”

Page 13: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Class Exercise:

For problems 3 (x2 vs. y2) from last class:• What type of model will likely fit this data?

(from last time)• Determine the full model including

parameter values. • Plot the data and the fitted curve on one

plot

For problem 2 (x1 vs. y1), repeat the above.

Page 14: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:
Page 15: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

x y

1 5

2 8

3 10

4 20

5 21

6 29

7 34

8 36

9 45

Please plot this data and determine: • the likely model • parameters (m&b)(data is available in FnDiscovery.mat)

plot resulting data and model

Page 16: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

A Reminder of Some Nomenclature:

y response (dependent variable) vector yi an individual response

x predictor (independent variable) vectorxi an individual predictor value

the predicted value (the fits) an individual predicted value (fit)

y

iy

Page 17: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

x

y

Experimental Data

Fit: y = 5.02*x - 1.97

Page 18: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Residuals:• What is left after subtracting model from data:

residuals = y – yhat

• Represents what is not fit by the model

• Ideal model should capture all systematic information

• Residuals should contain only random error

• Plot residuals and look for patterns

Page 19: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

What to look for in a residual plot:

1. Does the residual plot look correct? data should vary about zerosum of residuals must equal zero

2. Are there any patterns in the residuals?, e.g., curvature: high center, low ends or

low center, high ends

changes is variability: the spread of the data in the y direction should be constant

3. How big are the residuals?(what is the magnitude of the y axis)

Page 20: Empirical Model Building Ib: Objectives: By the end of this class you should be able to:

Thermocouple Calibration Data is it linear?

• Plot this data Does it look linear?

• Fit a linear model

• Determine the residualsPrepare a residuals plot

• Is it linear?

• (data is available in FnDiscovery.mat)

mV (mV) T(C)

0 0 0.3910 10.0000 0.7900 20.0000 1.1960 30.0000 1.6120 40.0000 2.0360 50.0000 2.4680 60.0000 2.9090 70.0000 3.3580 80.0000 3.8140 90.0000 4.2790 100.0000


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