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Regression Statistics
Thomas Knotts
Engineers and Regression
Engineers often:Regress data
Analysis Fit to theory Data reduction
Use the regression of others
Antoine Equation DIPPR 0 2 4 6 8 1012141618
-5
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X
Y
Linear RegressionThere are two classes of regressions
Linear Non-linear
“Linear” refers to the parameters, not the functional dependence of the independent variable
You can use the Mathcad function “linfit” on linear equations
Linear RegressionThere are two classes of regressions
Linear Non-linear
“Linear” refers to the parameters, not the functional dependence of the independent variable
You can use the Mathcad function “linfit” on linear equations
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Quiz1.
2.
3.
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5.
Straight Line Model
0 2 4 6 8 10 12 14 16 18-5
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25 residual (error)
“X” data
“Y” predicted
slope
Intercept
Straight Line Model
residual (error)
sum squared error
iX iY iY ie“X” data
“Y” data
intercept 0.92291455 slope 0.516173934
1 2.749032178 1.439088483 1.3099436942 3.719910224 2.362003033 1.3579071923 0.925995017 3.284917582 -2.358922574 2.623482686 4.207832132 -1.584349455 6.539797342 5.130746681 1.4090506616 6.779909177 6.053661231 0.7262479467 4.946150401 6.976575781 -2.030425388 9.674178069 7.89949033 1.7746877399 7.61959821 8.82240488 -1.2028066710 7.650020996 9.745319429 -2.0952984311 11.514 10.66823398 0.84576602112 13.18285068 11.59114853 1.59170215213 13.28173635 12.51406308 0.76767327514 13.60444592 13.43697763 0.1674682915 12.79535218 14.35989218 -1.5645416 17.82374778 15.28280673 2.54094105617 14.55068379 16.20572128 -1.65503748
42.76608602 SSE
“Y” predicted
The R2 StatisticA useful statistic but not definitive
Tells you how well the data fit the model.
It does not tell you if the model is correct.
How much of the distribution of the data about the mean is
described by the model.
Problems with R2
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f(x) = 0.500090909090909 x + 3.00009090909091R² = 0.666542459508775
X
Y
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13f(x) = 0.499909090909091 x + 3.00172727272727R² = 0.666707256898465
X
Y
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f(x) = 0.499727272727273 x + 3.00245454545455R² = 0.666324041066559
X
Y
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f(x) = 0.5 x + 3.00090909090909R² = 0.666242033727484
X
Y
Residuals Should Be Normally Distributed
3 5 7 9 11 13 15 17 193
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f(x) = 0.500090909090909 x + 3.00009090909091R² = 0.666542459508775
X
Y
1 2 3 4 5 6 7 8 9 10 11
-2.5-2
-1.5-1
-0.50
0.51
1.52
2.5
e
3 5 7 9 11 13 15 17 193
5
7
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f(x) = 0.499727272727273 x + 3.00245454545455R² = 0.666324041066559
X
Y
1 2 3 4 5 6 7 8 9 10 11
-1.5-1
-0.50
0.51
1.52
2.53
3.5
e
Residuals Should Be Normally Distributed
3 5 7 9 11 13 15 17 193
5
7
9
11
13f(x) = 0.499909090909091 x + 3.00172727272727R² = 0.666707256898465
X
Y
1 2 3 4 5 6 7 8 9 10 11
-2
-1.5
-1
-0.5
0
0.5
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1.5
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2.5
e
3 5 7 9 11 13 15 17 193
5
7
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f(x) = 0.5 x + 3.00090909090909R² = 0.666242033727484
X
Y1 2 3 4 5 6 7 8 9 10 11
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
e
Statistics on the Slope/Intercept
When you fit data to a straight line, the slope and intercept are only estimates of the true slope and intercept.
(1-a)100%Confidence Intervals
Standard Errors
Statistics on the Predicted Variable
0 2 4 6 8 10 12 14 16 18-5
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f(x) = 0.922914602046951 x + 0.516173934846863R² = 0.890424511795483
Data
Excel Trendline
Least Squares Fit
Lower 95% Confidence Region
Upper 95% Confidence Region
X
Y
Example On Excel
Generalized Linear Regression
Linear regression can be written in matrix form. X Y21 18624 21432 28847 42550 45559 53968 62274 67562 56250 45341 37030 274
Straight Line Model
Quadratic Model
Statistics with Matrices
ParameterConfidence Intervals
Predicted VariableConfidence Intervals
Standard Error of bi is the square root of the i-th diagonal term of the matrix
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