EGR 105
Foundations of Engineering I
Excel Part III
Curve-Fitting, Regression
Section 8 Fall 2013
Excel Part II Topics
•
•
•
•
•
Data Analysis Concepts
Regression Methods
Example Function Discovery
Regression Tools in Excel
Homework Assignment
Analysis of x-y Data
dependent
• Independent versus dependent
variables
y = f(x)
y
independent
x
Common Types of Plots
Example: Y=3X2
350
300
Cartesian
250
1000
log-log : log y-log x
Y
200
150
100
logY
100
50
10
0
0
2
4
6
8
10
X
350
1
Note!
300
1
logX
Semi-log : log x
250
10
Y
200
y = 3x2
150
log(y) = log(3) + 2*log(x)
100
Straight Line on log-log Plot!
50
0
1
10
logX
What About Other Values?
• Often have a limited set of data
• What if you want to know…
– Prediction of what occurred before data
– Prediction of what will occur after data
• Many real applications of this…
– Discuss this in a little while
Finding Other Values
• Interpolation
– Data between known points
– Need assume variation between points
– May be easier to do for closer points
data
points
Finding Other Values
• Extrapolation (requires assumptions)
– Data beyond the measured range
– Forecasting (looking ahead)
– Hindcasting (looking behind)
• Examples (apply equations or models)
– Sales
– Ocean waves
– Stock market
– The weather
– etc.
Stock Market
Forecasting – can require complex model(s)
Finding Other Values
• Regression – curve fitting of data
– Simple representation of data
– Understand workings of system
• Elements of system behavior are important
– How do they affect the overall system?
– How important is each one?
• Can represent these in model(s)
– Useful for prediction
Excel Part III Topics
•
•
•
•
•
Data Analysis Concepts
Regression Methods
Example Function Discovery
Regression Tools in Excel
Homework Assignment
Something Must Be In
There…Somewhere….
Curve-Fitting - Regression
• Useful for noisy or uncertain data
– n pairs of data (xi , yi)
• Choose a functional form y = f(x)
• polynomial
• exponential
• etc.
and evaluate parameters for a “close” fit
What Does “Close” Mean?
• Want a consistent rule to determine
• Common is the least squares fit (SSE):
y
(x3,y3)
e3
(x1,y1)
(x2,y2)
(x4,y4)
ei = yi – f(xi),
i =1,2,…,n
n
SSE = å ei
i =1
2
x
SSE
Quality of the Fit: R = 1 SST
2
n
SSE = å ei
2
y
i =1
n
SST = å ( yi - y ) 2
i =1
Notes:
y is the average y value
0  R2  1
-closer to 1 is a “better” fit
y=y
x
Coefficient of Determination
SSE
R = 1SST
2
• R2 = 1.0
– All of the data can be explained by the fit
• R2 = 0.0
– None of the data can be explained by the curve fit
(Note: R2 = is sometimes reported as a %)
Caution!!!
• A good fit statistically may not be the
correct fit
• Must always consider the physical
phenomenon you are attempting to “model”
• Does the fit to the data describe reality?
Linear Regression
• Functional choice y = m x + b
slope
intercept
• Squared errors sum to
SSE = å (yi - m xi - b )
2
i
• Set m and b derivatives to zero
¶ SSE
=0
¶m
¶ SSE
=0
¶b
Further Regression
Possibilities:
• Could force intercept:
y=mx+c
• Other two parameter ( a and b ) fits:
– Logarithmic:
– Exponential:
– Power function:
y = a ln x + b
y = a e bx
y=axb
• Other polynomials with more parameters:
– Parabola:
– Higher order:
y = a x2 + bx + c
y = a xk + bxk-1 + …
Excel Part III Topics
•
•
•
•
•
Data Analysis Concepts
Regression Methods
Example Function Discovery
Regression Tools in Excel
Homework Assignment
Example Function Discovery
(How to find the “best” relationship)
• Look for straight lines on log axes:
– linear on semilog x
 y = a ln x
– linear on semilog y
 y = a e bx
– linear on log log  y = a x b
• No rule for 2nd or higher order
polynomial fits
Excel Part III Topics
•
•
•
•
•
Data Analysis Concepts
Regression Methods
Example Function Discovery
Regression Tools in Excel
Homework Assignment
Excel’s Regression Tool
• Highlight your chart
• On chart menu, select “add trendline”
• Choose type:
– Linear, log, polynomial, exponential, power
• Set options:
–
–
–
–
Forecast = extrapolation
Select y intercept (use zero only if it applies)
Show R2 value on chart
Show equation of fit on chart
Linear & Quartic Curve Fit
Example
7
y = 0.996x
R² = 0.986
6
5
Y
4
3
2
1
X
0
0
1
2
3
4
5
6
7
7
6
y = 0.037x4 - 0.523x3 + 2.518x2 - 3.878x + 3.133
R² = 0.997
5
4
Y
Better fit but does it
make sense with
expected behavior?
3
2
1
0
0
1
2
3
4
5
6
7
X
Example Applications
• Look at some curve fitting examples
– Examine previous EGR 105 projects
• Pendulum
• Elastic bungee cord
Previous EGR 105 Project
• Discover how a pendulum’s timing is
impacted by the
– length of the string?
– mass of the bob?
1. Take experimental data
•
Use string, weights, rulers, and watches
2. Analyze data and “discover” relationships
Experimental Setup:
Length
Mass
One Team’s Results
length (inches)
time (sec)
121.5
114.0
105.0
97.0
85.0
79.0
67.5
58.5
50.0
43.0
13.0
13.73
3.5
3.4
3.3
3.1
2.9
2.8
2.6
2.4
2.3
2.1
1.2
mass (grams)
27.47 41.20 54.94
3.5
3.5
3.5
3.4
3.4
3.4
3.3
3.3
3.3
3.1
3.1
3.1
2.9
2.9
2.9
2.8
2.8
2.8
2.6
2.6
2.6
2.4
2.4
2.4
2.3
2.3
2.3
2.1
2.1
2.1
1.2
1.2
1.2
Mass
appears to
have no
impact, but
length
does
To determine the effect of length,
first plot the data
4.0
time (seconds)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
20.0
40.0
60.0
80.0
length (inches)
100.0
120.0
140.0
Try a linear fit
4.0
time (seconds)
3.5
3.0
2.5
2.0
1.5
y = 0.02x + 1.1692
2
R = 0.9776
1.0
0.5
0.0
0.0
20.0
40.0
60.0
80.0
length (inches)
100.0
120.0
140.0
Force a zero intercept (why?)
time (seconds)
4.5
4.0
3.5
3.0
2.5
2.0
1.5
y = 0.0332x
2
R = 0.4832
1.0
0.5
0.0
0.0
20.0
40.0
60.0
80.0
length (inches)
100.0
120.0
140.0
Try a quadratic polynomial fit
4.0
time (seconds)
3.5
3.0
2.5
2.0
1.5
2
y = -0.0002x + 0.0551x
2
R = 0.9117
1.0
0.5
0.0
0.0
20.0
40.0
60.0
80.0
length (inches)
100.0
120.0
140.0
Try a logarithmic fit
4.0
time (seconds)
3.5
3.0
2.5
2.0
1.5
y = 1.0349Ln(x) - 1.6506
2
R = 0.9609
1.0
0.5
0.0
0.0
20.0
40.0
60.0
80.0
length (inches)
100.0
120.0
140.0
Try a power function fit
4.0
time (seconds)
3.5
3.0
2.5
2.0
1.5
0.4774
y = 0.3504x
2
R = 0.9989
1.0
0.5
0.0
0.0
20.0
40.0
60.0
80.0
length (inches)
100.0
120.0
140.0
On log-log axes, nice straight line
time (seconds)
10.0
b
1.0
1.0
10.0
100.0
1000.0
length (inches)
Power Law Relation:
t = al b Þ log(t ) = log(a) + b log(l )
Question?
• Which one was the best fit here?
• Explain why
One More Example
• Another EGR 105 project
• Elastic bungee cord models
– Stretching of an elastic cord
• Here we have two models to consider
– Linear elastic (Hooke’s Law)
– Non-linear elastic (Cubic model)
Elastic Bungee Cord Models
Determined by Curve Fitting the Data
• Linear Model (Hooke’s Law): F ( s) = ks
• Nonlinear Cubic Model: F ( s) = k1s + k2 s 2 + k3 s 3
Force (lb)

eaxial =









Collected Data
Cubic Fit
Better and it Makes
Sense with the Physics
Linear Fit
Elongation
l - lo
=
= axial strain
Original Length
lo
Homework Assignment #5
• See Handout (Excel Part 3)
– Analysis of stress-strain data
– Plotting of data
– Determine equation for best fit to data
• Regression analysis
– Linear elastic model
– Cubic polynomial model
• Discussion of results
Remember to email submit using EGR105_5 in Subject Line!