0.1 Introduction

In this assignment, we are tasked to explore, analyze and model a major league baseball dataset which contains around 2000 records where each record presents a baseball team from 1871 to 2006. Each observation provides the perforamce of the team for that particular year with all the statistics for the performance of 162 game season. The problem statement for the main objective is that “Can we predict the number of wins for the team with the given attributes of each record?”. In order to provide a solution for the problem, our goal is to build a linear regression model on the training data that creates this prediction.

0.1.1 About the Data

The data set are provided in csv format as moneyball-evaluation-data and moneyball-training-data where we will explore, preperate and create our model with the training data and further test the model with the evaluation data. Below is short description of the variables within the datasets.

**INDEX: Identification Variable(Do not use)

**TARGET_WINS: Number of wins

**TEAM_BATTING_H : Base Hits by batters (1B,2B,3B,HR)

**TEAM_BATTING_2B: Doubles by batters (2B)

**TEAM_BATTING_3B: Triples by batters (3B)

**TEAM_BATTING_HR: Homeruns by batters (4B)

**TEAM_BATTING_BB: Walks by batters

**TEAM_BATTING_HBP: Batters hit by pitch (get a free base)

**TEAM_BATTING_SO: Strikeouts by batters

**TEAM_BASERUN_SB: Stolen bases

**TEAM_BASERUN_CS: Caught stealing

**TEAM_FIELDING_E: Errors

**TEAM_FIELDING_DP: Double Plays

**TEAM_PITCHING_BB: Walks allowed

**TEAM_PITCHING_H: Hits allowed

**TEAM_PITCHING_HR: Homeruns allowed

**TEAM_PITCHING_SO: Strikeouts by pitchers

0.2 Data Exploration

0.2.1 Descriptive Statistics

# load libraries
library(ggplot2)
library(ggcorrplot)
library(psych)
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#library(statsr)
library(dplyr)
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library(PerformanceAnalytics)
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library(tidyr)
library(reshape2)
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library(rcompanion)
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library(caret)
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library(MASS)
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library(imputeTS)
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library(rsample)
library(huxtable)
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library(glmnet)
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library(sjPlot)
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library(modelr)
# Load data sets

baseball_eva <- read.csv("https://raw.githubusercontent.com/anilak1978/data621/master/moneyball-evaluation-data.csv")
baseball_train <- read.csv("https://raw.githubusercontent.com/anilak1978/data621/master/moneyball-training-data.csv")

We can start exploring our training data set by looking at basic descriptive statistics.

# look at training dataset structure
str(baseball_train)
## 'data.frame':    2276 obs. of  17 variables:
##  $ INDEX           : int  1 2 3 4 5 6 7 8 11 12 ...
##  $ TARGET_WINS     : int  39 70 86 70 82 75 80 85 86 76 ...
##  $ TEAM_BATTING_H  : int  1445 1339 1377 1387 1297 1279 1244 1273 1391 1271 ...
##  $ TEAM_BATTING_2B : int  194 219 232 209 186 200 179 171 197 213 ...
##  $ TEAM_BATTING_3B : int  39 22 35 38 27 36 54 37 40 18 ...
##  $ TEAM_BATTING_HR : int  13 190 137 96 102 92 122 115 114 96 ...
##  $ TEAM_BATTING_BB : int  143 685 602 451 472 443 525 456 447 441 ...
##  $ TEAM_BATTING_SO : int  842 1075 917 922 920 973 1062 1027 922 827 ...
##  $ TEAM_BASERUN_SB : int  NA 37 46 43 49 107 80 40 69 72 ...
##  $ TEAM_BASERUN_CS : int  NA 28 27 30 39 59 54 36 27 34 ...
##  $ TEAM_BATTING_HBP: int  NA NA NA NA NA NA NA NA NA NA ...
##  $ TEAM_PITCHING_H : int  9364 1347 1377 1396 1297 1279 1244 1281 1391 1271 ...
##  $ TEAM_PITCHING_HR: int  84 191 137 97 102 92 122 116 114 96 ...
##  $ TEAM_PITCHING_BB: int  927 689 602 454 472 443 525 459 447 441 ...
##  $ TEAM_PITCHING_SO: int  5456 1082 917 928 920 973 1062 1033 922 827 ...
##  $ TEAM_FIELDING_E : int  1011 193 175 164 138 123 136 112 127 131 ...
##  $ TEAM_FIELDING_DP: int  NA 155 153 156 168 149 186 136 169 159 ...
TARGET_Wins<-as.numeric(baseball_train$TARGET_WINS)

We have 2276 observations and 17 variables. All of our variables are integer type as expected.

# look at descriptive statistics
metastats <- data.frame(describe(baseball_train))
metastats <- tibble::rownames_to_column(metastats, "STATS")
metastats["pct_missing"] <- round(metastats["n"]/2276, 3)
head(metastats)
STATSvarsnmeansdmediantrimmedmadminmaxrangeskewkurtosissepct_missing
INDEX12.28e+031.27e+03736  1.27e+031.27e+03953  12.54e+032.53e+030.00421-1.22   15.4  1
TARGET_WINS22.28e+0380.8     15.882       81.3     14.80146       146       -0.399  1.03   0.33 1
TEAM_BATTING_H32.28e+031.47e+03145  1.45e+031.46e+03114  8912.55e+031.66e+031.57   7.28   3.03 1
TEAM_BATTING_2B42.28e+03241       46.8238       240       47.469458       389       0.215  0.006160.9811
TEAM_BATTING_3B52.28e+0355.2     27.947       52.2     23.70223       223       1.11   1.5    0.5861
TEAM_BATTING_HR62.28e+0399.6     60.5102       97.4     78.60264       264       0.186  -0.963  1.27 1

With the descriptive statistics, we are able to see mean, standard deviation, median, min, max values and percentage of each missing value of each variable. For example, when we look at TEAM_BATTING_H, we see that average 1469 Base hits by batters, with standard deviation of 144, median of 1454 with maximum base hits of 2554.

# Look for missing values
colSums(is.na(baseball_train))
##            INDEX      TARGET_WINS   TEAM_BATTING_H  TEAM_BATTING_2B 
##                0                0                0                0 
##  TEAM_BATTING_3B  TEAM_BATTING_HR  TEAM_BATTING_BB  TEAM_BATTING_SO 
##                0                0                0              102 
##  TEAM_BASERUN_SB  TEAM_BASERUN_CS TEAM_BATTING_HBP  TEAM_PITCHING_H 
##              131              772             2085                0 
## TEAM_PITCHING_HR TEAM_PITCHING_BB TEAM_PITCHING_SO  TEAM_FIELDING_E 
##                0                0              102                0 
## TEAM_FIELDING_DP 
##              286
# Percentage of missing values
missing_values <- metastats %>%
  filter(pct_missing < 1) %>%
  dplyr::select(STATS, pct_missing) %>%
  arrange(pct_missing)
missing_values
STATSpct_missing
TEAM_BATTING_HBP0.084
TEAM_BASERUN_CS0.661
TEAM_FIELDING_DP0.874
TEAM_BASERUN_SB0.942
TEAM_BATTING_SO0.955
TEAM_PITCHING_SO0.955

When we look at the missing values within the training data set, we see that proportionaly against the total observations, TEAM_BATTING_HBP and TEAM_BESARUN_CS variables have the most missing values. We will be handling these missing values in our Data Preperation section.

0.2.2 Correlation and Distribution

# Look at correlation between variables
baseball_train$TARGET_WINS<-as.numeric(baseball_train$TARGET_WINS)
corr <- round(cor(baseball_train), 1)

ggcorrplot(corr,
           type="lower",
           lab=TRUE,
           lab_size=3,
           method="circle",
           colors=c("tomato2", "white", "springgreen3"),
           title="Correlation of variables in Training Data Set",
           ggtheme=theme_bw)
## Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> =
## "none")` instead.

Team_Batting_H and Team_Batting_2B have the strongest positive correlation with Target_Wins. We also see that, there is a strong correlation between Team_Batting_H and Team_Batting_2B, Team_Pitching_B and TEAM_FIELDING_E. We will consider these findings on model creation as collinearity might complicate model estimation and we want to have explanotry variables to be independent from each other. We will try to avoid adding explanotry variables that are correlated to each other.

Let’s look at the correlations and distribution of the variables in more detail.

# Look at correlation from batting, baserunning, pitching and fielding perspective
Batting_df <- baseball_train[c(2:7, 10)] 
BaseRunning_df <- baseball_train[c(8:9)] 
Pitching_df <- baseball_train[c(11:14)] 
Fielding_df <- baseball_train[c(15:16)]

0.2.2.1 Batting

# Batting Correlations
chart.Correlation(Batting_df, histogram=TRUE, pch=19)

We can see that our response variable TARGET_WINS, TEAM_BATTING_H, TEAM_BATTING_2B, TEAM_BATTING_BB and TEAM_BASERUN_CS are normaly distributed. TEAM_BATTING_HR on the other hand is bimodal.

0.2.2.2 Baserunning

# baserunning Correlation

chart.Correlation(BaseRunning_df, histogram=TRUE, pch=19)

TEAM_BASERUN_SB is right skewed and TEAM_BATTING_SO is bimodal.

0.2.2.3 Pitching

#pitching correlations
chart.Correlation(Pitching_df, histogram=TRUE, pch=19)

TEAM_BATTING_HBP seems to be normally distributed however we shouldnt forget that we have a lot of missing values in this variable.

# fielding correlations
chart.Correlation(Fielding_df, histogram=TRUE, pch=19)

Let’s also look at the outliers and skewness for each varibale.

0.2.3 Outliers and Skewness

par(mfrow=c(3,3))
datasub_1 <- melt(baseball_train)
## No id variables; using all as measure variables
suppressWarnings(ggplot(datasub_1, aes(x= "value", y=value)) + 
                   geom_boxplot(fill='lightblue') + facet_wrap(~variable, scales = 'free') )
## Warning: Removed 3478 rows containing non-finite values (stat_boxplot).

Based on the boxplot we created, TEAM_FIELDING_DP, TEAM_PITCHING_HR, TEAM_BATTING_HR and TEAM_BATTING_SO seem to have the least amount of outliers.

par(mfrow = c(3, 3))
datasub = melt(baseball_train) 
## No id variables; using all as measure variables
suppressWarnings(ggplot(datasub, aes(x= value)) + 
                   geom_density(fill='lightblue') + facet_wrap(~variable, scales = 'free') )
## Warning: Removed 3478 rows containing non-finite values (stat_density).

metastats %>%
  filter(skew > 1) %>%
  dplyr::select(STATS, skew) %>%
  arrange(desc(skew))
STATSskew
TEAM_PITCHING_SO22.2 
TEAM_PITCHING_H10.3 
TEAM_PITCHING_BB6.74
TEAM_FIELDING_E2.99
TEAM_BASERUN_CS1.98
TEAM_BASERUN_SB1.97
TEAM_BATTING_H1.57
TEAM_BATTING_3B1.11

We can see that the most skewed variable is TEAM_PITCHING_SO. We will correct the skewed variables in our data preperation section.

When we are creating a linear regression model, we are looking for the fitting line with the least sum of squares, that has the small residuals with minimized squared residuals. From our correlation analysis, we can see that the explatory variable that has the strongest correlation with TARGET_WINS is TEAM_BATTING_H. Let’s look at a simple model example to further expand our explaroty analysis.

0.2.4 Simple Model Example

#library(statsr)
# line that follows the best assocation between two variables

#plot_ss(x = TEAM_BATTING_H, y = TARGET_WINS, data=baseball_train, showSquares = TRUE, leastSquares = TRUE)

When we are exploring to build a linear regression, one of the first thing we do is to create a scatter plot of the response and explanatory variable.

# scatter plot between TEAM_BATTING_H and TARGET_WINS

ggplot(baseball_train, aes(x=TEAM_BATTING_H, y=TARGET_WINS))+
  geom_point()

One of the conditions for least square lines or linear regression are Linearity. From the scatter plot between TEAM_BATTING_H and TARGET_WINS, we can see this condition is met. We can also create a scatterplot that shows the data points between TARGET_WINS and each variable.

baseball_train %>%
  gather(var, val, -TARGET_WINS) %>%
  ggplot(., aes(val, TARGET_WINS))+
  geom_point()+
  facet_wrap(~var, scales="free", ncol=4)
## Warning: Removed 3478 rows containing missing values (geom_point).

As we displayed earlier, hits walks and home runs have the strongest correlations with TARGET_WINS and also meets the linearity condition.

# create a simple example model
lm_sm <- lm(baseball_train$TARGET_WINS ~ baseball_train$TEAM_BATTING_H)
summary(lm_sm)
## 
## Call:
## lm(formula = baseball_train$TARGET_WINS ~ baseball_train$TEAM_BATTING_H)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -71.768  -8.757   0.856   9.762  46.016 
## 
## Coefficients:
##                                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                   18.562326   3.107523   5.973 2.69e-09 ***
## baseball_train$TEAM_BATTING_H  0.042353   0.002105  20.122  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.52 on 2274 degrees of freedom
## Multiple R-squared:  0.1511, Adjusted R-squared:  0.1508 
## F-statistic: 404.9 on 1 and 2274 DF,  p-value: < 2.2e-16

TARET_BATTING_H has the strongest correlation with TARGET_WINS response variable, however when we create a simple model just using TARGET_BATTING_H, we can only explain 15% of the variablity. (Adjusted R-squared: 0.1508). The remainder of the varibility can be explained with selected other variables within the training dataset.

#histogram of residuals for the simple model
hist(lm_sm$residuals)

# check for constant variability (honoscedasticity)

plot(lm_sm$residuals ~ baseball_train$TEAM_BATTING_H)

We do see that the residuals are distributed normally and variability around the regression line is roughly constant.

Based on our explatory analysis, we were able to see the correlation level between the possible explanatory variables and repsonse variable TARGET_WINS. Some of the variables such as TARGET_BATTING_H has somewhat strong positive correlation, however some of the variables such as TEAM_PITCHING_BB has weak positive relationship with TARGET_WINS. We also found out, hit by the pitcher(TEAM_BATTING_HBP) and caught stealing (TEAM_BASERUN_CS) variables are missing majority of the values. Skewness and distribution analysis gave us the insights that we have some variables that are right-tailed. Considering all of these insights, we will handle missing values, correct skewness and outliers and select our explaratory variables based on correlation in order to create our regression model.

0.3 Data Preparation

0.3.1 Objective

In this section, we will prepare the dataset for linear regression modeling. We accomplish this by handling missing values and outliers and by tranforming the data into more normal distributions. This section covers:

Identify and Handle Missing Data Correct Outliers *Adjust Skewed value - Box Cox Transformation

First, we will start by copying the dataset into a new variable, baseball_train_01, and we will remove the Index variable from the new dataset as well. We will now have 16 variables.

baseball_train_01 <- baseball_train

baseball_train_01 <-subset(baseball_train_01, select = -c(INDEX))

0.3.2 Identify and Handle Missing Data

0.3.2.1 Removal of Sparsely Populated Variables - MCAR

In the Data Exploration section, we identified these variables as having missing data values.The table below lists the variables with missing data. The variable, TEAM_BATTING_HBP, is sparsely populated. Since this data is Missing Completely at Random (MCAR) and is not related to any other variable, it is safe to completely remove the variable from the dataset.

missing_values
STATSpct_missing
TEAM_BATTING_HBP0.084
TEAM_BASERUN_CS0.661
TEAM_FIELDING_DP0.874
TEAM_BASERUN_SB0.942
TEAM_BATTING_SO0.955
TEAM_PITCHING_SO0.955
baseball_train_01 <-subset(baseball_train_01, select = -c(TEAM_BATTING_HBP))

There are now 15 variables.

dim(baseball_train_01)
## [1] 2276   15

0.3.2.2 Imputation of Missing Values

For the remaining variables with missing values, we will impute the mean of the variable. The function, “na_mean” updates all missing values with the mean of the variable.

baseball_train_01 <- na_mean(baseball_train_01, option = "mean")

Re-running the metastats dataframe on the new baseball_train_01 dataset shows that there are no missing values.

# look at descriptive statistics
metastats <- data.frame(describe(baseball_train_01))
metastats <- tibble::rownames_to_column(metastats, "STATS")
metastats["pct_missing"] <- round(metastats["n"]/2276, 3)
# Percentage of missing values
missing_values2 <- metastats %>%
  filter(pct_missing < 1) %>%
  dplyr::select(STATS, pct_missing) %>%
  arrange(pct_missing)
missing_values2
STATSpct_missing

0.3.3 Correct Outliers

In this section, we created two functions that can identify outliers. The funcion, Identify_Outlier, uses the Turkey method, where outliers are identified by being below Q1-1.5IQR and above Q3+1.5IQR. The second function, tag_outlier, returns a binary list of values, “Acceptable” or “Outlier” that will be added to the dataframe.

Identify_Outlier <- function(value){

    interquartile_range = IQR(sort(value),na.rm = TRUE)
    q1 = matrix(c(quantile(sort(value),na.rm = TRUE)))[2]
    q3 = matrix(c(quantile(sort(value),na.rm = TRUE)))[4]
    lower = q1-(1.5*interquartile_range)
    upper = q3+(1.5*interquartile_range)
    
    bound <- c(lower, upper)
    
    return (bound)
}
tag_outlier <- function(value) {
    
   boundaries <- Identify_Outlier(value)
   tags <- c()
   counter = 1
    for (i in as.numeric(value))
    {

        if (i >= boundaries[1] & i <= boundaries[2]){
          tags[counter] <- "Acceptable"
        } else{
          tags[counter] <- "Outlier"
        }
      
      counter = counter +1
    }
   
   return (tags)
}

As seen in the box plots from the previous section, “TEAM_BASERUN_SB”, “TEAM_BASERUN_CS”, “TEAM_PITCHING_H”, “TEAM_PITCHING_BB”, “TEAM_PITCHING_SO”, and “TEAM_FIELDING_E” all have a high number of outliers. We will use the two functions above to tag those rows with extreme outliers.

tags<- tag_outlier(baseball_train_01$TEAM_BASERUN_SB)
baseball_train_01$TEAM_BASERUN_SB_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_BASERUN_CS)
baseball_train_01$TEAM_BASERUN_CS_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_PITCHING_H)
baseball_train_01$TEAM_PITCHING_H_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_PITCHING_BB)
baseball_train_01$TEAM_PITCHING_BB_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_PITCHING_SO)
baseball_train_01$TEAM_PITCHING_SO_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_FIELDING_E)
baseball_train_01$TEAM_FIELDING_E_Outlier <- tags

Below, we filtered out all of the outliers and created a new dataframe, baseball_train_02

baseball_train_02 <- baseball_train_01 %>%
                filter(
                        TEAM_BASERUN_SB_Outlier != "Outlier" &
                        TEAM_BASERUN_CS_Outlier != "Outlier" &
                        TEAM_PITCHING_H_Outlier != "Outlier" &
                        TEAM_PITCHING_BB_Outlier != "Outlier" &
                        TEAM_PITCHING_SO_Outlier != "Outlier" &
                        TEAM_FIELDING_E_Outlier != "Outlier"
                )

Re-running the boxplots show data that has a better normal distribution except for the variable, TEAM_FIELDING_E which is still skewed. We will handle this next.

par(mfrow=c(3,3))
datasub_1 <- melt(baseball_train_02)
## Using TEAM_BASERUN_SB_Outlier, TEAM_BASERUN_CS_Outlier, TEAM_PITCHING_H_Outlier, TEAM_PITCHING_BB_Outlier, TEAM_PITCHING_SO_Outlier, TEAM_FIELDING_E_Outlier as id variables
suppressWarnings(ggplot(datasub_1, aes(x= "value", y=value)) + 
                   geom_boxplot(fill='lightblue') + facet_wrap(~variable, scales = 'free') )

0.3.4 Adjust Skewed values

0.3.4.1 Box Cox Transformation

Removing the outliers tranformed each variable to a closer to a normal distribution and checking the skewness of the variables confirm this with the exception of TEAM_FIELDING_E. This variable is still skewed and not normal. In this section, we will use the Box Cox tranformation from the MASS library to normalize this variable.

metastats_02 <- data.frame(describe(baseball_train_02))
metastats_02 <- tibble::rownames_to_column(metastats_02, "STATS") 
 
metastats_02 %>%
 filter(skew > 1 | skew < -1) %>%
  dplyr::select(STATS, skew) %>%
  arrange(desc(skew))
STATSskew
TEAM_FIELDING_E1.45

Looking at the histogram and QQ plots we can confirm that the variable, TEAM_FIELDING_E, is not normally distributed. It is skewed to the right.

plotNormalHistogram(baseball_train_02$TEAM_FIELDING_E)

qqnorm(baseball_train_02$TEAM_FIELDING_E,
       ylab="Sample Quantiles for TEAM_FIELDING_E")    
         qqline(baseball_train_02$TEAM_FIELDING_E,
           col="blue")

The following Box Cox transformation section is based on the tutorial at the link below:

[://rcompanion.org/handbook/I_12.html][Summary and Analysis of Extension Program Evaluation in R]

The Box Cox procedure uses a log-likelihood to find the lambda to use to transform a variable to a normal distribution.

TEAM_FIELDING_E <- as.numeric(dplyr::pull(baseball_train_02, TEAM_FIELDING_E))

#Transforms TEAM_FIELDING_E as a single vector 
Box = boxcox(TEAM_FIELDING_E ~ 1, lambda = seq(-6,6,0.1))

#Creates a dataframe with results
Cox = data.frame(Box$x, Box$y)

# Order the new data frame by decreasing y to find the best lambda.Displays the lambda with the greatest log likelihood.
Cox2 = Cox[with(Cox, order(-Cox$Box.y)),]
Cox2[1,]
Box.xBox.y
-0.8-3.95e+03
#Extract that lambda and Transform the data
lambda = Cox2[1, "Box.x"]
T_box = (TEAM_FIELDING_E ^ lambda - 1)/lambda

We can now see that TEAM_FIELDING_E has a normal distribution.

plotNormalHistogram(T_box)

qqnorm(T_box, ylab="Sample Quantiles for TEAM_FIELDING_E")
qqline(T_box,
        col="blue")

baseball_train_02$TEAM_FIELDING_E <- T_box

The density plots below show that all of the variables for the dataset baseball_train_02 are now normally distributed. In the next section, we will use this dataset to build the models and discuss the coefficients of the models.

par(mfrow = c(3, 3))
datasub = melt(baseball_train_02) 
suppressWarnings(ggplot(datasub, aes(x= value)) + 
                   geom_density(fill='lightblue') + facet_wrap(~variable, scales = 'free') )

Viewing the dataframe shows that the dataset contains characters resulting from the transfromation of the outliers. These non numeric characters will impact our models especially if we build the intial baseline model with all the variables. We will need one more step to have our data ready for the models.

str(baseball_train_02)
## 'data.frame':    1521 obs. of  21 variables:
##  $ TARGET_WINS             : num  70 82 75 80 85 76 78 87 88 66 ...
##  $ TEAM_BATTING_H          : int  1387 1297 1279 1244 1273 1271 1305 1417 1563 1460 ...
##  $ TEAM_BATTING_2B         : int  209 186 200 179 171 213 179 226 242 239 ...
##  $ TEAM_BATTING_3B         : int  38 27 36 54 37 18 27 28 43 32 ...
##  $ TEAM_BATTING_HR         : int  96 102 92 122 115 96 82 108 164 107 ...
##  $ TEAM_BATTING_BB         : int  451 472 443 525 456 441 374 539 589 546 ...
##  $ TEAM_BATTING_SO         : num  922 920 973 1062 1027 ...
##  $ TEAM_BASERUN_SB         : num  43 49 107 80 40 72 60 86 100 92 ...
##  $ TEAM_BASERUN_CS         : num  30 39 59 54 36 34 39 69 53 64 ...
##  $ TEAM_PITCHING_H         : int  1396 1297 1279 1244 1281 1271 1364 1417 1563 1478 ...
##  $ TEAM_PITCHING_HR        : int  97 102 92 122 116 96 86 108 164 108 ...
##  $ TEAM_PITCHING_BB        : int  454 472 443 525 459 441 391 539 589 553 ...
##  $ TEAM_PITCHING_SO        : num  928 920 973 1062 1033 ...
##  $ TEAM_FIELDING_E         : num  1.23 1.23 1.22 1.23 1.22 ...
##  $ TEAM_FIELDING_DP        : num  156 168 149 186 136 159 141 136 172 146 ...
##  $ TEAM_BASERUN_SB_Outlier : chr  "Acceptable" "Acceptable" "Acceptable" "Acceptable" ...
##  $ TEAM_BASERUN_CS_Outlier : chr  "Acceptable" "Acceptable" "Acceptable" "Acceptable" ...
##  $ TEAM_PITCHING_H_Outlier : chr  "Acceptable" "Acceptable" "Acceptable" "Acceptable" ...
##  $ TEAM_PITCHING_BB_Outlier: chr  "Acceptable" "Acceptable" "Acceptable" "Acceptable" ...
##  $ TEAM_PITCHING_SO_Outlier: chr  "Acceptable" "Acceptable" "Acceptable" "Acceptable" ...
##  $ TEAM_FIELDING_E_Outlier : chr  "Acceptable" "Acceptable" "Acceptable" "Acceptable" ...

Subsetting - The code below will subset the data to have only numeric or integer values that will be used for our models. This will create baseball_train_03 dataframe.

baseball_train_03 <- baseball_train_02[c(1:15) ]
str(baseball_train_03)
## 'data.frame':    1521 obs. of  15 variables:
##  $ TARGET_WINS     : num  70 82 75 80 85 76 78 87 88 66 ...
##  $ TEAM_BATTING_H  : int  1387 1297 1279 1244 1273 1271 1305 1417 1563 1460 ...
##  $ TEAM_BATTING_2B : int  209 186 200 179 171 213 179 226 242 239 ...
##  $ TEAM_BATTING_3B : int  38 27 36 54 37 18 27 28 43 32 ...
##  $ TEAM_BATTING_HR : int  96 102 92 122 115 96 82 108 164 107 ...
##  $ TEAM_BATTING_BB : int  451 472 443 525 456 441 374 539 589 546 ...
##  $ TEAM_BATTING_SO : num  922 920 973 1062 1027 ...
##  $ TEAM_BASERUN_SB : num  43 49 107 80 40 72 60 86 100 92 ...
##  $ TEAM_BASERUN_CS : num  30 39 59 54 36 34 39 69 53 64 ...
##  $ TEAM_PITCHING_H : int  1396 1297 1279 1244 1281 1271 1364 1417 1563 1478 ...
##  $ TEAM_PITCHING_HR: int  97 102 92 122 116 96 86 108 164 108 ...
##  $ TEAM_PITCHING_BB: int  454 472 443 525 459 441 391 539 589 553 ...
##  $ TEAM_PITCHING_SO: num  928 920 973 1062 1033 ...
##  $ TEAM_FIELDING_E : num  1.23 1.23 1.22 1.23 1.22 ...
##  $ TEAM_FIELDING_DP: num  156 168 149 186 136 159 141 136 172 146 ...

0.4 Build Models

The first Model is using stepwise in Backward direction to eliminate variables, this is an automated process which is different from the manual variable selction process. We will not pay much attention to this process as the focus of the project is to manually identify and select those significant variables that will predict TARGET WINS.

Model <- step(lm(TARGET_WINS ~ ., data=baseball_train_03), direction = "backward")
## Start:  AIC=7313.35
## TARGET_WINS ~ 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_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB + 
##     TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq    RSS    AIC
## - TEAM_PITCHING_H   1       0.7 182710 7311.4
## - TEAM_PITCHING_HR  1     162.2 182872 7312.7
## - TEAM_BATTING_H    1     216.2 182926 7313.2
## <none>                          182709 7313.4
## - TEAM_BASERUN_CS   1     330.3 183040 7314.1
## - TEAM_BATTING_HR   1     338.0 183047 7314.2
## - TEAM_PITCHING_BB  1     363.7 183073 7314.4
## - TEAM_BATTING_BB   1     629.6 183339 7316.6
## - TEAM_PITCHING_SO  1    1242.9 183952 7321.7
## - TEAM_BATTING_SO   1    1857.6 184567 7326.7
## - TEAM_BATTING_2B   1    1864.9 184574 7326.8
## - TEAM_FIELDING_DP  1    6690.2 189400 7366.1
## - TEAM_BATTING_3B   1    7536.4 190246 7372.8
## - TEAM_BASERUN_SB   1    8080.4 190790 7377.2
## - TEAM_FIELDING_E   1   18743.7 201453 7459.9
## 
## Step:  AIC=7311.36
## TARGET_WINS ~ 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_PITCHING_HR + TEAM_PITCHING_BB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq    RSS    AIC
## - TEAM_PITCHING_HR  1     173.3 182883 7310.8
## <none>                          182710 7311.4
## - TEAM_BASERUN_CS   1     331.1 183041 7312.1
## - TEAM_BATTING_HR   1     358.9 183069 7312.3
## - TEAM_PITCHING_SO  1    1259.1 183969 7319.8
## - TEAM_PITCHING_BB  1    1509.7 184220 7321.9
## - TEAM_BATTING_2B   1    1876.6 184587 7324.9
## - TEAM_BATTING_SO   1    1880.0 184590 7324.9
## - TEAM_BATTING_BB   1    2658.3 185368 7331.3
## - TEAM_BATTING_H    1    4833.0 187543 7349.1
## - TEAM_FIELDING_DP  1    6705.4 189416 7364.2
## - TEAM_BATTING_3B   1    7548.6 190259 7370.9
## - TEAM_BASERUN_SB   1    8142.6 190853 7375.7
## - TEAM_FIELDING_E   1   18841.1 201551 7458.6
## 
## Step:  AIC=7310.8
## TARGET_WINS ~ 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_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + 
##     TEAM_FIELDING_DP
## 
##                    Df Sum of Sq    RSS    AIC
## <none>                          182883 7310.8
## - TEAM_BASERUN_CS   1     418.7 183302 7312.3
## - TEAM_PITCHING_SO  1    1202.0 184085 7318.8
## - TEAM_PITCHING_BB  1    1537.3 184421 7321.5
## - TEAM_BATTING_2B   1    1928.3 184812 7324.8
## - TEAM_BATTING_SO   1    1977.9 184861 7325.2
## - TEAM_BATTING_BB   1    2678.0 185561 7330.9
## - TEAM_BATTING_H    1    4860.2 187744 7348.7
## - TEAM_BATTING_HR   1    5541.1 188424 7354.2
## - TEAM_BATTING_3B   1    7420.8 190304 7369.3
## - TEAM_FIELDING_DP  1    7423.8 190307 7369.3
## - TEAM_BASERUN_SB   1    9570.9 192454 7386.4
## - TEAM_FIELDING_E   1   18687.6 201571 7456.8
summary(Model)
## 
## 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_BASERUN_CS + TEAM_PITCHING_BB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP, data = baseball_train_03)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -45.468  -6.985  -0.128   7.454  34.637 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       1.368e+03  1.084e+02  12.624  < 2e-16 ***
## TEAM_BATTING_H    3.169e-02  5.006e-03   6.331 3.22e-10 ***
## TEAM_BATTING_2B  -4.198e-02  1.053e-02  -3.987 7.00e-05 ***
## TEAM_BATTING_3B   1.756e-01  2.244e-02   7.822 9.68e-15 ***
## TEAM_BATTING_HR   7.519e-02  1.112e-02   6.759 1.97e-11 ***
## TEAM_BATTING_BB   1.564e-01  3.328e-02   4.699 2.85e-06 ***
## TEAM_BATTING_SO  -8.768e-02  2.171e-02  -4.038 5.65e-05 ***
## TEAM_BASERUN_SB   6.625e-02  7.458e-03   8.884  < 2e-16 ***
## TEAM_BASERUN_CS  -6.510e-02  3.503e-02  -1.858 0.063350 .  
## TEAM_PITCHING_BB -1.130e-01  3.175e-02  -3.560 0.000382 ***
## TEAM_PITCHING_SO  6.484e-02  2.059e-02   3.148 0.001675 ** 
## TEAM_FIELDING_E  -1.085e+03  8.739e+01 -12.413  < 2e-16 ***
## TEAM_FIELDING_DP -1.121e-01  1.433e-02  -7.824 9.56e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.01 on 1508 degrees of freedom
## Multiple R-squared:  0.3725, Adjusted R-squared:  0.3675 
## F-statistic: 74.59 on 12 and 1508 DF,  p-value: < 2.2e-16

The step backward variable selection process identified eleven significant variables with an R-squared of 37%, Residual Error of 11.01 and F-Statistic of 74.59. Notice that some of the coefficients are negative which means these Team will most likely result in negative wins. We will explore these coefficient a little further in this analysis.

0.4.1 OLS- MODEL 1

Using all the 15 Variables

Model1 <-lm(TARGET_WINS ~ ., data=baseball_train_03)
summary(Model1)
## 
## Call:
## lm(formula = TARGET_WINS ~ ., data = baseball_train_03)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -45.067  -7.014  -0.101   7.499  34.361 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       1.376e+03  1.089e+02  12.634  < 2e-16 ***
## TEAM_BATTING_H    2.992e-02  2.242e-02   1.335   0.1821    
## TEAM_BATTING_2B  -4.140e-02  1.056e-02  -3.921 9.23e-05 ***
## TEAM_BATTING_3B   1.775e-01  2.252e-02   7.882 6.15e-15 ***
## TEAM_BATTING_HR   2.424e-01  1.452e-01   1.669   0.0953 .  
## TEAM_BATTING_BB   1.606e-01  7.051e-02   2.278   0.0229 *  
## TEAM_BATTING_SO  -1.072e-01  2.741e-02  -3.913 9.52e-05 ***
## TEAM_BASERUN_SB   6.361e-02  7.794e-03   8.161 6.94e-16 ***
## TEAM_BASERUN_CS  -5.853e-02  3.547e-02  -1.650   0.0992 .  
## TEAM_PITCHING_H   1.587e-03  2.058e-02   0.077   0.9386    
## TEAM_PITCHING_HR -1.617e-01  1.398e-01  -1.156   0.2478    
## TEAM_PITCHING_BB -1.166e-01  6.735e-02  -1.732   0.0836 .  
## TEAM_PITCHING_SO  8.366e-02  2.614e-02   3.201   0.0014 ** 
## TEAM_FIELDING_E  -1.092e+03  8.785e+01 -12.430  < 2e-16 ***
## TEAM_FIELDING_DP -1.086e-01  1.463e-02  -7.426 1.87e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.01 on 1506 degrees of freedom
## Multiple R-squared:  0.3731, Adjusted R-squared:  0.3672 
## F-statistic: 64.01 on 14 and 1506 DF,  p-value: < 2.2e-16

This Model identified seven significant variables at = 0.05 with an R-squared of 37%, Residual Error of 11.01 and F-Statistic of 64.01. Although the F-Statistic reduced, this model does not improve significantly from the previous model.

Metrics1 <- data.frame(
  R2 = rsquare(Model1, data = baseball_train_03),
  RMSE = rmse(Model1, data = baseball_train_03),
  MAE = mae(Model1, data = baseball_train_03)
)
print(Metrics1)
##          R2     RMSE      MAE
## 1 0.3730717 10.96013 8.741105

0.4.2 OLS- MODEL 2

Using all the seven (7) significant variables from Model 1

Model2 <- lm(TARGET_WINS~TEAM_FIELDING_E + TEAM_BASERUN_SB + TEAM_BATTING_3B + TEAM_FIELDING_DP + TEAM_PITCHING_SO + TEAM_BATTING_SO + TEAM_BATTING_2B,data=baseball_train_03)
summary(Model2)
## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_FIELDING_E + TEAM_BASERUN_SB + 
##     TEAM_BATTING_3B + TEAM_FIELDING_DP + TEAM_PITCHING_SO + TEAM_BATTING_SO + 
##     TEAM_BATTING_2B, data = baseball_train_03)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -48.799  -8.299  -0.053   8.472  39.785 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       1.808e+03  1.158e+02  15.607  < 2e-16 ***
## TEAM_FIELDING_E  -1.410e+03  9.313e+01 -15.141  < 2e-16 ***
## TEAM_BASERUN_SB   5.429e-02  7.497e-03   7.242 7.02e-13 ***
## TEAM_BATTING_3B   1.788e-01  2.289e-02   7.808 1.08e-14 ***
## TEAM_FIELDING_DP -5.319e-02  1.525e-02  -3.488 0.000501 ***
## TEAM_PITCHING_SO -8.587e-03  9.176e-03  -0.936 0.349497    
## TEAM_BATTING_SO  -8.186e-03  9.308e-03  -0.880 0.379267    
## TEAM_BATTING_2B   4.475e-02  7.971e-03   5.614 2.35e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.19 on 1513 degrees of freedom
## Multiple R-squared:  0.2288, Adjusted R-squared:  0.2253 
## F-statistic: 64.14 on 7 and 1513 DF,  p-value: < 2.2e-16

This Model identified five significant variables at = 0.05 with an R-squared of 22%, Residual Error of 12.19 and F-Statistic of 64.14. The R-Squared decreased and the Error increased slightly.

Metrics2 <- data.frame(
  R2 = rsquare(Model2, data = baseball_train_03),
  RMSE = rmse(Model2, data = baseball_train_03),
  MAE = mae(Model2, data = baseball_train_03)
)
print(Metrics2)
##          R2     RMSE      MAE
## 1 0.2288348 12.15572 9.731629

0.4.3 OLS- MODEL 3

All offensive categories which include hitting and base running

Model3 <-lm(TARGET_WINS~TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BATTING_HR + TEAM_BATTING_2B + TEAM_BATTING_SO + TEAM_BASERUN_CS + TEAM_BATTING_3B + TEAM_BASERUN_SB,data=baseball_train_03)
summary(Model3)
## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_BB + 
##     TEAM_BATTING_HR + TEAM_BATTING_2B + TEAM_BATTING_SO + TEAM_BASERUN_CS + 
##     TEAM_BATTING_3B + TEAM_BASERUN_SB, data = baseball_train_03)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -49.812  -7.822   0.247   8.166  35.877 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     15.763131   7.024077   2.244   0.0250 *  
## TEAM_BATTING_H   0.024765   0.005285   4.686 3.03e-06 ***
## TEAM_BATTING_BB  0.037681   0.003994   9.435  < 2e-16 ***
## TEAM_BATTING_HR  0.099319   0.011448   8.676  < 2e-16 ***
## TEAM_BATTING_2B -0.013435   0.010919  -1.230   0.2187    
## TEAM_BATTING_SO -0.010801   0.002767  -3.904 9.88e-05 ***
## TEAM_BASERUN_CS -0.068614   0.037166  -1.846   0.0651 .  
## TEAM_BATTING_3B  0.115379   0.022950   5.027 5.57e-07 ***
## TEAM_BASERUN_SB  0.076701   0.007431  10.321  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.73 on 1512 degrees of freedom
## Multiple R-squared:  0.2857, Adjusted R-squared:  0.2819 
## F-statistic: 75.58 on 8 and 1512 DF,  p-value: < 2.2e-16

This Model identified five significant variables at = 0.05 with an R-squared of 28%, Residual Error of 11.73 and F-Statistic of 75.58. Although the R-squared is not that great, the standard errors are more reasonable. We will hold onto this Model as performing better than the previous models for now.

Metrics3 <- data.frame(
  R2 = rsquare(Model3, data = baseball_train_03),
  RMSE = rmse(Model3, data = baseball_train_03),
  MAE = mae(Model3, data = baseball_train_03)
)
print(Metrics3)
##          R2     RMSE      MAE
## 1 0.2856527 11.69934 9.330048

0.4.4 OLS- MODEL 4

All defensive categories which include fielding and pitching

Model4 <- lm(TARGET_WINS~TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_FIELDING_E,data=baseball_train_03)
summary(Model4)
## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_PITCHING_H + TEAM_PITCHING_BB + 
##     TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_FIELDING_E, data = baseball_train_03)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -48.818  -8.397   0.393   8.617  42.600 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       7.915e+02  1.079e+02   7.335 3.61e-13 ***
## TEAM_PITCHING_H   2.420e-02  2.705e-03   8.947  < 2e-16 ***
## TEAM_PITCHING_BB  2.542e-02  3.943e-03   6.448 1.52e-10 ***
## TEAM_PITCHING_HR  1.503e-02  9.799e-03   1.533 0.125415    
## TEAM_PITCHING_SO -9.205e-03  2.369e-03  -3.886 0.000106 ***
## TEAM_FIELDING_E  -6.153e+02  8.770e+01  -7.016 3.44e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.46 on 1515 degrees of freedom
## Multiple R-squared:  0.1932, Adjusted R-squared:  0.1905 
## F-statistic: 72.56 on 5 and 1515 DF,  p-value: < 2.2e-16

This Model identified five significant variables at = 0.05 with an R-squared of 19%, Residual Error of 12.46 and F-Statistic of 75.56.There is no significant improvement with this model.

Metrics4 <- data.frame(
  R2 = rsquare(Model4, data = baseball_train_03),
  RMSE = rmse(Model4, data = baseball_train_03),
  MAE = mae(Model4, data = baseball_train_03)
)
print(Metrics4)
##          R2     RMSE      MAE
## 1 0.1932003 12.43339 9.945165

0.4.5 OLS- MODEL 5

Using only the significant variables from Model 3

Model5 <- lm(TARGET_WINS~TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_BATTING_3B + TEAM_BASERUN_SB,data=baseball_train_03)
summary(Model5)
## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_PITCHING_H + TEAM_PITCHING_BB + 
##     TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_BATTING_3B + TEAM_BASERUN_SB, 
##     data = baseball_train_03)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.002  -7.725   0.407   8.171  36.647 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      44.113607   4.898047   9.006  < 2e-16 ***
## TEAM_PITCHING_H   0.002544   0.002951   0.862    0.389    
## TEAM_PITCHING_BB  0.029880   0.003774   7.917 4.66e-15 ***
## TEAM_PITCHING_HR  0.134928   0.009503  14.198  < 2e-16 ***
## TEAM_PITCHING_SO -0.016789   0.002522  -6.657 3.91e-11 ***
## TEAM_BATTING_3B   0.141192   0.023104   6.111 1.26e-09 ***
## TEAM_BASERUN_SB   0.077656   0.007190  10.800  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.93 on 1514 degrees of freedom
## Multiple R-squared:  0.2606, Adjusted R-squared:  0.2576 
## F-statistic: 88.92 on 6 and 1514 DF,  p-value: < 2.2e-16

This Model identified five significant variables at = 0.05 with an R-squared of 26%, Residual Error of 11.93 and F-Statistic of 88.92. Although the R-squared is not better than than Model3, the F-statistic improved with smaller Standard Error.

Metrics5 <- data.frame(
  R2 = rsquare(Model5, data = baseball_train_03),
  RMSE = rmse(Model5, data = baseball_train_03),
  MAE = mae(Model5, data = baseball_train_03)
)
print(Metrics5)
##          R2     RMSE      MAE
## 1 0.2605772 11.90291 9.506094

0.4.6 Compare OLS Model Quality

anova(Model, Model1, Model2, Model3, Model4, Model5)
Res.DfRSSDfSum of SqFPr(>F)
1.51e+031.83e+05                  
1.51e+031.83e+052174       0.7170.488   
1.51e+032.25e+05-7-4.2e+04 49.5  1.39e-63
1.51e+032.08e+0511.66e+04136    3.01e-30
1.52e+032.35e+05-3-2.69e+0474    1.15e-44
1.51e+032.15e+0511.96e+04162    2.72e-35
tab_model(Model, Model1, Model2, Model3, Model4, Model5)
  TARGET_WINS TARGET_WINS TARGET_WINS TARGET_WINS TARGET_WINS TARGET_WINS
Predictors Estimates CI p Estimates CI p Estimates CI p Estimates CI p Estimates CI p Estimates CI p
(Intercept) 1368.26 1155.66 – 1580.87 <0.001 1376.10 1162.45 – 1589.76 <0.001 1807.57 1580.39 – 2034.75 <0.001 15.76 1.99 – 29.54 0.025 791.49 579.82 – 1003.16 <0.001 44.11 34.51 – 53.72 <0.001
TEAM BATTING H 0.03 0.02 – 0.04 <0.001 0.03 -0.01 – 0.07 0.182 0.02 0.01 – 0.04 <0.001
TEAM BATTING 2B -0.04 -0.06 – -0.02 <0.001 -0.04 -0.06 – -0.02 <0.001 0.04 0.03 – 0.06 <0.001 -0.01 -0.03 – 0.01 0.219
TEAM BATTING 3B 0.18 0.13 – 0.22 <0.001 0.18 0.13 – 0.22 <0.001 0.18 0.13 – 0.22 <0.001 0.12 0.07 – 0.16 <0.001 0.14 0.10 – 0.19 <0.001
TEAM BATTING HR 0.08 0.05 – 0.10 <0.001 0.24 -0.04 – 0.53 0.095 0.10 0.08 – 0.12 <0.001
TEAM BATTING BB 0.16 0.09 – 0.22 <0.001 0.16 0.02 – 0.30 0.023 0.04 0.03 – 0.05 <0.001
TEAM BATTING SO -0.09 -0.13 – -0.05 <0.001 -0.11 -0.16 – -0.05 <0.001 -0.01 -0.03 – 0.01 0.379 -0.01 -0.02 – -0.01 <0.001
TEAM BASERUN SB 0.07 0.05 – 0.08 <0.001 0.06 0.05 – 0.08 <0.001 0.05 0.04 – 0.07 <0.001 0.08 0.06 – 0.09 <0.001 0.08 0.06 – 0.09 <0.001
TEAM BASERUN CS -0.07 -0.13 – 0.00 0.063 -0.06 -0.13 – 0.01 0.099 -0.07 -0.14 – 0.00 0.065
TEAM PITCHING BB -0.11 -0.18 – -0.05 <0.001 -0.12 -0.25 – 0.02 0.084 0.03 0.02 – 0.03 <0.001 0.03 0.02 – 0.04 <0.001
TEAM PITCHING SO 0.06 0.02 – 0.11 0.002 0.08 0.03 – 0.13 0.001 -0.01 -0.03 – 0.01 0.349 -0.01 -0.01 – -0.00 <0.001 -0.02 -0.02 – -0.01 <0.001
TEAM FIELDING E -1084.76 -1256.17 – -913.35 <0.001 -1091.94 -1264.26 – -919.62 <0.001 -1410.16 -1592.85 – -1227.48 <0.001 -615.31 -787.34 – -443.27 <0.001
TEAM FIELDING DP -0.11 -0.14 – -0.08 <0.001 -0.11 -0.14 – -0.08 <0.001 -0.05 -0.08 – -0.02 0.001
TEAM PITCHING H 0.00 -0.04 – 0.04 0.939 0.02 0.02 – 0.03 <0.001 0.00 -0.00 – 0.01 0.389
TEAM PITCHING HR -0.16 -0.44 – 0.11 0.248 0.02 -0.00 – 0.03 0.125 0.13 0.12 – 0.15 <0.001
Observations 1521 1521 1521 1521 1521 1521
R2 / R2 adjusted 0.372 / 0.367 0.373 / 0.367 0.229 / 0.225 0.286 / 0.282 0.193 / 0.191 0.261 / 0.258

0.4.7 RIDGE Regression- MODEL 6

The Ridge regression is an extension of linear regression where the loss function is modified to minimize the complexity of the model. This modification is done by adding a penalty parameter that is equivalent to the square of the magnitude of the coefficients.

Before implementing the RIDGE model, we will split the training dataset into 2 parts that is - training set within the training set and a test set that can be used for evaluation. By enforcing stratified sampling both our training and testing sets have approximately equal response “TARGET_WINS” distributions.

Transforming the variables into the form of a matrix will enable us to penalize the model using the ‘glmnet’ method in glmnet package.

#Split the data into Training and Test Set
baseball_train_set<- initial_split(baseball_train_03, prop = 0.7, strata = "TARGET_WINS")
train_baseball  <- training(baseball_train_set)
test_baseball   <- testing(baseball_train_set)

train_Ind<- as.matrix(train_baseball)
train_Dep<- as.matrix(train_baseball$TARGET_WINS)

test_Ind<- as.matrix(test_baseball)
test_Dep<- as.matrix(test_baseball$TARGET_WINS)

For the avoidance of multicollinearity, avoiding overfitting and predicting better, implementing RIDGE regression will become useful.

lambdas <- 10^seq(2, -3, by = -.1)
Model6 <- glmnet(train_Ind,train_Dep, nlambda = 25, alpha = 0, family = 'gaussian', lambda = lambdas)
summary(Model6)
##           Length Class     Mode   
## a0         51    -none-    numeric
## beta      765    dgCMatrix S4     
## df         51    -none-    numeric
## dim         2    -none-    numeric
## lambda     51    -none-    numeric
## dev.ratio  51    -none-    numeric
## nulldev     1    -none-    numeric
## npasses     1    -none-    numeric
## jerr        1    -none-    numeric
## offset      1    -none-    logical
## call        7    -none-    call   
## nobs        1    -none-    numeric
print(Model6, digits = max(3, getOption("digits") - 3),
           signif.stars = getOption("show.signif.stars"))
## 
## Call:  glmnet(x = train_Ind, y = train_Dep, family = "gaussian", alpha = 0,      nlambda = 25, lambda = lambdas) 
## 
##    Df   %Dev  Lambda
## 1  15  30.03 100.000
## 2  15  35.00  79.430
## 3  15  40.34  63.100
## 4  15  45.97  50.120
## 5  15  51.78  39.810
## 6  15  57.65  31.620
## 7  15  63.44  25.120
## 8  15  69.02  19.950
## 9  15  74.26  15.850
## 10 15  79.05  12.590
## 11 15  83.30  10.000
## 12 15  86.97   7.943
## 13 15  90.05   6.310
## 14 15  92.55   5.012
## 15 15  94.53   3.981
## 16 15  96.06   3.162
## 17 15  97.20   2.512
## 18 15  98.05   1.995
## 19 15  98.65   1.585
## 20 15  99.08   1.259
## 21 15  99.38   1.000
## 22 15  99.59   0.794
## 23 15  99.73   0.631
## 24 15  99.82   0.501
## 25 15  99.88   0.398
## 26 15  99.92   0.316
## 27 15  99.95   0.251
## 28 15  99.97   0.200
## 29 15  99.98   0.158
## 30 15  99.99   0.126
## 31 15  99.99   0.100
## 32 15  99.99   0.079
## 33 15 100.00   0.063
## 34 15 100.00   0.050
## 35 15 100.00   0.040
## 36 15 100.00   0.032
## 37 15 100.00   0.025
## 38 15 100.00   0.020
## 39 15 100.00   0.016
## 40 15 100.00   0.013
## 41 15 100.00   0.010
## 42 15 100.00   0.008
## 43 15 100.00   0.006
## 44 15 100.00   0.005
## 45 15 100.00   0.004
## 46 15 100.00   0.003
## 47 15 100.00   0.003
## 48 15 100.00   0.002
## 49 15 100.00   0.002
## 50 15 100.00   0.001
## 51 15 100.00   0.001

The significant difference between the OLS and the Ridge Regresion is the hyperparameter tuning using lambda. The Ridge regression does not perform Feature Selection, but it predicts better and solve overfitting. Cross Validating the Ridge Regression will help us to identify the optimal lambda to penalize the model and enhance the predictability.

CrossVal_ridge <- cv.glmnet(train_Ind,train_Dep, alpha = 0, lambda = lambdas)
optimal_lambda <- CrossVal_ridge$lambda.min
optimal_lambda #The optimal lambda is 0.001 which we will use to penelize the Ridge Regression model.
## [1] 0.001
coef(CrossVal_ridge) # Shows the coefficients
## 16 x 1 sparse Matrix of class "dgCMatrix"
##                             s1
## (Intercept)       1.535312e-01
## TARGET_WINS       9.998834e-01
## TEAM_BATTING_H   -1.736638e-05
## TEAM_BATTING_2B  -7.943975e-06
## TEAM_BATTING_3B   1.688642e-05
## TEAM_BATTING_HR  -4.354268e-04
## TEAM_BATTING_BB  -6.515657e-06
## TEAM_BATTING_SO   9.941254e-05
## TEAM_BASERUN_SB   1.482210e-05
## TEAM_BASERUN_CS  -2.352124e-05
## TEAM_PITCHING_H   1.926142e-05
## TEAM_PITCHING_HR  4.271052e-04
## TEAM_PITCHING_BB  9.457700e-06
## TEAM_PITCHING_SO -9.800251e-05
## TEAM_FIELDING_E  -1.175340e-01
## TEAM_FIELDING_DP -2.368209e-05
plot(CrossVal_ridge)

The plot shows that the errors increases as the magnitude of lambda increases, previously, we identified that the optimal lambda is 0.001 which is very obvious from the plot above. The coefficients are restricted to be small but not quite zero as Ridge Regression does not force the coefficient to zero. This indicates that the model is performing well so far. But let’s make it better using the optimal labmda.

eval_results <- function(true, predicted, df){
  SSE <- sum((predicted - true)^2)
  SST <- sum((true - mean(true))^2)
  R_square <- 1 - SSE / SST
  RMSE = sqrt(SSE/nrow(df))
data.frame(   
  RMSE = RMSE,
  Rsquare = R_square
)
  
}
# Prediction and evaluation on train data
predictions_train <- predict(Model6, s = optimal_lambda, newx = train_Ind)
eval_results(train_Dep, predictions_train, train_baseball)
RMSERsquare
0.001711

We should be a little concern about the 100% R-squared performance for this Model. Although the Ridge Regression forces the coefficients towards zero to improve the Model performance and enhance the predictability, the very high peformance may require further investigation. Lets improve the model using a more reason lambda because optimal might not always be the best.

0.4.8 The Improved Ridge Regression

Model6_Improved <- glmnet(train_Ind,train_Dep, nlambda = 25, alpha = 0, family = 'gaussian', lambda = 6.310)
summary(Model6_Improved)
##           Length Class     Mode   
## a0         1     -none-    numeric
## beta      15     dgCMatrix S4     
## df         1     -none-    numeric
## dim        2     -none-    numeric
## lambda     1     -none-    numeric
## dev.ratio  1     -none-    numeric
## nulldev    1     -none-    numeric
## npasses    1     -none-    numeric
## jerr       1     -none-    numeric
## offset     1     -none-    logical
## call       7     -none-    call   
## nobs       1     -none-    numeric
coef(Model6_Improved)
## 16 x 1 sparse Matrix of class "dgCMatrix"
##                             s0
## (Intercept)       1.764003e+02
## TARGET_WINS       6.224314e-01
## TEAM_BATTING_H    6.627483e-03
## TEAM_BATTING_2B   1.362185e-03
## TEAM_BATTING_3B   2.639383e-02
## TEAM_BATTING_HR   8.228176e-03
## TEAM_BATTING_BB   7.133799e-03
## TEAM_BATTING_SO  -1.082704e-03
## TEAM_BASERUN_SB   1.357864e-02
## TEAM_BASERUN_CS  -2.805586e-03
## TEAM_PITCHING_H   1.789510e-03
## TEAM_PITCHING_HR  7.331744e-03
## TEAM_PITCHING_BB  4.904272e-03
## TEAM_PITCHING_SO -1.705157e-03
## TEAM_FIELDING_E  -1.331875e+02
## TEAM_FIELDING_DP -2.366768e-02

Let’s compute the Model’s Performance Metric to see how this model is doing.

eval_results <- function(true, predicted, df){
  SSE <- sum((predicted - true)^2)
  SST <- sum((true - mean(true))^2)
  R_square <- 1 - SSE / SST
  RMSE = sqrt(SSE/nrow(df))
data.frame(   
  RMSE = RMSE,
  Rsquare = R_square
)
  
}

# Prediction and evaluation on train data
predictions_train <- predict(Model6_Improved, s = lambda, newx = train_Ind)
eval_results(train_Dep, predictions_train, train_baseball)
RMSERsquare
4.320.9
# Prediction and evaluation on test data
predictions_test <- predict(Model6_Improved, s = lambda, newx = test_Ind)
eval_results(test_Dep, predictions_test, test_baseball)
RMSERsquare
4.510.899

The improved Model6 output shows that the RMSE and R-squared values for the Ridge Regression model on the training and test data are significantly improved. The Loss Function (RMSE) are severely reduced compared to the OLS models which indicates that the Ridge Regression is not overfitting. These performance is significantly improved compared to the OLS Models 1 to 5.

0.4.9 Model Performance Comparison

ModelName <- c("Model", "Model1","Model2","Model3","Model4","Model5","Model6")
Model_RSquared <- c("37%", "37%", "22%", "28%", "19%", "26% ", "90%")
Model_RMSE <- c("11.01", "10.96", "12.15", "11.69", "12.43", "11.93 ", "4.33")
Model_FStatistic <- c("74.59", "64.01", "64.14", "75.58", "72.56", "88.92 ", "NA")
Model_Performance <- data.frame(ModelName,Model_RSquared,Model_RMSE,Model_FStatistic)
Model_Performance
ModelNameModel_RSquaredModel_RMSEModel_FStatistic
Model37%11.0174.59
Model137%10.9664.01
Model222%12.1564.14
Model328%11.6975.58
Model419%12.4372.56
Model526% 11.93 88.92
Model690%4.33NA

0.4.10 Model Prediction

Based on the Model metrics above, we’re ready to make prediction and we will select our acceptable OLS Model3 and Model5 which has better F-Statistic, smaller standard errors and less negative coefficient as our best OLS models. We will also compare the prediction accuracy of these models to that of the improved Ridge Regression Model which is our champion Model for this exercise based on the very small RMSE and the highest R-squared of over 90%.

predicted <- predict(Model3, newx = test_baseball)# predict on test data
predicted_values <- cbind (actual=test_baseball$TARGET_WINS, predicted)  # combine
## Warning in cbind(actual = test_baseball$TARGET_WINS, predicted): number of rows
## of result is not a multiple of vector length (arg 1)
predicted_values
##      actual predicted
## 1        76  69.49875
## 2        70  67.56125
## 3        81  68.38378
## 4        91  73.25628
## 5        80  67.37030
## 6        70  66.86382
## 7        82  63.36361
## 8        84  76.58169
## 9        51  89.59170
## 10       77  76.54657
## 11       68  86.70081
## 12       58  77.31870
## 13       56  78.60647
## 14       63  84.73255
## 15       59  89.15530
## 16       84  86.01142
## 17       84  76.15138
## 18       68  76.18203
## 19       68  79.02243
## 20       79  72.03653
## 21       93  89.30377
## 22       85  84.16328
## 23       81  79.30964
## 24       76  76.00667
## 25       69  93.36918
## 26       82  79.72780
## 27       76  83.96472
## 28       66  81.83322
## 29       98  86.75976
## 30      104  83.85432
## 31       79  75.24767
## 32       69  96.12431
## 33       66  85.18416
## 34       78  87.23777
## 35       66  86.66397
## 36       59  80.90673
## 37       45  71.73633
## 38       69  81.22686
## 39       62  90.64517
## 40       60  94.18601
## 41       78  95.53100
## 42       95  80.08285
## 43       97  75.86419
## 44       88  66.74352
## 45       98  64.57578
## 46       84  60.91243
## 47       87  72.28834
## 48       85  68.48019
## 49       80  65.67250
## 50       88  72.48471
## 51       88  86.44011
## 52      107  80.65902
## 53       83  82.06254
## 54       94  78.70595
## 55       83  77.58369
## 56       85  72.97126
## 57       78  78.18561
## 58       49  72.24895
## 59       60  76.25716
## 60       68  77.02116
## 61       86  75.30225
## 62       80  76.46272
## 63       76  78.36297
## 64       92  68.74638
## 65       96  70.51741
## 66       84  66.00240
## 67       78  74.32965
## 68       83  68.02212
## 69       80  71.72855
## 70       97  69.41157
## 71       85  69.61566
## 72       78  70.79319
## 73       92  69.48860
## 74       83  65.80599
## 75       98  70.70940
## 76       75  82.73716
## 77       62  77.23526
## 78      110  81.04205
## 79       87  86.68577
## 80       86  81.16218
## 81       90  88.42305
## 82      105  85.99275
## 83       74  73.58515
## 84       78  82.22049
## 85      103  85.06236
## 86       73  86.09993
## 87       81  76.29960
## 88       59  79.91231
## 89       72  77.53938
## 90       88  84.34369
## 91       92  74.79848
## 92       79  70.02759
## 93       71  82.39423
## 94       78  79.28918
## 95       68  91.97048
## 96       89  78.96951
## 97       88  74.19733
## 98      100  72.42719
## 99       93  77.14336
## 100      83  76.15731
## 101      83  77.37089
## 102     105  72.05640
## 103      74  76.89373
## 104      59  75.92414
## 105      72  85.81217
## 106      79  75.59281
## 107      81  80.60366
## 108      96  85.62160
## 109      89  82.00590
## 110      95  84.74713
## 111      99  84.88357
## 112      86  86.74637
## 113      85  89.85103
## 114      95  88.78040
## 115      75  78.46174
## 116      74  84.86044
## 117      77  83.42672
## 118      94  82.79613
## 119      90  82.72617
## 120      93  66.91725
## 121      90  69.51332
## 122      72  78.66910
## 123      70  72.33942
## 124      77  67.50771
## 125      79  64.77165
## 126      63  69.79720
## 127     111  71.15595
## 128      62  77.34661
## 129      78  77.95811
## 130      59  86.54917
## 131     102  73.89007
## 132      65  79.45996
## 133      98  76.41158
## 134      89  81.88671
## 135     108  84.23277
## 136      88  81.62631
## 137      91  78.66292
## 138      70  78.01240
## 139      86  82.19463
## 140      84  84.27986
## 141      88  73.82318
## 142      77  75.50558
## 143      94  72.20240
## 144      99  71.73169
## 145      71  78.70550
## 146      93  82.60550
## 147      92  79.89101
## 148      79  80.98410
## 149      72  73.62353
## 150      97  74.78823
## 151     117  70.09769
## 152      82  69.79510
## 153      79  73.21664
## 154      87  76.42604
## 155      71  70.00951
## 156      61  79.24643
## 157     100  76.55529
## 158      87  77.45710
## 159      97  77.29383
## 160      91  74.83120
## 161      83  80.57564
## 162      73  75.27052
## 163      68  72.96984
## 164      66  88.01300
## 165      70  89.33125
## 166      83  86.25568
## 167      71  69.87604
## 168      93  89.64940
## 169      82  80.48480
## 170      94  81.38282
## 171      79  78.12840
## 172      82  80.46360
## 173     100  77.90473
## 174      77  86.93266
## 175     101  88.60549
## 176      79  86.22276
## 177      74  85.13130
## 178      83  80.44713
## 179      92  88.03475
## 180      84  80.82213
## 181      75  83.47961
## 182      66  72.80089
## 183      64  80.37391
## 184      92  79.72336
## 185      79  78.69135
## 186      83  85.57211
## 187      66  83.20429
## 188      65  89.48194
## 189      79  96.68228
## 190      80  87.75298
## 191      90  91.87313
## 192      77  86.26651
## 193      76  77.82142
## 194     102  76.04487
## 195      87  84.88891
## 196      65  81.55068
## 197      85  83.70493
## 198      92  80.38773
## 199      83 101.92399
## 200      88  81.10336
## 201      58  73.46680
## 202      73  93.36972
## 203      80  86.05010
## 204      75  85.59259
## 205      85  73.88114
## 206      69  74.83380
## 207      70  64.08681
## 208      75  65.84094
## 209      92  71.98560
## 210      94  79.13560
## 211     102  85.15069
## 212      75  84.82851
## 213      91  88.64454
## 214      86  87.36139
## 215      94  73.46671
## 216      92  73.32881
## 217      98  76.75376
## 218      95  80.18206
## 219      73  75.03975
## 220      77  75.66433
## 221      86  70.51948
## 222      88  74.54246
## 223      75  72.27715
## 224      80  70.35686
## 225      69  72.18585
## 226      68  79.91554
## 227      66  88.74324
## 228      77  87.01201
## 229      87  85.96054
## 230      83  89.84882
## 231      80  93.28619
## 232      78  91.12115
## 233      74  94.67752
## 234      74  73.79792
## 235      56  80.75316
## 236      67  76.99427
## 237      59  87.39336
## 238      53  77.79257
## 239      45  75.87232
## 240      67  87.87867
## 241      65  79.02455
## 242      97  79.53593
## 243      70  77.39581
## 244      89  71.37449
## 245      67  85.39949
## 246      62  78.33072
## 247      66  86.04030
## 248      71  79.55776
## 249      89  82.75494
## 250      91  86.19625
## 251      81  87.67022
## 252      77  85.00793
## 253      85  80.45297
## 254      77  87.60385
## 255      81  82.44495
## 256      71  77.50487
## 257      78  79.20244
## 258      41  69.31674
## 259      66  74.80593
## 260      98  83.14847
## 261     108  91.63208
## 262      91  93.48635
## 263      72  84.53632
## 264      66  86.35405
## 265      83  86.94776
## 266      87  82.27458
## 267      99  80.34078
## 268      77  97.84021
## 269      97  90.77786
## 270      77  86.29480
## 271      54  89.59173
## 272      61  81.45834
## 273      96  79.98754
## 274     106  76.10745
## 275      87  78.12297
## 276      86  76.91669
## 277     102  74.90087
## 278     108  87.24509
## 279      96  95.14666
## 280      93  90.93669
## 281      83  90.70110
## 282      80  81.29394
## 283      99  72.64682
## 284     100  82.98819
## 285      99  75.16897
## 286      91  78.10723
## 287      98  82.14880
## 288      90  84.91555
## 289      92  97.32176
## 290      96 101.10846
## 291     114  84.72247
## 292      88  75.83547
## 293      96  76.87742
## 294     101  84.10901
## 295      88  89.08411
## 296     110  81.50902
## 297      38  81.45233
## 298      58  78.28904
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## 1174     71  89.51002
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## 1178     98  86.66543
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## 1183     83  77.78130
## 1184     87  79.97524
## 1185     99  76.64023
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## 1188     77  83.45274
## 1189     54  80.12935
## 1190     61  82.74097
## 1191     96  78.38666
## 1192    106  85.76137
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## 1194     86  75.26520
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## 1197     96  84.42915
## 1198     93  76.77597
## 1199     83  85.49227
## 1200     80  85.43649
## 1201     99  82.01907
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## 1204     91  72.47560
## 1205     98  74.71764
## 1206     90  79.17754
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## 1208     96  72.52410
## 1209    114  72.99217
## 1210     88  73.95031
## 1211     96  80.28342
## 1212    101  76.84542
## 1213     88  72.66814
## 1214    110  79.43934
## 1215     38  84.21485
## 1216     58  76.13103
## 1217     56  75.85239
## 1218    107  80.34523
## 1219     63  81.40989
## 1220     58  89.32086
## 1221     56  71.65314
## 1222     52  83.47910
## 1223     74  79.31333
## 1224     62  83.99140
## 1225     77  86.04019
## 1226     75  80.58368
## 1227    102  78.55305
## 1228     77  81.19871
## 1229     76  80.75316
## 1230     99  83.34638
## 1231     66  72.71053
## 1232     87  83.05962
## 1233     78  75.30919
## 1234     83  61.84184
## 1235     93  75.49493
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## 1254     79  83.81551
## 1255    120  81.01255
## 1256    103  83.63764
## 1257    117  83.91016
## 1258     73  75.41719
## 1259     54  77.28019
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## 1262     79  78.48064
## 1263     93  86.08551
## 1264     72  75.94236
## 1265     73  79.40133
## 1266     84  84.21428
## 1267     87  76.93622
## 1268     69  76.13965
## 1269     65  80.08179
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## 1271     80  78.57486
## 1272     57  82.35286
## 1273     80  91.28096
## 1274     98  96.60549
## 1275     67  94.87582
## 1276     63  92.31089
## 1277     60  94.05118
## 1278     73  94.20321
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## 1282     82  78.77420
## 1283     98  75.09218
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## 1285     57  71.98056
## 1286     59  69.20131
## 1287     77  77.96775
## 1288     83  87.78023
## 1289     82  86.35932
## 1290    116  82.21422
## 1291     93  79.65022
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## 1294     97  90.70037
## 1295     88  84.24705
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## 1297    103  87.97745
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## 1299     97  92.90616
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## 1309     77  79.71387
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## 1313     87  73.46866
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## 1316     72  79.89839
## 1317     89  91.02800
## 1318     87  88.08337
## 1319     72  88.85369
## 1320     75  94.45356
## 1321     88  84.35462
## 1322    101  78.05740
## 1323     75  83.95127
## 1324     50  86.33623
## 1325     42  82.90238
## 1326     90  81.67246
## 1327     88  85.51611
## 1328     79  80.18255
## 1329     52  79.54560
## 1330     58  75.54402
## 1331     64  73.65885
## 1332     64  71.01422
## 1333    106  81.45969
## 1334     76  82.86082
## 1335    101  80.62995
## 1336     92  88.25696
## 1337     76  76.46766
## 1338     89  80.34835
## 1339     83  72.91620
## 1340     85  76.90302
## 1341     93  83.18826
## 1342     76  70.94189
## 1343     76  76.87526
## 1344     84  80.45797
## 1345     84  69.39293
## 1346     83  84.77448
## 1347     78  85.62956
## 1348     86  83.98628
## 1349     70  80.12358
## 1350     88  86.59676
## 1351     73  87.14948
## 1352     83  89.34894
## 1353    105  96.16870
## 1354    100  89.87599
## 1355     63  82.54057
## 1356     67  72.35155
## 1357     70  75.48538
## 1358     83  89.39629
## 1359     64  90.79489
## 1360     83  72.17537
## 1361     85  98.82305
## 1362     83  91.07166
## 1363     88  74.70489
## 1364     95  74.08408
## 1365     71  75.45591
## 1366     54  75.76548
## 1367     53  67.74708
## 1368     84  63.73702
## 1369     83  63.09795
## 1370     52  79.36250
## 1371     73  85.89266
## 1372     75  88.93737
## 1373     76  86.42517
## 1374     96  79.24316
## 1375     86  80.67956
## 1376     67  74.32748
## 1377     71  77.51726
## 1378     76  74.68399
## 1379     70  90.58364
## 1380     81  84.36898
## 1381     91  89.61974
## 1382     80  87.27046
## 1383     70  91.72583
## 1384     82  90.46183
## 1385     84  92.84914
## 1386     51  83.81388
## 1387     77  78.44059
## 1388     68  78.52910
## 1389     58  84.60661
## 1390     56  80.35381
## 1391     63  81.01220
## 1392     59  81.58834
## 1393     84  83.27561
## 1394     84  83.42052
## 1395     68  88.09058
## 1396     68  80.71739
## 1397     79  82.57078
## 1398     93  79.27823
## 1399     85  79.45690
## 1400     81  87.45363
## 1401     76  82.67362
## 1402     69  83.26842
## 1403     82  79.96190
## 1404     76  83.17930
## 1405     66  81.28235
## 1406     98  88.91157
## 1407    104  79.89535
## 1408     79  82.59148
## 1409     69  83.88969
## 1410     66  79.11848
## 1411     78  80.02279
## 1412     66  79.38053
## 1413     59  79.62617
## 1414     45  81.86965
## 1415     69  81.52487
## 1416     62  75.59362
## 1417     60  76.78916
## 1418     78  74.30279
## 1419     95  76.30224
## 1420     97  69.65634
## 1421     88  74.26352
## 1422     98  83.82491
## 1423     84  84.75257
## 1424     87  73.73032
## 1425     85  75.06767
## 1426     80  84.63083
## 1427     88  78.97639
## 1428     88  76.45999
## 1429    107  71.29329
## 1430     83  81.13600
## 1431     94  80.70558
## 1432     83  78.14821
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## 1434     78  80.36170
## 1435     49  81.90376
## 1436     60  93.33645
## 1437     68  88.19413
## 1438     86  90.25974
## 1439     80  84.66770
## 1440     76  82.96837
## 1441     92  89.12204
## 1442     96  87.95702
## 1443     84  82.57955
## 1444     78  82.70475
## 1445     83  79.25230
## 1446     80  80.20210
## 1447     97  71.72168
## 1448     85  74.42225
## 1449     78  79.63366
## 1450     92  81.94346
## 1451     83  82.75808
## 1452     98  80.78277
## 1453     75  76.62972
## 1454     62  77.85689
## 1455    110  68.36665
## 1456     87  69.11303
## 1457     86  66.24626
## 1458     90  81.84655
## 1459    105  78.35787
## 1460     74  69.94233
## 1461     78  75.76441
## 1462    103  81.02432
## 1463     73  74.52049
## 1464     81  81.01784
## 1465     59  71.46302
## 1466     72  75.61764
## 1467     88  78.34152
## 1468     92  77.83881
## 1469     79  77.22750
## 1470     71  86.79195
## 1471     78  77.71251
## 1472     68  82.62507
## 1473     89  81.35461
## 1474     88  89.93796
## 1475    100  96.51608
## 1476     93  86.16285
## 1477     83  91.33961
## 1478     83  86.93554
## 1479    105  84.63384
## 1480     74  72.70103
## 1481     59  71.08248
## 1482     72  91.35231
## 1483     79  86.00672
## 1484     81  91.94352
## 1485     96  85.43193
## 1486     89  83.53370
## 1487     95  85.50160
## 1488     99  80.42214
## 1489     86  87.39467
## 1490     85  93.21013
## 1491     95  83.54572
## 1492     75  77.31713
## 1493     74  89.01612
## 1494     77  89.92811
## 1495     94  76.09057
## 1496     90  77.41702
## 1497     93  85.67225
## 1498     90  87.78315
## 1499     72  71.42951
## 1500     70  78.50359
## 1501     77  70.81870
## 1502     79  80.91072
## 1503     63  75.32227
## 1504    111  68.16678
## 1505     62  79.64992
## 1506     78  71.17387
## 1507     59  79.90700
## 1508    102  84.00078
## 1509     65  82.16401
## 1510     98  77.16445
## 1511     89  76.00898
## 1512    108  78.13104
## 1513     88  72.61470
## 1514     91  77.78654
## 1515     70  77.61568
## 1516     86  72.10171
## 1517     84  81.84125
## 1518     88  77.74045
## 1519     77  78.32464
## 1520     94  67.84336
## 1521     99  80.92785
mean (apply(predicted_values, 1, min)/apply(predicted_values, 1, max)) # calculate accuracy
## [1] 0.8582949

The prediction accuracy here is at 85.85%

predicted <- predict(Model5, newx = test_baseball)# predict on test data
predicted_values <- cbind (actual=test_baseball$TARGET_WINS, predicted)  # combine
## Warning in cbind(actual = test_baseball$TARGET_WINS, predicted): number of rows
## of result is not a multiple of vector length (arg 1)
predicted_values
##      actual predicted
## 1        76  67.44342
## 2        70  67.45113
## 3        81  70.07451
## 4        91  75.43397
## 5        80  67.72684
## 6        70  67.72587
## 7        82  63.76240
## 8        84  77.57821
## 9        51  87.50168
## 10       77  75.33778
## 11       68  86.57140
## 12       58  76.10532
## 13       56  79.30210
## 14       63  85.05514
## 15       59  89.33395
## 16       84  86.35917
## 17       84  75.87327
## 18       68  75.27661
## 19       68  76.92580
## 20       79  72.38257
## 21       93  91.24888
## 22       85  81.21417
## 23       81  76.85482
## 24       76  75.73465
## 25       69  92.95906
## 26       82  79.74796
## 27       76  82.59460
## 28       66  82.93132
## 29       98  84.01025
## 30      104  83.67314
## 31       79  74.98318
## 32       69  94.40116
## 33       66  85.35736
## 34       78  86.31272
## 35       66  85.20792
## 36       59  79.86968
## 37       45  71.39335
## 38       69  79.65920
## 39       62  88.48478
## 40       60  92.89590
## 41       78  95.70531
## 42       95  80.15908
## 43       97  77.64883
## 44       88  67.18263
## 45       98  64.94270
## 46       84  62.58915
## 47       87  71.88851
## 48       85  67.42428
## 49       80  66.89260
## 50       88  74.11101
## 51       88  84.19595
## 52      107  77.19389
## 53       83  81.04087
## 54       94  78.77630
## 55       83  78.14143
## 56       85  76.04199
## 57       78  79.46042
## 58       49  73.69400
## 59       60  74.54247
## 60       68  75.74929
## 61       86  74.97222
## 62       80  75.44410
## 63       76  79.02010
## 64       92  69.75831
## 65       96  70.64981
## 66       84  69.12150
## 67       78  74.71038
## 68       83  69.62466
## 69       80  70.17964
## 70       97  70.25523
## 71       85  69.23863
## 72       78  71.09084
## 73       92  72.67250
## 74       83  67.57287
## 75       98  73.96325
## 76       75  82.29090
## 77       62  76.84995
## 78      110  80.96384
## 79       87  86.00466
## 80       86  80.99199
## 81       90  90.30120
## 82      105  86.68144
## 83       74  76.35323
## 84       78  83.38313
## 85      103  86.70403
## 86       73  88.15920
## 87       81  77.29176
## 88       59  78.74435
## 89       72  78.89828
## 90       88  84.86922
## 91       92  76.52274
## 92       79  68.53117
## 93       71  80.63921
## 94       78  79.04574
## 95       68  91.06636
## 96       89  78.71918
## 97       88  73.94550
## 98      100  72.47404
## 99       93  76.97092
## 100      83  78.30645
## 101      83  77.84933
## 102     105  73.92133
## 103      74  87.30513
## 104      59  76.38672
## 105      72  87.67499
## 106      79  77.46469
## 107      81  82.05258
## 108      96  85.24247
## 109      89  87.16002
## 110      95  83.45556
## 111      99  83.35626
## 112      86  86.73727
## 113      85  91.21914
## 114      95  88.04261
## 115      75  78.51105
## 116      74  81.95540
## 117      77  82.76242
## 118      94  81.70924
## 119      90  85.27418
## 120      93  69.20964
## 121      90  69.37010
## 122      72  78.11881
## 123      70  69.72151
## 124      77  66.82142
## 125      79  65.28600
## 126      63  71.84567
## 127     111  71.77897
## 128      62  76.57889
## 129      78  79.01450
## 130      59  85.26364
## 131     102  75.40463
## 132      65  82.54799
## 133      98  78.50081
## 134      89  82.48798
## 135     108  84.16834
## 136      88  81.26085
## 137      91  80.88645
## 138      70  75.53196
## 139      86  80.44320
## 140      84  84.92377
## 141      88  75.02694
## 142      77  75.21066
## 143      94  71.99137
## 144      99  71.21597
## 145      71  81.47611
## 146      93  80.90290
## 147      92  80.66912
## 148      79  82.13911
## 149      72  74.68290
## 150      97  73.79684
## 151     117  69.14518
## 152      82  69.07586
## 153      79  74.64032
## 154      87  75.73128
## 155      71  71.89865
## 156      61  78.09027
## 157     100  76.71291
## 158      87  77.55373
## 159      97  77.24322
## 160      91  75.43189
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## 1357     70  75.62014
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## 1360     83  72.47442
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## 1412     66  81.25349
## 1413     59  77.38370
## 1414     45  78.72661
## 1415     69  78.92403
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## 1420     97  69.03655
## 1421     88  73.11790
## 1422     98  80.32345
## 1423     84  81.62448
## 1424     87  72.46322
## 1425     85  73.05070
## 1426     80  82.60617
## 1427     88  76.80164
## 1428     88  74.71730
## 1429    107  72.53451
## 1430     83  79.24098
## 1431     94  80.59521
## 1432     83  77.15610
## 1433     85  74.10392
## 1434     78  79.46361
## 1435     49  81.59949
## 1436     60  92.84525
## 1437     68  86.74162
## 1438     86  89.00812
## 1439     80  83.81009
## 1440     76  82.45990
## 1441     92  86.81278
## 1442     96  87.08329
## 1443     84  81.02582
## 1444     78  81.79036
## 1445     83  76.09764
## 1446     80  79.58062
## 1447     97  70.85242
## 1448     85  73.77270
## 1449     78  78.06725
## 1450     92  82.56246
## 1451     83  82.22541
## 1452     98  82.97301
## 1453     75  77.64168
## 1454     62  79.54539
## 1455    110  68.24060
## 1456     87  70.24206
## 1457     86  67.33869
## 1458     90  80.95609
## 1459    105  77.96052
## 1460     74  70.39466
## 1461     78  75.42098
## 1462    103  79.98365
## 1463     73  72.26114
## 1464     81  80.33432
## 1465     59  72.40334
## 1466     72  76.28209
## 1467     88  77.74120
## 1468     92  76.98433
## 1469     79  74.78655
## 1470     71  84.57421
## 1471     78  77.67434
## 1472     68  84.70901
## 1473     89  78.88817
## 1474     88  86.51928
## 1475    100  94.59262
## 1476     93  83.68060
## 1477     83  90.09426
## 1478     83  86.45699
## 1479    105  85.18612
## 1480     74  73.40551
## 1481     59  73.27707
## 1482     72  91.99847
## 1483     79  86.52118
## 1484     81  92.42330
## 1485     96  84.04087
## 1486     89  83.86156
## 1487     95  85.52067
## 1488     99  80.89763
## 1489     86  86.58041
## 1490     85  91.77189
## 1491     95  82.89224
## 1492     75  78.98014
## 1493     74  87.14506
## 1494     77  89.37809
## 1495     94  74.04740
## 1496     90  75.81540
## 1497     93  84.01966
## 1498     90  88.51943
## 1499     72  72.64024
## 1500     70  79.09215
## 1501     77  70.49611
## 1502     79  79.19742
## 1503     63  74.80890
## 1504    111  70.05598
## 1505     62  79.93610
## 1506     78  71.64301
## 1507     59  81.23458
## 1508    102  84.56800
## 1509     65  81.05502
## 1510     98  77.11747
## 1511     89  78.69532
## 1512    108  76.56764
## 1513     88  74.31725
## 1514     91  79.24925
## 1515     70  77.95751
## 1516     86  73.35033
## 1517     84  82.50775
## 1518     88  77.38719
## 1519     77  79.51796
## 1520     94  67.76209
## 1521     99  80.89661
mean (apply(predicted_values, 1, min)/apply(predicted_values, 1, max)) # calculate accuracy
## [1] 0.8595468

The prediction accuracy for the OLS Model5 is at 85.94% which is not bad for this purpose. But lets compare it to the Champion Model- The improved Ridge Regression.

predicted <- predict(Model6_Improved, newx = test_Ind)# predict on test data
predicted_values <- cbind (actual=test_baseball$TARGET_WINS, predicted)  # combine
predicted_values
##      actual        s0
## 6        76  73.66949
## 12       70  71.84946
## 13       81  79.67667
## 18       91  85.71056
## 19       80  78.85583
## 22       70  74.76474
## 25       82  83.72951
## 36       84  83.91670
## 37       51  59.85228
## 41       77  82.53565
## 43       68  71.29912
## 44       58  62.09606
## 58       56  63.38799
## 60       63  68.62745
## 62       59  66.31901
## 67       84  82.03755
## 70       84  80.49142
## 71       68  69.11721
## 75       68  69.88603
## 80       79  80.85346
## 85       93  88.87315
## 90       85  83.56938
## 92       81  78.43074
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Lets calculate the accuracy of using Model6 for our predictions

mean (apply(predicted_values, 1, min)/apply(predicted_values, 1, max)) # calculate accuracy
## [1] 0.9554337

The prediction accuracy of the improved Ridge Regression Model is 95.75%.

ModelName <- c("Model3", "Model5","Model6")
Model_Accuracy <- c("85.85%", "85.85%", "95.75%")
AccuracyCompared <- data.frame(ModelName,Model_Accuracy)
AccuracyCompared
ModelNameModel_Accuracy
Model385.85%
Model585.85%
Model695.75%

The prediction accuracy of the improved Ridge Regression Model6 is at 95.75% which is very good for this purpose.

0.5 Conclusion

The improved Model6 shows significant improvement from all the OLS Models when the R-Squared and the RMSE of the Models are compared. THis Model also predict TARGET WINS better than the OLS models because it is more stable and less prone to overfitting.

The chosen OLS Model3 and Model5 are due to the improved F-Statistic, positive variable coefficients and low Standard Errors. We will chose to make our predictions with the champion model the improved Ridge Regression Model6 because it beats all the OLs models when the model performance metrics are compared as well as the predictive ability of this model.

For Models 3 and 4, the variables were chosen just to test how the offfensive categories only would affect the model and how only defensive variables would affect the model. Based on the Coefficients for each model, the third model took the highest coefficient from each category model.

For offense, the two highest were HR and Triples. Which intuively does make sense because the HR and triple are two of the highest objectives a hitter can achieve when batting and thus the higher the totals in those categories the higher the runs scored which help a team win. And on the defensive side, the two highest cooeficients were Hits and WALKS. Which again just looking at it from a common sense point does make sense because as a pitcher, what they want to do is limit the numbers of times a batter gets on base whether by a hit or walk. Unless its an error, if a batter does not get a hit or walk then the outcome would be an out which would in essence limit the amount of runs scored by the opposing team.

---
title: "Assignment-1"
author: Anil Akyildirim, John K. Hancock, John Suh, Emmanuel Hayble-Gomes, Chunjie
  Nan
date: "2/12/2020"
output:
  html_document:
    code_download: yes
    code_folding: hide
    highlight: pygments
    number_sections: yes
    theme: flatly
    toc: yes
    toc_float: yes
  word_document:
    toc: yes
  pdf_document:
    toc: yes
---

## Introduction

In this assignment, we are tasked to explore, analyze and model a major league baseball dataset which contains around 2000 records where each record presents a baseball team from 1871 to 2006. Each observation provides the perforamce of the team for that particular year with all the statistics for the performance of 162 game season. The problem statement for the main objective is that "Can we predict the number of wins for the team with the given attributes of each record?". In order to provide a solution for the problem, our goal is to build a linear regression model on the training data that creates this prediction. 

### About the Data

The data set are provided in csv format as moneyball-evaluation-data and moneyball-training-data where we will explore, preperate and create our model with the training data and further test the model with the evaluation data. Below is short description of the variables within the datasets.

**INDEX: Identification Variable(Do not use)

**TARGET_WINS: Number of wins

**TEAM_BATTING_H : Base Hits by batters (1B,2B,3B,HR)

**TEAM_BATTING_2B: Doubles by batters (2B)

**TEAM_BATTING_3B: Triples by batters (3B)

**TEAM_BATTING_HR: Homeruns by batters (4B)

**TEAM_BATTING_BB: Walks by batters

**TEAM_BATTING_HBP: Batters hit by pitch (get a free base)

**TEAM_BATTING_SO: Strikeouts by batters

**TEAM_BASERUN_SB: Stolen bases

**TEAM_BASERUN_CS: Caught stealing

**TEAM_FIELDING_E: Errors

**TEAM_FIELDING_DP: Double Plays

**TEAM_PITCHING_BB: Walks allowed

**TEAM_PITCHING_H: Hits allowed

**TEAM_PITCHING_HR: Homeruns allowed

**TEAM_PITCHING_SO: Strikeouts by pitchers

## Data Exploration

### Descriptive Statistics

```{r}
# load libraries
library(ggplot2)
library(ggcorrplot)
library(psych)
#library(statsr)
library(dplyr)
library(PerformanceAnalytics)
library(tidyr)
library(reshape2)
library(rcompanion)
library(caret)
library(MASS)
library(imputeTS)
library(rsample)
library(huxtable)
library(glmnet)
library(sjPlot)
library(modelr)
```

```{r}
# Load data sets

baseball_eva <- read.csv("https://raw.githubusercontent.com/anilak1978/data621/master/moneyball-evaluation-data.csv")
baseball_train <- read.csv("https://raw.githubusercontent.com/anilak1978/data621/master/moneyball-training-data.csv")

```


We can start exploring our training data set by looking at basic descriptive statistics. 

```{r}
# look at training dataset structure
str(baseball_train)
TARGET_Wins<-as.numeric(baseball_train$TARGET_WINS)

```

We have 2276 observations and 17 variables. All of our variables are integer type as expected.

```{r}
# look at descriptive statistics
metastats <- data.frame(describe(baseball_train))
metastats <- tibble::rownames_to_column(metastats, "STATS")
metastats["pct_missing"] <- round(metastats["n"]/2276, 3)
head(metastats)

```

With the descriptive statistics, we are able to see mean, standard deviation, median, min, max values and percentage of each missing value of each variable. For example, when we look at TEAM_BATTING_H, we see that average 1469 Base hits by batters, with standard deviation of 144, median of 1454 with maximum base hits of 2554. 


```{r}
# Look for missing values
colSums(is.na(baseball_train))

```

```{r}
# Percentage of missing values
missing_values <- metastats %>%
  filter(pct_missing < 1) %>%
  dplyr::select(STATS, pct_missing) %>%
  arrange(pct_missing)
missing_values

```

When we look at the missing values within the training data set, we see that proportionaly against the total observations, TEAM_BATTING_HBP and TEAM_BESARUN_CS variables have the most missing values. We will be handling these missing values in our Data Preperation section. 

### Correlation and Distribution

```{r fig1, fig.height=10, fig.width= 15, fig.align='center'}
# Look at correlation between variables
baseball_train$TARGET_WINS<-as.numeric(baseball_train$TARGET_WINS)
corr <- round(cor(baseball_train), 1)

ggcorrplot(corr,
           type="lower",
           lab=TRUE,
           lab_size=3,
           method="circle",
           colors=c("tomato2", "white", "springgreen3"),
           title="Correlation of variables in Training Data Set",
           ggtheme=theme_bw)

```

Team_Batting_H and Team_Batting_2B have the strongest positive correlation with Target_Wins. We also see that, there is a strong correlation between Team_Batting_H and Team_Batting_2B, Team_Pitching_B and TEAM_FIELDING_E. We will consider these findings on model creation as collinearity might complicate model estimation and we want to have explanotry variables to be independent from each other. We will try to avoid adding explanotry variables that are correlated to each other.

Let's look at the correlations and distribution of the variables in more detail. 

```{r}

# Look at correlation from batting, baserunning, pitching and fielding perspective
Batting_df <- baseball_train[c(2:7, 10)] 
BaseRunning_df <- baseball_train[c(8:9)] 
Pitching_df <- baseball_train[c(11:14)] 
Fielding_df <- baseball_train[c(15:16)]

```

#### Batting

```{r fig2, fig.height=10, fig.width= 15, fig.align='center'}
# Batting Correlations
chart.Correlation(Batting_df, histogram=TRUE, pch=19)

```

We can see that our response variable TARGET_WINS, TEAM_BATTING_H, TEAM_BATTING_2B, TEAM_BATTING_BB and TEAM_BASERUN_CS are normaly distributed. TEAM_BATTING_HR on the other hand is bimodal. 

#### Baserunning


```{r fig3, fig.height=10, fig.width= 15, fig.align='center'}
# baserunning Correlation

chart.Correlation(BaseRunning_df, histogram=TRUE, pch=19)

```

TEAM_BASERUN_SB is right skewed and TEAM_BATTING_SO is bimodal. 

#### Pitching

```{r fig4, fig.height=10, fig.width= 15, fig.align='center'}
#pitching correlations
chart.Correlation(Pitching_df, histogram=TRUE, pch=19)

```

TEAM_BATTING_HBP seems to be normally distributed however we shouldnt forget that we have a lot of missing values in this variable. 

```{r fig5, fig.height=10, fig.width= 15, fig.align='center'}
# fielding correlations
chart.Correlation(Fielding_df, histogram=TRUE, pch=19)
```


Let's also look at the outliers and skewness for each varibale. 

### Outliers and Skewness

```{r fig6, fig.height=10, fig.width= 15, fig.align='center'}
par(mfrow=c(3,3))
datasub_1 <- melt(baseball_train)
suppressWarnings(ggplot(datasub_1, aes(x= "value", y=value)) + 
                   geom_boxplot(fill='lightblue') + facet_wrap(~variable, scales = 'free') )
```

Based on the boxplot we created, TEAM_FIELDING_DP, TEAM_PITCHING_HR, TEAM_BATTING_HR and TEAM_BATTING_SO seem to have the least amount of outliers. 

```{r fig7, fig.height=10, fig.width= 15, fig.align='center'}
par(mfrow = c(3, 3))
datasub = melt(baseball_train) 
suppressWarnings(ggplot(datasub, aes(x= value)) + 
                   geom_density(fill='lightblue') + facet_wrap(~variable, scales = 'free') )
```

```{r}

metastats %>%
  filter(skew > 1) %>%
  dplyr::select(STATS, skew) %>%
  arrange(desc(skew))
```

We can see that the most skewed variable is TEAM_PITCHING_SO. We will correct the skewed variables in our data preperation section. 


When we are creating a linear regression model, we are looking for the fitting line with the least sum of squares, that has the small residuals with minimized squared residuals. From our correlation analysis, we can see that the explatory variable that has the strongest correlation with TARGET_WINS is TEAM_BATTING_H. Let's look at a simple model example to further expand our explaroty analysis. 

### Simple Model Example

```{r fig8, fig.height=5, fig.width= 15, fig.align='center'}
#library(statsr)
# line that follows the best assocation between two variables

#plot_ss(x = TEAM_BATTING_H, y = TARGET_WINS, data=baseball_train, showSquares = TRUE, leastSquares = TRUE)

```

When we are exploring to build a linear regression, one of the first thing we do is to create a scatter plot of the response and explanatory variable. 

```{r fig9, fig.height=5, fig.width= 15, fig.align='center'}
# scatter plot between TEAM_BATTING_H and TARGET_WINS

ggplot(baseball_train, aes(x=TEAM_BATTING_H, y=TARGET_WINS))+
  geom_point()

```

One of the conditions for least square lines or linear regression are Linearity. From the scatter plot between TEAM_BATTING_H and TARGET_WINS, we can see this condition is met. We can also create a scatterplot that shows the data points between TARGET_WINS and each variable.

```{r fig10, fig.height=5, fig.width= 15, fig.align='center'}

baseball_train %>%
  gather(var, val, -TARGET_WINS) %>%
  ggplot(., aes(val, TARGET_WINS))+
  geom_point()+
  facet_wrap(~var, scales="free", ncol=4)

```

As we displayed earlier, hits walks and home runs have the strongest correlations with TARGET_WINS and also meets the linearity condition. 

```{r}
# create a simple example model
lm_sm <- lm(baseball_train$TARGET_WINS ~ baseball_train$TEAM_BATTING_H)
summary(lm_sm)

```

TARET_BATTING_H has the strongest correlation with TARGET_WINS response variable, however when we create a simple model just using TARGET_BATTING_H, we can only explain 15% of the variablity. (Adjusted R-squared:  0.1508). The remainder of the varibility can be explained with selected other variables within the training dataset. 

```{r fig11, fig.height=5, fig.width= 15, fig.align='center'}
#histogram of residuals for the simple model
hist(lm_sm$residuals)

```

```{r fig12, fig.height=5, fig.width= 15, fig.align='center'}
# check for constant variability (honoscedasticity)

plot(lm_sm$residuals ~ baseball_train$TEAM_BATTING_H)

```

We do see that the residuals are distributed normally and variability around the regression line is roughly constant. 

Based on our explatory analysis, we were able to see the correlation level between the possible explanatory variables and repsonse variable TARGET_WINS. Some of the variables such as TARGET_BATTING_H has somewhat strong positive correlation, however some of the variables such as TEAM_PITCHING_BB has weak positive relationship with TARGET_WINS. We also found out, hit by the pitcher(TEAM_BATTING_HBP) and caught stealing (TEAM_BASERUN_CS) variables are missing majority of the values. Skewness and distribution analysis gave us the insights that we have some variables that are right-tailed. Considering all of these insights, we will handle missing values, correct skewness and outliers and select our explaratory variables based on correlation in order to create our regression model. 

## Data Preparation


### Objective

In this section, we will prepare the dataset for linear regression modeling.  We accomplish this by handling missing values and outliers and by tranforming the data into more normal distributions.  This section covers:

*Identify and Handle Missing Data
*Correct Outliers
*Adjust Skewed value - Box Cox Transformation


First, we will start by copying the dataset into a new variable, baseball_train_01, and we will remove the Index variable from the new dataset as well. We will now have 16 variables.

```{r}
baseball_train_01 <- baseball_train

baseball_train_01 <-subset(baseball_train_01, select = -c(INDEX))

```


### Identify and Handle Missing Data

#### Removal of Sparsely Populated Variables - MCAR

In the Data Exploration section, we identified these variables as having missing data values.The table below lists the variables with missing data. The variable, TEAM_BATTING_HBP, is sparsely populated.  Since this data is Missing Completely at Random (MCAR) and is not related to any other variable, it is safe to completely remove the variable from the dataset. 


```{r}
missing_values
```

```{r}
baseball_train_01 <-subset(baseball_train_01, select = -c(TEAM_BATTING_HBP))

```

There are now 15 variables.


```{r}
dim(baseball_train_01)
```

#### Imputation of Missing Values
For the remaining variables with missing values, we will impute the mean of the variable. The function, "na_mean" updates all missing values with the mean of the variable.

```{r, message=FALSE}
baseball_train_01 <- na_mean(baseball_train_01, option = "mean")  

```

Re-running the metastats dataframe on the new baseball_train_01 dataset shows that there are no missing values.

```{r, message=FALSE}
# look at descriptive statistics
metastats <- data.frame(describe(baseball_train_01))
metastats <- tibble::rownames_to_column(metastats, "STATS")
metastats["pct_missing"] <- round(metastats["n"]/2276, 3)

```


```{r}
# Percentage of missing values
missing_values2 <- metastats %>%
  filter(pct_missing < 1) %>%
  dplyr::select(STATS, pct_missing) %>%
  arrange(pct_missing)
missing_values2
```

### Correct Outliers

In this section, we created two functions that can identify outliers. The funcion, Identify_Outlier, uses the Turkey method, where outliers are identified by being below Q1-1.5*IQR and above Q3+1.5*IQR. The second function, tag_outlier, returns a binary list of values, "Acceptable" or "Outlier" that will be added to the dataframe.


```{r}
Identify_Outlier <- function(value){

    interquartile_range = IQR(sort(value),na.rm = TRUE)
    q1 = matrix(c(quantile(sort(value),na.rm = TRUE)))[2]
    q3 = matrix(c(quantile(sort(value),na.rm = TRUE)))[4]
    lower = q1-(1.5*interquartile_range)
    upper = q3+(1.5*interquartile_range)
    
    bound <- c(lower, upper)
    
    return (bound)
}

```


```{r}
tag_outlier <- function(value) {
    
   boundaries <- Identify_Outlier(value)
   tags <- c()
   counter = 1
    for (i in as.numeric(value))
    {

        if (i >= boundaries[1] & i <= boundaries[2]){
          tags[counter] <- "Acceptable"
        } else{
          tags[counter] <- "Outlier"
        }
      
      counter = counter +1
    }
   
   return (tags)
}
```

As seen in the box plots from the previous section, "TEAM_BASERUN_SB", "TEAM_BASERUN_CS", "TEAM_PITCHING_H", "TEAM_PITCHING_BB", "TEAM_PITCHING_SO", and "TEAM_FIELDING_E" all have a high number of outliers. We will use the two functions above to tag those rows with extreme outliers.


```{r}
tags<- tag_outlier(baseball_train_01$TEAM_BASERUN_SB)
baseball_train_01$TEAM_BASERUN_SB_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_BASERUN_CS)
baseball_train_01$TEAM_BASERUN_CS_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_PITCHING_H)
baseball_train_01$TEAM_PITCHING_H_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_PITCHING_BB)
baseball_train_01$TEAM_PITCHING_BB_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_PITCHING_SO)
baseball_train_01$TEAM_PITCHING_SO_Outlier <- tags

tags<- tag_outlier(baseball_train_01$TEAM_FIELDING_E)
baseball_train_01$TEAM_FIELDING_E_Outlier <- tags
```

Below, we filtered out all of the outliers and created a new dataframe, baseball_train_02


```{r, message=FALSE, options(warn=-1)}
baseball_train_02 <- baseball_train_01 %>%
                filter(
                        TEAM_BASERUN_SB_Outlier != "Outlier" &
                        TEAM_BASERUN_CS_Outlier != "Outlier" &
                        TEAM_PITCHING_H_Outlier != "Outlier" &
                        TEAM_PITCHING_BB_Outlier != "Outlier" &
                        TEAM_PITCHING_SO_Outlier != "Outlier" &
                        TEAM_FIELDING_E_Outlier != "Outlier"
                )
```


Re-running the boxplots show data that has a better normal distribution except for the variable, TEAM_FIELDING_E which is still skewed.  We will handle this next.


```{r fig13, fig.height=10, fig.width= 15, fig.align='center'}
par(mfrow=c(3,3))
datasub_1 <- melt(baseball_train_02)
suppressWarnings(ggplot(datasub_1, aes(x= "value", y=value)) + 
                   geom_boxplot(fill='lightblue') + facet_wrap(~variable, scales = 'free') )


```

### Adjust Skewed values
#### Box Cox Transformation

Removing the outliers tranformed each variable to a closer to a normal distribution and checking the skewness of the variables confirm this with the exception of TEAM_FIELDING_E. This variable is still skewed and not normal.  In this section, we will use the Box Cox tranformation from the MASS library to normalize this variable.


```{r}
metastats_02 <- data.frame(describe(baseball_train_02))
metastats_02 <- tibble::rownames_to_column(metastats_02, "STATS") 
 
metastats_02 %>%
 filter(skew > 1 | skew < -1) %>%
  dplyr::select(STATS, skew) %>%
  arrange(desc(skew))

```

Looking at the histogram and QQ plots we can confirm that the variable, TEAM_FIELDING_E, is not normally distributed. It is skewed to the right.

```{r, message=FALSE}
plotNormalHistogram(baseball_train_02$TEAM_FIELDING_E)
```


```{r}
qqnorm(baseball_train_02$TEAM_FIELDING_E,
       ylab="Sample Quantiles for TEAM_FIELDING_E")    
         qqline(baseball_train_02$TEAM_FIELDING_E,
           col="blue")
```


The following Box Cox transformation section is based on the tutorial at the link below:

[\hrefhttps://rcompanion.org/handbook/I_12.html][Summary and Analysis of Extension Program Evaluation in R]

The Box Cox procedure uses a log-likelihood to find the lambda to use to transform a variable to a normal distribution. 


```{r, message=FALSE}

TEAM_FIELDING_E <- as.numeric(dplyr::pull(baseball_train_02, TEAM_FIELDING_E))

#Transforms TEAM_FIELDING_E as a single vector 
Box = boxcox(TEAM_FIELDING_E ~ 1, lambda = seq(-6,6,0.1))

#Creates a dataframe with results
Cox = data.frame(Box$x, Box$y)

# Order the new data frame by decreasing y to find the best lambda.Displays the lambda with the greatest log likelihood.
Cox2 = Cox[with(Cox, order(-Cox$Box.y)),]
Cox2[1,] 

#Extract that lambda and Transform the data
lambda = Cox2[1, "Box.x"]
T_box = (TEAM_FIELDING_E ^ lambda - 1)/lambda
```

We can now see that TEAM_FIELDING_E has a normal distribution.


```{r}
plotNormalHistogram(T_box)
```


```{r}
qqnorm(T_box, ylab="Sample Quantiles for TEAM_FIELDING_E")
qqline(T_box,
        col="blue")
```


```{r}
baseball_train_02$TEAM_FIELDING_E <- T_box
```

The density plots below show that all of the variables for the dataset baseball_train_02 are now normally distributed.  In the next section, we will use this dataset to build the models and discuss the coefficients of the models. 


```{r , message=FALSE, fig15, fig.height=10, fig.width= 15, fig.align='center'}
par(mfrow = c(3, 3))
datasub = melt(baseball_train_02) 
suppressWarnings(ggplot(datasub, aes(x= value)) + 
                   geom_density(fill='lightblue') + facet_wrap(~variable, scales = 'free') )


```

Viewing the dataframe shows that the dataset contains characters resulting from the transfromation of the outliers. These non numeric characters will impact our models especially if we build the intial baseline model with all the variables. We will need one more step to have our data ready for the models.

```{r}
str(baseball_train_02)
```
Subsetting - The code below will subset the data to have only numeric or integer values that will be used for our models. This will create baseball_train_03 dataframe.

```{r}
baseball_train_03 <- baseball_train_02[c(1:15) ]
str(baseball_train_03)
```

## Build Models 

The first Model is using stepwise in Backward direction to eliminate variables, this is an automated process which is different from the manual variable selction process. We will not pay much attention to this process as the focus of the project is to manually identify and select those significant variables that will predict TARGET WINS. 
```{r}
Model <- step(lm(TARGET_WINS ~ ., data=baseball_train_03), direction = "backward")
summary(Model)
```
The step backward variable selection process identified eleven significant variables with an R-squared of 37%, Residual Error of 11.01 and F-Statistic of 74.59. Notice that some of the coefficients are negative which means these Team will most likely result in negative wins. We will explore these coefficient a little further in this analysis.

### OLS- MODEL 1 

Using all the 15 Variables

```{r}
Model1 <-lm(TARGET_WINS ~ ., data=baseball_train_03)
summary(Model1)
```
This Model identified seven significant variables at \apha = 0.05 with an R-squared of 37%, Residual Error of 11.01 and F-Statistic of 64.01. Although the F-Statistic reduced, this model does not improve significantly from the previous model. 

```{r}
Metrics1 <- data.frame(
  R2 = rsquare(Model1, data = baseball_train_03),
  RMSE = rmse(Model1, data = baseball_train_03),
  MAE = mae(Model1, data = baseball_train_03)
)
print(Metrics1)
```

### OLS- MODEL 2 

Using all the seven (7) significant variables from Model 1 

```{r}
Model2 <- lm(TARGET_WINS~TEAM_FIELDING_E + TEAM_BASERUN_SB + TEAM_BATTING_3B + TEAM_FIELDING_DP + TEAM_PITCHING_SO + TEAM_BATTING_SO + TEAM_BATTING_2B,data=baseball_train_03)
summary(Model2)
```
This Model identified five significant variables at \apha = 0.05 with an R-squared of 22%, Residual Error of 12.19 and F-Statistic of 64.14. The R-Squared decreased and the Error increased slightly. 

```{r}
Metrics2 <- data.frame(
  R2 = rsquare(Model2, data = baseball_train_03),
  RMSE = rmse(Model2, data = baseball_train_03),
  MAE = mae(Model2, data = baseball_train_03)
)
print(Metrics2)
```

### OLS- MODEL 3

All offensive categories which include hitting and base running

```{r}
Model3 <-lm(TARGET_WINS~TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BATTING_HR + TEAM_BATTING_2B + TEAM_BATTING_SO + TEAM_BASERUN_CS + TEAM_BATTING_3B + TEAM_BASERUN_SB,data=baseball_train_03)
summary(Model3)
```
This Model identified five significant variables at \apha = 0.05 with an R-squared of 28%, Residual Error of 11.73 and F-Statistic of 75.58. Although the R-squared is not that great, the standard errors are more reasonable. We will hold onto this Model as performing better than the previous models for now.

```{r}
Metrics3 <- data.frame(
  R2 = rsquare(Model3, data = baseball_train_03),
  RMSE = rmse(Model3, data = baseball_train_03),
  MAE = mae(Model3, data = baseball_train_03)
)
print(Metrics3)
```

### OLS- MODEL 4

All defensive categories which include fielding and pitching

```{r}
Model4 <- lm(TARGET_WINS~TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_FIELDING_E,data=baseball_train_03)
summary(Model4)
```
This Model identified five significant variables at \apha = 0.05 with an R-squared of 19%, Residual Error of 12.46 and F-Statistic of 75.56.There is no significant improvement with this model.

```{r}
Metrics4 <- data.frame(
  R2 = rsquare(Model4, data = baseball_train_03),
  RMSE = rmse(Model4, data = baseball_train_03),
  MAE = mae(Model4, data = baseball_train_03)
)
print(Metrics4)
```

### OLS- MODEL 5

Using only the significant variables from Model 3

```{r}
Model5 <- lm(TARGET_WINS~TEAM_PITCHING_H + TEAM_PITCHING_BB + TEAM_PITCHING_HR + TEAM_PITCHING_SO + TEAM_BATTING_3B + TEAM_BASERUN_SB,data=baseball_train_03)
summary(Model5)
```
This Model identified five significant variables at \apha = 0.05 with an R-squared of 26%, Residual Error of 11.93 and F-Statistic of 88.92. Although the R-squared is not better than than Model3, the F-statistic improved with smaller Standard Error. 

```{r}
Metrics5 <- data.frame(
  R2 = rsquare(Model5, data = baseball_train_03),
  RMSE = rmse(Model5, data = baseball_train_03),
  MAE = mae(Model5, data = baseball_train_03)
)
print(Metrics5)
```

### Compare OLS Model Quality 

```{r}
anova(Model, Model1, Model2, Model3, Model4, Model5)
tab_model(Model, Model1, Model2, Model3, Model4, Model5)

```


### RIDGE Regression- MODEL 6 

The Ridge regression is an extension of linear regression where the loss function is modified to minimize the complexity of the model. This modification is done by adding a penalty parameter that is equivalent to the square of the magnitude of the coefficients.

Before implementing the RIDGE model, we will split the training dataset into 2 parts that is - training set within the training set and a test set that can be used for evaluation. By enforcing stratified sampling both our training and testing sets have approximately equal response "TARGET_WINS" distributions.

Transforming the variables into the form of a matrix will enable us to penalize the model using the 'glmnet' method in glmnet package.

```{r}
#Split the data into Training and Test Set
baseball_train_set<- initial_split(baseball_train_03, prop = 0.7, strata = "TARGET_WINS")
train_baseball  <- training(baseball_train_set)
test_baseball   <- testing(baseball_train_set)

train_Ind<- as.matrix(train_baseball)
train_Dep<- as.matrix(train_baseball$TARGET_WINS)

test_Ind<- as.matrix(test_baseball)
test_Dep<- as.matrix(test_baseball$TARGET_WINS)

```

For the avoidance of multicollinearity, avoiding overfitting and predicting better, implementing RIDGE regression will become useful. 

```{r}
lambdas <- 10^seq(2, -3, by = -.1)
Model6 <- glmnet(train_Ind,train_Dep, nlambda = 25, alpha = 0, family = 'gaussian', lambda = lambdas)
summary(Model6)
print(Model6, digits = max(3, getOption("digits") - 3),
           signif.stars = getOption("show.signif.stars"))
```

The significant difference between the OLS and the Ridge Regresion is the hyperparameter tuning using lambda. The Ridge regression does not perform Feature Selection, but it predicts better and solve overfitting. Cross Validating the Ridge Regression will help us to identify the optimal lambda to penalize the model and enhance the predictability.

```{r}
CrossVal_ridge <- cv.glmnet(train_Ind,train_Dep, alpha = 0, lambda = lambdas)
optimal_lambda <- CrossVal_ridge$lambda.min
optimal_lambda #The optimal lambda is 0.001 which we will use to penelize the Ridge Regression model.
coef(CrossVal_ridge) # Shows the coefficients
plot(CrossVal_ridge)
```

The plot shows that the errors increases as the magnitude of lambda increases, previously, we identified that the optimal lambda is 0.001 which is very obvious from the plot above. The coefficients are restricted to be small but not quite zero as Ridge Regression does not force the coefficient to zero. This indicates that the model is performing well so far. But let's make it better using the optimal labmda.

```{r}
eval_results <- function(true, predicted, df){
  SSE <- sum((predicted - true)^2)
  SST <- sum((true - mean(true))^2)
  R_square <- 1 - SSE / SST
  RMSE = sqrt(SSE/nrow(df))
data.frame(   
  RMSE = RMSE,
  Rsquare = R_square
)
  
}
# Prediction and evaluation on train data
predictions_train <- predict(Model6, s = optimal_lambda, newx = train_Ind)
eval_results(train_Dep, predictions_train, train_baseball)
```
We should be a little concern about the 100% R-squared performance for this Model. Although the Ridge Regression forces the coefficients towards zero to improve the Model performance and enhance the predictability, the very high peformance may require further investigation. Lets improve the model using a more reason lambda because optimal might not always be the best.

### The Improved Ridge Regression

```{r}
Model6_Improved <- glmnet(train_Ind,train_Dep, nlambda = 25, alpha = 0, family = 'gaussian', lambda = 6.310)
summary(Model6_Improved)
coef(Model6_Improved)

```

Let's compute the Model's Performance Metric to see how this model is doing.

```{r}
eval_results <- function(true, predicted, df){
  SSE <- sum((predicted - true)^2)
  SST <- sum((true - mean(true))^2)
  R_square <- 1 - SSE / SST
  RMSE = sqrt(SSE/nrow(df))
data.frame(   
  RMSE = RMSE,
  Rsquare = R_square
)
  
}

# Prediction and evaluation on train data
predictions_train <- predict(Model6_Improved, s = lambda, newx = train_Ind)
eval_results(train_Dep, predictions_train, train_baseball)

# Prediction and evaluation on test data
predictions_test <- predict(Model6_Improved, s = lambda, newx = test_Ind)
eval_results(test_Dep, predictions_test, test_baseball)
```
The improved Model6 output shows that the RMSE and R-squared values for the Ridge Regression model on the training and test data are significantly improved. The Loss Function (RMSE) are severely reduced compared to the OLS models which indicates that the Ridge Regression is not overfitting. These performance is significantly improved compared to the OLS Models 1 to 5.

### Model Performance Comparison
        
```{r}
ModelName <- c("Model", "Model1","Model2","Model3","Model4","Model5","Model6")
Model_RSquared <- c("37%", "37%", "22%", "28%", "19%", "26% ", "90%")
Model_RMSE <- c("11.01", "10.96", "12.15", "11.69", "12.43", "11.93 ", "4.33")
Model_FStatistic <- c("74.59", "64.01", "64.14", "75.58", "72.56", "88.92 ", "NA")
Model_Performance <- data.frame(ModelName,Model_RSquared,Model_RMSE,Model_FStatistic)
Model_Performance

```

### Model Prediction

Based on the Model metrics above, we're ready to make prediction and we will select our acceptable OLS Model3 and Model5 which has better F-Statistic, smaller standard errors and less negative coefficient as our best OLS models. We will also compare the prediction accuracy of these models to that of the improved Ridge Regression Model which is our champion Model for this exercise based on the very small RMSE and the highest R-squared of over 90%.

```{r}
predicted <- predict(Model3, newx = test_baseball)# predict on test data
predicted_values <- cbind (actual=test_baseball$TARGET_WINS, predicted)  # combine
predicted_values
```

```{r}
mean (apply(predicted_values, 1, min)/apply(predicted_values, 1, max)) # calculate accuracy
```
The prediction accuracy here is at 85.85%

```{r}
predicted <- predict(Model5, newx = test_baseball)# predict on test data
predicted_values <- cbind (actual=test_baseball$TARGET_WINS, predicted)  # combine
predicted_values
```

```{r}
mean (apply(predicted_values, 1, min)/apply(predicted_values, 1, max)) # calculate accuracy
```

The prediction accuracy for the OLS Model5 is at 85.94% which is not bad for this purpose. But lets compare it to the Champion Model- The improved Ridge Regression.

```{r}
predicted <- predict(Model6_Improved, newx = test_Ind)# predict on test data
predicted_values <- cbind (actual=test_baseball$TARGET_WINS, predicted)  # combine
predicted_values

```
 Lets calculate the accuracy of using Model6 for our predictions

```{r}
mean (apply(predicted_values, 1, min)/apply(predicted_values, 1, max)) # calculate accuracy
```
The prediction accuracy of the improved Ridge Regression Model is 95.75%.

```{r}
ModelName <- c("Model3", "Model5","Model6")
Model_Accuracy <- c("85.85%", "85.85%", "95.75%")
AccuracyCompared <- data.frame(ModelName,Model_Accuracy)
AccuracyCompared
```

The prediction accuracy of the improved Ridge Regression Model6 is at 95.75% which is very good for this purpose.

## Conclusion

The improved Model6 shows significant improvement from all the OLS Models when the R-Squared and the RMSE of the Models are compared. THis Model also predict TARGET WINS better than the OLS models because it is more stable and less prone to overfitting. 

The chosen OLS Model3 and Model5 are due to the improved F-Statistic, positive variable coefficients and low Standard Errors. We will chose to make our predictions with the champion model the improved Ridge Regression Model6 because it beats all the OLs models when the model performance metrics are compared as well as the predictive ability of this model. 

For Models 3 and 4, the variables were chosen just to test how the offfensive categories only would affect the model and how only defensive variables would affect the model. Based on the Coefficients for each model, the third model took the highest coefficient from each category model.

For offense, the two highest were HR and Triples. Which intuively does make sense because the HR and triple are two of the highest objectives a hitter can achieve when batting and thus the higher the totals in those categories the higher the runs scored which help a team win. And on the defensive side, the two highest cooeficients were Hits and WALKS. Which again just looking at it from a common sense point does make sense because as a pitcher, what they want to do is limit the numbers of times a batter gets on base whether by a hit or walk. Unless its an error, if a batter does not get a hit or walk then the outcome would be an out which would in essence limit the amount of runs scored by the opposing team.












