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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

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egr 105 foundations of engineering i

EGR 105 Foundations of Engineering I

Fall 2007 – week 7

Excel part 3 - regression

analysis of x y data
Analysis of x-y Data
  • Independent versus dependent variables

y

y = f(x)x

dependent

independent

finding other values
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
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

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):
quality of the fit

y

x

Quality of the Fit:

Notes: is the average y value

0 R2 1

closer to 1 is a “better” fit

linear regression
Linear Regression
  • Functional choicey = m x + b

slopeintercept

  • Squared errors sum to
  • Set m and b derivatives to zero
further regression possibilities
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 discovery or how to find the best relationship
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)