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EGR 105 Foundations of Engineering I
Fall 2007 – week 7
Excel part 3 - regression
EGR105 – Week 7 Topics
• Data analysis concepts
• Regression methods
• Function discovery by example
• Regression tools in Excel
• Assignment # 7
Analysis of x-y Data
• Independent versus dependent variables
y
y = f(x) xindependent
depe
nden
t
Finding Other Values
• Interpolation– Data between known points
• Regression – curve fitting– Simple representation of data– Understand workings of system – Useful for prediction
• Extrapolation– Data beyond the measured range
datapoints
EGR105 – Week 7 Topics
• Data analysis concepts
• Regression methods
• Function discovery by example
• Regression tools in Excel
• Assignment
EGR105 – Week 7 Topics
• Data analysis concepts
• Regression methods
• Function discovery by example
• Regression tools in Excel
• Assignment
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
• Common is the least squares fit (SSE):
(x1,y1) (x2,y2)
(x3,y3) (x4,y4)
x
y
e3
ei = yi – f(xi), i =1,2,…,n
n
1i
2ieSSE
sum
squa
red
erro
rs
Quality of the Fit:
Notes: is the average y value
0 R2 1closer to 1 is a “better” fit
SST
SSE12 R
n
1i
2ieSSE
n
yy1i
2i )(SST
x
y
yy
y
Linear Regression
• Functional choice y = m x + b slope
intercept
• Squared errors sum to
• Set m and b derivatives to zero
2SSE
iii bxmy
0SSE
0SSE
bm
Further Regression Possibilities:
• Could force intercept: y = m x + c• Other two parameter ( a and b ) fits:
– Logarithmic: y = a ln x + b– Exponential: y = a e bx
– Power function: y = a x b
• Other polynomials with more parameters:– Parabola: y = a x2 + bx + c– Higher order: y = a xk + bxk-1 + …
EGR105 – Week 7 topics
• Data analysis concepts
• Regression methods
• Function discovery by example
• Regression tools in Excel
• Assignment
Function Discoveryor
How to find the best relationship
• Look for straight lines on log axes: linear on semilog x y = a ln x + b 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 (not very useful toward real problems)
Previous EGR105 Project
Discover how a pendulum’s timing is impacted by the:
– length of the string?– mass of the bob?
1. Take experimental data – string, weights, rulers, and watches
2. Analyze data and “discover” relationships
Experimental Setup:
One Team’s Results:time (sec)
13.73 27.47 41.20 54.94121.5 3.5 3.5 3.5 3.5114.0 3.4 3.4 3.4 3.4105.0 3.3 3.3 3.3 3.3
97.0 3.1 3.1 3.1 3.185.0 2.9 2.9 2.9 2.979.0 2.8 2.8 2.8 2.867.5 2.6 2.6 2.6 2.658.5 2.4 2.4 2.4 2.450.0 2.3 2.3 2.3 2.343.0 2.1 2.1 2.1 2.113.0 1.2 1.2 1.2 1.2
mass (grams)
leng
th (
inch
es)
Mass appears to have no impact, but length does
To determine the effect of length, first plot the data:
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0
length (inches)
tim
e (
seco
nd
s)
Try a linear fit:
y = 0.02x + 1.1692
R2 = 0.9776
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0
length (inches)
tim
e (
seco
nd
s)
Force a zero intercept:
y = 0.0332x
R2 = 0.4832
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0
length (inches)
tim
e (
seco
nd
s)
Try a quadratic polynomial:
y = -0.0002x2 + 0.0551x
R2 = 0.9117
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0
length (inches)
tim
e (
seco
nd
s)
Try logarithmic:
y = 1.0349Ln(x) - 1.6506
R2 = 0.9609
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0
length (inches)
tim
e (
seco
nd
s)
Try power function:
y = 0.3504x0.4774
R2 = 0.9989
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0
length (inches)
tim
e (
seco
nd
s)
On log-log axes, a nice straight line:
1.0
10.0
1.0 10.0 100.0 1000.0
length (inches)
tim
e (
seco
nd
s)
EGR105 – Week 7 Topics
• Data analysis concepts
• Regression methods
• Function discovery by example
• Regression tools in Excel
• 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– Show R2 value on chart– Show equation on chart
EGR105 – Week 7 Topics
• Data analysis concepts
• Regression methods
• Function discovery by example
• Regression tools in Excel
• Assignment 7
