Regression II

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Regression II

• Engineering Experimental Design
• Winter 2003

2
Outline

• Comparing the slope (intercept) to an expected
  value
• Estimating a value from the slope or intercept
• Placing an uncertainty on a calculated
  (predicted) value of y
• Nonlinear regression using Matlab

3
Problem Vapor-Liquid Eq.

• The relative volatility, ?, is the molar ratio of
  components i and j in the gas phase divided by
  the molar ratio of the two components in the
  liquid phase.
• Assuming ? is a constant, it can be found
  experimentally from the mole fractions of one
  component in the gas phase (y) and the liquid
  phase using this equation
• y ?x / (1 (?-1)x)
• Taking both sides to the power 1 linearizes the
  equation
• Data for methanol in water is shown to the left
• x mole fraction methanol in liquid
• y mole fraction methanol in gas phase

4
Problem, ctd.

• Linearize the equation and perform least-squares
  linear regression using Excel.
• You should get a slope of roughly 1.24 and an
  intercept of roughly 0.11
• Evaluate how well the linearized equation fits
  the transformed data.
• Use r-squared, ANOVA, residual plot, and line fit
  plot

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Comparing the Slope or Intercept to an Expected
Value

• Excel gives you the t-statistic and confidence
  interval on the slope and intercept
• 95 significance level by default
• Can also specify other significance level
• If the expected value is included in the
  confidence interval, then the slope (intercept)
  is not significantly different from the expected
  value.
• If t-statistic lt t-critical, then the slope
  (intercept) is not significantly different from
  the expected value.
• To get t-critical, use table with n-2 (two less
  than number of data points) degrees of freedom

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Estimating a Value from the Slope or Intercept

• Use the confidence interval
• Example ln(y) A(Re) ln(B)
• Regression gives ln(B) 0.90 0.15
• eln(B) e0.9 2.4596
• ?(ex) (dex/dx) ?x ex ?x e0.9 ? 0.15 0.4
• Or
• ?(ex) exmax ex or ex exmin e1.05-e0.9 or
  e0.9-e0.75 0.4 or 0.3
• B 2.5 0.4

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Estimating a Value from the Slope or Intercept

• Example ln(y) A(Re) ln(B)
• Regression gives ln(B) 0.09 0.15
• eln(B) e0.09 1.094
• ?(ex) (dex/dx) ?x ex ?x e0.09 ? 0.15 0.16
• Or
• ?(ex) exmax ex or ex exmin e0.24-e0.9
  or e0.9-e-0.06 0.18 or 0.15
• B 1.00 0.16

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Why didnt I throw out the intercept?

• 0.09 0.15 spans zero, so why didnt I just set
  the intercept to zero and regress again?
• ln(y) A(Re) ln(B) is a linearized form of y
  B(Re)A
• When ln(B) 0, B 1.0
• You cannot evaluate the significance of B
  directly by looking at the size of ln(B)

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Back to the Problem

• Calculate the value of ? using the slope for the
  linearized equation.
• Calculate the value of ? using the intercept for
  the linearized equation.
• Do these values of ? agree at the 5
  significance level? What value of ? would you
  report?

10
Problem, ctd.

• In fact, ? is not constant for the methanol-water
  system over this whole range.
• The 1/x, 1/y transformation of the data weights
  points at higher temperature (low x and y values)
  very heavily in the least-squares regression.
• The slope gives a value of ? similar to the true
  values at high temperatures (low x and y)
• Linearization usually gives different values for
  parameters than nonlinear regression, because the
  data transformation weights points differently

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Placing an Uncertainty on a Predicted Value of y
Quick Dirty

• Under Regression Statistics Excel gives a
  Standard Error
• This is an estimate of the standard deviation of
  the normally-distributed random error in y
• As a first approximation, you can be 65 certain
  that if you go to the lab and measure a value of
  y, you will be within 1 standard error of the
  value you would predict from the regression
  equation

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Placing an Uncertainty on a Predicted Value of y
Right Way

• Check out the equations on pages 479, 487, 490
• To get Sxx, divide the standard error (se) for
  the regression by the standard error for the
  slope (sb) and then square the result.
• Find the mean of all the x-values.
• Calculate sy-hat
• Look up t-critical for your significance level
• Degrees of freedom samples - 2

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Prediction vs. Confidence Interval

• Look at Figure 11.11
• The confidence interval on the true value of y
  is narrower than the prediction interval, and
  narrower for values of x in the middle than for
  values on the ends
• The prediction interval is about the same for all
  values of x
• Usually, you want to predict a value of y for a
  given value of x
• This is why the quick-n-dirty method is not too
  bad for putting an uncertainty on a predicted y

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Back to the Problem

• Using the linear regression results, what mole
  fraction of methanol in the gas phase do you
  expect to measure for a liquid phase mole
  fraction of 0.30? What about for 0.60?
• Use the quick-n-dirty method for uncertainty
  first, assuming that 2 standard deviations covers
  95 of the data.
• Calculate the 95 prediction interval the right
  way for each y second

15
Nonlinear Regression with Matlab

• Refer to the solution to exam 1 for an example
• Use the function nlinfit
• Make one vector of dependent variable values and
  one of independent variable values
• Define your model equation as an inline function
• Dont forget the confidence interval on the
  relative volatility