STAT 3220 Unit 3
During periods of high electricity demand, especially during the hot summer months, the power output from a gas turbine engine can drop dramatically. One way to counter this drop in power is by cooling the inlet air to the gas turbine. An increasingly popular cooling method uses high pressure inlet fogging. The performance of a sample of 67 gas turbines augmented with high pressure inlet fogging was investigated in the Journal of Engineering for Gas Turbines and Power (January 2005). One measure of performance is heat rate (kilojoules per kilowatt per hour). Heat rates for the 67 gas turbines, saved in the gasturbine file.
Unit 3.2: Residuals and Assumptions
gasturbine<-read.delim("https://raw.githubusercontent.com/kvaranyak4/STAT3220/main/GASTURBINE.txt")
head(gasturbine)names(gasturbine)[1] "ENGINE" "SHAFTS" "RPM" "CPRATIO" "INLET.TEMP"
[6] "EXH.TEMP" "AIRFLOW" "POWER" "HEATRATE" Step 6: Check the Model Assumptions
Step 2: Hypothesize Relationship (Exploratory Data Analysis)
Remider we looked at multicollinearioty and variable screening to end up with a starting subset of predictors of: RPM, INLET.TEMP,
#updated model
gasmod2<-lm(HEATRATE~.-ENGINE-SHAFTS-POWER-CPRATIO,data=gasturbine)
# Syntax Note: We can use the . to indicate all the variables in the data frame
# And use the - to exclude a particular variable from the model
summary(gasmod2)
Call:
lm(formula = HEATRATE ~ . - ENGINE - SHAFTS - POWER - CPRATIO,
data = gasturbine)
Residuals:
Min 1Q Median 3Q Max
-1007.7 -290.5 -106.0 240.1 1414.8
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.362e+04 8.133e+02 16.744 < 2e-16 ***
RPM 8.882e-02 1.344e-02 6.608 1.02e-08 ***
INLET.TEMP -9.186e+00 7.704e-01 -11.923 < 2e-16 ***
EXH.TEMP 1.436e+01 2.260e+00 6.356 2.76e-08 ***
AIRFLOW -8.475e-01 4.370e-01 -1.939 0.057 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 455.1 on 62 degrees of freedom
Multiple R-squared: 0.9235, Adjusted R-squared: 0.9186
F-statistic: 187.1 on 4 and 62 DF, p-value: < 2.2e-16Try adding qualitative variable Engine
#updated model
gasmod3<-lm(HEATRATE~.-SHAFTS-POWER-CPRATIO,data=gasturbine)
# Syntax Note: We can use the . to indicate all the variables in the data frame
# And use the - to exclude a particular variable from the model
summary(gasmod3)
Call:
lm(formula = HEATRATE ~ . - SHAFTS - POWER - CPRATIO, data = gasturbine)
Residuals:
Min 1Q Median 3Q Max
-1002.38 -257.50 -60.08 252.72 1397.12
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.537e+04 1.284e+03 11.968 < 2e-16 ***
ENGINEAeroderiv -4.245e+02 2.806e+02 -1.513 0.1356
ENGINETraditional -3.772e+02 2.183e+02 -1.728 0.0891 .
RPM 9.065e-02 1.459e-02 6.214 5.38e-08 ***
INLET.TEMP -9.908e+00 9.161e-01 -10.816 1.01e-15 ***
EXH.TEMP 1.314e+01 2.377e+00 5.530 7.36e-07 ***
AIRFLOW -8.534e-01 4.338e-01 -1.967 0.0538 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 450.8 on 60 degrees of freedom
Multiple R-squared: 0.9274, Adjusted R-squared: 0.9201
F-statistic: 127.7 on 6 and 60 DF, p-value: < 2.2e-16Perform nested F test to determine if Engine type is significant.
redgasmod3<-lm(HEATRATE~.-ENGINE-SHAFTS-POWER-CPRATIO,data=gasturbine)
anova(redgasmod3,gasmod3)- Hypotheses:
- \(H_0: \beta_1=\beta_2=0\) (the engine type does not contribute to predicting heat rate)
- \(H_a:\beta_4, \beta_5 \neq 0\) (the species contributes to predicting heat rate)
- Distribution of test statistic: F 2 with 60 DF
- Test Statistic: F=1.603
- Pvalue: 0.2098
- Decision: 0.2098>0.05 -> FAIL TO REJECT H0
- Conclusion: The engine type is not significant at predicting heat rate. We will remove both dummy variables in the model and not test them individually.
For the sake of this example, we will not consider any further testing.
Step 4: Specify the distribution of the errors and find the estimate of the variance
Step 5: Evaluate the Utility of the model
Step 6: Check the Model Assumptions
First we will evaluate model assumptions
#There are a few options to view plots for assumptions
#Residuals Plots of explanatory variables vs residuals
residualPlots(gasmod3,tests=F)
#Residual vs Fitted and QQ plot
plot(gasmod3, which=c(1,2))
#histogram of residuals
hist(residuals(gasmod3))
- Lack of Fit Plot: Residual Plots
- Constant Variance: Residual Plots and Residual vs Fitted
- Normality: QQ Plot and Histogram of Residuals
- Independence: We do not have time series data
Then we will identify outliers and influential observations.
#Cooks Distance Thresholds
plot(gasmod3,which=4)
#Leverage vs Studentized Residuals
influencePlot(gasmod3,fill=F)
#We can use various functions to store and view these statistics for all observations
dffits(gasmod3) 1 2 3 4 5 6
0.760196099 0.484742791 -0.202119453 -0.458551481 -0.107150221 -0.029733726
7 8 9 10 11 12
-0.022548928 0.138806983 -0.008912956 -0.026512455 0.883790595 0.462687401
13 14 15 16 17 18
-0.060732559 -0.040992246 0.224449807 -0.123046747 -0.155064888 -0.183487684
19 20 21 22 23 24
-0.015050769 0.178189075 0.445846820 0.107259411 -0.055098516 -0.103878755
25 26 27 28 29 30
0.018100698 -0.007710417 -0.102430426 -0.056103363 -0.193438489 -0.197493559
31 32 33 34 35 36
-0.207168823 -0.794247664 0.055167417 -0.030580437 -0.293112289 1.135238201
37 38 39 40 41 42
0.085446025 0.099343488 -0.424322676 -0.458157215 -0.139615819 -0.129422848
43 44 45 46 47 48
-0.215621851 -0.116659552 0.810400278 0.025727980 0.846040408 0.183644637
49 50 51 52 53 54
-0.145327656 0.258925459 -0.041409259 0.495274858 0.256127721 -0.220523608
55 56 57 58 59 60
0.063572125 -0.199340056 -0.086775927 -0.336206852 0.168821930 -0.302900575
61 62 63 64 65 66
0.776485175 -0.699216883 -0.329478620 0.995300017 -0.358519481 0.545795996
67
-0.249149287 dfbetas(gasmod3) (Intercept) ENGINEAeroderiv ENGINETraditional RPM INLET.TEMP
1 -0.4535499243 0.0167439440 0.317180177 3.938248e-01 0.1965176870
2 0.3915692912 -0.2840894465 -0.235877400 1.782809e-01 -0.0740844654
3 0.0961433931 0.0074080582 -0.093668836 -1.111369e-01 -0.1170746327
4 -0.2774071543 0.2146974001 0.138460808 -4.837385e-02 0.0593733446
5 0.0731691716 -0.0180179579 -0.059564440 -5.350942e-03 -0.0323056489
6 0.0113656632 -0.0082633805 -0.018547530 3.539220e-03 -0.0145789601
7 0.0062129985 -0.0048764686 -0.013219338 -1.162223e-03 -0.0135147596
8 0.0133757319 0.0274485052 0.055812137 -7.762876e-03 0.0289248410
9 0.0009799309 -0.0033245107 -0.004462150 2.424455e-03 0.0006715006
10 0.0012932338 -0.0082886475 -0.012978012 -3.696201e-03 -0.0029625717
11 -0.0123943863 -0.2908018941 0.034210440 3.966546e-01 -0.0456822310
12 -0.0036620829 -0.0474799774 -0.002062140 -1.930113e-01 -0.2213322739
13 0.0477937569 -0.0131807399 -0.039592299 -1.734180e-02 -0.0368240576
14 -0.0149832061 0.0099935533 0.002551843 5.366802e-03 -0.0036521416
15 0.1313305676 -0.0462874245 -0.086412964 -1.280964e-01 -0.1733892418
16 0.0552535976 -0.0432585116 -0.052577665 8.330807e-02 0.0229444459
17 0.0266782395 -0.0537208621 -0.055724722 9.450724e-02 0.0634771006
18 0.0288759242 -0.0702760154 -0.074494525 7.660335e-02 0.0702299359
19 0.0062248536 -0.0075777093 -0.010396932 6.109760e-04 -0.0015021213
20 -0.0759357844 0.0008412255 0.087294404 8.111878e-02 0.1319065697
21 -0.1460586844 0.1191047370 0.133584778 -3.346838e-01 -0.0497176797
22 -0.0275126025 0.0275993461 0.054844838 -3.614848e-02 0.0271570301
23 0.0071677611 -0.0202522339 -0.027739156 1.510395e-02 0.0058105336
24 0.0374982623 -0.0480830807 -0.058146665 4.637035e-02 0.0213650092
25 -0.0044763132 0.0073417912 0.011701421 -1.015495e-03 0.0042674656
26 -0.0018006339 -0.0013472437 -0.001929857 2.257055e-04 0.0013319221
27 0.0112514145 -0.0375726776 -0.048227696 9.076031e-03 0.0152878319
28 -0.0241955852 0.0279787368 0.020532149 -2.272197e-02 0.0274493744
29 -0.1627086434 0.1161310699 0.103750410 -6.850409e-02 0.0506679177
30 -0.0025525613 0.0594362970 -0.012372040 -5.339859e-02 0.0064318234
31 0.0211130553 0.0503469195 -0.045499356 -9.955372e-02 -0.0768971635
32 -0.0448072255 0.2149531002 -0.128394688 -5.266997e-01 -0.5018550732
33 0.0290556398 -0.0196906407 -0.009561385 6.672910e-03 0.0106649412
34 -0.0054372832 0.0055156016 -0.002956651 -9.909128e-05 -0.0100234437
35 -0.0130971643 0.0573362366 -0.061774019 -1.101131e-01 -0.1642393772
36 0.3228113257 -0.0263005519 -0.258400386 -7.999446e-01 -0.9716864002
37 0.0180815706 0.0134951327 0.010758803 -4.178265e-02 -0.0445846341
38 0.0585762335 -0.0063267033 -0.012704594 -1.401378e-02 -0.0474052060
39 -0.1966438156 0.1649359223 0.066467067 -5.656278e-02 -0.0450794837
40 -0.1239033000 0.2440325603 0.279959173 5.549717e-02 -0.0318658861
41 0.0069030482 0.0360677900 0.061925112 7.299118e-02 0.0455837507
42 0.0085612808 0.0254047045 0.048642943 3.815236e-02 0.0386796632
43 0.0968070776 -0.0124275031 0.009109390 8.024291e-02 0.0353591512
44 -0.0150629668 0.0196585692 0.035845023 -9.703150e-03 0.0331577279
45 0.0910556837 -0.1393077827 -0.045523501 4.885240e-01 0.4152678357
46 -0.0093241803 0.0024221334 0.003677005 6.106168e-03 0.0056616828
47 0.1356881026 -0.3905633714 -0.481810392 -2.443774e-01 -0.0615592126
48 0.0373132314 -0.0916703593 -0.111863502 -4.483104e-02 -0.0142594672
49 0.0304264979 0.0157242805 0.037235788 4.461512e-02 0.0343471466
50 -0.0669050659 -0.0160171372 0.011162221 9.393020e-02 0.1288797539
51 -0.0086760187 0.0094867982 0.012646285 -1.169426e-02 0.0023785949
52 -0.0018688689 -0.0433994260 0.005787565 2.748199e-01 0.2030320743
53 0.1160875703 -0.1506162485 -0.170904553 4.540871e-03 0.0327828718
54 0.0264751584 0.0550668550 0.085205917 1.015311e-01 0.0384182200
55 0.0373202574 -0.0361073931 -0.046685184 5.643887e-03 -0.0082357370
56 -0.0265172134 0.0727437488 0.092023418 -4.973490e-03 -0.0127555308
57 0.0043079474 0.0216131136 0.031086648 8.759746e-03 0.0022490996
58 -0.1211446207 0.1105331955 0.206079477 8.946833e-02 0.2126689083
59 -0.0433021915 -0.0124230796 0.005613581 5.952930e-02 0.0877642966
60 -0.0934189979 0.0829915950 0.123591896 -5.807800e-02 0.0698530282
61 0.2819978706 0.0022374921 -0.264923268 3.779586e-01 -0.2417315553
62 0.0436283804 -0.2189487851 0.043492562 -2.262431e-01 0.1611606021
63 0.0167039636 -0.1913930169 0.013100275 3.795333e-02 0.0729760773
64 0.0440009760 0.4820952976 0.157861070 1.115376e-02 0.5450767970
65 -0.0761241224 -0.1848000724 0.018848460 1.239129e-01 -0.0408620003
66 -0.3050670451 0.4319674273 0.218933117 -7.325678e-02 0.1591430104
67 0.0252085113 -0.1607132597 0.002572737 1.034296e-01 0.0808525882
EXH.TEMP AIRFLOW
1 0.2296344844 -0.001764431
2 -0.3226764693 0.127169860
3 0.0163179443 0.022343955
4 0.2021781858 0.033626555
5 -0.0455820200 0.048050440
6 0.0011619152 0.007721298
7 0.0059494329 0.002262205
8 -0.0431702299 0.040097008
9 -0.0012787190 -0.002415159
10 0.0039932789 -0.017598497
11 0.0367257879 -0.015001787
12 0.2399491776 -0.234712343
13 -0.0134781465 0.018467025
14 0.0151607893 0.012682091
15 0.0424715587 -0.037362967
16 -0.0849660197 0.071445954
17 -0.0875277591 0.017895379
18 -0.0871565457 -0.037731980
19 -0.0035851133 -0.006460154
20 -0.0453584335 -0.033514126
21 0.2399282078 -0.335231712
22 0.0080300551 -0.036502840
23 -0.0105440758 -0.013942912
24 -0.0553371116 -0.002782808
25 -0.0003839523 0.005845356
26 0.0011207448 -0.004681146
27 -0.0168529480 -0.054946469
28 0.0011779680 -0.015070421
29 0.1170779487 -0.059057370
30 -0.0058848168 0.035683677
31 0.0496714033 0.023347699
32 0.5200453184 -0.080793285
33 -0.0360513597 -0.004608631
34 0.0125833155 0.009802877
35 0.1580002310 0.032232409
36 0.6358697215 -0.290633117
37 0.0230630042 0.010801334
38 -0.0187905258 0.051650950
39 0.2162612880 0.062062610
40 0.0854578279 0.245525735
41 -0.0693213117 0.093771200
42 -0.0547506225 0.019285151
43 -0.1452165765 0.070369879
44 -0.0132155479 -0.059338171
45 -0.5130448855 0.519556368
46 0.0031545599 0.010938454
47 0.0562696492 -0.590679228
48 0.0044587989 -0.117353276
49 -0.0733397385 0.028225244
50 -0.0527184589 0.079995026
51 0.0077204671 -0.025875603
52 -0.2164704094 0.343959391
53 -0.1140757033 -0.078097081
54 -0.0950226035 0.156908152
55 -0.0254005492 0.011933315
56 0.0227443752 -0.004984691
57 -0.0131908682 0.005460117
58 -0.0911498217 -0.049446953
59 -0.0368867380 0.045357851
60 0.0358202972 -0.181797200
61 -0.1117080138 0.349597036
62 -0.1526398283 -0.151325887
63 -0.0857270389 0.028858134
64 -0.5237985830 -0.001945503
65 0.0932263398 0.059656021
66 0.1684668241 -0.170453367
67 -0.1085547000 0.066224326#We can subset by the observations of interest
dffits(gasmod3)[c(11,36,61,64)] 11 36 61 64
0.8837906 1.1352382 0.7764852 0.9953000 Next Steps to resolve
Variable transformation: We can directly use a funciton of the response variable within lm() or add a new variable to the table
#Create new variable
gasturbine$sqrty<-sqrt(gasturbine$HEATRATE)
#remember to remove original
gasmodsqrt<-lm(sqrty~.-ENGINE-SHAFTS-POWER-CPRATIO-HEATRATE,data=gasturbine)
summary(gasmodsqrt)
Call:
lm(formula = sqrty ~ . - ENGINE - SHAFTS - POWER - CPRATIO -
HEATRATE, data = gasturbine)
Residuals:
Min 1Q Median 3Q Max
-4.3749 -1.3484 -0.3872 1.1073 6.1302
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.172e+02 3.616e+00 32.409 < 2e-16 ***
RPM 3.631e-04 5.976e-05 6.076 8.28e-08 ***
INLET.TEMP -4.344e-02 3.425e-03 -12.683 < 2e-16 ***
EXH.TEMP 6.904e-02 1.005e-02 6.872 3.58e-09 ***
AIRFLOW -5.240e-03 1.943e-03 -2.697 0.009 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.023 on 62 degrees of freedom
Multiple R-squared: 0.9289, Adjusted R-squared: 0.9243
F-statistic: 202.6 on 4 and 62 DF, p-value: < 2.2e-16#Or directly transform in the lm() function
#remember to remove transformation
gasmodsqrt2<-lm(sqrt(HEATRATE)~.-ENGINE-SHAFTS-POWER-CPRATIO-sqrty,data=gasturbine)
summary(gasmodsqrt2)
Call:
lm(formula = sqrt(HEATRATE) ~ . - ENGINE - SHAFTS - POWER - CPRATIO -
sqrty, data = gasturbine)
Residuals:
Min 1Q Median 3Q Max
-4.3749 -1.3484 -0.3872 1.1073 6.1302
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.172e+02 3.616e+00 32.409 < 2e-16 ***
RPM 3.631e-04 5.976e-05 6.076 8.28e-08 ***
INLET.TEMP -4.344e-02 3.425e-03 -12.683 < 2e-16 ***
EXH.TEMP 6.904e-02 1.005e-02 6.872 3.58e-09 ***
AIRFLOW -5.240e-03 1.943e-03 -2.697 0.009 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.023 on 62 degrees of freedom
Multiple R-squared: 0.9289, Adjusted R-squared: 0.9243
F-statistic: 202.6 on 4 and 62 DF, p-value: < 2.2e-16Removing observations from analyis by subsetting the data
#We can remove particular observations
subsetgas<-gasturbine[-c(11,36,61,64),]
gasmod2obs<-lm(HEATRATE~.-ENGINE-SHAFTS-POWER-CPRATIO-sqrty,data=subsetgas)
# Syntax Note: We can use the . to indicate all the variables in the data frame
# And use the - to exclude a particular variable from the model
summary(gasmod2obs)
Call:
lm(formula = HEATRATE ~ . - ENGINE - SHAFTS - POWER - CPRATIO -
sqrty, data = subsetgas)
Residuals:
Min 1Q Median 3Q Max
-957.18 -214.19 -93.73 233.18 1030.40
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.325e+04 7.218e+02 18.361 < 2e-16 ***
RPM 8.526e-02 1.370e-02 6.226 5.77e-08 ***
INLET.TEMP -8.347e+00 8.048e-01 -10.372 7.88e-15 ***
EXH.TEMP 1.320e+01 2.316e+00 5.698 4.26e-07 ***
AIRFLOW -9.429e-01 4.004e-01 -2.355 0.0219 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 391.6 on 58 degrees of freedom
Multiple R-squared: 0.9231, Adjusted R-squared: 0.9178
F-statistic: 174 on 4 and 58 DF, p-value: < 2.2e-16