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) x independentdependent

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 data points

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 (x i, y i ) 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): (x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) (x 4,y 4 ) x y e3e3 e i = y i – f(x i ), i =1,2,…,n sum squared errors

Quality of the Fit: Notes: is the average y value 0  R 2  1 closer to 1 is a “better” fit x y

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 x 2 + bx + c – Higher order:y = a x k + bx k-1 + …

EGR105 – Week 7 topics Data analysis concepts Regression methods Function discovery by example Regression tools in Excel Assignment

Function Discovery or 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 2 nd 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: 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:

Try a quadratic polynomial:

Try logarithmic:

Try power function:

On log-log axes, a nice straight line:

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 R 2 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