Supervised Learning
Sharang Nimbalkar, Salil Tari, Rinisha Marar, Prashant Bhosale
2020-04-06
Predicting Playoff Qualifications for NBA Teams
Executive Summary
Summary
The National Basketball Association (NBA) is a men’s professional basketball league in North America. It is one of the four major professional sports leagues in the United States and Canada and is widely considered to be the premier men’s professional basketball league in the world. The NBA playoffs are a best-of-seven elimination tournament annually held after the National Basketball Association’s regular season to determine the league’s champion.
The purpose of this project is to create a regression model that predicts which teams would qualify for the NBA Playoffs based on historical performance statistics. We have obtained the NBA data for period 1980-2011 from Kaggle.com. This model will take into consideration several factors which will give a team a probability of making it through to the playoffs and the dataset has several observations/factors which will help us in making these predictions.
This report is divided into several chapters and it starts with explanation of the different covariates such as number of different types of field goals attempted and made, free throws attempted and made, number of different types of rebounds, assists, steals, blocks, turnovers, etc.
The report progresses with different parts of regression model-building process such as model specification, parameter estimation, model adequacy checking, and model validation.
The report concludes with the final model which will predict the possibility of a team to qualify for the playoffs.
Exploratory Data Analysis
Exploratory Data Analysis
We loaded the NBA dataset for training purpose into R. This data was checked for missing values and inconsistency. Using domain knowledge, we have already identified the covariates for analysis purpose. We have performed exploratory data analysis to analyze the data set to summarize its main characteristics.
Dataset dimensions
## [1] 835 20Column Names
## [1] "SeasonEnd" "Team" "Playoffs" "W" "PTS" "oppPTS"
## [7] "FG" "FGA" "X2P" "X2PA" "X3P" "X3PA"
## [13] "FT" "FTA" "ORB" "DRB" "AST" "STL"
## [19] "BLK" "TOV"The statistics are about whether a team qualified for the playoffs in that season, how many matches the team won, how many points the team scored and other attributes that are listed below. From the available attributes, we are initially considering the following covariates:
- 2PA: 2 Point Field Goals Attempted
- 3PA: 3 Point Field Goals Attempted
- FTA: Free Throws Attempted
- ORB: Offensive Rebounds
- DRB: Defensive Rebounds
- AST: Assists
- STL: Steals
- BLK: Blocks
- TOV: Turnovers
The dataset we have is pretty clean and we didn’t find any null/missing values hence there were no imputations done.
Correlation between the variables
We determine the correlation between all the variables in the dataset. If the value is displayed in dark blue it implies that the covariates are positively corelated and if the vale is displayed in dark red it implies that the covariates are negatively corelated.
Histograms
We check the overall spread of the dataset using the histograms. We determine, whether the data is spread evenly or has a positive or negative skew to it.
- Inference
When we observed all the histograms, we’re getting symmetric or roughly symmetric shapes, hence there isn’t much skewness in the data of the significant variables that we’ve selected to build our model.
Box Plot
Using the box plot we have tried to identify the outliers in the data set. We have used two colours to show the teams that will qualify for the playoffs. The box plot in green shows success, that the team has quailifed for the playoffs and the box plot in yellow shows failure, that the team could not qualify for the next round.
- Inference
Similarly, when we checked the box plots for all the significant covariates, we observed that the shape of box plots was symmetric or roughly symmetric with minimal outliers. Hence, we can conclude that the data in our significant covariates followed a normal distribution.
Plotting the data
We plot the number of wins with the most significant covariates.
- Inference
After plotting the number of wins against points scored by a team, points scored by the opponents, and the difference of these 2 covariates, we found out that, number of wins has better correlation with points differential than that of number of points scored by a team or their opponents.
Model Building
Model Building
Our idea is to create 3 different models for calculating number of wins, number of points by the team in consideration, and number of points scored by its opponents.
We also confirmed the variable selction using AIC and BIC criteria. Based on the result of both AIC and BIC criteria, the covariates that we had choosen in the above model remained same. Hence, we can conclude that the variables that we’ve identified are good to move further ahead in the modeling process.
Following is the R code and summary of AIC and BIC criteria that we checked:
We have only considered variables which are important to build the model. Following are the list of variables with which we have started the backward selection technique.
##
## Call:
## lm(formula = W ~ PTSdiff, data = Nba)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.7393 -2.1018 -0.0672 2.0265 10.6026
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.100e+01 1.059e-01 387.0 <2e-16 ***
## PTSdiff 3.259e-02 2.793e-04 116.7 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.061 on 833 degrees of freedom
## Multiple R-squared: 0.9423, Adjusted R-squared: 0.9423
## F-statistic: 1.361e+04 on 1 and 833 DF, p-value: < 2.2e-16We get the following equation.
W = 41 + 0.0326 * PTSdiff
Linear Model 1 - Points Model
Backward Selection
##
## Call:
## lm(formula = PTS ~ X2PA + X3PA + FTA + AST + ORB + DRB + TOV +
## STL + BLK, data = Nba.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -512.75 -127.46 4.74 124.93 573.76
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.964e+03 2.548e+02 -7.708 5.65e-14 ***
## X2PA 1.037e+00 3.568e-02 29.060 < 2e-16 ***
## X3PA 1.262e+00 4.713e-02 26.786 < 2e-16 ***
## FTA 1.123e+00 4.113e-02 27.308 < 2e-16 ***
## AST 8.913e-01 5.170e-02 17.240 < 2e-16 ***
## ORB -9.212e-01 9.638e-02 -9.558 < 2e-16 ***
## DRB 3.015e-03 7.598e-02 0.040 0.9684
## TOV -2.289e-02 7.523e-02 -0.304 0.7611
## STL -2.425e-01 1.101e-01 -2.201 0.0281 *
## BLK -7.056e-03 1.106e-01 -0.064 0.9492
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 189.4 on 574 degrees of freedom
## Multiple R-squared: 0.8949, Adjusted R-squared: 0.8932
## F-statistic: 543 on 9 and 574 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = PTS ~ X2PA + X3PA + FTA + AST + ORB + DRB + STL +
## BLK, data = Nba.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -510.28 -129.72 5.65 124.45 575.18
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.993e+03 2.364e+02 -8.430 2.79e-16 ***
## X2PA 1.038e+00 3.556e-02 29.184 < 2e-16 ***
## X3PA 1.267e+00 4.491e-02 28.204 < 2e-16 ***
## FTA 1.121e+00 4.072e-02 27.537 < 2e-16 ***
## AST 8.915e-01 5.166e-02 17.258 < 2e-16 ***
## ORB -9.244e-01 9.571e-02 -9.658 < 2e-16 ***
## DRB 4.184e-03 7.582e-02 0.055 0.9560
## STL -2.484e-01 1.083e-01 -2.294 0.0222 *
## BLK -1.134e-02 1.097e-01 -0.103 0.9177
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 189.3 on 575 degrees of freedom
## Multiple R-squared: 0.8949, Adjusted R-squared: 0.8934
## F-statistic: 611.9 on 8 and 575 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = PTS ~ X2PA + X3PA + FTA + AST + ORB + STL + BLK,
## data = Nba.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -509.61 -129.50 5.55 124.72 575.97
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.986e+03 2.032e+02 -9.774 <2e-16 ***
## X2PA 1.038e+00 3.418e-02 30.373 <2e-16 ***
## X3PA 1.267e+00 4.207e-02 30.130 <2e-16 ***
## FTA 1.122e+00 4.040e-02 27.766 <2e-16 ***
## AST 8.919e-01 5.100e-02 17.487 <2e-16 ***
## ORB -9.257e-01 9.269e-02 -9.988 <2e-16 ***
## STL -2.506e-01 1.005e-01 -2.493 0.0129 *
## BLK -9.037e-03 1.013e-01 -0.089 0.9290
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 189.1 on 576 degrees of freedom
## Multiple R-squared: 0.8949, Adjusted R-squared: 0.8936
## F-statistic: 700.5 on 7 and 576 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = PTS ~ X2PA + X3PA + FTA + AST + ORB + STL, data = Nba.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -510.07 -129.69 5.45 123.88 574.72
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.990e+03 1.978e+02 -10.063 <2e-16 ***
## X2PA 1.039e+00 3.386e-02 30.677 <2e-16 ***
## X3PA 1.268e+00 4.153e-02 30.537 <2e-16 ***
## FTA 1.121e+00 4.021e-02 27.891 <2e-16 ***
## AST 8.914e-01 5.061e-02 17.613 <2e-16 ***
## ORB -9.267e-01 9.193e-02 -10.080 <2e-16 ***
## STL -2.504e-01 1.004e-01 -2.494 0.0129 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 189 on 577 degrees of freedom
## Multiple R-squared: 0.8949, Adjusted R-squared: 0.8938
## F-statistic: 818.6 on 6 and 577 DF, p-value: < 2.2e-16The final variables which we have selected for our Points model are X2PA, X3PA, FTA, AST, ORB and STL.
Forward Selection
## Single term additions
##
## Model:
## PTS ~ 1
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 195977838 7432.6
## X2PA 1 99352922 96624916 7021.6 598.4316 < 2.2e-16 ***
## X3PA 1 52396167 143581671 7252.9 212.3848 < 2.2e-16 ***
## FTA 1 82767605 113210233 7114.1 425.4982 < 2.2e-16 ***
## AST 1 109859692 86118146 6954.4 742.4491 < 2.2e-16 ***
## ORB 1 49087464 146890373 7266.2 194.4913 < 2.2e-16 ***
## DRB 1 1561939 194415898 7429.9 4.6758 0.031 *
## TOV 1 31699839 164277998 7331.5 112.3054 < 2.2e-16 ***
## STL 1 33787416 162190422 7324.1 121.2419 < 2.2e-16 ***
## BLK 1 5546264 190431574 7417.8 16.9506 4.392e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Single term additions
##
## Model:
## PTS ~ X2PA
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 96624916 7021.6
## X3PA 1 24979952 71644963 6848.9 202.5732 < 2.2e-16 ***
## FTA 1 20855031 75769884 6881.6 159.9154 < 2.2e-16 ***
## AST 1 26016833 70608083 6840.4 214.0800 < 2.2e-16 ***
## ORB 1 881394 95743521 7018.3 5.3486 0.021088 *
## DRB 1 6911241 89713675 6980.3 44.7583 5.252e-11 ***
## TOV 1 1330384 95294532 7015.5 8.1112 0.004555 **
## STL 1 1419988 95204927 7015.0 8.6657 0.003372 **
## BLK 1 4080 96620836 7023.6 0.0245 0.875594
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Single term additions
##
## Model:
## PTS ~ X2PA + X3PA
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 71644963 6848.9
## FTA 1 29263874 42381089 6544.3 400.4863 < 2.2e-16 ***
## AST 1 22603750 49041213 6629.6 267.3297 < 2.2e-16 ***
## ORB 1 4585823 67059140 6812.3 39.6632 5.965e-10 ***
## DRB 1 3210553 68434410 6824.1 27.2103 2.544e-07 ***
## TOV 1 222221 71422742 6849.1 1.8046 0.17969
## STL 1 313060 71331903 6848.4 2.5455 0.11115
## BLK 1 597793 71047170 6846.0 4.8801 0.02756 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Single term additions
##
## Model:
## PTS ~ X2PA + X3PA + FTA
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 42381089 6544.3
## AST 1 17354957 25026133 6238.7 401.5211 < 2.2e-16 ***
## ORB 1 10683352 31697737 6376.7 195.1452 < 2.2e-16 ***
## DRB 1 2198003 40183086 6515.2 31.6711 2.847e-08 ***
## TOV 1 401203 41979886 6540.8 5.5335 0.01899 *
## STL 1 146978 42234111 6544.3 2.0150 0.15629
## BLK 1 8314 42372775 6546.2 0.1136 0.73620
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Single term additions
##
## Model:
## PTS ~ X2PA + X3PA + FTA + AST
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 25026133 6238.7
## ORB 1 4202347 20823786 6133.3 116.6434 < 2.2e-16 ***
## DRB 1 458766 24567367 6229.9 10.7935 0.00108 **
## TOV 1 278193 24747939 6234.1 6.4973 0.01106 *
## STL 1 796259 24229874 6221.8 18.9946 1.552e-05 ***
## BLK 1 60493 24965640 6239.3 1.4005 0.23712
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Single term additions
##
## Model:
## PTS ~ X2PA + X3PA + FTA + AST + ORB
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 20823786 6133.3
## DRB 1 28813 20794973 6134.5 0.7995 0.37163
## TOV 1 24081 20799705 6134.6 0.6680 0.41408
## STL 1 222037 20601749 6129.1 6.2187 0.01292 *
## BLK 1 21 20823765 6135.3 0.0006 0.98058
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Single term additions
##
## Model:
## PTS ~ X2PA + X3PA + FTA + AST + ORB + STL
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 20601749 6129.1
## DRB 1 10.5 20601739 6131.1 0.0003 0.9863
## TOV 1 3558.4 20598191 6131.0 0.0995 0.7525
## BLK 1 284.5 20601465 6131.0 0.0080 0.9290The final variables after using the Forward Selection method are same as those we got after using the backward selection method.
Linear Model 2 - Opposition Points Model
Model 2 AIC Selection
## Start: AIC=8511.67
## oppPTS ~ X2PA + X3PA + FTA + AST + ORB + DRB + TOV + STL + BLK
##
## Df Sum of Sq RSS AIC
## - AST 1 26114 21819163 8510.7
## - BLK 1 43569 21836618 8511.3
## <none> 21793049 8511.7
## - STL 1 13185641 34978690 8904.8
## - ORB 1 16184366 37977415 8973.4
## - TOV 1 16586642 38379691 8982.2
## - DRB 1 19228733 41021782 9037.8
## - FTA 1 19834232 41627280 9050.1
## - X3PA 1 78995378 100788426 9788.4
## - X2PA 1 104139976 125933025 9974.4
##
## Step: AIC=8510.67
## oppPTS ~ X2PA + X3PA + FTA + ORB + DRB + TOV + STL + BLK
##
## Df Sum of Sq RSS AIC
## - BLK 1 47213 21866376 8510.5
## <none> 21819163 8510.7
## + AST 1 26114 21793049 8511.7
## - STL 1 14347741 36166903 8930.6
## - TOV 1 16622373 38441536 8981.6
## - ORB 1 17211444 39030606 8994.3
## - FTA 1 19996810 41815973 9051.8
## - DRB 1 20066509 41885671 9053.2
## - X3PA 1 79951060 101770223 9794.5
## - X2PA 1 124299917 146119080 10096.5
##
## Step: AIC=8510.48
## oppPTS ~ X2PA + X3PA + FTA + ORB + DRB + TOV + STL
##
## Df Sum of Sq RSS AIC
## <none> 21866376 8510.5
## + BLK 1 47213 21819163 8510.7
## + AST 1 29757 21836618 8511.3
## - STL 1 14717721 36584097 8938.2
## - TOV 1 16586932 38453307 8979.8
## - ORB 1 18095365 39961741 9012.0
## - FTA 1 19985322 41851697 9050.5
## - DRB 1 24139548 46005923 9129.6
## - X3PA 1 84414035 106280410 9828.7
## - X2PA 1 130175808 152042184 10127.7##
## Call:
## lm(formula = oppPTS ~ X2PA + X3PA + FTA + ORB + DRB + TOV + STL,
## data = NBA)
##
## Coefficients:
## (Intercept) X2PA X3PA FTA ORB DRB
## 140.6485 1.6217 1.8463 0.8056 -1.6858 -1.4910
## TOV STL
## 1.3384 -1.8285## Start: AIC=7453.49
## oppPTS ~ 1
##
## Df Sum of Sq RSS AIC
## + X2PA 1 115220295 87897909 6966.3
## + TOV 1 63949743 139168461 7234.7
## + X3PA 1 62546369 140571835 7240.5
## + ORB 1 61752270 141365934 7243.8
## + FTA 1 57186574 145931630 7262.4
## + AST 1 55881080 147237124 7267.6
## + STL 1 20136861 182981343 7394.5
## + DRB 1 7808437 195309767 7432.6
## <none> 203118204 7453.5
## + BLK 1 371537 202746667 7454.4
##
## Step: AIC=6966.32
## oppPTS ~ X2PA
##
## Df Sum of Sq RSS AIC
## + X3PA 1 25913903 61984007 6764.3
## + FTA 1 5279126 82618783 6932.2
## + BLK 1 3651602 84246307 6943.5
## + DRB 1 1816935 86080974 6956.1
## + TOV 1 1715315 86182594 6956.8
## + STL 1 539682 87358227 6964.7
## <none> 87897909 6966.3
## + ORB 1 276437 87621472 6966.5
## + AST 1 114508 87783401 6967.6
## - X2PA 1 115220295 203118204 7453.5
##
## Step: AIC=6764.33
## oppPTS ~ X2PA + X3PA
##
## Df Sum of Sq RSS AIC
## + FTA 1 9779066 52204941 6666.1
## + TOV 1 9626349 52357657 6667.8
## + DRB 1 5117153 56866854 6716.0
## + ORB 1 3023781 58960226 6737.1
## + STL 1 1944510 60039497 6747.7
## + BLK 1 1459852 60524154 6752.4
## <none> 61984007 6764.3
## + AST 1 724 61983283 6766.3
## - X3PA 1 25913903 87897909 6966.3
## - X2PA 1 78587828 140571835 7240.5
##
## Step: AIC=6666.06
## oppPTS ~ X2PA + X3PA + FTA
##
## Df Sum of Sq RSS AIC
## + TOV 1 6367732 45837208 6592.1
## + DRB 1 5984683 46220258 6597.0
## + ORB 1 5761960 46442980 6599.8
## + STL 1 3864721 48340220 6623.1
## + BLK 1 2610890 49594050 6638.1
## <none> 52204941 6666.1
## + AST 1 148368 52056572 6666.4
## - FTA 1 9779066 61984007 6764.3
## - X3PA 1 30413843 82618783 6932.2
## - X2PA 1 72690013 124894954 7173.5
##
## Step: AIC=6592.09
## oppPTS ~ X2PA + X3PA + FTA + TOV
##
## Df Sum of Sq RSS AIC
## + ORB 1 8447416 37389792 6475.1
## + STL 1 6863832 38973377 6499.4
## + DRB 1 4668287 41168921 6531.4
## + BLK 1 3794236 42042972 6543.6
## <none> 45837208 6592.1
## + AST 1 103041 45734168 6592.8
## - TOV 1 6367732 52204941 6666.1
## - FTA 1 6520449 52357657 6667.8
## - X3PA 1 36166879 82004088 6929.8
## - X2PA 1 71652642 117489850 7139.8
##
## Step: AIC=6475.13
## oppPTS ~ X2PA + X3PA + FTA + TOV + ORB
##
## Df Sum of Sq RSS AIC
## + DRB 1 9601575 27788217 6303.8
## + STL 1 5194250 32195542 6389.8
## + BLK 1 3113154 34276638 6426.4
## + AST 1 1732849 35656943 6449.4
## <none> 37389792 6475.1
## - ORB 1 8447416 45837208 6592.1
## - FTA 1 8918978 46308770 6598.1
## - TOV 1 9053188 46442980 6599.8
## - X3PA 1 43685104 81074896 6925.1
## - X2PA 1 74321780 111711572 7112.3
##
## Step: AIC=6303.81
## oppPTS ~ X2PA + X3PA + FTA + TOV + ORB + DRB
##
## Df Sum of Sq RSS AIC
## + STL 1 11854495 15933722 5981.0
## + AST 1 927997 26860220 6286.0
## + BLK 1 385905 27402312 6297.6
## <none> 27788217 6303.8
## - TOV 1 7752957 35541174 6445.5
## - DRB 1 9601575 37389792 6475.1
## - FTA 1 11301026 39089243 6501.1
## - ORB 1 13380704 41168921 6531.4
## - X3PA 1 50859832 78648049 6909.4
## - X2PA 1 83172186 110960403 7110.4
##
## Step: AIC=5981
## oppPTS ~ X2PA + X3PA + FTA + TOV + ORB + DRB + STL
##
## Df Sum of Sq RSS AIC
## + BLK 1 64719 15869002 5980.6
## <none> 15933722 5981.0
## + AST 1 30573 15903148 5981.9
## - TOV 1 11605948 27539670 6298.6
## - STL 1 11854495 27788217 6303.8
## - ORB 1 12578729 28512450 6318.8
## - FTA 1 14857565 30791287 6363.7
## - DRB 1 16261820 32195542 6389.8
## - X3PA 1 60276315 76210037 6893.0
## - X2PA 1 94355627 110289349 7108.9
##
## Step: AIC=5980.63
## oppPTS ~ X2PA + X3PA + FTA + TOV + ORB + DRB + STL + BLK
##
## Df Sum of Sq RSS AIC
## <none> 15869002 5980.6
## - BLK 1 64719 15933722 5981.0
## + AST 1 26387 15842615 5981.7
## - STL 1 11533309 27402312 6297.6
## - TOV 1 11638243 27507245 6299.9
## - ORB 1 11864146 27733148 6304.7
## - DRB 1 12976522 28845524 6327.6
## - FTA 1 14896711 30765713 6365.3
## - X3PA 1 56808807 72677809 6867.3
## - X2PA 1 89337007 105206009 7083.3##
## Call:
## lm(formula = oppPTS ~ X2PA + X3PA + FTA + TOV + ORB + DRB + STL +
## BLK, data = Nba.train)
##
## Coefficients:
## (Intercept) X2PA X3PA FTA TOV ORB
## -53.3581 1.6314 1.8659 0.8314 1.3548 -1.6728
## DRB STL BLK
## -1.4280 -1.9241 -0.1484##
## Call:
## lm(formula = oppPTS ~ X2PA + X3PA + FTA + ORB + DRB + TOV + STL,
## data = Nba.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -553.86 -117.29 15.89 119.24 459.78
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -31.56190 222.71769 -0.142 0.887
## X2PA 1.64047 0.02809 58.403 <2e-16 ***
## X3PA 1.87910 0.04026 46.679 <2e-16 ***
## FTA 0.83009 0.03582 23.175 <2e-16 ***
## ORB -1.69486 0.07948 -21.324 <2e-16 ***
## DRB -1.46788 0.06054 -24.246 <2e-16 ***
## TOV 1.34200 0.06552 20.483 <2e-16 ***
## STL -1.93953 0.09369 -20.701 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 166.3 on 576 degrees of freedom
## Multiple R-squared: 0.9216, Adjusted R-squared: 0.9206
## F-statistic: 966.7 on 7 and 576 DF, p-value: < 2.2e-16Using the AIC selection technique,the final variables which we have selected for our model are X2PA, X3PA, FTA,ORB, DRB, TOV, STL and BLK
Model Checking
QQ plot for Pts model
Model Validation- Points Model
## Analysis of Variance Table
##
## Response: PTS
## Df Sum Sq Mean Sq F value Pr(>F)
## X2PA 1 1.41e+08 1.41e+08 4115.97 <2e-16 ***
## X3PA 1 3.67e+07 3.67e+07 1070.41 <2e-16 ***
## FTA 1 4.48e+07 4.48e+07 1304.62 <2e-16 ***
## AST 1 2.34e+07 2.34e+07 681.59 <2e-16 ***
## ORB 1 6.69e+06 6.69e+06 194.83 <2e-16 ***
## STL 1 2.53e+05 2.53e+05 7.38 0.0067 **
## Residuals 828 2.84e+07 3.43e+04
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Warning in cv.lm(data = NBA1, PointsReg4, m = 3, seed = 123):
##
## As there is >1 explanatory variable, cross-validation
## predicted values for a fold are not a linear function
## of corresponding overall predicted values. Lines that
## are shown for the different folds are approximate
##
## fold 1
## Observations in test set: 278
## 5 15 16 19 23 24 25 30 35 38 39 45 46
## Predicted 9130 8891 9185 8488 8414 8821 8494 8789 9116 8996 8992.65 8890 8170
## cvpred 9134 8903 9214 8485 8397 8826 8481 8775 9145 9019 9014.38 8903 8145
## PTS 8878 8949 9114 8820 8604 9008 8937 9006 9276 9156 9019.00 8662 8281
## CV residual -256 46 -100 335 207 182 456 231 131 137 4.62 -241 136
## 48 50 51 53 56 59 61 62 63 64 66
## Predicted 8535 8832 10145 8654 8677.0 8732.28 8919 8644.5 8788 9297.5 8837.8
## cvpred 8513 8823 10136 8645 8688.6 8754.53 8937 8663.1 8787 9303.7 8838.9
## PTS 8743 8575 10371 9092 8785.0 8746.00 9119 8705.0 9006 9272.0 8795.0
## CV residual 230 -248 235 447 96.4 -8.53 182 41.9 219 -31.7 -43.9
## 67 72 74 80 81 89 90 91 92 94 96
## Predicted 8787 7835 10146.9 9009 8749.01 9159 9164 8334 8259.2 8735 8983.3
## cvpred 8784 7823 10165.1 9055 8749.78 9163 9175 8346 8235.2 8727 8978.2
## PTS 9094 7964 10105.0 9433 8740.00 9019 8938 8134 8321.0 8501 9052.0
## CV residual 310 141 -60.1 378 -9.78 -144 -237 -212 85.8 -226 73.8
## 97 98 104 105 110 115 124 126 133 134 137
## Predicted 10378 9631 8616.8 8978.6 9812.1 8670.2 8947.2 8437 9381 9260 8758
## cvpred 10398 9640 8612.7 8990.6 9823.6 8664.8 8931.4 8421 9392 9271 8780
## PTS 10147 9602 8666.0 9019.0 9862.0 8743.0 8879.0 8784 9469 9412 8655
## CV residual -251 -38 53.3 28.4 38.4 78.2 -52.4 363 77 141 -125
## 142 144 147 148 150 152 155 157 160 162 164
## Predicted 9367 9387.6 8627 9028 9349 8380 9447.4 9099.06 8512.9 9038 8721
## cvpred 9357 9410.9 8633 9014 9357 8378 9456.3 9111.93 8499.5 9055 8708
## PTS 9453 9363.0 8519 8907 9390 8094 9436.0 9120.00 8442.0 9237 8558
## CV residual 96 -47.9 -114 -107 33 -284 -20.3 8.07 -57.5 182 -150
## 165 168 173 174 175 178 184 186 189 190 195
## Predicted 9366 9204.5 9148 8927.3 8572.8 9727.1 8730 8499 9753 8828 9188.6
## cvpred 9362 9207.7 9164 8914.7 8570.2 9731.4 8722 8507 9795 8806 9199.3
## PTS 9567 9188.0 9052 8893.0 8508.0 9667.0 8844 8609 9573 8957 9250.0
## CV residual 205 -19.7 -112 -21.7 -62.2 -64.4 122 102 -222 151 50.7
## 197 200 201 202 204 206 210 212 213 214
## Predicted 8252.7 9028 9595.8 8938.7 9129.0 8621.3 8586 8439.3 9942 8519
## cvpred 8247.3 9048 9604.4 8937.8 9147.7 8608.3 8608 8429.5 9984 8501
## PTS 8235.0 8901 9518.0 8855.0 9135.0 8653.0 8726 8484.0 9675 8740
## CV residual -12.3 -147 -86.4 -82.8 -12.7 44.7 118 54.5 -309 239
## 216 218 221 222 224 228 232 235 238 242 250 263
## Predicted 8920.5 8972 8854.0 8473.4 9099.7 8810 8511 8671 9713 8675 9388 8586
## cvpred 8927.7 8998 8872.3 8479.5 9105.1 8838 8498 8702 9757 8668 9373 8597
## PTS 8897.0 8712 8932.0 8506.0 9174.0 8652 8901 8977 9397 8962 9090 8343
## CV residual -30.7 -286 59.7 26.5 68.9 -186 403 275 -360 294 -283 -254
## 264 270 271 272 273 274 276 281 282 284 285 288
## Predicted 8101.6 8811 8600 8487 8546 8310 8434.0 8253 8790.1 8697 8521 9148
## cvpred 8097.8 8825 8607 8492 8577 8317 8443.3 8256 8787.7 8695 8525 9184
## PTS 8195.0 8491 8717 8349 8727 8169 8455.0 7928 8782.0 8527 8313 8980
## CV residual 97.2 -334 110 -143 150 -148 11.7 -328 -5.7 -168 -212 -204
## 291 292 311 313 318 321 322 324 326 332 335
## Predicted 8020.81 8029 8798.5 8758.3 8381 8949.4 8408 8873 8417.2 8806 9090
## cvpred 8008.89 8029 8802.9 8790.4 8379 8955.5 8430 8909 8398.8 8819 9107
## PTS 8007.00 8176 8877.0 8814.0 8141 9014.0 8531 8783 8495.0 8556 8847
## CV residual -1.89 147 74.1 23.6 -238 58.5 101 -126 96.2 -263 -260
## 342 343 344 345 346 349 351 352 354 355 357
## Predicted 8843 7952 8355.5 7787.20 8287.5 8342.5 8725 8409 8251 8196 8126.4
## cvpred 8873 7978 8365.5 7797.46 8273.8 8373.9 8760 8436 8272 8185 8147.9
## PTS 8732 8033 8296.0 7801.00 8221.0 8292.0 8447 8233 7949 7930 8076.0
## CV residual -141 55 -69.5 3.54 -52.8 -81.9 -313 -203 -323 -255 -71.9
## 359 360 365 368 374 375 377 379 381 385 386
## Predicted 8205 8878.6 8573 8252 8001.1 8813 7926 8657.1 8085 9059.5 7646
## cvpred 8210 8907.5 8584 8245 8023.8 8828 7921 8694.8 8086 9081.8 7634
## PTS 8033 8876.0 8354 8428 8053.0 8667 8136 8616.0 8146 9091.0 7820
## CV residual -177 -31.5 -230 183 29.2 -161 215 -78.8 60 9.2 186
## 388 391 395 400 402 403 408 412 414 416
## Predicted 8425.4 8857 8576.0 8159 8282.7 8423.2 7654 8578.6 8633.3 8159.3
## cvpred 8435.3 8889 8583.4 8151 8289.1 8446.4 7653 8601.2 8629.3 8152.7
## PTS 8451.0 9055 8495.0 8013 8334.0 8404.0 7837 8571.0 8552.0 8163.0
## CV residual 15.7 166 -88.4 -138 44.9 -42.4 184 -30.2 -77.3 10.3
## 419 433 435 437 442 443 446 449 450 461 464
## Predicted 7857 7838.79 8310 7519 8341 8515 7371.9 8338 7490 8049.2 8805
## cvpred 7880 7822.17 8323 7506 8337 8538 7372.7 8349 7510 8064.4 8808
## PTS 7994 7819.00 8200 7818 8215 8431 7418.0 8454 7313 8099.0 8652
## CV residual 114 -3.17 -123 312 -122 -107 45.3 105 -197 34.6 -156
## 469 474 478 481 482 485 488 489 490 492
## Predicted 7488.5 7958 8270.6 7681.3 8211.0 8307 8504.0 8335 8199.8 7655
## cvpred 7497.7 7953 8254.1 7659.9 8228.4 8317 8494.5 8351 8207.1 7652
## PTS 7509.0 7637 8279.0 7735.0 8146.0 7950 8483.0 7834 8156.0 7546
## CV residual 11.3 -316 24.9 75.1 -82.4 -367 -11.5 -517 -51.1 -106
## 494 502 505 506 508 516 522 523 527 529 530 531
## Predicted 7766.2 7883 8334 8295 7646.0 8332 8393 7472 7144 7823.8 8093 7737.7
## cvpred 7778.3 7891 8358 8311 7636.7 8347 8388 7476 7135 7813.2 8118 7753.9
## PTS 7739.0 7991 8125 7968 7702.0 7918 8251 7289 7275 7763.0 7710 7824.0
## CV residual -39.3 100 -233 -343 65.3 -429 -137 -187 140 -50.2 -408 70.1
## 532 533 534 537 539 540 543 545 546 552 554 556
## Predicted 8429 7996.1 7918 7860 7752.7 8160 7946 7834 7753.4 7677 7904.4 8133
## cvpred 8458 7979.3 7927 7866 7758.6 8196 7949 7851 7764.6 7685 7924.9 8161
## PTS 8343 7886.0 7978 7522 7711.0 7901 7812 7559 7735.0 7372 7996.0 7889
## CV residual -115 -93.3 51 -344 -47.6 -295 -137 -292 -29.6 -313 71.1 -272
## 559 560 561 567 568 573 580 583 585 587
## Predicted 7487.7 7766.76 7903.45 7609.67 7684.5 7406 8130 8133.6 7777.6 8164
## cvpred 7479.5 7792.34 7927.78 7604.28 7680.9 7414 8148 8145.9 7788.1 8178
## PTS 7461.0 7802.00 7925.00 7609.00 7714.0 6901 7995 8046.0 7699.0 8078
## CV residual -18.5 9.66 -2.78 4.72 33.1 -513 -153 -99.9 -89.1 -100
## 589 592 597 600 601 602 606 610 611 619 622 623
## Predicted 7866.0 7732 7680.3 7906 8425 8092 7633 7507 8278 7369.7 7876.9 7234
## cvpred 7875.5 7722 7680.7 7901 8457 8101 7645 7504 8274 7378.5 7906.3 7239
## PTS 7834.0 7856 7611.0 7619 8626 7972 7493 7402 8039 7440.0 7964.0 7006
## CV residual -41.5 134 -69.7 -282 169 -129 -152 -102 -235 61.5 57.7 -233
## 629 635 638 640 641 642 643 644 645 647 649
## Predicted 7960 7845.6 8109.7 8321 8043.6 7911.7 7868 7622 7891.3 8299 7613.26
## cvpred 7961 7847.7 8105.8 8301 8024.6 7912.9 7872 7624 7890.8 8314 7611.09
## PTS 7745 7796.0 8095.0 8327 7973.0 7934.0 7496 7252 7977.0 8128 7621.00
## CV residual -216 -51.7 -10.8 26 -51.6 21.1 -376 -372 86.2 -186 9.91
## 652 653 654 658 660 662 667 668 670 672 674
## Predicted 7781 8219.9 7794 8272 7962.6 8488 7924.6 8159 8114 7578.1 7683.5
## cvpred 7772 8226.8 7772 8291 7945.1 8495 7906.2 8164 8098 7571.1 7663.1
## PTS 8113 8178.0 7625 7943 8002.0 8227 7970.0 8154 8191 7522.0 7611.0
## CV residual 341 -48.8 -147 -348 56.9 -268 63.8 -10 93 -49.1 -52.1
## 675 677 681 682 686 689 691 696 697 700
## Predicted 7768 8167.37 7635 8128 7681.2 8111.3 7776 7954 7898.3 7757.94
## cvpred 7723 8155.76 7634 8135 7663.7 8118.8 7770 7951 7883.8 7756.03
## PTS 7842 8147.00 7837 8411 7680.0 8100.0 8201 7840 7843.0 7759.00
## CV residual 119 -8.76 203 276 16.3 -18.8 431 -111 -40.8 2.97
## 702 703 707 708 711 720 722 732 740 743 753 763
## Predicted 7894.3 8295 7759.0 8705 7929 7929 9288 7684 8156 8060 7643.3 7947.5
## cvpred 7895.1 8298 7743.2 8742 7937 7928 9312 7695 8149 8086 7634.7 7950.6
## PTS 7877.0 8001 7785.0 9037 8079 7903 9074 7839 8408 8215 7727.0 8044.0
## CV residual -18.1 -297 41.8 295 142 -25 -238 144 259 129 92.3 93.4
## 765 767 770 771 772 776 779 784 786 792
## Predicted 8582.2 8210.7 7697 8208.1 7765 8101 8063.4 8870.1 8470 8042.79
## cvpred 8613.3 8210.1 7696 8206.6 7774 8114 8058.5 8904.1 8485 8044.94
## PTS 8627.0 8286.0 8153 8248.0 7958 8338 7993.0 8922.0 8263 8051.00
## CV residual 13.7 75.9 457 41.4 184 224 -65.5 17.9 -222 6.06
## 795 796 802 805 809 815 819 823 824 825 827 830
## Predicted 8275.2 8066 8168 7742 8009.5 8718.0 7930 7897 7675 8588 8024 7722
## cvpred 8305.5 8062 8173 7730 8017.2 8740.1 7944 7891 7679 8610 8027 7740
## PTS 8373.0 8322 8312 7892 8087.0 8685.0 8195 7722 7784 8734 8135 7896
## CV residual 67.5 260 139 162 69.8 -55.1 251 -169 105 124 108 156
##
## Sum of squares = 9025347 Mean square = 32465 n = 278
##
## fold 2
## Observations in test set: 279
## 1 2 4 7 9 10 17 18 20 21 26
## Predicted 8532.3 9152 9363.6 8335 9082.7 8926.3 8153 9756.0 8793 8047 8578
## cvpred 8548.8 9127 9306.7 8299 9040.6 8884.7 8146 9715.1 8769 8038 8545
## PTS 8573.0 9303 9360.0 8493 9119.0 8860.0 8402 9788.0 8897 8394 8670
## CV residual 24.2 176 53.3 194 78.4 -24.7 256 72.9 128 356 125
## 27 31 33 34 36 41 42 52 57 58 65 71
## Predicted 8637 8721 8844.1 8902 8954 9027 8788.6 9052.8 9064 8669 8560 9256
## cvpred 8658 8704 8823.3 8865 8930 9036 8764.1 9045.4 9019 8663 8560 9258
## PTS 8322 8878 8769.0 9117 8768 9209 8737.0 9112.0 9400 8890 8896 9102
## CV residual -336 174 -54.3 252 -162 173 -27.1 66.6 381 227 336 -156
## 73 76 82 83 85 88 93 95 100 103 106 109
## Predicted 9301.9 8666 8555 8297 8793.51 8826 8932 8442.5 8970 9141 8583 8963
## cvpred 9299.6 8628 8534 8304 8772.37 8790 8925 8436.7 8946 9119 8585 8959
## PTS 9243.0 8908 8672 8198 8776.00 8903 9194 8386.0 9071 9478 8763 9277
## CV residual -56.6 280 138 -106 3.63 113 269 -50.7 125 359 178 318
## 112 114 117 118 123 125 131 135 141 146 151
## Predicted 9012 8584 8687 9131 8913 9378.0 8932 8663 8909.7 9442.9 8950.0
## cvpred 9013 8591 8703 9126 8899 9375.8 8949 8665 8898.3 9414.3 8933.2
## PTS 8865 8423 8916 8903 9118 9413.0 9261 8376 8836.0 9379.0 8949.0
## CV residual -148 -168 213 -223 219 37.2 312 -289 -62.3 -35.3 15.8
## 156 158 161 167 172 179 180 181 187 188 191
## Predicted 9018.2 8545.6 8930.9 8974 9443 9190 9012 9259.0 8567.0 8774 8797.98
## cvpred 9015.3 8537.7 8944.9 8982 9430 9197 8984 9252.5 8571.8 8785 8766.95
## PTS 8924.0 8564.0 9024.0 9118 9656 9095 8882 9325.0 8566.0 8960 8771.00
## CV residual -91.3 26.3 79.1 136 226 -102 -102 72.5 -5.8 175 4.05
## 192 193 198 207 208 211 215 217 226 230 236 239
## Predicted 8904.1 8387 8642.3 8947 8835 8892.4 9478 8583 9448 8754 8590 8458.1
## cvpred 8894.2 8369 8626.3 8948 8835 8882.8 9421 8590 9406 8780 8586 8474.7
## PTS 8935.0 8581 8655.0 9102 8958 8923.0 9558 8767 9395 8588 8411 8556.0
## CV residual 40.8 212 28.7 154 123 40.2 137 177 -11 -192 -175 81.3
## 243 244 246 249 252 253 257 259 262 266
## Predicted 8536.0 9005.5 8733 8799.8 9342.2 9461.8 8713.7 8893 8583 8112.4
## cvpred 8535.1 9003.4 8705 8792.4 9341.6 9450.3 8727.4 8886 8555 8138.5
## PTS 8509.0 9079.0 8691 8879.0 9423.0 9365.0 8760.0 9003 9024 8205.0
## CV residual -26.1 75.6 -14 86.6 81.4 -85.3 32.6 117 469 66.5
## 268 269 277 279 280 283 286 287 289 290
## Predicted 8590 9069.1 8781.9 9347.39 9434.49 8379 8755.96 8394 8728 8902
## cvpred 8562 9065.5 8782.7 9346.12 9415.88 8354 8717.53 8371 8714 8895
## PTS 8753 9159.0 8684.0 9348.00 9407.00 8744 8711.00 8745 9011 8926
## CV residual 191 93.5 -98.7 1.88 -8.88 390 -6.53 374 297 31
## 293 295 297 300 302 304 306 307 308 309
## Predicted 8055.5 8330.1 8549.0 8473 8421 8421.9 9125.8 9076.9 8650.2 8457
## cvpred 8072.3 8319.4 8526.3 8441 8395 8418.8 9111.4 9059.9 8622.6 8450
## PTS 8113.0 8366.0 8440.0 8609 8641 8330.0 9194.0 9135.0 8549.0 8524
## CV residual 40.7 46.6 -86.3 168 246 -88.8 82.6 75.1 -73.6 74
## 310 312 315 319 320 325 327 331 333 334
## Predicted 8390 8541 9041.70 8535.2 8280 8584.1 8424.7 8619.46 9093 9121
## cvpred 8379 8504 9031.45 8531.9 8272 8569.8 8393.1 8644.08 9082 9103
## PTS 8737 8395 9030.00 8626.0 8252 8546.0 8392.0 8652.00 9298 8898
## CV residual 358 -109 -1.45 94.1 -20 -23.8 -1.1 7.92 216 -205
## 336 337 338 340 341 358 361 362 366 367
## Predicted 8626.4 8435 8787.6 8278 8291.00 8629.9 9040 8650 8154.7 7847.5
## cvpred 8633.4 8404 8789.5 8248 8271.31 8632.6 9020 8644 8150.3 7835.6
## PTS 8652.0 8884 8709.0 8318 8267.00 8666.0 8795 8291 8229.0 7921.0
## CV residual 18.6 480 -80.5 70 -4.31 33.4 -225 -353 78.7 85.4
## 371 378 380 383 384 387 390 393 394 399 401 404
## Predicted 7324.0 8166 8077 8298 8173 9083.57 8604 8152 7692 8633 7638 8010.7
## cvpred 7348.2 8144 8078 8322 8186 9067.92 8616 8138 7685 8605 7676 8054.3
## PTS 7417.0 7927 8293 8042 8054 9073.00 8742 8242 8059 8409 7822 8144.0
## CV residual 68.8 -217 215 -280 -132 5.08 126 104 374 -196 146 89.7
## 405 406 407 410 411 415 421 423 426 427 428 429
## Predicted 7831 8336 7876.20 7850 7822 8207.6 7697 7432 8170 7011 7513.3 8211
## cvpred 7822 8322 7900.58 7877 7821 8226.7 7693 7438 8152 7031 7519.6 8215
## PTS 8153 8438 7909.00 7684 7971 8145.0 7362 7774 8458 7173 7431.0 8020
## CV residual 331 116 8.42 -193 150 -81.7 -331 336 306 142 -88.6 -195
## 431 436 440 444 448 451 454 455 458 459
## Predicted 8046 7670.9 7699.4 7960 7809.8 7994 7902.29 7952.8 7530 7695.15
## cvpred 8061 7682.8 7722.6 7978 7805.7 7985 7919.01 7953.7 7549 7719.17
## PTS 8171 7776.0 7819.0 8114 7829.0 8147 7923.00 7931.0 7300 7721.00
## CV residual 110 93.2 96.4 136 23.3 162 3.99 -22.7 -249 1.83
## 460 463 468 470 471 473 475 479 484 486 487 491
## Predicted 7627 7833.6 8112.1 7567 7803 7805 7654 8030 7351 8203 8447 8237.1
## cvpred 7633 7832.9 8116.6 7593 7828 7836 7696 8046 7380 8170 8445 8230.8
## PTS 7237 7865.0 8170.0 7387 7651 7734 7587 7923 6952 8316 8115 8306.0
## CV residual -396 32.1 53.4 -206 -177 -102 -109 -123 -428 146 -330 75.2
## 500 501 507 510 511 512 513 514 515 517 520 526 528
## Predicted 8052 8234 8039 7457.86 8079 7963 7615 7655 8120 8185 7784 7897 8223
## cvpred 8049 8213 8056 7453.62 8069 7974 7617 7666 8103 8188 7797 7900 8210
## PTS 7771 8111 7914 7459.00 7759 7539 7181 7561 8239 7837 7591 7552 7992
## CV residual -278 -102 -142 5.38 -310 -435 -436 -105 136 -351 -206 -348 -218
## 536 542 547 550 553 555 564 566 569 572 574 578 586
## Predicted 8137 7754 8238 7860 7270 8029 7765 8113 7939 8284 7621 7812 7855.0
## cvpred 8160 7758 8250 7876 7278 8022 7741 8138 7919 8245 7639 7832 7828.9
## PTS 7959 7335 8009 7846 7150 8145 8014 7871 7599 8444 7492 7693 7860.0
## CV residual -201 -423 -241 -30 -128 123 273 -267 -320 199 -147 -139 31.1
## 588 595 596 599 603 605 607 612 613 616 617 627
## Predicted 8021.2 7970 7601 7828 7525 7236 7913 7731.0 7785 7912 7418 8272.8
## cvpred 8014.2 8003 7616 7815 7545 7248 7940 7726.8 7778 7913 7435 8279.8
## PTS 7941.0 7762 7502 7355 7388 7362 7771 7753.0 7401 7711 7215 8304.0
## CV residual -73.2 -241 -114 -460 -157 114 -169 26.2 -377 -202 -220 24.2
## 628 630 631 632 636 637 639 648 651 655 656 657
## Predicted 8109 8049 8190 8357 7620.4 7964 7666.62 8773 7913 8349 7802 7978.9
## cvpred 8101 8058 8189 8356 7642.9 7993 7670.83 8750 7926 8353 7825 7994.9
## PTS 7729 7914 8405 8161 7626.0 7849 7661.00 9054 7888 8241 7972 8033.0
## CV residual -372 -144 216 -195 -16.9 -144 -9.83 304 -38 -112 147 38.1
## 659 664 666 669 671 676 678 683 684 685 694
## Predicted 8193 8382 7768.7 7552.2 8074.6 7525 8676 8108 7883 8276.1 8834.8
## cvpred 8189 8375 7787.2 7571.9 8079.1 7566 8615 8108 7938 8288.9 8792.9
## PTS 8020 8076 7699.0 7558.0 8020.0 7784 8886 8287 7573 8336.0 8737.0
## CV residual -169 -299 -88.2 -13.9 -59.1 218 271 179 -365 47.1 -55.9
## 698 699 704 705 712 714 715 718 719 721 724 733
## Predicted 8497 8279.0 7692 7880.0 7863 8436 8580.3 8100 8191 8017 8977 8283
## cvpred 8493 8298.9 7699 7917.1 7854 8467 8579.5 8100 8181 8030 8918 8303
## PTS 8474 8331.0 7833 7994.0 8130 8322 8556.0 7959 7981 8234 9105 7857
## CV residual -19 32.1 134 76.9 276 -145 -23.5 -141 -200 204 187 -446
## 734 736 737 742 744 748 757 758 759 760 762 773
## Predicted 7917 8427 7854.2 8150 8567 7646.9 7942 8531 7481 7949 8042 8089.5
## cvpred 7890 8435 7856.1 8134 8566 7660.2 7929 8504 7495 7927 8040 8079.4
## PTS 8270 8567 7923.0 7999 8710 7677.0 7799 8768 7698 8061 8019 8121.0
## CV residual 380 132 66.9 -135 144 16.8 -130 264 203 134 -21 41.6
## 777 778 780 781 787 788 790 791 793 794 797 799
## Predicted 8109.6 7836.6 8220 8256 7912.1 8222 7691 8081.8 7749 8027 8163 8579
## cvpred 8103.8 7850.1 8228 8232 7911.6 8205 7688 8048.2 7753 7992 8173 8578
## PTS 8136.0 7813.0 8373 8364 7849.0 8339 7914 8009.0 7575 8220 8426 9039
## CV residual 32.2 -37.1 145 132 -62.6 134 226 -39.2 -178 228 253 461
## 800 803 806 807 810 814 816 817 818 820 826 828
## Predicted 7823 8340 7729.7 7814 8188 8211 8164.6 8243 8125 8055 8384 8131.30
## cvpred 7837 8345 7727.6 7806 8187 8160 8152.4 8249 8116 8076 8388 8109.08
## PTS 8045 8534 7790.0 7913 7827 8477 8183.0 8089 8321 8369 8596 8119.00
## CV residual 208 189 62.4 107 -360 317 30.6 -160 205 293 208 9.92
## 829 831 833 834 835
## Predicted 8557.9 8119.8 8055.5 8179 7993.237
## cvpred 8534.8 8105.9 8050.9 8176 7976.881
## PTS 8611.0 8151.0 8124.0 8153 7977.000
## CV residual 76.2 45.1 73.1 -23 0.119
##
## Sum of squares = 10172554 Mean square = 36461 n = 279
##
## fold 3
## Observations in test set: 278
## 3 6 8 11 12 13 14 22 28 29 32
## Predicted 8906 8775 8918 8995 8935.4 9091 9237.0 9050 10146 8193.1 8853.2
## cvpred 8905 8801 8921 9001 8944.7 9109 9254.4 9066 10187 8202.6 8855.5
## PTS 8813 8933 9084 9438 9025.0 8879 9344.0 8773 9986 8174.0 8827.0
## CV residual -92 132 163 437 80.3 -230 89.6 -293 -201 -28.6 -28.5
## 37 40 43 44 47 49 54 55 60 68 69
## Predicted 8779.7 8989.9 8622 8163 8999 8513.8 8491 8415.7 8542 8515 8380.4
## cvpred 8797.6 9000.8 8635 8172 9007 8534.5 8495 8429.6 8547 8531 8391.4
## PTS 8849.0 9080.0 8531 8301 9180 8463.0 8680 8379.0 8707 8485 8335.0
## CV residual 51.4 79.2 -104 129 173 -71.5 185 -50.6 160 -46 -56.4
## 70 75 77 78 79 84 86 87 99 101 102
## Predicted 9129.3 9368 8420 8755 9243.7 8859 8796.48 9180 8922.8 8650.1 9117
## cvpred 9141.2 9385 8439 8772 9257.7 8863 8804.88 9183 8944.5 8660.6 9123
## PTS 9191.0 9239 8145 8911 9328.0 9191 8808.00 9375 9008.0 8566.0 9023
## CV residual 49.8 -146 -294 139 70.3 328 3.12 192 63.5 -94.6 -100
## 107 108 111 113 116 119 120 121 122 127 128
## Predicted 8688 8869 8777 9450.4 9126 9044 9802.8 9481.85 8929.3 9260 8982
## cvpred 8685 8877 8787 9460.9 9133 9055 9829.9 9500.65 8957.3 9255 8984
## PTS 8838 9101 9077 9428.0 9412 9116 9841.0 9508.00 9052.0 9696 9090
## CV residual 153 224 290 -32.9 279 61 11.1 7.35 94.7 441 106
## 129 130 132 136 138 139 140 143 145 149 153 154
## Predicted 8901 8634.1 9017 9136 8848.0 9215 8840 9542 9196.0 9326 8900 9211
## cvpred 8912 8645.1 9020 9151 8849.4 9220 8846 9575 9217.8 9327 8899 9214
## PTS 8975 8627.0 8858 8937 8906.0 9359 8962 9410 9299.0 9618 9051 9023
## CV residual 63 -18.1 -162 -214 56.6 139 116 -165 81.2 291 152 -191
## 159 163 166 169 170 171 176 177 182 183 185 194 196
## Predicted 9229 8621.9 9924 8990 8841 8960 8529 9237 9287 8748 9096 8541 8865
## cvpred 9237 8618.6 9949 8992 8851 8975 8526 9238 9300 8772 9085 8554 8865
## PTS 8871 8596.0 9569 8765 8698 8566 8729 9111 8844 8690 9315 8103 8697
## CV residual -366 -22.6 -380 -227 -153 -409 203 -127 -456 -82 230 -451 -168
## 199 203 205 209 219 220 223 225 227 229 231 233
## Predicted 8620.3 9476 9139 8982 9241 8172 9240 9722.4 8738 8967 9045 8981.5
## cvpred 8615.4 9487 9134 8987 9240 8169 9249 9741.9 8752 8969 9066 8975.2
## PTS 8667.0 9314 8899 8566 9406 8016 9567 9727.0 8651 9196 8879 9023.0
## CV residual 51.6 -173 -235 -421 166 -153 318 -14.9 -101 227 -187 47.8
## 234 237 240 241 245 247 248 251 254 255 256 258
## Predicted 8597 8209 9619 8824.7 8394 7956 8322 8774 8506 8717.46 8377 9003
## cvpred 8611 8218 9648 8824.3 8406 7955 8351 8783 8515 8720.65 8380 9018
## PTS 8232 8384 9534 8752.0 8247 7803 8208 9039 8341 8718.00 8769 8832
## CV residual -379 166 -114 -72.3 -159 -152 -143 256 -174 -2.65 389 -186
## 260 261 265 267 275 278 294 296 298 299 301
## Predicted 8577 8562 10172 9441 8510.0 8551.4 9406 9190.1 8264.3 8420 8379
## cvpred 8583 8571 10228 9463 8523.6 8566.4 9425 9178.2 8272.6 8429 8390
## PTS 9145 8428 9828 9564 8441.0 8641.0 9732 9197.0 8229.0 8608 8237
## CV residual 562 -143 -400 101 -82.6 74.6 307 18.8 -43.6 179 -153
## 303 305 314 316 317 323 328 329 330 339 347
## Predicted 8303.2 8145 8236 8343 8795.0 8797.3 8197 8308 8569 8607 7933.20
## cvpred 8290.2 8149 8244 8334 8785.8 8786.2 8189 8308 8559 8611 7942.05
## PTS 8328.0 8358 8502 8625 8832.0 8836.0 8046 8431 8328 8353 7949.00
## CV residual 37.8 209 258 291 46.2 49.8 -143 123 -231 -258 6.95
## 348 350 353 356 363 364 369 370 372 373 376 382
## Predicted 8876.1 8176 8236 8666 7972 8569 8232.9 8097 8553.3 8077 8375 7961
## cvpred 8869.9 8155 8227 8667 7957 8561 8214.8 8081 8540.8 8061 8358 7949
## PTS 8844.0 8280 8475 8461 8202 8687 8249.0 8325 8462.0 8309 8491 7726
## CV residual -25.9 125 248 -206 245 126 34.2 244 -78.8 248 133 -223
## 389 392 396 397 398 409 413 417 418 420 422 424
## Predicted 8024.2 8582 8244 8186 7236 8123.5 7875 8499.25 8423 8308 8099 8400
## cvpred 8008.3 8559 8231 8164 7204 8113.9 7864 8485.82 8401 8281 8095 8403
## PTS 8056.0 8726 8431 8625 7473 8024.0 7746 8477.00 8572 8404 8408 8248
## CV residual 47.7 167 200 461 269 -89.9 -118 -8.82 171 123 313 -155
## 425 430 432 434 438 439 441 445 447 452 453
## Predicted 7945 7352 8161 7881.0 8019 8014.2 7685.8 7868.0 8170 7640 7964.7
## cvpred 7919 7337 8137 7876.7 8005 8003.5 7668.8 7855.5 8147 7628 7958.7
## PTS 8108 7723 8248 7969.0 7882 7974.0 7719.0 7908.0 8274 7860 7864.0
## CV residual 189 386 111 92.3 -123 -29.5 50.2 52.5 127 232 -94.7
## 456 457 462 465 466 467 472 476 477 480 483 493
## Predicted 7696.2 7517.1 7819 7711.4 7576 8545 8229 8150 7906 8199 8591 8527
## cvpred 7675.1 7524.3 7801 7688.5 7569 8537 8212 8130 7899 8192 8584 8521
## PTS 7585.0 7494.0 7874 7787.0 7748 8290 8166 8246 7781 7969 8072 8267
## CV residual -90.1 -30.3 73 98.5 179 -247 -46 116 -118 -223 -512 -254
## 495 496 497 498 499 503 504 509 518 519 521 524
## Predicted 8112 8205 8122.6 7416 8174.8 8789 7946 7958.2 8107 7857 7677.7 8084
## cvpred 8111 8194 8128.1 7420 8184.3 8798 7937 7954.2 8106 7856 7676.3 8081
## PTS 8300 8079 8036.0 7555 8206.0 8607 7886 7921.0 7584 7972 7581.0 8260
## CV residual 189 -115 -92.1 135 21.7 -191 -51 -33.2 -522 116 -95.3 179
## 525 535 538 541 544 548 549 551 557 558 562 563
## Predicted 8060.4 8339 7793 7854 8441 7587.9 8190 8415 7674 8270.2 8592 7888.0
## cvpred 8052.6 8331 7789 7845 8448 7588.9 8184 8411 7670 8273.7 8596 7883.7
## PTS 7982.0 8007 7645 7700 8629 7572.0 7938 8304 7514 8240.0 8578 7932.0
## CV residual -70.6 -324 -144 -145 181 -16.9 -246 -107 -156 -33.7 -18 48.3
## 565 570 571 575 576 577 579 581 582 584 590 591
## Predicted 7553 7996 7806 8351.2 7544 8276 8372 7235 8090.3 8136 7755.6 8422.4
## cvpred 7542 7996 7803 8346.5 7540 8266 8365 7237 8088.3 8128 7740.6 8426.1
## PTS 7494 7786 7495 8400.0 7688 7940 8230 7016 8158.0 7820 7803.0 8342.0
## CV residual -48 -210 -308 53.5 148 -326 -135 -221 69.7 -308 62.4 -84.1
## 593 594 598 604 608 609 614 615 618 620 621 624
## Predicted 7675 7620 7912.7 7763 8478 8167 7759 7449.4 7684.5 8537.7 7646 7376
## cvpred 7668 7621 7908.3 7754 8470 8154 7744 7443.1 7692.6 8524.4 7635 7353
## PTS 7555 7453 7811.0 7649 8052 7930 7529 7542.0 7723.0 8433.0 7501 7271
## CV residual -113 -168 -97.3 -105 -418 -224 -215 98.9 30.4 -91.4 -134 -82
## 625 626 633 634 646 650 661 663 665 673 679
## Predicted 7739 7657.1 7798 8379 8068.1 8496 7697 8044.5 7522 8135 7311.9
## cvpred 7735 7655.6 7781 8380 8075.7 8493 7694 8025.6 7515 8120 7309.8
## PTS 7528 7605.0 7653 8094 8160.0 8505 8130 7941.0 7387 7691 7285.0
## CV residual -207 -50.6 -128 -286 84.3 12 436 -84.6 -128 -429 -24.8
## 680 687 688 690 692 693 695 701 706 709 710
## Predicted 8209.2 7905.4 8226 8039 8771 7822.4 7916.7 8032 7604 7534 8502
## cvpred 8204.3 7903.7 8223 8034 8769 7812.3 7909.5 8030 7595 7533 8510
## PTS 8106.0 7857.0 7945 7934 8639 7872.0 7953.0 8172 7771 7717 8303
## CV residual -98.3 -46.7 -278 -100 -130 59.7 43.5 142 176 184 -207
## 713 716 717 723 725 726 727 728 729 730 731
## Predicted 8153.71 8082 8031 7903 7913.2 8697 8063 8831.9 8259.9 7675 7986.7
## cvpred 8154.96 8074 8017 7891 7904.7 8702 8066 8828.5 8272.1 7675 7981.4
## PTS 8157.00 8054 8245 7992 7931.0 8527 7692 8904.0 8257.0 7491 7956.0
## CV residual 2.04 -20 228 101 26.3 -175 -374 75.5 -15.1 -184 -25.4
## 735 738 739 741 745 746 747 749 750 751 752 754
## Predicted 7987 8832 7792.5 7728 7905 7943 7925 8211 7864 8088 8509.9 8847.3
## cvpred 7993 8828 7786.7 7724 7904 7939 7910 8218 7858 8089 8504.7 8867.1
## PTS 7948 9026 7820.0 7820 8104 8046 8275 8378 8223 8343 8555.0 8905.0
## CV residual -45 198 33.3 96 200 107 365 160 365 254 50.3 37.9
## 755 756 761 764 766 768 769 774 775 782 783
## Predicted 7847 8463 8218.9 7572 7929.2 7865 8539 8542.2 7804.2 8650.8 7669.4
## cvpred 7844 8476 8217.7 7569 7934.8 7860 8536 8533.3 7807.4 8658.3 7666.6
## PTS 8070 8617 8142.0 7857 7952.0 7987 8974 8492.0 7877.0 8729.0 7709.0
## CV residual 226 141 -75.7 288 17.2 127 438 -41.3 69.6 70.7 42.4
## 785 789 798 801 804 808 811 812 813 821 822 832
## Predicted 8437.7 8078 7817 8146.9 8598 7762 8083 8732 7812 7667 8184.1 8235
## cvpred 8440.2 8092 7818 8156.2 8582 7756 8069 8736 7811 7673 8191.5 8232
## PTS 8395.0 8404 8014 8200.0 8547 7650 8220 8811 7951 7534 8288.0 8502
## CV residual -45.2 312 196 43.8 -35 -106 151 75 140 -139 96.5 270
##
## Sum of squares = 10324557 Mean square = 37139 n = 278
##
## Overall (Sum over all 278 folds)
## ms
## 35356## [1] 35356MSPE for our Points Model
## [1] 29522458The Press value for our model.
## [1] 0.895Prediction R^2 value for our model.
## [1] 0.895Adjusted R^2 value for our model.
We have taken a test data, which is 30% of our total data to test our model.
## fit lwr upr
## 1 8532 8484 8581
## 13 9080 9041 9120
## 14 9225 9180 9270
## 17 8153 8111 8196
## 18 9741 9695 9787
## 19 8487 8438 8536## [1] 31261MSPE value.
## [1] 7846412Press Value
## [1] 0.908Predicted R-square value
QQ plot for Opposition Pts model
#### Model Validation for Opp Pts model.
##
## Call:
## lm(formula = oppPTS ~ X2PA + X3PA + FTA + AST + ORB + DRB + TOV +
## STL + BLK, data = NBA)
##
## Residuals:
## Min 1Q Median 3Q Max
## -574 -112 9 116 467
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 114.0949 178.2704 0.64 0.52
## X2PA 1.6263 0.0259 62.79 <2e-16 ***
## X3PA 1.8413 0.0337 54.69 <2e-16 ***
## FTA 0.8098 0.0296 27.40 <2e-16 ***
## AST -0.0383 0.0385 -0.99 0.32
## ORB -1.6897 0.0683 -24.75 <2e-16 ***
## DRB -1.4553 0.0539 -26.98 <2e-16 ***
## TOV 1.3430 0.0536 25.06 <2e-16 ***
## STL -1.7971 0.0804 -22.34 <2e-16 ***
## BLK -0.0988 0.0769 -1.28 0.20
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 163 on 825 degrees of freedom
## Multiple R-squared: 0.924, Adjusted R-squared: 0.923
## F-statistic: 1.12e+03 on 9 and 825 DF, p-value: <2e-16## Analysis of Variance Table
##
## Response: oppPTS
## Df Sum Sq Mean Sq F value Pr(>F)
## X2PA 1 1.65e+08 1.65e+08 6238.03 < 2e-16 ***
## X3PA 1 3.56e+07 3.56e+07 1348.41 < 2e-16 ***
## FTA 1 1.48e+07 1.48e+07 561.10 < 2e-16 ***
## AST 1 2.98e+05 2.98e+05 11.27 0.00082 ***
## ORB 1 1.05e+07 1.05e+07 397.11 < 2e-16 ***
## DRB 1 1.53e+07 1.53e+07 578.66 < 2e-16 ***
## TOV 1 1.13e+07 1.13e+07 428.32 < 2e-16 ***
## STL 1 1.35e+07 1.35e+07 509.36 < 2e-16 ***
## BLK 1 4.36e+04 4.36e+04 1.65 0.19941
## Residuals 825 2.18e+07 2.64e+04
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Warning in cv.lm(data = NBA, PointsRegopp, m = 3, seed = 123):
##
## As there is >1 explanatory variable, cross-validation
## predicted values for a fold are not a linear function
## of corresponding overall predicted values. Lines that
## are shown for the different folds are approximate
##
## fold 1
## Observations in test set: 278
## 5 15 16 19 23 24 25 30 35 38 39 45
## Predicted 9084 8628 8985 9279.4 8821.7 8898 8590 9098.5 8987 8543.4 8950 9067
## cvpred 9072 8639 8983 9257.6 8807.7 8898 8580 9059.3 8989 8558.1 8940 9072
## oppPTS 9240 8603 8819 9160.0 8858.0 8526 8775 9103.0 8680 8512.0 8567 8661
## CV residual 168 -36 -164 -97.6 50.3 -372 195 43.7 -309 -46.1 -373 -411
## 46 48 50 51 53 56 59 61 62 63 64
## Predicted 8323.9 8616 9167 10169 8903 9158 8850 8561.0 8521 8891.5 9147.8
## cvpred 8327.5 8602 9146 10159 8895 9160 8849 8583.4 8525 8889.6 9157.1
## oppPTS 8237.0 8909 8938 10328 9007 9039 8690 8649.0 8422 8957.0 9083.0
## CV residual -90.5 307 -208 169 112 -121 -159 65.6 -103 67.4 -74.1
## 66 67 72 74 80 81 89 90 91 92 94
## Predicted 8614 9278 8488.0 10075.6 8835 8778 8996 9267.6 8617 8573 8993.1
## cvpred 8629 9281 8476.8 10074.6 8838 8789 8995 9279.4 8615 8580 8980.9
## oppPTS 8456 9558 8574.0 10054.0 8978 8379 8756 9282.0 8145 8427 8926.0
## CV residual -173 277 97.2 -20.6 140 -410 -239 2.6 -470 -153 -54.9
## 96 97 98 104 105 110 115 124 126 133 134 137
## Predicted 9276 10256.2 9497 8634 9118 9652 8943.2 9199 9061.3 9312 9031 8951
## cvpred 9270 10256.1 9487 8633 9110 9651 8943.5 9188 9053.4 9310 9028 8968
## oppPTS 9017 10237.0 9308 8325 8929 9884 8862.0 9388 9152.0 9190 9337 8677
## CV residual -253 -19.1 -179 -308 -181 233 -81.5 200 98.6 -120 309 -291
## 142 144 147 148 150 152 155 157 160 162 164
## Predicted 9313.4 9236.9 8545 9394.4 9005 8978 9353.23 9053 8882 8700.6 9030
## cvpred 9310.9 9234.2 8546 9386.1 9005 8968 9340.66 9046 8911 8716.8 9018
## oppPTS 9363.0 9267.0 8792 9475.0 8649 8554 9349.00 9272 8590 8692.0 8871
## CV residual 52.1 32.8 246 88.9 -356 -414 8.34 226 -321 -24.8 -147
## 165 168 173 174 175 178 184 186 189 190
## Predicted 9018.8 9277 8930 9260.3 9038.79 9410.9 8536.21 8105 9486 8716
## cvpred 9016.2 9266 8934 9236.4 9027.55 9390.3 8540.38 8116 9503 8708
## oppPTS 9050.0 9380 8731 9307.0 9022.00 9410.0 8549.00 8330 9239 8533
## CV residual 33.8 114 -203 70.6 -5.55 19.7 8.62 214 -264 -175
## 195 197 200 201 202 204 206 210 212 213 214
## Predicted 8730.4 8729 9065 9163.26 9143 9076.0 8863 8338 8579.8 9734 8397
## cvpred 8728.7 8720 9057 9150.36 9132 9060.4 8863 8344 8584.4 9749 8388
## oppPTS 8771.0 8900 9268 9147.00 9327 8966.0 8716 8608 8583.0 9536 8264
## CV residual 42.3 180 211 -3.36 195 -94.4 -147 264 -1.4 -213 -124
## 216 218 221 222 224 228 232 235 238 242 250
## Predicted 8775.1 9255 8766 8709 8908 9233.3 8656 8479 9359.8 8875.3 9528
## cvpred 8771.5 9258 8759 8707 8900 9215.5 8638 8482 9375.2 8865.2 9503
## oppPTS 8819.0 9525 8636 9027 9051 9249.0 8817 8710 9281.0 8949.0 9821
## CV residual 47.5 267 -123 320 151 33.5 179 228 -94.2 83.8 318
## 263 264 270 271 272 273 274 276 281 282
## Predicted 8467.6 8399 8697.8 8316 8873.6 8476.1 8516.5 8324 8515.2 8354.2
## cvpred 8470.6 8409 8698.1 8313 8855.1 8467.8 8519.7 8320 8519.2 8366.2
## oppPTS 8545.0 8570 8774.0 8164 8840.0 8524.0 8491.0 8474 8484.0 8412.0
## CV residual 74.4 161 75.9 -149 -15.1 56.2 -28.7 154 -35.2 45.8
## 284 285 288 291 292 311 313 318 321 322 324 326
## Predicted 8231.5 8820 8960 8445 8625 8125 8622 9189 9081.5 8035 8609 8610.55
## cvpred 8242.9 8821 8963 8437 8623 8132 8593 9164 9057.3 8044 8617 8592.24
## oppPTS 8254.0 8721 9300 8634 8821 8353 8885 9387 9095.0 8184 8754 8589.00
## CV residual 11.1 -100 337 197 198 221 292 223 37.7 140 137 -3.24
## 332 335 342 343 344 345 346 349 351 352 354
## Predicted 8859 9175.6 8538 7742.0 7933 8474.8 8198 7946.4 8670 8577.5 8383
## cvpred 8865 9160.3 8542 7736.9 7933 8451.3 8215 7956.1 8663 8573.5 8366
## oppPTS 9029 9107.0 8750 7780.0 7966 8514.0 8099 7938.0 8916 8585.0 8480
## CV residual 164 -53.3 208 43.1 33 62.7 -116 -18.1 253 11.5 114
## 355 357 359 360 365 368 374 375 377 379 381
## Predicted 8431.9 7634 8447 8366 7974.3 8516.7 8337 8993 7742 8553.2 8522.9
## cvpred 8421.4 7623 8451 8352 7975.2 8494.7 8325 8953 7720 8547.6 8491.3
## oppPTS 8498.0 7503 8658 8479 8008.0 8582.0 8651 9111 7833 8634.0 8504.0
## CV residual 76.6 -120 207 127 32.8 87.3 326 158 113 86.4 12.7
## 385 386 388 391 395 400 402 403 408 412 414
## Predicted 8515.1 8167.2 8265 8200 8725.9 8174.2 8272 8257.8 8248.1 8194 8227
## cvpred 8492.6 8162.8 8245 8187 8711.7 8173.7 8249 8250.6 8227.6 8181 8218
## oppPTS 8512.0 8236.0 8138 8384 8774.0 8235.0 8453 8261.0 8272.0 8115 8525
## CV residual 19.4 73.2 -107 197 62.3 61.3 204 10.4 44.4 -66 307
## 416 419 433 435 437 442 443 446 449 450 461
## Predicted 8254 8259 7842.3 7950.8 7853 8560 8321.1 7767 7587 7871 8106.7
## cvpred 8225 8239 7820.9 7947.6 7844 8522 8310.9 7748 7587 7857 8080.5
## oppPTS 8385 8610 7739.0 7850.0 7973 8751 8377.0 8064 7733 8152 8161.0
## CV residual 160 371 -81.9 -97.6 129 229 66.1 316 146 295 80.5
## 464 469 474 478 481 482 485 488 489 490 492
## Predicted 8085.6 7500 8177.4 7623 7989 8108 8227 8513 8278 8272.4 8416.4
## cvpred 8075.1 7491 8167.4 7615 7990 8097 8220 8501 8259 8256.4 8420.6
## oppPTS 8017.0 7307 8095.0 7743 8176 8208 8237 8365 8512 8227.0 8491.0
## CV residual -58.1 -184 -72.4 128 186 111 17 -136 253 -29.4 70.4
## 494 502 505 506 508 516 522 523 527 529 530
## Predicted 7475.0 7518 8230 8015.1 8033 8080.9 8012.7 7466 7378 7470.2 7778
## cvpred 7488.8 7518 8221 8022.5 8018 8085.8 8007.5 7471 7392 7473.6 7793
## oppPTS 7484.0 7466 8047 7981.0 8163 8120.0 7974.0 7101 7059 7412.0 7529
## CV residual -4.8 -52 -174 -41.5 145 34.2 -33.5 -370 -333 -61.6 -264
## 531 532 533 534 537 539 540 543 545 546 552
## Predicted 7446.3 7915.7 7349 7965.8 8008.2 7937 7796.8 7853 8137.3 7769 7856
## cvpred 7449.4 7933.9 7371 7964.4 8008.6 7937 7800.2 7859 8129.5 7782 7857
## oppPTS 7480.0 7866.0 7250 7976.0 7992.0 8058 7720.0 8085 8036.0 7560 7982
## CV residual 30.6 -67.9 -121 11.6 -16.6 121 -80.2 226 -93.5 -222 125
## 554 556 559 560 561 567 568 573 580 583 585 587
## Predicted 7813 7770 7226.8 7906.5 7598.1 7676 7772 7760 8144 7808.1 7665 8274
## cvpred 7820 7780 7230.2 7911.2 7590.9 7681 7776 7748 8152 7826.7 7663 8266
## oppPTS 8014 7548 7330.0 7858.0 7680.0 7724 8006 7580 8260 7876.0 7526 8067
## CV residual 194 -232 99.8 -53.2 89.1 43 230 -168 108 49.3 -137 -199
## 589 592 597 600 601 602 606 610 611 619 622 623
## Predicted 7940 7592 7852 7668 8055 8026 7277 7601 8072 7468.4 8085.8 7507
## cvpred 7942 7598 7847 7681 8059 8032 7280 7599 8073 7467.3 8074.5 7520
## oppPTS 7741 7412 7992 7834 8262 7884 7021 7359 7952 7544.0 8016.0 7253
## CV residual -201 -186 145 153 203 -148 -259 -240 -121 76.7 -58.5 -267
## 629 635 638 640 641 642 643 644 645 647 649
## Predicted 8042 7635 8192 7848.1 8129.8 7764.8 7841 8049 8065 8186.46 7834
## cvpred 8027 7639 8169 7855.5 8121.8 7774.1 7842 8029 8054 8187.35 7836
## oppPTS 7658 7465 8338 7792.0 8218.0 7815.0 7619 7832 8177 8189.00 7949
## CV residual -369 -174 169 -63.5 96.2 40.9 -223 -197 123 1.65 113
## 652 653 654 658 660 662 667 668 670 672
## Predicted 8088 8267.2 7987.71 8297.1 7868.8 8184.0 7691 8003.9 7833.1 7746.8
## cvpred 8063 8266.9 7966.08 8298.6 7863.4 8195.8 7719 7995.8 7837.9 7764.7
## oppPTS 7925 8311.0 7975.00 8271.0 7819.0 8208.0 7841 7949.0 7874.0 7676.0
## CV residual -138 44.1 8.92 -27.6 -44.4 12.2 122 -46.8 36.1 -88.7
## 674 675 677 681 682 686 689 691 696 697 700 702
## Predicted 7791 8176 8147 7531 8531 8025.4 7958 7756 8186 7764 7737.2 8129.3
## cvpred 7797 8143 8159 7543 8505 8019.1 7959 7758 8176 7774 7741.9 8129.2
## oppPTS 7842 8367 8307 7278 8659 8070.0 7689 7609 8040 7881 7834.0 8178.0
## CV residual 45 224 148 -265 154 50.9 -270 -149 -136 107 92.1 48.8
## 703 707 708 711 720 722 732 740 743 753 763
## Predicted 8157.3 8024.60 8643 7670 7823 8600 8076 8444 7897.2 7663.0 8107
## cvpred 8137.7 8027.19 8645 7681 7817 8617 8077 8442 7914.5 7672.9 8107
## oppPTS 8064.0 8033.00 8438 7388 7932 8770 8395 8593 7977.0 7767.0 8244
## CV residual -73.7 5.81 -207 -293 115 153 318 151 62.5 94.1 137
## 765 767 770 771 772 776 779 784 786 792 795
## Predicted 8826 7885 7718.73 8753 7690.8 7975.9 8008.4 9087 8610 8466 8555
## cvpred 8806 7887 7705.82 8744 7709.9 7984.5 8030.3 9097 8618 8464 8547
## oppPTS 8841 7737 7715.00 8966 7650.0 7956.0 8127.0 9217 8510 8838 8686
## CV residual 35 -150 9.18 222 -59.9 -28.5 96.7 120 -108 374 139
## 796 802 805 809 815 819 823 824 825 827 830
## Predicted 7895 8021 8248.7 7640 8540.6 7903.0 8253.4 7579 8664.31 7894 7876
## cvpred 7907 8024 8248.9 7648 8532.4 7924.7 8250.2 7595 8667.04 7883 7868
## oppPTS 8036 7895 8284.0 7487 8506.0 8003.0 8234.0 7711 8670.00 7687 7771
## CV residual 129 -129 35.1 -161 -26.4 78.3 -16.2 116 2.96 -196 -97
##
## Sum of squares = 7910667 Mean square = 28456 n = 278
##
## fold 2
## Observations in test set: 279
## 1 2 4 7 9 10 17 18 20 21 26 27
## Predicted 8361 9051 9696 8586 9113.5 8865 8498.8 9807.96 8780 8528 9328 9216
## cvpred 8359 9068 9682 8577 9103.1 8846 8502.5 9814.76 8782 8532 9336 9249
## oppPTS 8334 8664 9332 8853 9176.0 8603 8469.0 9819.00 8515 8887 9068 9011
## CV residual -25 -404 -350 276 72.9 -243 -33.5 4.24 -267 355 -268 -238
## 31 33 34 36 41 42 52 57 58 65 71
## Predicted 8961 8776.05 8751.1 9421 8851 9177 9387 8755 8590 9388.0 9534.2
## cvpred 8962 8770.23 8734.4 9420 8843 9187 9392 8733 8598 9416.1 9569.2
## oppPTS 8851 8768.00 8802.0 9262 8973 8867 9187 9001 8441 9502.0 9503.0
## CV residual -111 -2.23 67.6 -158 130 -320 -205 268 -157 85.9 -66.2
## 73 76 82 83 85 88 93 95 100 103 106 109
## Predicted 9470 9149.7 8688 8571 8641 9495 8801 8695.3 9335.5 8979 8792 9071.8
## cvpred 9490 9144.1 8695 8579 8633 9510 8799 8692.3 9338.3 8985 8802 9075.5
## oppPTS 9277 9205.0 8445 7997 8361 9299 8656 8735.0 9324.0 9170 8448 8986.0
## CV residual -213 60.9 -250 -582 -272 -211 -143 42.7 -14.3 185 -354 -89.5
## 112 114 117 118 123 125 131 135 141 146 151
## Predicted 9027 8811 8896.8 9206.4 9144 9571.8 8747 8929 9091.9 9224.4 9015.1
## cvpred 9031 8834 8898.2 9208.4 9153 9608.7 8754 8943 9091.4 9213.5 9027.3
## oppPTS 8879 8660 8985.0 9129.0 8977 9632.0 8925 8822 9071.0 9165.0 9112.0
## CV residual -152 -174 86.8 -79.4 -176 23.3 171 -121 -20.4 -48.5 84.7
## 156 158 161 167 172 179 180 181 187 188 191
## Predicted 9251.4 8812 8416.0 8695 8868.2 9364.1 8954 9262.45 8584.0 8467 9241
## cvpred 9272.4 8821 8416.9 8697 8865.3 9391.8 8965 9291.14 8573.9 8474 9240
## oppPTS 9176.0 8572 8431.0 8836 8893.0 9359.0 9300 9287.00 8504.0 8602 9453
## CV residual -96.4 -249 14.1 139 27.7 -32.8 335 -4.14 -69.9 128 213
## 192 193 198 207 208 211 215 217 226 230
## Predicted 8804.3 8608.6 8999 8630.6 8674 8281.7 9554.8 8897 9399 8198.7
## cvpred 8788.4 8607.8 9014 8625.3 8678 8255.6 9529.1 8925 9410 8198.6
## oppPTS 8821.0 8646.0 8695 8699.0 8863 8300.0 9583.0 9109 9275 8176.0
## CV residual 32.6 38.2 -319 73.7 185 44.4 53.9 184 -135 -22.6
## 236 239 243 244 246 249 252 253 257 259 262
## Predicted 8435.424 8444 8624 8558.8 8750.56 8644 8784.6 9079 8277 8727 8139
## cvpred 8436.446 8457 8614 8564.3 8746.95 8654 8768.5 9091 8282 8729 8112
## oppPTS 8436.000 8057 8787 8523.0 8755.00 8763 8841.0 8847 8367 8940 8278
## CV residual -0.446 -400 173 -41.3 8.05 109 72.5 -244 85 211 166
## 266 268 269 277 279 280 283 286 287 289 290 293
## Predicted 8178 8493.8 8997 9127 8767.8 8920 8511 8432 8132 8182.76 8382 8073
## cvpred 8189 8503.9 9013 9155 8753.8 8931 8508 8430 8100 8161.61 8353 8086
## oppPTS 7937 8466.0 9191 9010 8811.0 8695 8643 8834 8448 8155.00 8479 7946
## CV residual -252 -37.9 178 -145 57.2 -236 135 404 348 -6.61 126 -140
## 295 297 300 302 304 306 307 308 309 310 312 315
## Predicted 8350 8230 8570 8588 8868.3 8679.2 8683 8879 8069 8495.0 8752.2 8879
## cvpred 8353 8203 8589 8588 8880.5 8656.4 8696 8852 8045 8494.1 8730.1 8875
## oppPTS 8507 8352 8749 8780 8897.0 8707.0 8539 9046 8252 8583.0 8761.0 9050
## CV residual 154 149 160 192 16.5 50.6 -157 194 207 88.9 30.9 175
## 319 320 325 327 331 333 334 336 337 338 340
## Predicted 8700 8473 8470 8573 8567.4 8708.9 8713.5 8381.5 8232.9 8229 7692
## cvpred 8690 8501 8458 8573 8593.6 8720.7 8704.5 8366.6 8208.9 8223 7677
## oppPTS 8769 8366 8650 8698 8544.0 8752.0 8643.0 8433.0 8304.0 8531 7886
## CV residual 79 -135 192 125 -49.6 31.3 -61.5 66.4 95.1 308 209
## 341 358 361 362 366 367 371 378 380 383 384
## Predicted 8439 8465 8687 8715.7 8654 7929 7441.1 8671.62 8348.4 8071 7940
## cvpred 8416 8490 8685 8729.9 8670 7944 7455.1 8671.44 8375.5 8101 7975
## oppPTS 8618 8347 8579 8764.0 8834 7816 7366.0 8678.00 8427.0 8299 7799
## CV residual 202 -143 -106 34.1 164 -128 -89.1 6.56 51.5 198 -176
## 387 390 393 394 399 401 404 405 406 407 410
## Predicted 8674.9 8017 8718.2 7972.9 8840 7728 8019 8389.4 8072.41 7933 7742
## cvpred 8702.3 8023 8732.2 7999.3 8901 7755 8061 8404.1 8068.98 7971 7757
## oppPTS 8755.0 8253 8701.0 7959.0 8811 7617 7878 8448.0 8073.00 7792 8031
## CV residual 52.7 230 -31.2 -40.3 -90 -138 -183 43.9 4.02 -179 274
## 411 415 421 423 426 427 428 429 431 436 440
## Predicted 7829.0 8125 8145.5 7491 7591 7182 7898.3 8522.9 8420.2 7656 7656
## cvpred 7836.1 8163 8159.1 7512 7613 7201 7925.2 8563.1 8467.3 7689 7695
## oppPTS 7781.0 7952 8180.0 7328 7572 7022 7952.0 8535.0 8557.0 7326 7563
## CV residual -55.1 -211 20.9 -184 -41 -179 26.8 -28.1 89.7 -363 -132
## 444 448 451 454 455 458 459 460 463 468 470
## Predicted 7907 8071.62 7902 7652 7582 8045 7661.7 7843 8447.54 8198 7756
## cvpred 7935 8094.09 7904 7667 7596 8055 7666.7 7857 8464.51 8213 7772
## oppPTS 7772 8085.00 8014 7759 7348 8266 7592.0 7985 8469.00 8041 7475
## CV residual -163 -9.09 110 92 -248 211 -74.7 128 4.49 -172 -297
## 471 473 475 479 484 486 487 491 500 501 507 510
## Predicted 7770.1 7732 7328 8205 7783 8483 8375.5 8074 7740.1 7887 7688 7864.5
## cvpred 7776.4 7759 7337 8241 7824 8472 8383.4 8076 7731.3 7899 7690 7868.8
## oppPTS 7847.0 7619 7260 8522 7723 8363 8289.0 7929 7661.0 7683 7548 7886.0
## CV residual 70.6 -140 -77 281 -101 -109 -94.4 -147 -70.3 -216 -142 17.2
## 511 512 513 514 515 517 520 526 528 536 542
## Predicted 8070 7.37e+03 7852.1 7815.5 8071 8200 7659.4 8155 8228 7736 7887
## cvpred 8084 7.37e+03 7856.5 7813.5 8050 8206 7660.8 8147 8231 7745 7888
## oppPTS 7934 7.37e+03 7927.0 7909.0 7888 7976 7607.0 7966 7911 7574 8035
## CV residual -150 5.94e-02 70.5 95.5 -162 -230 -53.8 -181 -320 -171 147
## 547 550 553 555 564 566 569 572 574 578 586 588
## Predicted 8276 8010 7387 7855.0 7725.7 7679 7903 8001 7604 7994.0 8231 7646
## cvpred 8294 8017 7378 7856.1 7723.3 7689 7918 7975 7605 8005.8 8246 7634
## oppPTS 8452 7884 7276 7868.0 7766.0 7798 7631 7806 7190 8031.0 7971 7752
## CV residual 158 -133 -102 11.9 42.7 109 -287 -169 -415 25.2 -275 118
## 595 596 599 603 605 607 612 613 616 617 627
## Predicted 7475 7588.3 7896.9 7291 7378 8077 7351.6 7407 8328 7522 8201.3
## cvpred 7488 7569.5 7910.3 7279 7393 8107 7324.5 7409 8338 7531 8209.6
## oppPTS 7566 7585.0 7876.0 6909 7220 8147 7303.0 7196 8287 7419 8233.0
## CV residual 78 15.5 -34.3 -370 -173 40 -21.5 -213 -51 -112 23.4
## 628 630 631 632 636 637 639 648 651 655 656
## Predicted 8167.2 7874.3 7888.9 7887 7664 7768 7654 8518.3 7619 8314.0 8228
## cvpred 8156.3 7870.3 7869.1 7871 7677 7775 7653 8534.6 7616 8328.3 8252
## oppPTS 8220.0 7849.0 7934.0 7995 7564 7912 7473 8470.0 7248 8268.0 8362
## CV residual 63.7 -21.3 64.9 124 -113 137 -180 -64.6 -368 -60.3 110
## 657 659 664 666 669 671 676 678 683 684 685
## Predicted 8223.6 8291 8410 7660 7565 8100.0 7771.9 8594 8556.8 7805.3 8360
## cvpred 8240.1 8312 8432 7678 7570 8120.6 7777.2 8600 8572.9 7830.9 8370
## oppPTS 8159.0 7968 8187 7544 7255 8105.0 7872.0 8431 8532.0 7789.0 8184
## CV residual -81.1 -344 -245 -134 -315 -15.6 94.8 -169 -40.9 -41.9 -186
## 694 698 699 704 705 712 714 715 718 719
## Predicted 8837.1 8511.8 8611 7985.1 8107.0 8279.7 8111.0 8647.5 8145 8170.0
## cvpred 8849.4 8533.9 8634 7998.8 8159.5 8300.4 8145.5 8658.8 8152 8174.6
## oppPTS 8765.0 8480.0 8753 7962.0 8228.0 8367.0 8087.0 8598.0 8318 8234.0
## CV residual -84.4 -53.9 119 -36.8 68.5 66.6 -58.5 -60.8 166 59.4
## 721 724 733 734 736 737 742 744 748 757 758 759
## Predicted 7759 8893.9 8115 7747.7 8245 7799.8 8409 8037 7866 8353 8079.7 7949
## cvpred 7760 8898.6 8143 7741.8 8274 7790.8 8408 8045 7886 8356 8071.9 7945
## oppPTS 7862 8924.0 8272 7837.0 8119 7889.0 8717 8146 7781 8518 8140.0 8146
## CV residual 102 25.4 129 95.2 -155 98.2 309 101 -105 162 68.1 201
## 760 762 773 777 778 780 781 787 788 790 791
## Predicted 7979.9 8380.5 8164 7840.4 7837 7760 7955 8121 7986.3 7648.1 8195
## cvpred 7959.5 8403.8 8158 7837.1 7842 7764 7938 8139 7983.4 7630.9 8200
## oppPTS 8040.0 8422.0 8352 7836.0 7693 7838 8141 8370 7952.0 7727.0 7870
## CV residual 80.5 18.2 194 -1.1 -149 74 203 231 -31.4 96.1 -330
## 793 794 797 799 800 803 806 807 810 814 816
## Predicted 8172 8178 7930 8553.2 7863.1 8431 7913.7 7506.5 8403 8434 8282.5
## cvpred 8170 8178 7953 8581.9 7868.7 8436 7921.2 7491.4 8415 8433 8288.4
## oppPTS 8323 8422 7812 8637.0 7774.0 8680 7857.0 7473.0 8566 8668 8271.0
## CV residual 153 244 -141 55.1 -94.7 244 -64.2 -18.4 151 235 -17.4
## 817 818 820 826 828 829 831 833 834 835
## Predicted 8325.28 7795.6 7768.3 8058 7918.3 8701.2 8440 8370 8177 8406
## cvpred 8355.49 7793.8 7768.5 8046 7902.5 8721.7 8454 8378 8173 8394
## oppPTS 8346.00 7820.0 7757.0 8285 7996.0 8684.0 8589 8639 8303 8584
## CV residual -9.49 26.2 -11.5 239 93.5 -37.7 135 261 130 190
##
## Sum of squares = 7714830 Mean square = 27652 n = 279
##
## fold 3
## Observations in test set: 278
## 3 6 8 11 12 13 14 22 28 29 32 37
## Predicted 8967.5 9471 9265 8782 9016 9151 9484.6 9058.2 10219 8934 8829 9074
## cvpred 8959.3 9478 9274 8774 9024 9155 9495.5 9053.1 10232 8949 8844 9088
## oppPTS 9035.0 9609 9070 8954 8702 8975 9438.0 8982.0 10025 8692 8712 8716
## CV residual 75.7 131 -204 180 -322 -180 -57.5 -71.1 -207 -257 -132 -372
## 40 43 44 47 49 54 55 60 68 69 70 75
## Predicted 9122 8969 8577 8972 9191.7 8730 8611.1 8900.9 8595 8763 9009 9543
## cvpred 9129 8972 8580 8978 9208.5 8744 8622.3 8905.8 8592 8771 9013 9551
## oppPTS 9007 8666 8784 8657 9161.0 8683 8532.0 8926.0 8413 8413 8752 9272
## CV residual -122 -306 204 -321 -47.5 -61 -90.3 20.2 -179 -358 -261 -279
## 77 78 79 84 86 87 99 101 102 107 108 111
## Predicted 9354 9319.8 9530 8571.9 8855 8926 9436 8714 9304 8572.6 9138 9243.9
## cvpred 9361 9328.9 9532 8575.2 8860 8921 9456 8716 9310 8581.2 9142 9246.7
## oppPTS 9096 9391.0 9209 8562.0 8633 9075 9287 8961 9144 8658.0 9028 9344.0
## CV residual -265 62.1 -323 -13.2 -227 154 -169 245 -166 76.8 -114 97.3
## 113 116 119 120 121 122 127 128 129 130 132
## Predicted 9264.4 8749 9109 9736 9370.7 9602.8 8944 8921 8923.1 9163 9077.5
## cvpred 9267.3 8745 9115 9757 9377.8 9622.6 8935 8930 8935.2 9175 9070.5
## oppPTS 9335.0 8867 8938 9641 9304.0 9654.0 9093 8528 8956.0 9007 9031.0
## CV residual 67.7 122 -177 -116 -73.8 31.4 158 -402 20.8 -168 -39.5
## 136 138 139 140 143 145 149 153 154 159 163
## Predicted 9168 8708.79 8655.7 9198.9 9680 9378 9041.5 8682 9241.7 9039 8384
## cvpred 9171 8714.42 8646.5 9206.4 9699 9390 9034.6 8691 9232.9 9045 8392
## oppPTS 8946 8712.00 8587.0 9274.0 9303 9582 8983.0 8858 9268.0 8901 8523
## CV residual -225 -2.42 -59.5 67.6 -396 192 -51.6 167 35.1 -144 131
## 166 169 170 171 176 177 182 183 185 194 196
## Predicted 9723 8823 8698.9 9496.8 8507 9180 9009 8922 8737.7 8956.9 8728.5
## cvpred 9741 8827 8701.2 9513.5 8515 9180 9022 8950 8730.2 8965.9 8733.8
## oppPTS 9640 8683 8751.0 9503.0 8745 9311 8811 8802 8828.0 8949.0 8653.0
## CV residual -101 -144 49.8 -10.5 230 131 -211 -148 97.8 -16.9 -80.8
## 199 203 205 209 219 220 223 225 227 229 231
## Predicted 8642 9491 8614.0 9221.6 8630 8892.7 9494 9271 9053.3 8953.9 9220
## cvpred 8649 9499 8612.9 9228.1 8625 8895.8 9503 9272 9053.5 8970.4 9232
## oppPTS 8785 9714 8597.0 9265.0 8818 8937.0 9258 9096 9106.0 8958.0 9056
## CV residual 136 215 -15.9 36.9 193 41.2 -245 -176 52.5 -12.4 -176
## 233 234 237 240 241 245 247 248 251 254
## Predicted 8409 8819.6 8226 9713 8390 8956 8166.6 8831.172 8592.5 8837.1
## cvpred 8397 8829.2 8233 9722 8388 8962 8188.5 8853.474 8600.2 8837.9
## oppPTS 8696 8873.0 8378 9791 8632 9044 8150.0 8853.000 8630.0 8756.0
## CV residual 299 43.8 145 69 244 82 -38.5 -0.474 29.8 -81.9
## 255 256 258 260 261 265 267 275 278 294 296
## Predicted 8339.3 8700 9069 8201 8758.5 10480 9502.2 8845.3 8446 9254 8667
## cvpred 8348.2 8710 9076 8199 8769.3 10499 9514.7 8862.7 8452 9264 8655
## oppPTS 8432.0 8684 9009 8668 8858.0 10723 9430.0 8811.0 8656 9412 9042
## CV residual 83.8 -26 -67 469 88.7 224 -84.7 -51.7 204 148 387
## 298 299 301 303 305 314 316 317 323 328 329
## Predicted 8247.3 8737 8827.8 7988.9 8248 8209 8005.9 8243.7 8472 8500 8190
## cvpred 8269.3 8742 8848.1 7985.5 8259 8216 8019.9 8242.9 8470 8503 8196
## oppPTS 8319.0 8953 8815.0 8009.0 8462 8429 8109.0 8303.0 8697 8684 8328
## CV residual 49.7 211 -33.1 23.5 203 213 89.1 60.1 227 181 132
## 330 339 347 348 350 353 356 363 364 369
## Predicted 7923.7 8771 8506.7 8643.6 7878 8256.31 8162 7581 7910.9 8060.3
## cvpred 7914.7 8774 8507.7 8646.5 7874 8255.23 8171 7576 7926.1 8050.2
## oppPTS 7823.0 8930 8587.0 8701.0 7997 8256.00 8281 7771 7942.0 7980.0
## CV residual -91.7 156 79.3 54.5 123 0.77 110 195 15.9 -70.2
## 370 372 373 376 382 389 392 396 397 398 409
## Predicted 7946.0 8660 8025 8155 8497.8 8190.3 7863 8489.10 7747 7317 8370.8
## cvpred 7946.6 8657 8024 8145 8499.4 8186.8 7856 8482.77 7745 7319 8371.3
## oppPTS 7929.0 8700 8240 8317 8464.0 8138.0 8071 8478.00 7621 7261 8463.0
## CV residual -17.6 43 216 172 -35.4 -48.8 215 -4.77 -124 -58 91.7
## 413 417 418 420 422 424 425 430 432 434
## Predicted 8446 7933.5 7888.9 7810.5 8459 8724 7880.8 7353.7 7859.1 8199.4
## cvpred 8447 7926.1 7878.3 7806.1 8459 8731 7872.2 7352.3 7837.3 8205.7
## oppPTS 8566 7960.0 7933.0 7864.0 8321 8849 7955.0 7293.0 7881.0 8162.0
## CV residual 119 33.9 54.7 57.9 -138 118 82.8 -59.3 43.7 -43.7
## 438 439 441 445 447 452 453 456 457 462 465
## Predicted 8046 8148 7926 8178.29 7650.35 7498.6 8161.0 7473 8060.2 7592 7506
## cvpred 8049 8153 7926 8175.01 7652.79 7502.3 8178.1 7471 8069.4 7589 7500
## oppPTS 8003 8348 7748 8185.00 7644.00 7572.0 8079.0 7361 7995.0 7375 7383
## CV residual -46 195 -178 9.99 -8.79 69.7 -99.1 -110 -74.4 -214 -117
## 466 467 472 476 477 480 483 493 495 496 497
## Predicted 7865.5 8184 7799.1 7830 8289 8043 7795.2 7809 8218.9 7909.0 8232
## cvpred 7869.7 8186 7794.1 7833 8306 8048 7789.2 7804 8219.9 7907.1 8241
## oppPTS 7905.0 8232 7741.0 7658 8541 7921 7853.0 7566 8282.0 7872.0 8121
## CV residual 35.3 46 -53.1 -175 235 -127 63.8 -238 62.1 -35.1 -120
## 498 499 503 504 509 518 519 521 524 525 535 538
## Predicted 7711 8302 8589 7475.7 8132.5 8190 7907 7923.8 7911.7 7776 7986 8078
## cvpred 7702 8302 8580 7468.3 8128.4 8197 7899 7916.4 7907.7 7776 7987 8076
## oppPTS 7435 8150 8368 7399.0 8190.0 8326 7784 7818.0 7942.0 7871 7822 8192
## CV residual -267 -152 -212 -69.3 61.6 129 -115 -98.4 34.3 95 -165 116
## 541 544 548 549 551 557 558 562 563 565 570
## Predicted 7557.0 8244.4 7992.0 7888.2 7912 7899.2 8291 7924.0 7622 7628 8179
## cvpred 7557.3 8240.9 7992.1 7879.9 7910 7888.5 8290 7918.6 7620 7633 8171
## oppPTS 7621.0 8280.0 7973.0 7916.0 7720 7841.0 8111 7954.0 7423 7530 8207
## CV residual 63.7 39.1 -19.1 36.1 -190 -47.5 -179 35.4 -197 -103 36
## 571 575 576 577 579 581 582 584 590 591
## Predicted 8074 8278 7627.3 7685.3 8076.5 7543 8092.7 7701 7628.1 7873.2
## cvpred 8068 8277 7621.8 7678.5 8072.1 7539 8091.7 7696 7623.3 7862.8
## oppPTS 8284 8493 7567.0 7654.0 8039.0 7430 8139.0 7392 7589.0 7809.0
## CV residual 216 216 -54.8 -24.5 -33.1 -109 47.3 -304 -34.3 -53.8
## 593 594 598 604 608 609 614 615 618 620
## Predicted 7606.7 7966.3 7906.1 7730.0 7918 7813 7697 7577.3 7928.8 8014.9
## cvpred 7605.1 7968.9 7897.5 7721.7 7914 7818 7691 7566.9 7930.4 8005.1
## oppPTS 7565.0 7934.0 7930.0 7709.0 7732 7730 7537 7663.0 8030.0 8022.0
## CV residual -40.1 -34.9 32.5 -12.7 -182 -88 -154 96.1 99.6 16.9
## 621 624 625 626 633 634 646 650 661 663 665 673
## Predicted 7186 7601 7990.57 8077 7464 8404 8141 8206 7646.2 7691 7475.0 7786
## cvpred 7180 7602 7988.64 8077 7461 8404 8140 8204 7651.7 7695 7461.9 7771
## oppPTS 6909 7371 7990.00 8401 7336 8271 8344 8328 7632.0 7394 7517.0 7579
## CV residual -271 -231 1.36 324 -125 -133 204 124 -19.7 -301 55.1 -192
## 679 680 687 688 690 692 693 695 701 706 709
## Predicted 7968.8 8127 8273 8344.3 7892 8540.5 7675 7718 8527.5 7844 8060.41
## cvpred 7973.7 8117 8267 8342.6 7886 8531.6 7679 7700 8521.7 7837 8060.05
## oppPTS 8060.0 7980 8137 8252.0 7620 8506.0 7531 7555 8531.0 7707 8069.00
## CV residual 86.3 -137 -130 -90.6 -266 -25.6 -148 -145 9.3 -130 8.95
## 710 713 716 717 723 725 726 727 728 729 730 731
## Predicted 8539.1 8133.4 7981 7542 7593 7635.2 8643.8 8092 8195 8660 8049 8292
## cvpred 8535.4 8124.7 7978 7527 7595 7623.3 8630.2 8087 8182 8648 8044 8289
## oppPTS 8451.0 8076.0 8203 7404 7386 7547.0 8642.0 8288 8309 8766 8200 8521
## CV residual -84.4 -48.7 225 -123 -209 -76.3 11.8 201 127 118 156 232
## 735 738 739 741 745 746 747 749 750 751 752
## Predicted 8376 8502 7985.4 7595 7979 7887.8 7646.8 8322.3 7586.6 7890 8149
## cvpred 8368 8483 7978.4 7581 7980 7885.8 7630.8 8321.9 7579.9 7884 8145
## oppPTS 8490 8612 7900.0 7427 8131 7917.0 7659.0 8401.0 7491.0 8181 8275
## CV residual 122 129 -78.4 -154 151 31.2 28.2 79.1 -88.9 297 130
## 754 755 756 761 764 766 768 769 774 775 782 783
## Predicted 8954 7765.46 8552 8343 7576 8187 7874 8504 8135 8094 8375.1 7997
## cvpred 8958 7749.55 8542 8341 7571 8182 7880 8494 8132 8097 8374.9 8003
## oppPTS 9212 7742.00 8708 8231 7730 8452 7981 8816 8275 8489 8394.0 8128
## CV residual 254 -7.55 166 -110 159 270 101 322 143 392 19.1 125
## 785 789 798 801 804 808 811 812 813 821 822 832
## Predicted 8617 8275 7961 8436 7972 7903.4 7802.2 8368 8133 7792 8614 7971.9
## cvpred 8608 8278 7960 8430 7958 7896.8 7782.2 8355 8138 7790 8605 7959.8
## oppPTS 8425 8528 8334 8558 8109 7978.0 7873.0 8421 8246 7603 8832 8034.0
## CV residual -183 250 374 128 151 81.2 90.8 66 108 -187 227 74.2
##
## Sum of squares = 6828900 Mean square = 24564 n = 278
##
## Overall (Sum over all 278 folds)
## ms
## 26891## [1] 26891MSPE Value for opposition points model.
## [1] 22454397Press value.
## [1] 0.922Predicted R^2 value.
## [1] 0.924Adjusted R^2 value.
We have used the sample test data which is generated randomly to test our model.
## [1] 112359MSPE
## [1] 28202013Press Value
## [1] 0.907Predicted R^2 value.
Logistic Model
Here we have to write logistics content
##
## Call:
## lm(formula = W ~ PTSdiff, data = Nba)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.739 -2.102 -0.067 2.026 10.603
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.10e+01 1.06e-01 387 <2e-16 ***
## PTSdiff 3.26e-02 2.79e-04 117 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.06 on 833 degrees of freedom
## Multiple R-squared: 0.942, Adjusted R-squared: 0.942
## F-statistic: 1.36e+04 on 1 and 833 DF, p-value: <2e-16##
## Call:
## glm(formula = Playoffs ~ W, family = "binomial", data = Nbalogitrain2)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.8941 -0.0908 0.0131 0.1749 2.9447
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -18.4806 1.6214 -11.4 <2e-16 ***
## W 0.4719 0.0406 11.6 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1138.8 on 834 degrees of freedom
## Residual deviance: 308.9 on 833 degrees of freedom
## AIC: 312.9
##
## Number of Fisher Scoring iterations: 8## [1] 0.48Optimal_pcut value is 0.48.
Here, we can observe that our model correctly predicted 768 out of the 835 observations from the test dataset which is a success rate of 91.97%
Based on the result of above code, we observed that the optimal p-cut value is 0.48 which corresponds to minimum number of wins required to qualify for the playoffs = 39.
## pred_value2
## actual_value2 0 1
## 0 308 47
## 1 20 460Final Equation
For calculating minimum number of wins required to qualify the playoffs, we will create a 4th model - logistic regression model.
Following are the results of the steps we’ve taken to create these models:
Model for calculating number of wins: W = 41 + 0.0326 * PTSdiffW = 41 + 0.0326 * PTSdiff 39 = 41 + 0.0326 * PTSdiff We get PTSdiff = -61.35
Conclusion:
Thus, we can conclude that to qualify for the playoffs, a team needs to:
Win at least 39 matches
Have a points difference of at least -61.35
Combining all the models together.
Diff2<-(predictionfinal[,1]-predictionfinal2[,1])
Playoffsdif<-data.frame(NBA_test$Playoffs,Diff2)
Playoffsdif$Predictedplayoffs[Playoffsdif$Diff2 > -61] <- 1
Playoffsdif$Predictedplayoffs[Playoffsdif$Diff2 < -61] <- 0
actual_value3=NBA_test$Playoffs
pred_value3=Playoffsdif$Predictedplayoffs
confusion_matrix3=table(actual_value3,pred_value3)
confusion_matrix3## pred_value3
## actual_value3 0 1
## 0 79 23
## 1 28 121Out of the 251 results, our model got 200 of them right, which is an accuracy of 79%.
CONCLUSION
From the tests conducted, we conclude that an NBA Team needs to win at least 39 games with a point difference (Points Scored -Opposition Points) of at least greater than -61.35 to make it to the playoffs.
Decision Tree
Decision Tree
A decision tree is an effective machine learning modeling technique for regression and classification problems. A decision tree provides a sequential, hierarchical decision about the response variable based on the predictor data. A hierarchical model is defined by a series of questions that lead to a value when applied to a reading. Once the model is ready, it acts like a protocol in a series of “If then” conditions that produce a particular result from the provided data.
CART stands for classification and regression tree.
- Types
- Regression tree: response variable Y is numerical
- Classification tree: response variable Y is categorical
The decision tree models are classified as Classification trees when the target variable uses a discrete set of values and Regression trees when the response variable is binary. In our case the response variable Playoff has only two outcomes, that is whether team will qualify for the playoffs or not, hence we form a Classification tree. The goal of the decision tree is to select the optical choice at the end of each node. The decision to split at each node is made according to the metric known as purity. A node is 100% impure if split evenly 50/50 and 100% pure when the entire data belongs to a single class. In order to achieve the best model, impurity should be avoided.
Splitting the data in Training and Test data
The in-sample and out-of-sample prediction for regression tree:
set.seed(13439960)
sample_index <- sample(nrow(Nba),nrow(Nba)*0.70)
Nba.train = Nba[sample_index,]
Nba.test = Nba[-sample_index,]In-sample prediction \[sample_index <- sample(nrow(Nba),nrow(Nba)*0.70)\]
Out-of-sample prediction \[Nba.train = Nba[sample_index,]\] \[Nba.test = Nba[-sample_index,]\]
Dimensions of the Training data
dim(Nba.train)## [1] 584 21We observe that the Training sample consists of 0.7*835=584 rows and 20 variables/columns
Dimensions of the Test data
dim(Nba.test)## [1] 251 21We observe that the testing sample consists of 0.3*835=251 rows and 20 variables/columns
library(rpart)
library(rpart.plot)Classification tree with 12 predictors
The classification trees are a bit complicated, due to the presence of asymmetric cost function. We select the weight for the false negatives (predicting 0 when truth is 1) and the false positives (predicting 1 when truth is 0).
In our case we make the assumption that false negative cost equals the false positive cost therefore select symmetric cost function.
Also, In the rpart() funcation, the cp(complexity parameter) argument is one of the parameters that is used to control the compexity of the tree. The overall R-square value must increase by cp at each step and smaller the cp value, the larger (complex) tree rpart will attempt to fit. We have taken the default value for cp as 0.001 and would change this value after pruning.
Nba.rpart <- rpart(formula = Playoffs ~ W + PTS + oppPTS + X2PA + X3PA + FTA + AST + ORB + DRB + TOV + STL + BLK, data = Nba.train, method = "class",cp=0.001) # Method=class is req if the response var is not declared as factors
prp(Nba.rpart, extra = 1,fallen.leaves = TRUE)
In-sample performance
pred.train <- predict(Nba.rpart, type = "class")
table(Nba.train$Playoffs, pred.train, dnn = c("True", "Predicted"))## Predicted
## True 0 1
## 0 235 18
## 1 16 315Misclassification rate :
sum(ifelse(pred.train != Nba.train$Playoffs,1,0))/dim(Nba.train)[1]## [1] 0.0582Out of Sample performance
pred.test <- predict(Nba.rpart, Nba.test, type = "class")
table(Nba.test$Playoffs, pred.test, dnn = c("True", "Predicted"))## Predicted
## True 0 1
## 0 94 8
## 1 19 130Misclassification rate :
sum(ifelse(pred.test != Nba.test$Playoffs,1,0))/dim(Nba.test)[1]## [1] 0.108ROC and AUC
- ROC: In order to determine the overall measure of goodness of classification, we use the Receiver Operating Characteristic (ROC) curve. Rather than using an overall misclassification rate, it employs two measures – true positive fraction (TPF) sensitivity and false positive fraction (FPF) 1 - specificity
Nba.test.prob.rpart = predict(Nba.rpart,Nba.test, type="prob")
library(ROCR)
pred = prediction(Nba.test.prob.rpart[,2], Nba.test$Playoffs)
perf = performance(pred, "tpr", "fpr")
plot(perf, colorize=TRUE)
- AUC: Area under the curve, as per the industry standards any value of AUC above .70 is considered a good measure.
slot(performance(pred, "auc"), "y.values")[[1]]## [1] 0.963Prune the tree
We use pruning to avoid overfitting the training data. It is a technique that reduces the size of decision trees by removing sections of the tree that provide little value to classify instances. Pruning reduces the complexity of the final classifier, and hence improves predictive accuracy by the reduction of overfitting.
- Computing the cp value
plotcp(Nba.rpart)
Nba.rpart.pruned <- prune(Nba.rpart, cp = 0.098)
Nba.rpart.pruned <- rpart(formula = Playoffs ~ W + PTS + oppPTS + X2PA + X3PA + FTA + AST + ORB + DRB + TOV + STL + BLK, data = Nba.train, method = "class",cp=0.098)While selecting the best cp(Complexity parameter) we plot the cp graph and select the first cp value below the horizontal line. In our case that value turns out to be 0.098
Note: Method = class is required, if the response variable is not declared as factors
- Forming the tree with new cp value
We substitute the value of cp = 0.098 in the equation and plot the optimal tree.
We observe that the new tree with cp = 0.098 has optimal length of 2 nodes.
After Pruning the tree
We now compute misclassification rate for the In-sample and Out-of-sample data and determine the ROC and AUC values.
Our aim here is to compare both the misclassification values before and after pruning.
In-sample performance
pred.train <- predict(Nba.rpart.pruned, type = "class")
table(Nba.train$Playoffs, pred.train, dnn = c("True", "Predicted"))## Predicted
## True 0 1
## 0 219 34
## 1 14 317Misclassification rate :
sum(ifelse(pred.train != Nba.train$Playoffs,1,0))/dim(Nba.train)[1]## [1] 0.0822Out of Sample performance
pred.test <- predict(Nba.rpart.pruned, Nba.test, type = "class")
table(Nba.test$Playoffs, pred.test, dnn = c("True", "Predicted"))## Predicted
## True 0 1
## 0 89 13
## 1 6 143Misclassification rate :
sum(ifelse(pred.test != Nba.test$Playoffs,1,0))/dim(Nba.test)[1]## [1] 0.0757ROC and AUC
Now computing the Receiver Operating Characteristic (ROC) curve and Area under the curve (AUC) values for the new pruned tree.
ROC:
Nba.test.prob.rpart = predict(Nba.rpart.pruned,Nba.test, type="prob")
library(ROCR)
pred = prediction(Nba.test.prob.rpart[,2], Nba.test$Playoffs)
perf = performance(pred, "tpr", "fpr")
plot(perf, colorize=TRUE)
AUC:
slot(performance(pred, "auc"), "y.values")[[1]]## [1] 0.916Conclusion:
- Before Pruning
- The Misclassification rate for In-sample data: 0.05821918
- The Misclassification rate for In-sample data: 0.1075697
- Area under curve value: 0.9632188
- After Pruning
- The Misclassification rate for In-sample data: 0.08219178
- The Misclassification rate for In-sample data: 0.07569721
- Area under curve value: 0.9161403
We observe that even though the AUC value before pruning seems better than the one after pruning, the missclassification rate after pruning is more constant and classifies better. Therefore, we can conclude that pruning the tree helps.
Random Forest
Random forest with 12 predictor variables
Random Forest is an advanced tree model which is basically an extension of the Bagging model. As compared to Bagging, Random Forest performs better in terms of prediction accuracy. Random decision forest corrects for the decisiion tree’s habit of overfitting to the training data set. Random forest builds lots of bushy trees and then averages them and reduces the variance of the bushy trees by averaging.The idea of random forest is to select (randomly) n out of the total p predictors as candidate variables and by doing so, it decorrelates the tree which in turn reduces the variance when we aggregate the trees. Along with a reduction in the variance, it does not compromise the Bias (i.e.it does not increase the Bias).
library(randomForest)
num_of_predictors <- sqrt(12)
Nba.rf <- randomForest(Playoffs~W + PTS + oppPTS + X2PA + X3PA + FTA + AST + ORB + DRB + TOV + STL + BLK, data = Nba.train, family = gaussian, mtry = floor(num_of_predictors),ntree = 500,importance=TRUE)
Nba.rf##
## Call:
## randomForest(formula = Playoffs ~ W + PTS + oppPTS + X2PA + X3PA + FTA + AST + ORB + DRB + TOV + STL + BLK, data = Nba.train, family = gaussian, mtry = floor(num_of_predictors), ntree = 500, importance = TRUE)
## Type of random forest: regression
## Number of trees: 500
## No. of variables tried at each split: 3
##
## Mean of squared residuals: 0.0659
## % Var explained: 73.2The most important tuning parameter in random Forest is the variable mtry which is the number of predictors selected at each split of each tree. By default it is total no of predictors p/3 for a regression problem and square root of total predictors p for a classification problem. In our case, we can see from the summary it chooses 3 random predictor variables out of the 12 predictor variables during every split of each tree thus decorrelating the trees.The argument importance = TRUE allows us to view the variable importance. ntree specifies the number of trees to fit the model and by default it is 500. From the summary we can see that the MSR is nothing but the Out of bag error and it comes to be 0.06454635 and each observation is predicted using the average of trees that didn’t include it and hence these are de-biased estimates of prediction error.
varImpPlot(Nba.rf)
We observe that the most important variable in our equation is Win followed by oppPts and so on.
The fitted random Forest saves the OOB error for each ntree value ranging from 1 to 500 and it can be seen how the OOB error changes with the different values of ntree.
We observe that the OOB error decreases with an increase in the value of ntree.
Prediction on training sample i.e. In sample performance
NBA.rf.pred.train <- predict(Nba.rf, Nba.train)
NBA.rf.pred.train_classified <- as.numeric(NBA.rf.pred.train > 0.48)
table(Nba.train$Playoffs, NBA.rf.pred.train_classified, dnn=c("Truth","Predicted"))## Predicted
## Truth 0 1
## 0 253 0
## 1 0 331The in-sample mean squared error is observed to be 0.01398344 which is very less than the in sample performances obtained from the linear and the classification model.
sum(ifelse(NBA.rf.pred.train_classified != Nba.train$Playoffs,1,0))/dim(Nba.test)[1] #0.003984064 : Symmetric misclassification rate## [1] 0In sample ROC and AUC
library(ROCR)
pred = prediction(NBA.rf.pred.train_classified, Nba.train$Playoffs)
perf = performance(pred, "tpr", "fpr")
plot(perf, colorize=TRUE)
slot(performance(pred, "auc"), "y.values")[[1]] ## [1] 1The In-sample AUC value is: 0.9980237 which implies that our model has an accuracy rate of 99%
Prediction on testing sample i.e. Out of sample performance
NBA.rf.pred.test <- predict(Nba.rf, Nba.test)
NBA.rf.pred.test_classified <- as.numeric(NBA.rf.pred.test> 0.48)
table(Nba.test$Playoffs, NBA.rf.pred.test_classified, dnn=c("Truth","Predicted"))## Predicted
## Truth 0 1
## 0 93 9
## 1 9 140The out of sample error is 0.05468076 which is again the best value obtained as compared to the out of sample performance values from the linear and classification tree model.
sum(ifelse(NBA.rf.pred.test_classified != Nba.test$Playoffs,1,0))/dim(Nba.test)[1] #0.07569721 : Symmetric misclassification rate## [1] 0.0717Out-of-sample ROC and AUC
pred = prediction(NBA.rf.pred.test_classified, Nba.test$Playoffs)
perf = performance(pred, "tpr", "fpr")
plot(perf, colorize=TRUE)
slot(performance(pred, "auc"), "y.values")[[1]]## [1] 0.926The Out-of-sample AUC value is: 0.920779 which implies that our model has an accuracy rate of 92%
Comparison
We compare all the 4 techniques: Logistics regression model, Classification tree, Pruned Classification Tree and Random Forest, which we have used in our analysis.
Conclusion
We observe that the misclassification rate for the Random forest is minimum, making it the most suitable model for our prediction.
Using these techniques the NBA managers can scout the players during the off-season, who are suitable to the club’s style of play. They can concentrate on certain key areas that need to be worked on to gain an advantage over other NBA clubs.