EGR 105 Foundations of Engineering I

Excel Part III Curve-Fitting, Regression

Section 8 Fall 2013

EGR 105 Foundations of Engineering I

Excel Part II Topics

• Data Analysis Concepts • Regression Methods• Example Function Discovery• Regression Tools in Excel• Homework Assignment

Analysis of x-y Data• Independent versus dependent

variables

y

y = f(x) xindependent

depe

nden

t

Common Types of Plots Example: Y=3X2

log(y) = log(3) + 2*log(x)y = 3x2

Straight Line on log-log Plot!

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X

Y

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logX

Y

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logX

logY

Cartesian

Semi-log : log x

log-log : log y-log x

Note!

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

datapoints

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 MarketForecasting – 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):

(x1,y1) (x2,y2)

(x3,y3) (x4,y4)

x

ye3

ei = yi – f(xi), i =1,2,…,n

sum

squa

red

erro

rs

Quality of the Fit:

Notes: is the average y value0 R2 1-closer to 1 is a “better” fit

x

y

Coefficient of Determination

• 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

• Set m and b derivatives to zero

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 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

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f(x) = 0.0375 x⁴ − 0.523148 x³ + 2.518056 x² − 3.878439 x + 3.133333R² = 0.997526534200979

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f(x) = 0.996703296703297 xR² = 0.997473121604204

Better fit but does it make sense with expected behavior?

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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:

Mass

Length

One Team’s Results

Mass appears to have no impact, but length does

To determine the effect of length, first plot the data

Try a linear fit

Force a zero intercept (why?)

Try a quadratic polynomial fit

Try a logarithmic fit

Try a power function fit

On log-log axes, nice straight line

Power Law Relation:

b

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): • Nonlinear Cubic Model:

Linear Fit

Cubic Fit Better and it Makes Sense with the Physics

Force (lb)

Collected Data

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

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