EGR 105 Foundations of Engineering I

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# EGR 105 Foundations of Engineering I - PowerPoint PPT Presentation

EGR 105 Foundations of Engineering I. Fall 2007 – week 7 Excel part 3 - regression. Analysis of x-y Data. Independent versus dependent variables y y = f(x) x. dependent. independent. Finding Other Values. Interpolation

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### EGR 105 Foundations of Engineering I

Fall 2007 – week 7

Excel part 3 - regression

Analysis of x-y Data
• Independent versus dependent variables

y

y = f(x)x

dependent

independent

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

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

y

(x3,y3)

(x4,y4)

(x1,y1)

(x2,y2)

e3

ei= yi – f(xi), i =1,2,…,n

x

What Does “close” Mean?

errors

squared

sum

• Want a consistent rule
• Common is the least squares fit (SSE):

y

x

Quality of the Fit:

Notes: is the average y value

0 R2 1

closer to 1 is a “better” fit

Linear Regression
• Functional choicey = m x + b

slopeintercept

• 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 + …
Function Discoveryor How to find the best relationship
• Look for straight lines on log axes:

àlinear on semilog x y = a ln x + b

àlinear on semilog y y = ae bx

àlinear on log log y = ax b

• No rule for 2nd or higher order polynomial fits (not very useful toward real problems)