Empirical Model Building Ib: Objectives: By the end of this class you should be able to: Determine...
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![Page 1: 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.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/1.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/2.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/3.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/4.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/5.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/6.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/7.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/8.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/9.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/10.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/11.jpg)
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
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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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/13.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/14.jpg)
![Page 15: 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.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/15.jpg)
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
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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
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0 1 2 3 4 5 6 7 8 9 100
5
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15
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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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/18.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/19.jpg)
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: Determine the coefficients for any of the basic two parameter.](https://reader036.fdocuments.in/reader036/viewer/2022082712/56649e2c5503460f94b1b3a9/html5/thumbnails/20.jpg)
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