[email protected] ENGR-25_Plot_Model-4.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25:...

[email protected] • ENGR-25_Plot_Model-4.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Chp6 MATLAB

Regression



Learning Goals cont

Use Regression Analysis as quantified by the “Least Squares” Method• Calculate–Sum-of-Squared Errors (SSE or J)The Squared Errors are Called “Residuals”

– “Best Fit” CoEfficients ( and )–Sum-of-Squares About the Mean (SSM or S)–CoEfficient of Determination (r2)

• Scale Data if Needed–Creates more Meaningful Spacing



Learning Goals cont

Build Math Models for Physical Data using “nth” Degree Polynomials

Use MATLAB’s “Basic Fitting” Utility to find Math models for Plotted Data



Scatter on Plots in XY-Plane A scatter plot usually

shows how an EXPLANATORY, or independent, variable affects a RESPONSE, or Dependent Variable

Sometimes the SHAPE of the scatter reveals a relationship

Shown Below is a Conceptual Scatter plot that could Relate the RESPONSE to some EXCITITATION



Linear Fit by Guessing The previous plot

looks sort of Linear We could use a

Ruler to draw a y = mx+b line thru the data

But • which Line is

BETTER?• and WHY?



Least Squares Curve Fitting

In a Previous Example polyfit(x,y,1) returned the Values of m & b• How does PolyFit Make these Calcs?• How Good is the fitted Line Compared to

the Data?

polyfit, as do most other curve fitters, uses the “Least Squares” Criterion



Least Squares

y

kk yx ,

hbmxy kL

m

byx kL

x

To make a Good Fit, MINIMIZE the |GUESS − data| distance by one of

22

2

2

yx

yxh

ybmxy

xm

byx

kk

kk

data

Best Guess-y

Best Guess-x



Least Squares cont

MATLAB polyfit Minimizes the VERTICAL distances; i.e.:

n

kkk

n

kk

ybmxJ

yJ

1

2

1

2

Note that The Function J contains two Variables; m & b

Recall from MTH1 that to MINIMIZE a Function set the 1st (partial) Derivative(s) equal to Zero



Least Squares cont

To Minimize J, take

0

0

1

2

1

2

n

kkk

n

kkk

ybmxbb

J

ybmxmm

J

The Two Partials yield Two LINEAR Eqns in m & b

The two eqns can be solved EXACTLY for m & b

the Book on pg 271 gives a good example

Remember, at this point m & b are UNKNOWN

x y

0 2

5 6

10 11



Least Squares cont

In This Case 2

22

1110

6520

bm

bmbmJ

Solving theseEqns for m & b yields• m = 9/10• b = 11/6

This produces the best fit Line

Taking ∂J/∂m = 0, and ∂J/∂b = 0 yields

x y

0 2

5 6

10 11

38630

28030250

bm

bm

6

11

10

9 xy



Goodness of Fit

The Distance from The Best-Fit Line to the Actual Data Point is called the RESIDUAL

For the Vertical Distance the Residual is just δy

If the Sum of the Residuals were ZERO, then the Line would Fit Perfectly

Thus J, after finding m & b, is an Indication of the Goodness of Fit

2kk ybmxy

n

kkk

n

kk

ybmxJ

yJ

1

2

1

2



Goodness of Fit cont

Now J is an indication of Fit, but we Might want to SCALE it relative to the MAGNITUDE of the Data• For example

consider–DataSet1 with x&y

values in the MILLIONS–DataSet2 with x&y

values in the single digits

• In this case we would expect J1 >> J2

To remove the affect of Absolute Magnitude, Scale J against the Data Set mean; e.g• mean1 = 730 000• mean2 = 4.91



Goodness of Fit cont

The Mean-Scaling Quantity is the Actual-Data Relative to the Actual-Mean

Finally the Scaled Fit-Metric, “r-squared’

2

1

n

kk yyS

n

kk

n

kkk

yy

ybmxr

S

Jr

1

2

1

2

2

2

1

1

As before the Squaring Ensures that all Terms in the sum are POSITIVE



r2 = Coeff of Determination

The r2 Value is Also Called the COEFFICIENT OF DETERMINATION

SJr 12• J Sum of Residual (errors)

– May be Zero or Positive

• S Data-to-Mean Scaling Factor– Always Positive if >1 Data-Pt and

data not “perfectly Horizontal”

If J = 0, then there is NO Distance Between the calculated Line and Data

Thus if J = 0, then r2 = 1; so r2 = 1 (or 100%)indicates a PERFECT FIT



Meaning of r2

The COEFFICIENT OF DETERMINATION

n

kk

n

kkk

yy

ybmxr

1

2

1

2

2 Has This Meaning

The coefficient of determination tells you what proportion of the variation between the data points is explained or accounted for by the best line fitted to the points. It indicates how close the points are to the line.



Norm of Residuals

MATLAB uses the Norm of Residuals as a Measure of Goodness of Fit

The Norm of Residuals, NR, is simply the SqRt of J:

n

kkkR

n

kkLR

ybmxN

yyJN

1

2

1

2

Thus r2 in Terms of NR:

As a Measure of Goodness of Fit as the FIT Approaches Perfection J→0 so:

0

12

RN

r

SNr R22 1



NR vs r2 → SJSNr R 11 22

Notice that r2 is a RELATIVE Measure → it’s NORMALIZED to the WORST CASE Value of J which is S• Thus r2 can be expressed at

PERCENTAGE withOUT Units

NR is an ABSOLUTE measure that Technically Requires that it be stated with SAME UNITS as the dependent variable, y

%2 r

etc. V, Teslas, F, m, Sec, RN



Data Scaling

Data-Scaling is a SubTopic of DIMENSIONAL ANALYIS (DA)• DA is Covered in 3rd Yr ME/CE Courses– I Learned it in a Fluid Mechanics Course

For Our Purposes we will cover only SCALING

Sometimes we Collect Data with a SMALL Variation Relative to the Magnitude of the MEAN• Leads to a

SENSITIVE Analysis; e.g. This Data is Noisy During Analysis

x y

8974 7313

8971 7309

8969 7310



Data Scaling - Normalization

The Significance of ANY Data Set Can be Improved by Normalizing

Normalize Scale Data such that the Values run:• 0 →1• 0% → 100%

Steps to Normalization1. Find the MAX &

MIN values in the Data Set; e.g.,• zmax & zmin

2. Calculate the Data Range, RD

• RD = (zmax – zmin)

3. Calc the Individual Data Differences Relative to the MIN• Δzk = zk - zmin



Data Scaling – Normailzation cont

4. Finally, Scale the Δzk relative to RD

• Ψk = Δzk /RD

5. Scale the corresponding “y” values in the Same Manner to produce say, Φk

6. Plot Φk vs Ψk on x & y scales that Run from 0→1

Example– Do Frogs Croak More on WARM Nites?

Temperature (ºF)

Croaks/Hr

88.6 20.0

71.6 16.0

93.3 19.8

84.3 18.4

80.6 17.1

75.2 15.5

69.7 14.7

82.0 17.1

69.4 15.4

83.3 16.2

78.6 15.0

82.6 17.2

80.6 16.0

83.5 17.0

76.3 14.1



Normalization Example

Normalize• T → Θ• CPH → Ω

minmax

min

minmax

min

CPHCPH

CPHCPH

TT

TT

kk

kk

Now Compare Plots• CPH vs T• Ω vs Θ

http://www.fs.fed.us/psw/topics/wildlife/herp/images/pope_plot2.jpg



Plots Compared

T-CPH Plot Ω-Θ Plot

65 70 75 80 85 90 9514

15

16

17

18

19

20Frog Croaking in the Evening - 2045hrs

Temp (°F)

Cro

aks

Per

Ho

ur

(CP

H)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Frog Croaking in the Evening - 2045hrs

Theta (Normalized Temp)

Om

ege

(Nor

mal

ized

CP

H)

• The Θ-Ω Plot Fully Utilizes Both Axes



Basic Fitting

Use MATLAB’s AutoMatic Fitting Utility to Find The Best Line for the the Frog Croaking Data 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Theta

Om

ega

Frog Croaking in the Evening - 2045hrs

SEE: Demo_Frog_Croak_BasicFit_1110.m



All Done for Today

CroakingFrog



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Appendix 6972 23 xxxxf



Linear Regression Tutorial Minimize Height

Error δy See File ENGR-

25_Linear_Regression_Tutorial_1309.pptx



Altitude of Right Triangle

The Area of RIGHT Triangle

y

x

L

h

yxA 21 The Area of an ARBITRARY

Triangle

hLA 21

By Pythagoras for Rt-Triangle

22 yxL



Altitude of Right Triangle cont

y

x

L h

hyxyx 222121

Solving for h

22 yx

yxh

Lx 21 Base&Base Equating the A=½·Base·Hgt noting that



Normalized Plot>> T = [69.4, 69.7, 71.6, 75.2, 76.3, 78.6, 80.6, 80.6, 82, 82.6, 83.3, 83.5, 84.3, 88.6, 93.3];>> CPH = [15.4, 14.7, 16, 15.5, 14.1, 15, 17.1, 16, 17.1, 17.2, 16.2, 17, 18.4, 20, 19.8];>> Tmax = max(T);>> Tmin = min(T);>> CPHmax = max(CPH);>> CPHmin = min(CPH);>> Rtemp = Tmax - Tmin;>> Rcroak = CPHmax - CPHmin;>> DelT = T - Tmin;>> DelCPH = CPH - CPHmin;>> Theta = DelT/Rtemp;DelCPH = CPH - CPHmin;>> Omega = DelCPH/Rcroak;>> plot(T, CPH,), grid>> plot(Theta,Omega), grid



Start Basic Fitting Interface 1

FIRST → Plot the DATA




Expand Dialog Box

Goodness of Fit; smaller is

Better




Result Chk by

polyfit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Theta

Om

ega

Frog Croaking in the Evening - 2045hrs

y = 0.8737*x + 0.04286

Croak Data

linear Fit

>> p = polyfit(Theta,Omega,1)p = 0.8737 0.0429



Caveat



Greek Letters in Plots

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Frog Croaking Frequency

Croak Data

Linear Fit

= 0.8737 + 0.04286



Plot “Discoverables”% "Discoverable" Functions Displayed% Bruce Mayer, PE • ENGR25 • 15Jul09%x = linspace(-5, 5);ye = exp(x);ypp = x.^2;ypm = x.^(-2);% plot all 3 on a single grapheplot(x,ye, x,ypp, x,ypm),grid,legend('ye', 'ypp', 'ypm')disp('Showing MultiGraph Plot - Hit ANY KEY to continue')pause%% PLot Side-by-Sidesubplot(1,3,1)plot(x,ye), gridsubplot(1,3,2)plot(x,ypp), gridsubplot(1,3,3)plot(x,ypm), grid



% "Discoverable" Functions Displayed% Bruce Mayer, PE • ENGR25 • 15Jul09%x = linspace(-5, 5);ye = exp(x);ypp = x.^2;ypm = x.^(-2);% plot all 3 on a single grapheplot(x,ye, x,ypp, x,ypm),grid,legend('ye', 'ypp', 'ypm')disp('Showing MultiGraph Plot - Hit ANY KEY to continue')pause%% PLot Side-by-Sidesubplot(1,3,1)plot(x,ye), gridsubplot(1,3,2)plot(x,ypp), gridsubplot(1,3,3)plot(x,ypm), grid

[email protected] ENGR-25_Plot_Model-4.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25:...

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