Assignment One: Baseball

Data Exploration

This assignment reviews a baseball dataset, which looks at teams across a number of different features. We’ll be building a number of linear regression models and comparing their efficacy.

To start, we will explore the dataset: its shape, composition, and any information that may help with future data processing and model building.

## Warning: package 'corrplot' was built under R version 4.1.2
## Warning: package 'caret' was built under R version 4.1.2
## Warning: package 'kableExtra' was built under R version 4.1.2
## Warning: package 'caTools' was built under R version 4.1.2
## Warning: package 'car' was built under R version 4.1.2
## Warning: package 'carData' was built under R version 4.1.2
## Warning: package 'ggResidpanel' was built under R version 4.1.2
#eval <- getURL("https://raw.githubusercontent.com/cmm6/data608/main/moneyball-evaluation-data.csv",.opts=curlOptions(followlocation = TRUE)) 
#train <- getURL("https://raw.githubusercontent.com/cmm6/data608/main/moneyball-training-data.csv",.opts=curlOptions(followlocation = TRUE)) 
baseball_eval <- read.csv("https://raw.githubusercontent.com/cmm6/data608/main/moneyball-evaluation-data.csv", header=TRUE, sep = ",")
baseball_training <- read.csv("https://raw.githubusercontent.com/cmm6/data608/main/moneyball-training-data.csv", header=TRUE, sep = ",")

baseball_eval <- subset(baseball_eval, select = -c(INDEX) )
baseball_training = subset(baseball_training, select = -c(INDEX) )

print(dim(baseball_training))
## [1] 2276   16
print(head(baseball_training))
##   TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## 1          39           1445             194              39              13
## 2          70           1339             219              22             190
## 3          86           1377             232              35             137
## 4          70           1387             209              38              96
## 5          82           1297             186              27             102
## 6          75           1279             200              36              92
##   TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_BASERUN_CS
## 1             143             842              NA              NA
## 2             685            1075              37              28
## 3             602             917              46              27
## 4             451             922              43              30
## 5             472             920              49              39
## 6             443             973             107              59
##   TEAM_BATTING_HBP TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB
## 1               NA            9364               84              927
## 2               NA            1347              191              689
## 3               NA            1377              137              602
## 4               NA            1396               97              454
## 5               NA            1297              102              472
## 6               NA            1279               92              443
##   TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## 1             5456            1011               NA
## 2             1082             193              155
## 3              917             175              153
## 4              928             164              156
## 5              920             138              168
## 6              973             123              149

We’ve dropped the index, which the data dictionary confirms is irrelevant to the target variable. As such, the evaluation set has 2276 observations, with 16 columns. The first, Target_Wins is the target of future linear regression modeling.

First, we’ll use summary() to get a sense of the type and values for each field.

Right away, we can see several fields have NA values, including the majority of TEAM_BATTING_HBP. We can try different ways to handle these in later model development, e.g. removing entirely vs. replacing with mean, etc.

summary(baseball_training)
##   TARGET_WINS     TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B 
##  Min.   :  0.00   Min.   : 891   Min.   : 69.0   Min.   :  0.00  
##  1st Qu.: 71.00   1st Qu.:1383   1st Qu.:208.0   1st Qu.: 34.00  
##  Median : 82.00   Median :1454   Median :238.0   Median : 47.00  
##  Mean   : 80.79   Mean   :1469   Mean   :241.2   Mean   : 55.25  
##  3rd Qu.: 92.00   3rd Qu.:1537   3rd Qu.:273.0   3rd Qu.: 72.00  
##  Max.   :146.00   Max.   :2554   Max.   :458.0   Max.   :223.00  
##                                                                  
##  TEAM_BATTING_HR  TEAM_BATTING_BB TEAM_BATTING_SO  TEAM_BASERUN_SB
##  Min.   :  0.00   Min.   :  0.0   Min.   :   0.0   Min.   :  0.0  
##  1st Qu.: 42.00   1st Qu.:451.0   1st Qu.: 548.0   1st Qu.: 66.0  
##  Median :102.00   Median :512.0   Median : 750.0   Median :101.0  
##  Mean   : 99.61   Mean   :501.6   Mean   : 735.6   Mean   :124.8  
##  3rd Qu.:147.00   3rd Qu.:580.0   3rd Qu.: 930.0   3rd Qu.:156.0  
##  Max.   :264.00   Max.   :878.0   Max.   :1399.0   Max.   :697.0  
##                                   NA's   :102      NA's   :131    
##  TEAM_BASERUN_CS TEAM_BATTING_HBP TEAM_PITCHING_H TEAM_PITCHING_HR
##  Min.   :  0.0   Min.   :29.00    Min.   : 1137   Min.   :  0.0   
##  1st Qu.: 38.0   1st Qu.:50.50    1st Qu.: 1419   1st Qu.: 50.0   
##  Median : 49.0   Median :58.00    Median : 1518   Median :107.0   
##  Mean   : 52.8   Mean   :59.36    Mean   : 1779   Mean   :105.7   
##  3rd Qu.: 62.0   3rd Qu.:67.00    3rd Qu.: 1682   3rd Qu.:150.0   
##  Max.   :201.0   Max.   :95.00    Max.   :30132   Max.   :343.0   
##  NA's   :772     NA's   :2085                                     
##  TEAM_PITCHING_BB TEAM_PITCHING_SO  TEAM_FIELDING_E  TEAM_FIELDING_DP
##  Min.   :   0.0   Min.   :    0.0   Min.   :  65.0   Min.   : 52.0   
##  1st Qu.: 476.0   1st Qu.:  615.0   1st Qu.: 127.0   1st Qu.:131.0   
##  Median : 536.5   Median :  813.5   Median : 159.0   Median :149.0   
##  Mean   : 553.0   Mean   :  817.7   Mean   : 246.5   Mean   :146.4   
##  3rd Qu.: 611.0   3rd Qu.:  968.0   3rd Qu.: 249.2   3rd Qu.:164.0   
##  Max.   :3645.0   Max.   :19278.0   Max.   :1898.0   Max.   :228.0   
##                   NA's   :102                        NA's   :286

One of the key requirements of linear regression is a linear relationship between the explanatory and target variables. Digging into these relationships using scatter plot, it looks like TEAM_BATTING_SO, TEAM_BATTING_BB, and TEAM_PITCHING_SO have clear clustering around 0 and may not be linear.

pairs(baseball_training, lower.panel = NULL, cex = 0.4, cex.labels=0.5)

Digging further in with scatterplots versus the target, as well as box plots, there is clear clustering of some values around 0 for both TEAM_BATTING_BB and TEAM_BATTING_SO, and nearly all TEAM_PITCHING_SO values are 0. The boxplot for the latter shows a few outliers, but otherwise the majority of the data tightly clustered at low value.

boxplot(baseball_training$TEAM_BATTING_BB)

plot(baseball_training$TEAM_BATTING_BB,baseball_training$TARGET_WINS)

boxplot(baseball_training$TEAM_BATTING_SO)

plot(baseball_training$TEAM_BATTING_SO,baseball_training$TARGET_WINS)

boxplot(baseball_training$TEAM_PITCHING_SO)

plot(baseball_training$TEAM_PITCHING_SO,baseball_training$TARGET_WINS)

These features and other nonlinearity can be kept in mind in executing the model.

Also valauble is building a correlation matrix. This serves two purposes - to show which features correlate highly to the Target variable, and to reduce potential for collinearity if two features are not offering distinct value to the model. NA values are omitted.

training_cor <- cor(na.omit(baseball_training))
corrplot(training_cor, method = 'number',number.cex=7/ncol(baseball_training))

In terms of collinearity, there is large correlation between: * TEAM_PITCHING_H and TEAM_BATTING_H * TEAM_PITCHING_HR and TEAM_BATTING_HR * TEAM_PITCHING_BB and TEAM_BATTING_BB * TEAM_PITCHING_SO and TEAM_BATTING_SO

In the final model, we can explore keeping just 1 of each of these pairs.

In terms of correlation to the TARGET_WINS, TEAM_BATTING_H, TEAM_BATTING_BB, TEAM_PITCHING_H, and TEAM_PITCHING_BB have highest positive correlation. These are also fields with mutual correlation, which is helpful to note going into the model development and data preparation.

Finally, we can create a baseline model, that takes every variable unadjusted. This can act as a baseline to outperform as we iterate with more elaborate models.

lm_baseline <- lm(TARGET_WINS~.,baseball_training)
summary(lm_baseline)
## 
## Call:
## lm(formula = TARGET_WINS ~ ., data = baseball_training)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.8708  -5.6564  -0.0599   5.2545  22.9274 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      60.28826   19.67842   3.064  0.00253 ** 
## TEAM_BATTING_H    1.91348    2.76139   0.693  0.48927    
## TEAM_BATTING_2B   0.02639    0.03029   0.871  0.38484    
## TEAM_BATTING_3B  -0.10118    0.07751  -1.305  0.19348    
## TEAM_BATTING_HR  -4.84371   10.50851  -0.461  0.64542    
## TEAM_BATTING_BB  -4.45969    3.63624  -1.226  0.22167    
## TEAM_BATTING_SO   0.34196    2.59876   0.132  0.89546    
## TEAM_BASERUN_SB   0.03304    0.02867   1.152  0.25071    
## TEAM_BASERUN_CS  -0.01104    0.07143  -0.155  0.87730    
## TEAM_BATTING_HBP  0.08247    0.04960   1.663  0.09815 .  
## TEAM_PITCHING_H  -1.89096    2.76095  -0.685  0.49432    
## TEAM_PITCHING_HR  4.93043   10.50664   0.469  0.63946    
## TEAM_PITCHING_BB  4.51089    3.63372   1.241  0.21612    
## TEAM_PITCHING_SO -0.37364    2.59705  -0.144  0.88577    
## TEAM_FIELDING_E  -0.17204    0.04140  -4.155 5.08e-05 ***
## TEAM_FIELDING_DP -0.10819    0.03654  -2.961  0.00349 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.467 on 175 degrees of freedom
##   (2085 observations deleted due to missingness)
## Multiple R-squared:  0.5501, Adjusted R-squared:  0.5116 
## F-statistic: 14.27 on 15 and 175 DF,  p-value: < 2.2e-16

Data Preparation

For Data preparation, there are two things we want to focus on, imputing missing values and making sure the data is ready for the models. The first item we can tackle is finding and replacing missing values. First, we want to find the percentage of missing values. The largest is TEAM_BATTING_HBP with 91% missing data while the second largest variables is TEAM_BASERUN_CS with 33% missing. We also have TEAM_FIELDING_DP with 12% missing, TEAM_BASERUN_SB with roughly 5% and both TEAM_PITCHING_SO and TEAM_BATTING_SO with 4.5% each.

For Team Batting HBP, I was originally going to fill in the NAs with 0 since there is a limited chance that it occurs and seems more like a data input error (the min is 29 but if it is not common should be 0), But since it is an issue that relates to how the games are recorded (this was not recoreded in the early days of baseball) we will get the mean data instead. And for TEAM_FIELDING_DP, I will do the same since it seems to have the same error. For all the other NAs it would make sense to fill in with the mean since there is not too much missing data or the min is 0.0.

Now, another thing we want to check for is if there are any issues with collinearity.

training_cor <- cor(na.omit(baseball_training))
corrplot(training_cor,method = 'color' ,order = 'hclust', addrect = 2)

From this heatmap we can see that there is a pairwise relationship between TEAM_PITCHING_HR and TEAM_BATTING_HR (close to 1), but since they are integral to our data, we will not get rid of either.

Finally, I will add several new columns, One for Predicted runs for season, Team Fielding, and Team Pitching

We need to do the same for baseball eval

And now, we are ready to split the data and build the model.

set.seed(678)

split <- sample.split(baseball_training$TARGET_WINS, SplitRatio = 0.8)
training_set <- subset(baseball_training, split == TRUE)
test_set <- subset(baseball_training, split == FALSE)

We split the data into train (80%) and test data (20%)

Build Models

Model 1

lm2 <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
           TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + 
           TEAM_BASERUN_SB +  TEAM_PITCHING_H +
            TEAM_PITCHING_BB + TEAM_PITCHING_SO +
           TEAM_FIELDING_E + TEAM_FIELDING_DP, data = training_set)

summary(lm2)
## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + 
##     TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + 
##     TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP, data = training_set)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -49.627  -8.472   0.049   8.515  56.544 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      22.2410579  5.9344238   3.748 0.000184 ***
## TEAM_BATTING_H    0.0490852  0.0041243  11.902  < 2e-16 ***
## TEAM_BATTING_2B  -0.0155517  0.0103227  -1.507 0.132100    
## TEAM_BATTING_3B   0.0658366  0.0183529   3.587 0.000343 ***
## TEAM_BATTING_HR   0.0568688  0.0110121   5.164 2.68e-07 ***
## TEAM_BATTING_BB   0.0075975  0.0056821   1.337 0.181357    
## TEAM_BATTING_SO  -0.0062301  0.0028160  -2.212 0.027066 *  
## TEAM_BASERUN_SB   0.0244755  0.0048265   5.071 4.36e-07 ***
## TEAM_PITCHING_H  -0.0009467  0.0003790  -2.498 0.012590 *  
## TEAM_PITCHING_BB  0.0023327  0.0036950   0.631 0.527908    
## TEAM_PITCHING_SO  0.0025050  0.0008933   2.804 0.005100 ** 
## TEAM_FIELDING_E  -0.0199673  0.0026315  -7.588 5.17e-14 ***
## TEAM_FIELDING_DP -0.1234703  0.0146783  -8.412  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.23 on 1813 degrees of freedom
## Multiple R-squared:  0.3232, Adjusted R-squared:  0.3187 
## F-statistic: 72.15 on 12 and 1813 DF,  p-value: < 2.2e-16
vif(lm2)
##   TEAM_BATTING_H  TEAM_BATTING_2B  TEAM_BATTING_3B  TEAM_BATTING_HR 
##         3.780657         2.473079         2.888639         4.667396 
##  TEAM_BATTING_BB  TEAM_BATTING_SO  TEAM_BASERUN_SB  TEAM_PITCHING_H 
##         5.480579         5.016713         1.804216         3.561706 
## TEAM_PITCHING_BB TEAM_PITCHING_SO  TEAM_FIELDING_E TEAM_FIELDING_DP 
##         4.540903         2.903665         4.161496         1.333660

The most common way to detect multicollinearity is by using the variance inflation factor (VIF), which measures the correlation and strength of correlation between the predictor variables in a regression model.

A value greater than 5 indicates potentially severe correlation between a given predictor variable and other predictor variables in the model.

From our earlier analysis , we already noticed that a correlation exist between TEAM_PITCHING_H * TEAM_BATTING_H and TEAM_BATTING_BB * TEAM_PITCHING_BB and TEAM_PITCHING_SO * TEAM_BATTING_SO

Model Diagnostics 

resid_panel(lm2, plots='default', smoother = TRUE)
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'

par(mfrow=c(2,2))
plot(lm2)

Model 2

In this model we are going to remove the correlated predictors

lm3 <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
           TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + 
           TEAM_BASERUN_SB + TEAM_FIELDING_E, data = training_set)


summary(lm3) 
## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + 
##     TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + 
##     TEAM_BASERUN_SB + TEAM_FIELDING_E, data = training_set)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.078  -9.154   0.002   8.508  60.798 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      8.0105200  5.7444624   1.394  0.16334    
## TEAM_BATTING_H   0.0460224  0.0041572  11.071  < 2e-16 ***
## TEAM_BATTING_2B -0.0134959  0.0103968  -1.298  0.19442    
## TEAM_BATTING_3B  0.0813442  0.0183731   4.427 1.01e-05 ***
## TEAM_BATTING_HR  0.0352787  0.0109966   3.208  0.00136 ** 
## TEAM_BATTING_BB  0.0033577  0.0037405   0.898  0.36949    
## TEAM_BATTING_SO  0.0005925  0.0025871   0.229  0.81889    
## TEAM_BASERUN_SB  0.0323915  0.0047867   6.767 1.77e-11 ***
## TEAM_FIELDING_E -0.0229460  0.0021954 -10.452  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.53 on 1817 degrees of freedom
## Multiple R-squared:  0.2909, Adjusted R-squared:  0.2877 
## F-statistic: 93.15 on 8 and 1817 DF,  p-value: < 2.2e-16
vif(lm3)
##  TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR TEAM_BATTING_BB 
##        3.674111        2.399495        2.769016        4.451733        2.271714 
## TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_FIELDING_E 
##        4.050009        1.697371        2.770376

Model Diagnostics 

resid_panel(lm3, plots='default', smoother = TRUE)
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'

par(mfrow=c(2,2))
plot(lm3)

Model 3

We will do further clean up based on p-value

lm4 <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B +
           TEAM_BATTING_HR +  TEAM_BASERUN_SB + TEAM_FIELDING_E, data = training_set)


summary(lm4) 
## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + 
##     TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BASERUN_SB + TEAM_FIELDING_E, 
##     data = training_set)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -52.943  -9.046  -0.028   8.581  60.883 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     10.376668   3.450126   3.008  0.00267 ** 
## TEAM_BATTING_H   0.045480   0.003481  13.067  < 2e-16 ***
## TEAM_BATTING_2B -0.012592   0.010003  -1.259  0.20827    
## TEAM_BATTING_3B  0.082751   0.018220   4.542 5.94e-06 ***
## TEAM_BATTING_HR  0.038935   0.008380   4.647 3.62e-06 ***
## TEAM_BASERUN_SB  0.033813   0.004331   7.806 9.86e-15 ***
## TEAM_FIELDING_E -0.024096   0.001771 -13.603  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.52 on 1819 degrees of freedom
## Multiple R-squared:  0.2905, Adjusted R-squared:  0.2882 
## F-statistic: 124.2 on 6 and 1819 DF,  p-value: < 2.2e-16
vif(lm4)
##  TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR TEAM_BASERUN_SB 
##        2.577105        2.222817        2.724804        2.586646        1.390762 
## TEAM_FIELDING_E 
##        1.804790

Model Diagnostics 

resid_panel(lm4, plots='default', smoother = TRUE)
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'

par(mfrow=c(2,2))
plot(lm4)

Select Models

In order to select the best model, we will look at the evaluation metrics (RSE, R-Squared, Adj. R-Squared, F-Statistic, and AIC) for all three models and compare them.

Extract Model Evaluation Metrics
Model 1

# extract the rse, r.squared, adj.r.squared, F-statistic, and AIC for model 1
model1_rse <- round(summary(lm2)$sigma, 4) 
model1_r_squared <- round(summary(lm2)$r.squared, 4)
model1_adj_r_squared <- round(summary(lm2)$adj.r.squared, 4)
model1_f_statistic <- round(summary(lm2)$fstatistic[1], 4)
model1_aic <- round(AIC(lm2), 4)

model1_metrics <- c(model1_rse, model1_r_squared, model1_adj_r_squared, model1_f_statistic,
                    model1_aic)

Model 2

# extract the rse, r.squared, adj.r.squared, F-statistic, and AIC for model 2
model2_rse <- round(summary(lm3)$sigma, 4) 
model2_r_squared <- round(summary(lm3)$r.squared, 4)
model2_adj_r_squared <- round(summary(lm3)$adj.r.squared, 4)
model2_f_statistic <- round(summary(lm3)$fstatistic[1], 4)
model2_aic <- round(AIC(lm3), 4)

model2_metrics <- c(model2_rse, model2_r_squared, model2_adj_r_squared, model2_f_statistic,
                    model2_aic)

Model 3

# extract the rse, r.squared, adj.r.squared, F-statistic, and AIC for model 3
model3_rse <- round(summary(lm4)$sigma, 4) 
model3_r_squared <- round(summary(lm4)$r.squared, 4)
model3_adj_r_squared <- round(summary(lm4)$adj.r.squared, 4)
model3_f_statistic <- round(summary(lm4)$fstatistic[1], 4)
model3_aic <- round(AIC(lm4), 4)

model3_metrics <- c(model3_rse, model3_r_squared, model3_adj_r_squared, model3_f_statistic,
                    model3_aic)

Combine all metrics

metrics <- data.frame(model1_metrics, model2_metrics, model3_metrics)
metrics_rownames <- c("RSE", "RSquared", "Adj-RSquared", "F-Statistic", "AIC")
metrics_headers <- c("Model1", "Model2", "Model3")
rownames(metrics) <- metrics_rownames
colnames(metrics) <- metrics_headers
metrics <- metrics %>% kbl() %>% kable_styling()
metrics
Model1 Model2 Model3
RSE 13.2276 13.5252 13.5207
RSquared 0.3232 0.2909 0.2905
Adj-RSquared 0.3187 0.2877 0.2882
F-Statistic 72.1506 93.1534 124.1515
AIC 14627.4980 14704.7697 14701.5802


The model diagnostic plots for all three models appear to be fairly similar. The Q-Q plot shows that the distribution is nearly normal and the residual vs fitted plot also shows no specific patter to worry about. Also, looking at the metrics of all three models, we can see that the values are fairly close. The RSE values are around 13 for all three models and the R-Square and Adj. R-Squared are about 30% for all three models. Furthermore, the AIC (Akaike Information Criteria) for all models are fairly close as well. However, the F-Statistic for Model3 is well above those for models 1 and 2. We can see that as we kept improving the model by selecting features whose p-values are significant, other model metrics remain fairly similar but the F-Statistic improved significantly from model1 to model3. Hence, we select model3 since it has a higher F-Statistic and it’s a simpler model than the others and contains less features that are almost all statistically significant.

Summary

Now that we have the model selected, let’s run it against the training data and see how accurate all of them are to make sure that model three is the best is by looking at the RSME. Then run it against the data we have in the Eval Training Set.

test_predictions = predict(lm2, newdata=test_set, interval ="predict")


test_set_1 <- cbind(test_set,test_predictions)

# RMSE
paste0("The Root Sqaure Mean Error for model one is: ", round(sqrt(mean((test_set_1$TARGET_WINS - test_set_1$fit)^2)),2))
## [1] "The Root Sqaure Mean Error for model one is: 12.3"
test_predictions = predict(lm3, newdata=test_set, interval ="predict")

test_set_2 <- cbind(test_set,test_predictions)

# RMSE
paste0("The Root Sqaure Mean Error for model two is: ", round(sqrt(mean((test_set_2$TARGET_WINS - test_set_2$fit)^2)),2))
## [1] "The Root Sqaure Mean Error for model two is: 12.5"
test_predictions = predict(lm4, newdata=test_set, interval ="predict")

test_set_3 <- cbind(test_set,test_predictions)

# RMSE
paste0("The Root Sqaure Mean Error for model three is: ", round(sqrt(mean((test_set_3$TARGET_WINS - test_set_3$fit)^2)),2))
## [1] "The Root Sqaure Mean Error for model three is: 12.51"

Linear Model One has the lowest RSME, but overall they are not too far apart, meaning they will perform similar so we will still work with Linear Model Three.

test_predictions = predict(lm4, newdata=baseball_eval, interval ="predict")

baseball_eval_final <- cbind(test_predictions,baseball_eval)

Here is the model run with Baseball Eva. Fit is the predicted value, while lwr and upr are the values that fall within a 95% confidence degree.

baseball_eval_final %>%
  kbl() %>%
  kable_paper("hover", full_width = F, html_font = "Times New Roman", font_size = 10) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"))
fit lwr upr TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_BASERUN_CS TEAM_BATTING_HBP TEAM_PITCHING_H TEAM_PITCHING_HR TEAM_PITCHING_BB TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP PRED_RUNS TEAM_FIELDING
67.90646 41.350848 94.46208 1209 170 33 83 447 1080.0000 62.0000 50.00000 62.36842 1209 83 447 1080.000 140 156.000 907.3684 16.000
68.40512 41.839095 94.97114 1221 151 29 88 516 929.0000 54.0000 39.00000 62.36842 1221 88 516 929.000 135 164.000 939.3684 29.000
75.77338 49.214303 102.33247 1395 183 29 93 509 816.0000 59.0000 47.00000 62.36842 1395 93 509 816.000 156 153.000 982.3684 -3.000
87.08601 60.533402 113.63861 1539 309 29 159 486 914.0000 148.0000 57.00000 42.00000 1539 159 486 914.000 124 154.000 1230.0000 30.000
68.70025 42.145847 95.25464 1445 203 68 5 95 416.0000 123.7033 52.31977 62.36842 3902 14 257 1123.000 616 130.000 771.3914 -486.000
67.66162 41.094911 94.22833 1431 236 53 10 215 377.0000 123.7033 52.31977 62.36842 2793 20 420 736.000 572 105.000 957.3914 -467.000
79.18169 52.572600 105.79078 1430 219 55 37 568 527.0000 365.0000 52.31977 62.36842 1544 40 613 569.000 490 146.057 1403.6882 -343.943
74.48895 47.913359 101.06453 1385 158 42 33 356 609.0000 185.0000 52.31977 62.36842 1626 39 418 715.000 328 104.000 950.6882 -224.000
72.38324 45.820956 98.94553 1259 177 78 23 466 689.0000 150.0000 52.31977 62.36842 1342 25 497 734.000 226 132.000 1039.6882 -94.000
74.30083 47.753170 100.84848 1397 212 42 58 452 584.0000 52.0000 52.31977 62.36842 1489 62 482 622.000 184 145.000 960.6882 -39.000
74.81810 48.264837 101.37137 1427 243 40 50 495 640.0000 64.0000 52.31977 62.36842 1501 53 521 673.000 200 183.000 1032.6882 -17.000
84.35026 57.800401 110.90012 1496 239 55 164 462 670.0000 48.0000 28.00000 62.36842 1574 173 486 705.000 150 178.000 1082.3684 28.000
81.85578 55.276351 108.43522 1420 223 57 186 511 751.0000 31.0000 21.00000 62.36842 1494 196 538 790.000 137 167.000 1118.3684 30.000
80.42991 53.878734 106.98109 1460 232 22 176 503 680.0000 27.0000 8.00000 62.36842 1536 185 529 715.000 125 160.000 1056.3684 35.000
78.62753 52.069829 105.18523 1411 195 22 141 485 665.0000 59.0000 48.00000 62.36842 1411 141 485 665.000 115 114.000 1012.3684 -1.000
80.02598 53.464345 106.58761 1434 192 30 153 434 747.0000 57.0000 46.00000 62.36842 1434 153 434 747.000 146 180.000 974.3684 34.000
72.80680 46.265018 99.34858 1297 204 22 130 491 1008.0000 84.0000 55.00000 62.36842 1313 132 497 1021.000 154 126.000 1054.3684 -28.000
80.92890 54.389187 107.46861 1446 284 25 166 565 1041.0000 77.0000 39.00000 55.00000 1464 168 572 1054.000 115 172.000 1218.0000 57.000
68.74721 42.189647 95.30478 1276 162 52 17 383 709.3361 138.0000 52.31977 62.36842 1351 18 406 799.668 301 83.000 889.6882 -218.000
92.24734 65.691883 118.80279 1715 322 72 116 527 397.0000 90.0000 83.00000 62.36842 1816 123 558 420.000 232 174.000 1303.3684 -58.000
81.92984 55.363319 108.49636 1520 295 68 49 628 459.0000 77.0000 49.00000 62.36842 1620 52 669 489.000 166 158.000 1269.3684 -8.000
84.39483 57.839352 110.95030 1597 291 38 98 629 563.0000 54.0000 43.00000 62.36842 1702 104 670 600.000 155 174.000 1256.3684 19.000
79.90711 53.361572 106.45265 1453 256 67 105 653 651.0000 40.0000 41.00000 62.36842 1559 113 701 698.000 179 153.000 1272.3684 -26.000
73.71380 47.172487 100.25511 1378 225 26 118 533 677.0000 18.0000 36.00000 62.36842 1450 124 561 712.000 160 174.000 1046.3684 14.000
84.21949 57.678895 110.76008 1516 277 24 152 431 902.0000 89.0000 36.00000 54.00000 1516 152 431 902.000 105 164.000 1063.0000 59.000
87.19068 60.637675 113.74369 1556 288 20 164 474 878.0000 121.0000 32.00000 73.00000 1556 164 474 878.000 102 156.000 1172.0000 54.000
53.36989 26.628296 80.11148 1499 183 28 3 83 0.0000 123.7033 52.31977 62.36842 5167 10 286 0.000 1224 146.057 738.3914 -1077.943
76.81684 50.265541 103.36815 1464 263 58 47 385 479.0000 63.0000 66.00000 62.36842 1540 49 405 504.000 232 146.000 964.3684 -86.000
81.49095 54.898005 108.08389 1558 318 66 32 634 439.0000 83.0000 64.00000 62.36842 1639 34 667 462.000 218 130.000 1292.3684 -88.000
76.03297 49.426308 102.63963 1502 308 36 39 432 602.0000 45.0000 46.00000 62.36842 1601 42 460 642.000 199 135.000 996.3684 -64.000
86.87193 60.322680 113.42118 1596 320 58 130 718 596.0000 70.0000 54.00000 62.36842 1679 137 755 627.000 178 146.000 1449.3684 -32.000
84.99018 58.457558 111.52281 1546 260 59 110 630 541.0000 72.0000 65.00000 62.36842 1648 117 671 577.000 167 166.000 1299.3684 -1.000
82.70763 56.169930 109.24534 1516 282 53 115 723 695.0000 47.0000 38.00000 62.36842 1595 121 761 731.000 146 174.000 1358.3684 28.000
83.67289 57.133747 110.21204 1550 275 47 146 765 723.0000 29.0000 20.00000 62.36842 1631 154 805 761.000 178 177.000 1384.3684 -1.000
81.32920 54.786096 107.87231 1447 260 54 148 532 935.0000 39.0000 33.00000 62.36842 1465 150 539 947.000 130 154.000 1135.3684 24.000
81.30176 54.747265 107.85626 1450 252 28 203 594 855.0000 50.0000 48.00000 62.36842 1450 203 594 855.000 156 131.000 1237.3684 -25.000
75.72511 49.191270 102.25896 1347 239 36 130 546 897.0000 69.0000 31.00000 62.36842 1408 136 571 938.000 136 147.000 1138.3684 11.000
90.08979 63.511035 116.66855 1561 260 56 214 531 911.0000 66.0000 47.00000 62.36842 1571 215 534 917.000 133 163.000 1239.3684 30.000
84.70025 58.142547 111.25795 1578 252 26 135 567 780.0000 48.0000 47.00000 62.36842 2367 203 851 1170.000 137 162.000 1421.3684 25.000
88.40230 61.845815 114.95878 1598 259 45 181 500 842.0000 38.0000 25.00000 62.36842 1598 181 500 842.000 143 128.000 1110.3684 -15.000
80.04253 53.480584 106.60448 1497 322 21 145 599 711.0000 41.0000 34.00000 62.36842 1506 146 603 715.000 130 147.000 1228.3684 17.000
85.84309 59.294736 112.39145 1569 310 39 124 623 728.0000 65.0000 36.00000 62.36842 1569 124 623 728.000 93 123.000 1259.3684 30.000
29.18663 2.252219 56.12103 1119 118 33 7 37 0.0000 123.7033 52.31977 62.36842 4120 26 136 0.000 1568 146.057 532.3914 -1421.943
100.22825 73.507943 126.94855 1609 196 120 62 781 599.0000 536.0000 52.31977 62.36842 1931 74 937 719.000 470 146.057 1965.6882 -323.943
89.23984 62.607181 115.87249 1514 175 70 80 615 612.0000 392.0000 52.31977 62.36842 1803 95 733 729.000 413 146.057 1564.6882 -266.943
92.24573 65.666171 118.82528 1657 237 119 41 593 334.0000 325.0000 52.31977 62.36842 2114 52 756 426.000 537 146.057 1592.6882 -390.943
97.46780 70.858826 124.07677 1746 213 106 69 526 429.0000 324.0000 52.31977 62.36842 2176 86 655 535.000 500 146.057 1481.6882 -353.943
74.62464 48.077078 101.17221 1319 224 70 56 416 677.0000 176.0000 131.00000 62.36842 1397 59 440 717.000 284 100.000 1159.3684 -184.000
70.86662 44.311333 97.42191 1293 204 70 18 437 630.0000 134.0000 52.31977 62.36842 1360 19 460 663.000 281 127.000 1000.6882 -154.000
77.57368 51.031722 104.11565 1420 235 70 36 450 443.0000 121.0000 136.00000 62.36842 1494 38 473 466.000 237 118.000 1133.3684 -119.000
80.04953 53.507420 106.59165 1496 269 54 76 412 500.0000 55.0000 52.31977 62.36842 1574 80 433 526.000 177 171.000 1001.6882 -6.000
85.62533 59.054371 112.19629 1625 289 38 80 517 486.0000 72.0000 52.31977 62.36842 1709 84 544 511.000 154 164.000 1137.6882 10.000
78.87609 52.337103 105.41508 1391 239 50 145 499 1041.0000 70.0000 49.00000 62.36842 1391 145 499 1041.000 162 147.000 1114.3684 -15.000
74.94712 48.401519 101.49272 1319 203 43 130 415 854.0000 41.0000 30.00000 62.36842 1319 130 415 854.000 119 149.000 924.3684 30.000
77.40231 50.870833 103.93379 1411 251 35 107 471 912.0000 93.0000 64.00000 62.36842 1411 107 471 912.000 174 149.000 1083.3684 -25.000
79.47383 52.940251 106.00741 1420 221 41 104 417 816.0000 77.0000 51.00000 62.36842 1420 104 417 816.000 114 142.000 973.3684 28.000
90.00851 63.432941 116.58409 1552 206 106 38 566 401.0000 334.0000 52.31977 62.36842 1849 45 674 478.000 411 119.000 1472.6882 -292.000
74.22821 47.663136 100.79329 1280 203 72 15 392 616.0000 227.0000 52.31977 62.36842 1346 16 412 648.000 250 100.000 1043.6882 -150.000
63.65799 37.068826 90.24716 1120 122 61 7 427 709.3361 194.0000 52.31977 62.36842 1186 7 452 799.668 332 106.000 950.6882 -226.000
78.91953 52.370422 105.46864 1390 183 84 18 445 709.3361 216.0000 52.31977 62.36842 1462 19 468 799.668 304 107.000 1083.6882 -197.000
86.77523 60.223689 113.32677 1554 252 81 29 494 414.0000 174.0000 52.31977 62.36842 1798 34 572 479.000 200 134.000 1222.6882 -66.000
77.99621 51.445187 104.54722 1410 218 69 45 738 627.0000 65.0000 58.00000 62.36842 1483 47 776 660.000 142 189.000 1293.3684 47.000
84.70522 58.166860 111.24358 1507 262 28 159 573 907.0000 107.0000 52.00000 62.36842 1516 160 577 913.000 126 132.000 1247.3684 6.000
85.37853 58.805571 111.95149 1481 284 19 242 499 1030.0000 78.0000 51.00000 63.00000 1481 242 499 1030.000 100 167.000 1236.0000 67.000
85.19424 58.638887 111.74960 1450 253 23 200 435 1002.0000 137.0000 67.00000 79.00000 1450 200 435 1002.000 94 166.000 1194.0000 72.000
97.60697 70.956176 124.25776 1637 260 93 26 487 288.0000 446.0000 52.31977 62.36842 2088 33 621 367.000 321 146.057 1560.6882 -174.943
77.07514 50.538980 103.61130 1436 202 82 44 376 681.0000 160.0000 52.31977 62.36842 1674 51 438 794.000 414 119.000 1040.6882 -295.000
84.73351 58.182891 111.28413 1600 218 89 21 344 538.0000 152.0000 52.31977 62.36842 1851 24 398 623.000 373 137.000 992.6882 -236.000
78.88174 52.312859 105.45063 1348 168 76 23 506 709.3361 296.0000 52.31977 62.36842 1427 24 536 799.668 327 127.000 1213.6882 -200.000
87.19457 60.616352 113.77279 1460 191 111 22 612 629.0000 306.0000 52.31977 62.36842 1546 23 648 666.000 314 114.000 1392.6882 -200.000
89.93102 63.305242 116.55680 1621 255 126 37 478 350.0000 54.0000 52.31977 62.36842 1705 39 503 368.000 193 168.000 1089.6882 -25.000
77.01948 50.462422 103.57654 1433 241 49 45 468 501.0000 52.0000 52.31977 62.36842 1507 47 492 527.000 127 203.000 993.6882 76.000
81.14025 54.598206 107.68230 1440 232 48 155 586 679.0000 49.0000 32.00000 62.36842 1515 163 616 714.000 144 204.000 1194.3684 60.000
85.55268 58.946344 112.15902 1479 211 34 232 555 799.0000 47.0000 23.00000 62.36842 1556 244 584 841.000 119 155.000 1193.3684 36.000
84.00933 57.461042 110.55762 1573 281 36 106 379 938.0000 59.0000 55.00000 62.36842 1573 106 379 938.000 144 144.000 978.3684 0.000
87.17014 60.608026 113.73226 1558 224 42 171 474 1042.0000 79.0000 56.00000 62.36842 1558 171 474 1042.000 168 158.000 1108.3684 -10.000
81.57644 55.043069 108.10981 1385 225 46 130 637 961.0000 147.0000 66.00000 62.36842 1457 137 670 1011.000 116 150.000 1346.3684 34.000
81.56494 55.026766 108.10312 1419 250 27 164 488 1006.0000 124.0000 56.00000 62.36842 1419 164 488 1006.000 125 131.000 1171.3684 6.000
71.85034 45.295755 98.40493 1284 198 61 19 383 709.3361 186.0000 52.31977 62.36842 1351 20 403 799.668 270 100.000 981.6882 -170.000
78.16604 51.613431 104.71865 1403 200 68 10 390 709.3361 201.0000 52.31977 62.36842 1495 11 416 799.668 262 119.000 1009.6882 -143.000
85.65305 59.064226 112.24187 1631 358 48 105 553 455.0000 55.0000 34.00000 62.36842 1716 110 582 479.000 179 173.000 1244.3684 -6.000
90.25997 63.677723 116.84222 1666 343 82 98 487 600.0000 67.0000 57.00000 62.36842 1764 104 516 635.000 184 156.000 1225.3684 -28.000
97.55760 70.951952 124.16325 1804 376 86 129 541 494.0000 69.0000 56.00000 62.36842 1898 136 569 520.000 191 162.000 1347.3684 -29.000
81.35558 54.805333 107.90583 1534 284 53 74 539 624.0000 50.0000 44.00000 62.36842 1614 78 567 656.000 173 202.000 1134.3684 29.000
82.59983 56.049817 109.14985 1472 222 52 156 659 788.0000 48.0000 41.00000 62.36842 1548 164 693 829.000 163 148.000 1274.3684 -15.000
81.02543 54.473502 107.57735 1489 229 21 134 467 603.0000 61.0000 26.00000 62.36842 1566 141 491 634.000 133 174.000 1024.3684 41.000
78.80465 52.251314 105.35799 1367 198 21 156 506 857.0000 109.0000 46.00000 62.36842 1367 156 506 857.000 114 127.000 1098.3684 13.000
82.67668 56.139054 109.21431 1485 222 46 101 534 692.0000 88.0000 88.00000 62.36842 1494 102 537 696.000 131 146.000 1144.3684 15.000
82.88689 56.341477 109.43230 1458 225 32 109 651 625.0000 151.0000 68.00000 62.36842 1458 109 651 625.000 123 129.000 1298.3684 6.000
88.13140 61.559678 114.70312 1530 334 30 198 630 1061.0000 143.0000 60.00000 62.36842 1530 198 630 1061.000 110 146.000 1457.3684 36.000
79.16406 52.593132 105.73499 1421 160 72 30 523 508.0000 289.0000 52.31977 62.36842 1731 37 637 619.000 445 146.057 1302.6882 -298.943
87.04854 60.296301 113.80077 1869 301 122 58 347 127.0000 399.0000 52.31977 62.36842 10814 336 2008 735.000 1261 146.057 3002.6882 -1114.943
74.07883 47.532179 100.62549 1400 169 66 26 431 344.0000 156.0000 52.31977 62.36842 1680 31 517 413.000 398 133.000 1048.6882 -265.000
83.19515 56.639925 109.75037 1494 193 81 12 340 709.3361 207.0000 52.31977 62.36842 1614 13 367 799.668 285 85.000 974.6882 -200.000
82.56287 55.984272 109.14147 1449 223 62 20 423 709.3361 298.0000 52.31977 62.36842 1544 21 451 799.668 286 93.000 1168.6882 -193.000
79.99250 53.440140 106.54486 1385 200 76 29 483 709.3361 262.0000 52.31977 62.36842 1457 31 508 799.668 296 83.000 1189.6882 -213.000
84.44415 57.885204 111.00310 1443 218 99 24 716 554.0000 254.0000 154.00000 62.36842 1518 25 753 583.000 271 113.000 1564.3684 -158.000
98.45938 71.863494 125.05526 1825 284 106 61 616 398.0000 101.0000 94.00000 62.36842 1932 65 652 421.000 245 113.000 1360.3684 -132.000
88.14264 61.566570 114.71871 1627 296 95 38 630 445.0000 93.0000 76.00000 62.36842 1712 40 663 468.000 207 159.000 1323.3684 -48.000
91.01234 64.437446 117.58723 1623 299 106 54 622 445.0000 149.0000 77.00000 62.36842 1718 57 659 471.000 221 183.000 1406.3684 -38.000
84.44106 57.874322 111.00781 1556 298 82 60 500 550.0000 72.0000 53.00000 62.36842 1637 63 526 579.000 187 176.000 1153.3684 -11.000
73.15025 46.610615 99.68988 1381 228 39 80 535 501.0000 41.0000 42.00000 62.36842 1453 84 563 527.000 203 149.000 1055.3684 -54.000
83.72013 57.179361 110.26090 1556 272 46 114 532 634.0000 32.0000 37.00000 62.36842 1637 120 560 667.000 138 157.000 1123.3684 19.000
79.45605 52.900106 106.01200 1416 206 32 168 610 775.0000 36.0000 18.00000 62.36842 1490 177 642 815.000 130 138.000 1164.3684 8.000
80.77281 54.219494 107.32612 1413 257 21 204 546 1268.0000 87.0000 50.00000 62.36842 1413 204 546 1268.000 135 157.000 1227.3684 22.000
73.79022 47.210661 100.36979 1504 253 102 33 262 482.0000 123.7033 52.31977 62.36842 2901 64 505 930.000 652 154.000 1131.3914 -498.000
56.21510 29.595510 82.83469 1193 165 68 45 299 1011.0000 123.7033 52.31977 62.36842 1726 65 432 1462.000 743 146.057 948.3914 -596.943
82.20348 55.641577 108.76538 1461 325 30 166 470 1145.0000 89.0000 40.00000 67.00000 1461 166 470 1145.000 103 174.000 1187.0000 71.000
83.81973 57.271581 110.36788 1458 294 36 187 590 999.0000 89.0000 30.00000 61.00000 1458 187 590 999.000 101 136.000 1287.0000 35.000
62.24563 35.654134 88.83712 1295 237 64 25 360 814.0000 129.0000 52.31977 62.36842 1734 33 482 1090.000 609 146.057 1051.6882 -462.943
83.04656 56.511637 109.58149 1431 263 58 118 591 675.0000 155.0000 75.00000 62.36842 1431 118 591 675.000 155 151.000 1322.3684 -4.000
85.73302 59.165928 112.30012 1469 305 59 98 498 644.0000 216.0000 84.00000 62.36842 1469 98 498 644.000 150 153.000 1322.3684 3.000
93.51336 66.965130 120.06160 1633 266 59 115 508 709.0000 185.0000 43.00000 62.36842 1633 115 508 709.000 141 150.000 1238.3684 9.000
90.94204 64.404116 117.47996 1603 295 58 132 442 758.0000 133.0000 48.00000 62.36842 1603 132 442 758.000 127 140.000 1170.3684 13.000
83.95211 57.422812 110.48141 1487 269 52 117 400 832.0000 106.0000 64.00000 62.36842 1487 117 400 832.000 129 157.000 1070.3684 28.000
81.67109 55.106905 108.23527 1474 318 44 101 501 884.0000 108.0000 62.00000 62.36842 1483 102 504 889.000 123 162.000 1199.3684 39.000
90.65737 64.118655 117.19608 1594 296 52 152 538 938.0000 128.0000 39.00000 62.36842 1604 153 541 944.000 126 190.000 1270.3684 64.000
81.66964 55.125915 108.21336 1415 285 42 140 524 921.0000 140.0000 65.00000 52.00000 1415 140 524 921.000 130 153.000 1248.0000 23.000
78.95534 52.413308 105.49737 1445 289 34 126 424 1008.0000 53.0000 33.00000 63.00000 1445 126 424 1008.000 125 163.000 1022.0000 38.000
76.69830 50.089855 103.30674 1362 199 81 29 408 508.0000 386.0000 52.31977 62.36842 1576 34 472 588.000 581 146.057 1281.6882 -434.943
91.10848 64.534158 117.68281 1572 195 106 30 522 288.0000 297.0000 52.31977 62.36842 1721 33 571 315.000 344 146.057 1313.6882 -197.943
67.09385 40.525875 93.66182 1209 168 56 16 435 709.3361 217.0000 52.31977 62.36842 1280 17 461 799.668 363 92.000 1032.6882 -271.000
68.94708 42.385870 95.50829 1242 155 69 20 368 709.3361 132.0000 52.31977 62.36842 1359 22 403 799.668 287 103.000 893.6882 -184.000
63.09859 36.500985 89.69620 1098 116 63 29 340 709.3361 119.0000 52.31977 62.36842 1155 31 358 799.668 254 69.000 799.6882 -185.000
71.17337 44.608707 97.73803 1235 175 77 26 457 743.0000 159.0000 52.31977 62.36842 1299 27 481 782.000 246 131.000 1032.6882 -115.000
88.80027 62.241990 115.35855 1651 247 80 59 357 335.0000 83.0000 63.00000 62.36842 1737 62 376 352.000 219 146.000 970.3684 -73.000
90.57611 63.999565 117.15265 1712 265 85 68 463 406.0000 39.0000 32.00000 62.36842 1813 72 490 430.000 221 138.000 1041.3684 -83.000
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