EGR 105 Foundations of Engineering I

Fall 2008 – Session 4Excel – Plotting, Curve-Fitting, Regression

EGR105 – Session 4 Topics

• Review of Basic Plotting• Data Analysis Concepts • Regression Methods• Example Function Discovery• Regression Tools in Excel• Homework Assignment

Analysis of x-y Data

• Independent versus dependent variables

independent

depe

nden

ty = f (x)

x

y

Simple PlottingGenerate X and Y data to Plot

Common Types of Plots: Y=3X2

logy = log3 + 2logxy = 3x2

Straight Line on log-log Plot!

Normal

Semi-log: log x

log-log: log y-log x

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

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• 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 value0 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 + …

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

Linear & Quartic Curve Fit Example

Better fit but does it make sense with expected behavior?

Y

Y

X

X

Example Function DiscoveryHow 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

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:

)log()log()log( lbatalt b

1.0

10.0

1.0 10.0 100.0 1000.0

length (inches)

tim

e (

seco

nd

s)

Power Law Relation:

b

Elastic Bungee Cord Models Determined by Curve Fitting the Data

• Linear Model (Hooke’s Law): • Nonlinear Cubic Model: 3

32

21)( sksksksF kssF )(

Linear Fit

Cubic Fit Better and it Makes Sense with the Physics

Force (lb)

sl

ll

LengthOriginal

Elongation

o

o

Collected Data

Homework AssignmentSee passed out sheet or course web site