Empirical Model Building Ib: Objectives: By the end of this class you should be able to:
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Transcript of 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
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
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x
y
Linear Graph
0 10 20 3010
-1
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x
y
Semilog Graph
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10-1
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x
y
Log-Log Graph
2. Here are the plots for another dataset. Name the model and write its equation for this case
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Linear Graph
0 5 10 15 20 25 3010
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Semilog Graph
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10-1
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x
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Log-Log Graph
How would you
define the Best Fit
line?
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F o r c e ( lb s . )
Leng
th In
crea
se (i
n.)
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
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)
Fitting a Linear Equation in Matrix Form
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64.11
15.11
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9.5
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m
b
XY
>> B = X\Y
Matrix Equation:
The Full Matrices
MATAB Syntax for finding the parameter matrix
Linear Equation in Matrix Form
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08.0
64.11
15.11
47.01
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3.8
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XY ˆ
fits: >> yhat = X*B
residuals: >> res = Y - X*B
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
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
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
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”
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.
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
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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Experimental Data
Fit: y = 5.02*x - 1.97
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
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)
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