predicting-age-of-abalone-using-regression

Introduction

Abalone is a shellfish considered a delicacy in many parts of the world. An excellent source of iron and pantothenic acid, and a nutritious food resource and farming in Australia, America and East Asia. 100 grams of abalone yields more than 20% recommended daily intake of these nutrients. The economic value of abalone is positively correlated with its age. Therefore, to detect the age of abalone accurately is important for both farmers and customers to determine its price. However, the current technology to decide the age is quite costly and inefficient. Farmers usually cut the shells and count the rings through microscopes to estimate the abalones age. Telling the age of abalone is therefore difficult mainly because their size depends not only on their age, but on the availability of food as well. Moreover, abalone sometimes form the so-called 'stunted' populations which have their growth characteristics very different from other abalone populations This complex method increases the cost and limits its popularity. Our goal in this report is to find out the best indicators to forecast the rings, then the age of abalones.

Dataset

Background

This dataset comes from an original (non-machine-learning) study and received in December 1995:
- Warwick J Nash, Tracy L Sellers, Simon R Talbot, Andrew J Cawthorn and Wes B Ford (1994)
- "The Population Biology of Abalone (Haliotis species) in Tasmania. I. Blacklip Abalone (H. rubra) from the North Coast and Islands of Bass Strait",
- Sea Fisheries Division, Technical Report No. 48 (ISSN 1034-3288).
- Dataset can be found on UIC Machine learning repository site
- Citation:
  - There are more than 30 papers that cites this data set. Please find the full list at UIC Machine learning repository site

Description

From the original data examples with missing values were removed (the majority having the predicted value missing), and the ranges of the continuous values have been scaled for use with an ANN (by dividing by 200). For the purpose of this analysis, we will scale those variables back to its original form by multiplying by 200.
Total number of observations in dataset: 4176
Total number of variables in dataset : 8
Metadata and attribute information:
- Given is the attribute name, attribute type, the measurement unit and a brief description. The number of rings is the value to predict as a continuous value.

Variable List

Name	Data Type	Measurement	Description
Sex	categorical (factor)		M, F, and I (Infant)
Length	continuous	mm	Longest shell measurement
Diameter	continuous	mm	perpendicular to length
Height	continuous	mm	with meat in shell
Whole weight	continuous	grams	whole abalone
Shucked weight	continuous	grams	weight of meat
Viscera weight	continuous	grams	gut weight (after bleeding)
Shell weight	continuous	grams	after being dried
Rings	continuous		+1.5 gives the age in years

Interest

Predicting the age of abalone from physical measurements:
- The age of abalone is determined by cutting the shell through the cone, staining it, and counting the number of rings through a microscope -- a boring and time-consuming task. Other measurements, which are easier to obtain, are used to predict the age. Further information, such as weather patterns and location (hence food availability) may be required to solve the problem.
Apply different regression techniques:
- We are interested in performing various regression techniques such as additive models, interactions, polynomial transformations of the variables etc to be able to predict and assess the accuracy of our prediction.
Beyond regression models:
- Is there any other type of machine learning methodology which can predict the age more accurately than using regression model ?
Implementation in real application:
- Understand whether the data set & the regression models are sufficient to predict the age of abalone accurately enough so that it can be used in real application.

Below is the analysis

"Predicting the age of abalone using regression"

Methods

#Run these commands to intall all the required packages for this reportt
install.packages("readr")
install.packages("dplyr")
install.packages("car")
install.packages("lmtest")
install.packages("ggplot2")
install.packages("GGally")
install.packages("gridExtra")
install.packages("MASS")
install.packages("leaps")
install.packages("glmnet")
install.packages("caret")
install.packages("gbm")

#loading all necessary packages 
library(readr)
library(dplyr)
library(car)
library(lmtest)
library(ggplot2)
library(GGally)
library(gridExtra)
library(MASS)
library(leaps)
library(glmnet)
library(caret)
library(gbm)

Data summary and statistics

abalone <- read_csv("Abalone_data.csv")

is.factor(abalone$Sex)

## [1] FALSE

abalone$Sex <- as.factor(abalone$Sex)
str(abalone, give.attr = FALSE)

## Classes 'tbl_df', 'tbl' and 'data.frame':	4176 obs. of  9 variables:
##  $ Sex           : Factor w/ 3 levels "F","I","M": 3 1 3 2 2 1 1 3 1 1 ...
##  $ Length        : int  70 106 88 66 85 106 109 95 110 105 ...
##  $ Diameter      : int  53 84 73 51 60 83 85 74 88 76 ...
##  $ Height        : int  18 27 25 16 19 30 25 25 30 28 ...
##  $ Whole_weight  : num  45.1 135.4 103.2 41 70.3 ...
##  $ Shucked_weight: num  19.9 51.3 43.1 17.9 28.2 47.4 58.8 43.3 62.9 38.8 ...
##  $ Viscera_weight: num  9.7 28.3 22.8 7.9 15.5 28.3 29.9 22.5 30.2 29.5 ...
##  $ Shell_weight  : num  14 42 31 11 24 66 52 33 64 42 ...
##  $ Rings         : int  7 9 10 7 8 20 16 9 19 14 ...

library(knitr)
kable(abalone[1:10,], digits = 4,format = 'markdown')

Sex	Length	Diameter	Height	Whole_weight	Shucked_weight	Viscera_weight	Shell_weight	Rings
M	70	53	18	45.1	19.9	9.7	14	7
F	106	84	27	135.4	51.3	28.3	42	9
M	88	73	25	103.2	43.1	22.8	31	10
I	66	51	16	41.0	17.9	7.9	11	7
I	85	60	19	70.3	28.2	15.5	24	8
F	106	83	30	155.5	47.4	28.3	66	20
F	109	85	25	153.6	58.8	29.9	52	16
M	95	74	25	101.9	43.3	22.5	33	9
F	110	88	30	178.9	62.9	30.2	64	19
F	105	76	28	121.3	38.8	29.5	42	14

Categorial Variables

Sex	Number of observations
M	1527
F	1307
I	1342

Numeric Variables

	Length	Diameter	Height	Whole Weight	Shucked Weight	Viscera Weight	Shell Weight	Rings
Min	15.0	11.00	0.0000	0.4	0.20	0.10	0.30	1
Median	109.0	85.00	28.00	159.9	67.20	34.20	46.80	9
Mean	104.8	81.58	27.91	165.8	71.88	36.12	47.77	9.932
Max	163.0	130.00	226.00	565.1	297.60	152.00	201.00	29
Cor	0.557	0.5750	0.5581	0.5408	0.4212	0.5043	0.6280	1.000

Looking at the dataset summary, we can see that data is quite evenly distributed between the three factor levels of male, female and infant.
Also from we see that there are four different measure of weight i.e. Whole_weight, Shucked_weight, Viscera_weight and Shell.weight. Whole_weight is linear function of other weight predictors with Unknown mass of water/blood lost from shucking process. Also we observed that min value of predictor Height is 0. Practically this is not possible, we will investigate these observations to look closely.

Response variable

The dependent variable Rings is included in the dataset. It was measured as the number of rings observed after cutting and examining an abalone. Although it does not denote the age of a given abalone directly, it determines it more-or-less perfectly. The age of an abalone equals Rings + 1.5 . Since this relationship holds reliably, Rings will be considered as the dependent variable. The number of rings measured in the data ranges from 1 to 29 and most of the abalone have between 5 and 15 rings. The distribution is slightly positively skewed as well but this does not pose any specific problems for the further analysis. (see plot below)

####Pair Plot

ggpairs(abalone, aes(colour = Sex, alpha = 0.8), title="Pairs plot for abalone dataset") + 
  theme_grey(base_size = 8)

####Observations from Pair Plot :

First thing to note here is high correlation in data. There seems to be high multicollinearity between the predictors. for example correlation between Diameter and Length is extremely high (about 98.7).
Similarly Whole_weight seems to be highly correlated with other weight predictors and is the sum of Shucked_weight, Viscera_weight and Shell_weight.
Secondly, the distributions of predictor Sex with factor level value of female and male are very similar with respect to all other predictors.
The shape of distribution is also significantly similar for factor levels of female and male.
We could think about redefining this feature to define gender as infant vs non-infant (where non-infant = female and male both).
Most of the abalones rings are between 5 and 15.

Data quality

Addition of variables: We will update the abalone dataset to create new variable named Infant which will have values based on original value from Sex variable. It will have value of I, when Sex variable is I and NI otherwise.

abalone['Infant'] <- ifelse(abalone$Sex == 'I','I','NI')
abalone$Infant <- as.factor(abalone$Infant)
abalone$Sex <- as.factor(abalone$Sex)

Also we observed that min value of predictor Height is 0. Practically this is not possible, we will investigate these observations to look closely.

#Data quality check for height being 0
kable(abalone[abalone$Height == 0,], digits = 4,format = 'markdown')

Sex	Length	Diameter	Height	Whole_weight	Shucked_weight	Viscera_weight	Shell_weight	Rings	Infant
I	86	68	0	85.6	41.3	17.2	23.0	8	I
I	63	46	0	26.8	11.5	5.7	70.1	6	I

We see that there are two observations for which Height might not be recorded properly as other predictors seems to have valid values. Also if we look at the predictor Whole_weight, we see that these values are really small compare to rest of the observation and below first quantile. This tells us that this might not be a data error therefore we can not exclude these from our dataset.
We will also add a new variable named weight.diff.We can see in the summary that there are four different measure of weight i.e. Whole_weight, Shucked_weight, Viscera_weight and Shell.weight. Whole_weight is linear function of other weight predictors with unknown mass of water/blood lost from shucking process.

abalone$weight.diff <- abalone$Whole_weight - (abalone$Viscera_weight + abalone$Shucked_weight + abalone$Shell_weight)
str(abalone, give.attr = FALSE)

## Classes 'tbl_df', 'tbl' and 'data.frame':	4176 obs. of  11 variables:
##  $ Sex           : Factor w/ 3 levels "F","I","M": 3 1 3 2 2 1 1 3 1 1 ...
##  $ Length        : int  70 106 88 66 85 106 109 95 110 105 ...
##  $ Diameter      : int  53 84 73 51 60 83 85 74 88 76 ...
##  $ Height        : int  18 27 25 16 19 30 25 25 30 28 ...
##  $ Whole_weight  : num  45.1 135.4 103.2 41 70.3 ...
##  $ Shucked_weight: num  19.9 51.3 43.1 17.9 28.2 47.4 58.8 43.3 62.9 38.8 ...
##  $ Viscera_weight: num  9.7 28.3 22.8 7.9 15.5 28.3 29.9 22.5 30.2 29.5 ...
##  $ Shell_weight  : num  14 42 31 11 24 66 52 33 64 42 ...
##  $ Rings         : int  7 9 10 7 8 20 16 9 19 14 ...
##  $ Infant        : Factor w/ 2 levels "I","NI": 2 2 2 1 1 2 2 2 2 2 ...
##  $ weight.diff   : num  1.5 13.8 6.3 4.2 2.6 ...

We see that variable Whole_weight should be the linear function of variables Shucked_weight, Viscersa_weight and Shell_weight, and we could write it as Whole_weight = Shucked_weight + Viscera_weight+ Shell_weight + unknown mass of water/blood lost from shucking process.
However when we calculated difference between Whole_weight and other weight variables we see that there are 153 observations that are violating this. i.e. this seems to be a illogical and could be an error while recording data.
If we plot the histogram of the newly added weight.diff variable, we can see that there are observations when weight.diff is negative.

#Identify observations that are not recorded properly
#Histogram of illogical observations
ggplot(abalone, aes(x=weight.diff)) +
 geom_histogram(colour="dodgerblue",fill=rgb(1,.54,0,.7), bins = 30) +
 scale_y_continuous(name="count") + 
  labs(title="Histogram of abalone with weight difference less than zero")

To make sure, lets take a look at few of these records.

nrow(abalone[abalone$weight.diff < 0,])

## [1] 153

kable(head(abalone[abalone$weight.diff < 0,], n=10), digits = 4,format = 'markdown')

Sex	Length	Diameter	Height	Whole_weight	Shucked_weight	Viscera_weight	Shell_weight	Rings	Infant	weight.diff
I	48	35	9	14.0	6.3	4.7	4.0	5	I	-1.0
I	41	30	11	8.4	5.1	3.0	2.4	5	I	-2.1
I	42	30	10	8.4	3.5	2.5	3.0	4	I	-0.6
I	78	59	19	40.6	17.5	9.0	15.0	7	I	-0.9
M	94	74	24	115.9	58.6	45.4	28.0	9	NI	-16.1
M	90	69	21	82.3	36.0	22.5	27.0	7	NI	-3.2
M	101	81	22	125.0	61.0	32.0	35.0	9	NI	-3.0
F	87	79	21	72.7	27.2	19.6	26.0	9	NI	-0.1
M	93	72	21	86.2	34.4	21.4	35.0	9	NI	-4.6
I	72	56	16	35.1	16.2	10.1	14.0	6	I	-5.2

Note that there are total 153 observations that has combined weight of more than Whole_weight. When we looked at 10 such observations, it seems that other values are correct and there is no similarity so we are sure that this might be an data entry error. Therefore we will keep these observations for further analysis.
We will start with diving our dataset in train and test. The dataset will be splitted between train and test with the ratio of 70/30 with random selected observations.

####Train & Test Split

set.seed(42)
#Splitting dataset in train and test using 70/30 method
indexes <- sample(1:nrow(abalone), size = 0.3 * nrow(abalone))
abalone_train <- abalone[-indexes,]
abalone_test <- abalone[indexes,]

###Additive Multiple Linear Regression Model

We have started with fitting an additive model with all variables and will look at the significance of the parameters. Based on that we will modify our model. For now we will work with original value for variable Sex which have factor levels of F, I and M.

abalone_add <- lm(Rings ~ Sex+Length+Diameter+Height+ Whole_weight
               +Shucked_weight+Viscera_weight
               +Shell_weight,data = abalone_train)
summary(abalone_add)

## 
## Call:
## lm(formula = Rings ~ Sex + Length + Diameter + Height + Whole_weight + 
##     Shucked_weight + Viscera_weight + Shell_weight, data = abalone_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.9157 -1.2965 -0.3030  0.8309 12.0132 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     3.946559   0.339445  11.627  < 2e-16 ***
## SexI           -0.883507   0.120215  -7.349 2.57e-13 ***
## SexM            0.010608   0.098708   0.107    0.914    
## Length         -0.013912   0.010830  -1.285    0.199    
## Diameter        0.074949   0.013414   5.587 2.52e-08 ***
## Height          0.045139   0.008130   5.552 3.08e-08 ***
## Whole_weight    0.038447   0.004365   8.809  < 2e-16 ***
## Shucked_weight -0.088469   0.004812 -18.385  < 2e-16 ***
## Viscera_weight -0.050926   0.007771  -6.554 6.61e-11 ***
## Shell_weight    0.044969   0.006636   6.776 1.49e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.164 on 2914 degrees of freedom
## Multiple R-squared:  0.5341,	Adjusted R-squared:  0.5327 
## F-statistic: 371.2 on 9 and 2914 DF,  p-value: < 2.2e-16

In first additive model, note that factor level female is reference level for Sex variable.
After fitting the additive model with all predictors we can see that test statistics showing all variables as significant except Length. As we saw previously from pairs plot that Length and Diameter predictors are highly correlated. We also see that different weights predictors are also significant even though they should be linear function of each other.

####RMSE Score:

kable(compare_rmse(abalone_add,"Additive Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Additive Model	2.1601	2.2729

####Multicollinearity:

We will calculate variance inflation factor in order to find presence of Multicollinearity issue with the dataset.

faraway::vif(abalone_add)

##           SexI           SexM         Length       Diameter         Height 
##       1.982824       1.400405      42.874862      44.907429       3.114493 
##   Whole_weight Shucked_weight Viscera_weight   Shell_weight 
##     118.819537      29.579686      18.591058      22.182789

We looked that variable inflation factor for all variable and it appears that all predictors have multicollinearity issue except Sex and Height as we saw previously in pairs plot. Predictor Whole_weight has highest VIF value as it is a linear function of other weight.

####Partial correlation coefficient between Whole_weight & Rings : We will first calculate the partial correlation coefficient of the Whole_weight variable & the response variable (Rings).

#check variabolity in high collinearity variables
whole_weight_fit <- lm(Whole_weight ~ Sex + Length + Diameter + Height + Shucked_weight + Viscera_weight + Shell_weight, data=abalone_train)
 
abalone_add_without_whole_weight <- lm(Rings ~ Sex + Length + Diameter + Height
               + Shucked_weight + Viscera_weight + Shell_weight,data = abalone_train)

The correlation of these two residuals are close to zero, it means that the variation of Rings that is unexplained by predictors Sex,Length,Diameter,Height,Shucked_weight,Viscera_weight and Shell_weight shows very little correlation with the variation of Whole_weight that is not explained by Sex,Length,Diameter,Height,Shucked_weight,Viscera_weight and Shell_weight. Thus adding Whole_weight to the model would likely be of little benefit.

####Variable added plot

Similarly a variable added plot visualizes these residuals against each other. It is also helpful to regress the residuals of the response against the residuals of the predictor and add the regression line to the plot.

cor(resid(whole_weight_fit),resid(abalone_add_without_whole_weight))

## [1] 0.1610522

create_varplot(abalone_add_without_whole_weight,whole_weight_fit)

####Variance inflation factor of the additive model without the Whole_weight

Still the VIF of the Diameter and Length is high.

faraway::vif(abalone_add_without_whole_weight)

##           SexI           SexM         Length       Diameter         Height 
##       1.980330       1.400403      42.865399      44.886218       3.114448 
## Shucked_weight Viscera_weight   Shell_weight 
##       8.702278      11.022660       7.642714

####Partial correlation coefficient between Diameter & Rings

We will now calculate the partial correlation coefficient of the Diameter variable & the response variable (Rings) without having the Whole_weight variable in the model.

diameter_fit <- lm(Diameter ~ Sex + Length + Height + Shucked_weight + Viscera_weight + Shell_weight, data=abalone_train)
 
abalone_add_small <- lm(Rings ~ Sex + Length + Height + Shucked_weight + Viscera_weight + Shell_weight,data = abalone_train)

The correlation of these two residuals are close to zero, it means that the variation of Rings that is unexplained by Sex,Length,Height,Shucked_weight,Viscera_weight and Shell_weight shows very little correlation with the variation of Diameter that is not explained by Sex,Length,Height,Shucked_weight,Viscera_weight and Shell_weight. Thus adding Diameter to the model would likely be of little benefit.

cor(resid(diameter_fit),resid(abalone_add_small))

## [1] 0.1050962

create_varplot(abalone_add_small,diameter_fit)

Variance inflation factor of the additive model without the Whole_weight & Diameter

faraway::vif(abalone_add_small)

##           SexI           SexM         Length         Height Shucked_weight 
##       1.944920       1.396606       7.731242       3.080630       8.701861 
## Viscera_weight   Shell_weight 
##      11.007482       7.271741

Now the VIF is much lower. We will use the both the abalone_add & abalone_add_small for our analysis.

RMSE Score of abalone_add_small

kable(compare_rmse(abalone_add_small,"Additive Small Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Additive Small Model	2.2009	2.3276

Anova F Test

anova(abalone_add_small,abalone_add)

## Analysis of Variance Table
## 
## Model 1: Rings ~ Sex + Length + Height + Shucked_weight + Viscera_weight + 
##     Shell_weight
## Model 2: Rings ~ Sex + Length + Diameter + Height + Whole_weight + Shucked_weight + 
##     Viscera_weight + Shell_weight
##   Res.Df   RSS Df Sum of Sq      F    Pr(>F)    
## 1   2916 14163                                  
## 2   2914 13644  2    519.74 55.504 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

As per the Anova Test, we can reject the null hypothesis.
Now lets try to run AIC and BIC on the additive model for parameter selection.

#Running AIC and BIC on additive model
abalone_model_add_aic <- step(abalone_add, direction="backward", trace=0)
summary(abalone_model_add_aic)

## 
## Call:
## lm(formula = Rings ~ Sex + Diameter + Height + Whole_weight + 
##     Shucked_weight + Viscera_weight + Shell_weight, data = abalone_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.8654 -1.2936 -0.3086  0.8431 11.9523 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     3.809653   0.322317  11.820  < 2e-16 ***
## SexI           -0.895511   0.119865  -7.471 1.05e-13 ***
## SexM            0.006839   0.098676   0.069    0.945    
## Diameter        0.059348   0.005697  10.417  < 2e-16 ***
## Height          0.044974   0.008130   5.532 3.45e-08 ***
## Whole_weight    0.038530   0.004365   8.828  < 2e-16 ***
## Shucked_weight -0.088919   0.004800 -18.526  < 2e-16 ***
## Viscera_weight -0.051921   0.007733  -6.714 2.26e-11 ***
## Shell_weight    0.045291   0.006632   6.829 1.04e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.164 on 2915 degrees of freedom
## Multiple R-squared:  0.5338,	Adjusted R-squared:  0.5326 
## F-statistic: 417.3 on 8 and 2915 DF,  p-value: < 2.2e-16

n <- length(resid(abalone_add))
abalone_model_add_bic <- step(abalone_add, direction="backward", k=log(n), trace=0)
summary(abalone_model_add_bic)

## 
## Call:
## lm(formula = Rings ~ Sex + Diameter + Height + Whole_weight + 
##     Shucked_weight + Viscera_weight + Shell_weight, data = abalone_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.8654 -1.2936 -0.3086  0.8431 11.9523 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     3.809653   0.322317  11.820  < 2e-16 ***
## SexI           -0.895511   0.119865  -7.471 1.05e-13 ***
## SexM            0.006839   0.098676   0.069    0.945    
## Diameter        0.059348   0.005697  10.417  < 2e-16 ***
## Height          0.044974   0.008130   5.532 3.45e-08 ***
## Whole_weight    0.038530   0.004365   8.828  < 2e-16 ***
## Shucked_weight -0.088919   0.004800 -18.526  < 2e-16 ***
## Viscera_weight -0.051921   0.007733  -6.714 2.26e-11 ***
## Shell_weight    0.045291   0.006632   6.829 1.04e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.164 on 2915 degrees of freedom
## Multiple R-squared:  0.5338,	Adjusted R-squared:  0.5326 
## F-statistic: 417.3 on 8 and 2915 DF,  p-value: < 2.2e-16

We ran AIC AND BIC method using best model (abalone_add) from the previous Anova F Test. both AIC and BIC selected the same model without the Length predictor.
We selected the model from BIC (Since both models are same) and will plot fitted vs residuals and qq normal plots.

Additive Model Assumptions

model_assumptions(abalone_model_add_bic,"abalone_model_add_bic")

Here we see that fitted vs residuals plots suggests that model is violating the constant variation and appears to have a pattern where residuals are increasing as fitted values are increasing.
Normal QQ plot also has fat tail which are indication that errors are may not be normally distributed.
This is an indication that we might need to need to look to improve this model. Even though multicollinearity does not have effect on prediction, this looks like an extreme case where there is a clear pattern that violates model assumptions. This could be achieved by several techniques for example variable selection and transformation. Lets take a look at the variable selection method to see which variables could be used to create another model that does not violate constant variation and normality assumptions.

Exhaustive search

#exhaustive search 
all_abalone_add <- summary(regsubsets(Rings ~ Sex + Length + Diameter + Height + Whole_weight + Shell_weight + Shucked_weight + Viscera_weight , data=abalone_train))

kable(all_abalone_add$which, digits = 4,format = 'markdown')

(Intercept)	SexI	SexM	Length	Diameter	Height	Whole_weight	Shell_weight	Shucked_weight	Viscera_weight
TRUE	FALSE	FALSE	FALSE	FALSE	FALSE	FALSE	TRUE	FALSE	FALSE
TRUE	FALSE	FALSE	FALSE	FALSE	FALSE	FALSE	TRUE	TRUE	FALSE
TRUE	FALSE	FALSE	FALSE	TRUE	FALSE	FALSE	TRUE	TRUE	FALSE
TRUE	TRUE	FALSE	FALSE	TRUE	FALSE	FALSE	TRUE	TRUE	FALSE
TRUE	TRUE	FALSE	FALSE	TRUE	FALSE	TRUE	TRUE	TRUE	FALSE
TRUE	TRUE	FALSE	FALSE	TRUE	FALSE	TRUE	TRUE	TRUE	TRUE
TRUE	TRUE	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE
TRUE	TRUE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

var_num=rep(0,8)
for(i in c(1:8)){
  var_num[i]=sum(all_abalone_add$which[i,])-1
}

par(mfrow = c(1, 2),oma=c(0,0,0,0))
plot(var_num,all_abalone_add$rss,pch = 20, 
     col = "dodgerblue", type = "b", cex = 2,
     xlab = '# of Predictors (Including Dummy Variable)')
plot(var_num,all_abalone_add$adjr2,pch = 20, col = "dodgerblue", 
     xlab = '# of Predictors (Including Dummy Variable)',type = "b", cex = 2)

(bestrj2 <- which.max(all_abalone_add$adjr2))

## [1] 8

all_abalone_add$which[bestrj2,]

##    (Intercept)           SexI           SexM         Length       Diameter 
##           TRUE           TRUE          FALSE           TRUE           TRUE 
##         Height   Whole_weight   Shell_weight Shucked_weight Viscera_weight 
##           TRUE           TRUE           TRUE           TRUE           TRUE

#Plotting model parameter vs AIC plot
p <- length(coef(abalone_add))
n <- length(resid(abalone_add))

abalone_mod_aic <- n * log(all_abalone_add$rss / n) + 2 * (2:p)
abalone_mod_aic

## [1] 5278.462 4898.917 4703.464 4623.878 4589.702 4550.064 4521.532 4521.883
## [9] 5294.462

plot(abalone_mod_aic ~ I(2:p), ylab = "AIC", xlab = "p, number of parameters", 
     pch = 20, col = "dodgerblue", type = "b", cex = 2,
     main = "AIC vs Model Complexity")

Now we are seeing some interesting results here. previously we saw that t test showing some predictors as non-significant but when we performed the exhaustive search, it is suggesting that we do need all the predictors to create the model with lowest AIC value. As it can be seen from the plot that AIC goes down and is the least with the size of model with 8 parameters. We will again use all predictors from dataset to create model and look for variables transformation techniques.
Next in order to stabilize the constant variation, we will perform some response and predictor transformations.

###Response Transformation

####Box-Cox Transformations

One of the way to stabilize the variance is by using log transforming the response. In order to get the correct order of it, we used boxcox method which is suggesting to use $\lambda$ value of 0. As the at the value of 0, the log-likelyhood maximizes and the intervals are extremely close. Hence we will use transformation of form log(Rings) for our additive model.

boxcox(abalone_add,  plotit = TRUE, lambda = seq(-0.1, 0.1, by = 0.1))

####Additive Model with log response transformations

abalone_add_log <- lm(log(Rings) ~ Sex + Length + Diameter + Height + Whole_weight + Shucked_weight + Viscera_weight + Shell_weight,data = abalone_train)
summary(abalone_add_log)

## 
## Call:
## lm(formula = log(Rings) ~ Sex + Length + Diameter + Height + 
##     Whole_weight + Shucked_weight + Viscera_weight + Shell_weight, 
##     data = abalone_train)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.36706 -0.13063 -0.01448  0.10838  0.77172 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     1.3265359  0.0317433  41.789  < 2e-16 ***
## SexI           -0.0955745  0.0112420  -8.502  < 2e-16 ***
## SexM            0.0050846  0.0092308   0.551    0.582    
## Length          0.0019691  0.0010127   1.944    0.052 .  
## Diameter        0.0087565  0.0012544   6.981 3.62e-12 ***
## Height          0.0051007  0.0007603   6.709 2.35e-11 ***
## Whole_weight    0.0026152  0.0004082   6.407 1.72e-10 ***
## Shucked_weight -0.0075288  0.0004500 -16.731  < 2e-16 ***
## Viscera_weight -0.0043683  0.0007267  -6.011 2.07e-09 ***
## Shell_weight    0.0031137  0.0006206   5.017 5.56e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2023 on 2914 degrees of freedom
## Multiple R-squared:  0.5979,	Adjusted R-squared:  0.5966 
## F-statistic: 481.4 on 9 and 2914 DF,  p-value: < 2.2e-16

After log transforming the response, we see that t test was significant, it also increased the adjusted r squared value from previous additive model. We also see that almost all of the predictors are significant here in this model. lets check the assumptions.

Model Assumptions

Below the Fitted versus Residuals Plot & Q-Q Plots shows that the results have improved a lot after applying log transformation on the response variable.

model_assumptions(abalone_add_log,"abalone_add_log")

RMSE Score

kable(log_rmse(abalone_add_log,"Additive Log Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Additive Log Model	2.2392	2.3403

However we dont see any improvement in the RMSE Score. Though model assumptions now looks much better. the constant variance issue seems to be improved and qq plot is also looking good.
Next, we will perform some transformations on predictors and evaluate the models and will see of this helps further for prediction accuracy.

###Predictor Transformations

####Regression Analysis

In order for us to perform any predictors transformations, let first see how each of the predictors are related to response. The transformation will depend on it shape of the data and relationship between predictors and response.

par(mfrow = c(3, 3),oma=c(0,0,0,0))
grid.arrange(
create_scatter_plot(abalone_train$Length,abalone_train$Rings,"Length","Rings"),
create_scatter_plot(abalone_train$Diameter,abalone_train$Rings,"Diameter","Rings"),
create_scatter_plot(abalone_train$Height,abalone_train$Rings,"Height","Rings"),
create_scatter_plot(abalone_train$Whole_weight,abalone_train$Rings,"Whole_weight","Rings"),
create_scatter_plot(abalone_train$Shell_weight,abalone_train$Rings,"Shell_weight","Rings"),
create_scatter_plot(abalone_train$Shucked_weight,abalone_train$Rings,"Shucked_weight","Rings"),
create_scatter_plot(abalone_train$Viscera_weight,abalone_train$Rings,"Viscera_weight","Rings")
)

We can see the relationship with Rings and predictors Length,Diameter, Height is almost linear. Also we can see that relationship between weight predictors are not really linear but could benefit from polynomial transformations. So lets create a model using higher order polynomial of all weight predictors Whole_weight,Viscera_weight,Shucked_weight and Shell_weight.

Polynomials

The Model assumptions are same after using 2nd order term in the model.

abalone_add_poly2 <- lm(log(Rings) ~  Diameter + Length + Height + poly(Whole_weight,2) + poly(Viscera_weight,2) + poly(Shucked_weight,2) + poly(Shell_weight,2) + Sex, data=abalone_train)

model_assumptions(abalone_add_poly2,"Poly2 Log Model")

RMSE Score

kable(log_rmse(abalone_add_poly2,"Poly2 Log Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Poly2 Log Model	2.1494	2.2672

Here we have performed some variable transformations. First we log transformed the response as suggested by boxcox method and used polynomial transformations for weight predictors as suggested in pairs plot. after fitting the model, we see that rmse is lower than previous models and it also has better constant variation and Q-Q plots compared to previously fitted additive models. Since we have performed a polynomial transformations with the degree of 2, lets try to fit another model with degree of 3 and check the significance.

abalone_add_poly3 <- lm(log(Rings) ~  Diameter + Length + Height + poly(Whole_weight,3) + poly(Viscera_weight,3) + poly(Shucked_weight,3) + poly(Shell_weight,3) + Sex, data=abalone_train)

RMSE Score

kable(log_rmse(abalone_add_poly3,"Poly3 Log Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Poly3 Log Model	2.1067	2.2566

ANOVA F Test

anova(abalone_add_poly2,abalone_add_poly3)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     2) + poly(Viscera_weight, 2) + poly(Shucked_weight, 2) + 
##     poly(Shell_weight, 2) + Sex
## Model 2: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     3) + poly(Viscera_weight, 3) + poly(Shucked_weight, 3) + 
##     poly(Shell_weight, 3) + Sex
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1   2910 112.59                                  
## 2   2906 108.74  4    3.8548 25.754 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

F test has low p-value suggesting that transformation with degree 3 is significant and we saw that rmse went down compare to polynomial model with degree 2. Lets try to fit the model of degree 4 and check the significance.

abalone_add_poly4 <- lm(log(Rings) ~  Diameter + Length + Height + poly(Whole_weight,4) + poly(Viscera_weight,4) + poly(Shucked_weight,4) + poly(Shell_weight,4) + Sex, data=abalone_train)

RMSE Score

kable(log_rmse(abalone_add_poly4,"Poly4 Log Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Poly4 Log Model	2.0883	2.2281

ANOVA F Test

anova(abalone_add_poly3,abalone_add_poly4)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     3) + poly(Viscera_weight, 3) + poly(Shucked_weight, 3) + 
##     poly(Shell_weight, 3) + Sex
## Model 2: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + 
##     poly(Shell_weight, 4) + Sex
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1   2906 108.74                                  
## 2   2902 106.36  4    2.3786 16.225 3.781e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Again we see that it was significant. lets try to fit the model with degree of 5.

abalone_add_poly5 <- lm(log(Rings) ~  Diameter + Length + Height + poly(Whole_weight,5) + poly(Viscera_weight,5) + poly(Shucked_weight,5) + poly(Shell_weight,5) + Sex, data=abalone_train)

RMSE Score

kable(log_rmse(abalone_add_poly5,"Poly5 Log Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Poly5 Log Model	2.077	2.2247

ANOVA F Test

anova(abalone_add_poly4,abalone_add_poly5)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + 
##     poly(Shell_weight, 4) + Sex
## Model 2: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     5) + poly(Viscera_weight, 5) + poly(Shucked_weight, 5) + 
##     poly(Shell_weight, 5) + Sex
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1   2902 106.36                                  
## 2   2898 104.56  4    1.8034 12.496 4.429e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We again see that test was significant with lower rmse. lets try to fit the model with degree of 6.

abalone_add_poly6 <- lm(log(Rings) ~  Diameter + Length + Height + poly(Whole_weight,6) + poly(Viscera_weight,6) + poly(Shucked_weight,6) + poly(Shell_weight,6) + Sex, data=abalone_train)

RMSE Score

kable(log_rmse(abalone_add_poly6,"Poly6 Log Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Poly6 Log Model	2.0628	2.2336

ANOVA F Test

anova(abalone_add_poly5,abalone_add_poly6)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     5) + poly(Viscera_weight, 5) + poly(Shucked_weight, 5) + 
##     poly(Shell_weight, 5) + Sex
## Model 2: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     6) + poly(Viscera_weight, 6) + poly(Shucked_weight, 6) + 
##     poly(Shell_weight, 6) + Sex
##   Res.Df    RSS Df Sum of Sq     F    Pr(>F)    
## 1   2898 104.56                                 
## 2   2894 103.32  4    1.2334 8.637 6.301e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Now after fitting with polynomial degree of 6 we see that even though F test suggest that it was a significant, the test RMSE went up. This here is an indication that we may have now started overfitting the data i.e. our model is fitting data really closely, which we do not want to be happen.
Before this we see that there is a very slight difference between the test and train RMSE with polynomial degree of 5 and 4. test RMSE is almost the same. Therefore we are willing to sacrifice very minute improvement (third decimal point) in RMSE over the simpler model. Hence we select the model with the polynomial degree of 4 which is model abalone_add_poly4.

num_poly=seq(1,10,by = 1)
train_rmse=rep(0,length(num_poly))
test_rmse=rep(0,length(num_poly))

for(d in num_poly){
  abalone_add_poly=test_model(d)  
  rmse=get_log_rmse(abalone_add_poly)
  train_rmse[d]=rmse$train
  test_rmse[d]=rmse$test
}

plot(train_rmse,
       type = "o",
       col = "red", 
       xlab = "Models with # Higher order term",
       ylab = "Error (RMSE)", 
       main = "Train vs Test RMSE",
      ylim=c(min(as.vector(train_rmse)), 
              max(as.vector(test_rmse))),
       cex.lab=1.3,
       cex.axis=1.3)
  
lines(test_rmse, 
        type = "o", 
        col = "blue")
  
  legend('topright', 
         c("Train error", "Test error"), 
         lty = c(1,1), 
         lwd = c(2.5, 2.5),
         col = c("red", "blue"))

We see that there is a very slight difference between the test and train RMSE with polynomial degree of 5 and 4. test RMSE is almost the same. Therefore we are willing to sacrifice very minute improvement in RMSE over the simpler model. Hence we select the model with the polynomial degree of 4 which is model abalone_add_poly4.
Since we have selected the model lets run AIC and BIC method to select appropriate model further and see if we can make further improvements.
Now lets calculate and compare the RMSE for higher order term and plot that for both Train & Test data.

AIC and BIC on poly additive model:

Since we have selected the model lets run AIC and BIC method to select appropriate model further.

abalone_model_add_aic <- step(abalone_add_poly4, direction="backward", trace=FALSE)

n <- length(resid(abalone_add_poly4))
abalone_model_add_bic <- step(abalone_add_poly4, direction="backward", k=log(n), trace=FALSE)

####Compare AIC vs BIC Model Parameters

abalone_model_add_aic$call[2]

## (log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + 
##     poly(Shell_weight, 4) + Sex)()

abalone_model_add_bic$call[2]

## (log(Rings) ~ Diameter + Height + poly(Whole_weight, 4) + poly(Viscera_weight, 
##     4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex)()

####Anove F Test

anova(abalone_model_add_bic,abalone_model_add_aic)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Diameter + Height + poly(Whole_weight, 4) + poly(Viscera_weight, 
##     4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex
## Model 2: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + 
##     poly(Shell_weight, 4) + Sex
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1   2903 106.46                           
## 2   2902 106.36  1  0.096697 2.6384 0.1044

The model selected by BIC does not have predictor Length in it. The p-value of the Anova F Test is large, so we fail to reject the null hypothesis. The abalone_model_add_bic model is significant,so we will move forward with it and will check model assumptions.

####Model Assumptions (AIC & BIC):

model_assumptions(abalone_model_add_aic,"abalone_model_add_aic")

model_assumptions(abalone_model_add_bic,"abalone_model_add_bic")

Both constant variance and normality looks good in this case.

RMSE Score - AIC

kable(log_rmse(abalone_model_add_aic,paste("Additive Model - Degree 4 - AIC ")), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Additive Model - Degree 4 - AIC	2.0883	2.2281

RMSE Score - BIC

kable(log_rmse(abalone_model_add_bic,paste("Additive Model - Degree 4 - BIC")), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Additive Model - Degree 4 - BIC	2.0913	2.227

After doing variable selection using AIC and BIC, we selected the model from BIC and checked the t statistics and assumptions. Interestingly BIC model is dropping few predictors but also has similar test RMSE as original model we started with (model with polynomial degree of 4). Which suggests that we could drop some of the variables and still maintain lower RMSE. This bring us to the next modification and introduction of interaction terms between the variables selected by BIC model above.
The model assumptions from BIC model also looks better.
Next we will introduce interaction terms and will try to fit the model with predictors suggested by BIC method.

###Interaction Model

abalone_int <- lm(log(Rings) ~  Height + Diameter + poly(Whole_weight, 4) +
  poly(Viscera_weight, 4) + poly(Shucked_weight,4) + poly(Shell_weight, 4) + Sex + Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight,  4):Sex, data=abalone_train)

####RMSE Score

kable(log_rmse(abalone_int,"Interation Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Interation Model	2.0452	2.2041

ANOVA F Test

anova(abalone_add_poly4,abalone_int)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + 
##     poly(Shell_weight, 4) + Sex
## Model 2: log(Rings) ~ Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 
##     4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex + 
##     Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Sex
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1   2902 106.36                                  
## 2   2891  98.59 11    7.7702 20.714 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

After fitting the interaction model and performing F test with best additive model we see that test suggest that interaction model is a significant model, with improved adjusted r squared value. Also RMSE got lower Since it is better explaining the variability, we for now will choose the interaction model and will try to run AIC and BIC on interaction model.
At the same time, we would compare interaction model with several degrees to calculate and compare the RMSE for higher order term and plot that for both Train & Test data.

num_poly=seq(1,6,by = 1)
train_rmse_int=rep(0,length(num_poly))
test_rmse_int=rep(0,length(num_poly))

for(d in num_poly){
  abalone_int_poly=test_int_model(d)
  
  rmse=get_log_rmse(abalone_int_poly)
  train_rmse_int[d]=rmse$train
  test_rmse_int[d]=rmse$test
}

plot(train_rmse_int,
       type = "o",
       col = "red", 
       xlab = "Models with # Higer order term",
       ylab = "Error (RMSE)", 
       main = "Train vs Test RMSE",
      ylim=c(min(as.vector(train_rmse_int)), 
              max(as.vector(test_rmse_int))),
       cex.lab=1.3,
       cex.axis=1.3)
  
  lines(test_rmse_int, 
        type = "o", 
        col = "blue")
  
  legend('topright', 
         c("Train error", "Test error"), 
         lty = c(1,1), 
         lwd = c(2.5, 2.5),
         col = c("red", "blue"))

We can see that that RMSE is getting lower and lower as polynomial degree increases. Though this improvement in RMSE is so minute for this analysis that we can ignore this improvement over model simplicity. Considering this we can see that model with polynomial degree of 4 is performing better so we will move forward with that model.

AIC and BIC on Interaction model

abalone_model_int_aic <- step(abalone_int, direction="backward", trace=FALSE)

n <- length(resid(abalone_int))
abalone_model_int_bic <- step(abalone_int, direction="backward", k=log(n), trace=FALSE)

abalone_model_int_aic$call[2]

## (log(Rings) ~ Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 
##     4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex + 
##     Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Sex)()

abalone_model_int_bic$call[2]

## (log(Rings) ~ Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 
##     4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex + 
##     Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Sex)()

RMSE Score - BIC/AIC

kable(log_rmse(abalone_model_int_bic,paste("Interaction Model - Degree 4 - BIC")), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Interaction Model - Degree 4 - BIC	2.0452	2.2041

After running AIC and BIC on our interaction model, we see that model selected the same model. Since this is one of the best model we have seen so far with reasonable complexity, we will consider this as one of the candidate models for our comparison as the best fitted model for this analysis.
Talking about candidate models, during the exhaustive search we have seen that the lowest AIC comes with model when we use all predictors. We can try to build a model with all predictors with the interaction and polynomial degree and see how it stands compare to our first selected candidate model. So lets fit a model with all predictors.

abalone_int_full <- lm(log(Rings) ~ Length + Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex + Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Sex, data = abalone_train)

During the initial data analysis, we found that distribution of categorical variable Sex with factor level of female and male is extremely similar. Because of this we decided to merged these two factor level in one and have total factor level to be two infant and non-infant. We created new variable Infant. here non-infant represents female and male both. we performed analysis from this approach as well (which can be found on appendix section of this report).
Lets looks at Infant model analysis and see how this model stands up against our model of choice above.

Infant Model Analysis

Remember we talked about taking a different approach for this analysis. We introduced a new categorical predictor names Infant. We used existing categorical predictors Sex with 3 factor level and create a new with 2 factor levels. We did this because we identified similar distribution on 2 factor levels from original categorical predictor female and male. New factors levels now are I (infant = female and male combined) and NI (non infant).
The analysis with this new categorical is exactly the same as analysis above so we will move fast with this analysis with minimum explanation and details.

Additive Infant Models

abalone_add_inf <- lm(Rings ~ Infant+Length+Diameter+Height+ Whole_weight
               +Shucked_weight+Viscera_weight
               +Shell_weight,data = abalone_train)

summary(abalone_add_inf)

## 
## Call:
## lm(formula = Rings ~ Infant + Length + Diameter + Height + Whole_weight + 
##     Shucked_weight + Viscera_weight + Shell_weight, data = abalone_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.9199 -1.2927 -0.2992  0.8336 12.0077 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     3.064833   0.308712   9.928  < 2e-16 ***
## InfantNI        0.889654   0.105720   8.415  < 2e-16 ***
## Length         -0.013877   0.010823  -1.282      0.2    
## Diameter        0.074874   0.013393   5.590 2.47e-08 ***
## Height          0.045127   0.008128   5.552 3.08e-08 ***
## Whole_weight    0.038447   0.004364   8.810  < 2e-16 ***
## Shucked_weight -0.088435   0.004800 -18.422  < 2e-16 ***
## Viscera_weight -0.050967   0.007760  -6.568 6.02e-11 ***
## Shell_weight    0.044964   0.006635   6.777 1.48e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.163 on 2915 degrees of freedom
## Multiple R-squared:  0.5341,	Adjusted R-squared:  0.5328 
## F-statistic: 417.7 on 8 and 2915 DF,  p-value: < 2.2e-16

RMSE Score

rmse  <- function(actual, predicted) {
  sqrt(mean((actual - predicted) ^ 2))
}

kable(compare_rmse(abalone_add_inf,"Additive Infant Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Additive Infant Model	2.1601	2.2731

Additive Model Assumptions

model_assumptions(abalone_add_inf,"abalone_add_inf")

Box-Cox Transformations

boxcox(abalone_add_inf,  plotit = TRUE, lambda = seq(-0.1, 0.1, by = 0.1))

Additive Infant Model with log response transformations

abalone_add_log_inf <- lm(log(Rings) ~ Infant + Length + Diameter + Height + Whole_weight + Shucked_weight + Viscera_weight + Shell_weight,data = abalone_train)
summary(abalone_add_log)

## 
## Call:
## lm(formula = log(Rings) ~ Sex + Length + Diameter + Height + 
##     Whole_weight + Shucked_weight + Viscera_weight + Shell_weight, 
##     data = abalone_train)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.36706 -0.13063 -0.01448  0.10838  0.77172 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     1.3265359  0.0317433  41.789  < 2e-16 ***
## SexI           -0.0955745  0.0112420  -8.502  < 2e-16 ***
## SexM            0.0050846  0.0092308   0.551    0.582    
## Length          0.0019691  0.0010127   1.944    0.052 .  
## Diameter        0.0087565  0.0012544   6.981 3.62e-12 ***
## Height          0.0051007  0.0007603   6.709 2.35e-11 ***
## Whole_weight    0.0026152  0.0004082   6.407 1.72e-10 ***
## Shucked_weight -0.0075288  0.0004500 -16.731  < 2e-16 ***
## Viscera_weight -0.0043683  0.0007267  -6.011 2.07e-09 ***
## Shell_weight    0.0031137  0.0006206   5.017 5.56e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2023 on 2914 degrees of freedom
## Multiple R-squared:  0.5979,	Adjusted R-squared:  0.5966 
## F-statistic: 481.4 on 9 and 2914 DF,  p-value: < 2.2e-16

RMSE Score

kable(log_rmse(abalone_add_log_inf,"Additive Infant Model (Log)"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Additive Infant Model (Log)	2.2395	2.3412

Additive Model Assumptions

model_assumptions(abalone_add_log_inf,"abalone_add_log_inf")

Polynomial Model

abalone_add_poly3_int <- lm(log(Rings) ~  Diameter + Length + Height + poly(Whole_weight,3) + poly(Viscera_weight,3) + poly(Shucked_weight,3) + poly(Shell_weight,3) + Infant, data=abalone_train)

RMSE Score

kable(log_rmse(abalone_add_poly3_int,"Polynomial Infant Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Polynomial Infant Model	2.1068	2.2568

Polynomial Model Assumptions

model_assumptions(abalone_add_poly3_int,"abalone_add_poly3_int")

Anova F Test

anova(abalone_add_log_inf,abalone_add_poly3_int)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Infant + Length + Diameter + Height + Whole_weight + 
##     Shucked_weight + Viscera_weight + Shell_weight
## Model 2: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     3) + poly(Viscera_weight, 3) + poly(Shucked_weight, 3) + 
##     poly(Shell_weight, 3) + Infant
##   Res.Df    RSS Df Sum of Sq     F    Pr(>F)    
## 1   2915 119.33                                 
## 2   2907 108.74  8    10.587 35.38 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interaction Model

abalone_int_infant <- lm(log(Rings) ~  Height + Diameter + poly(Whole_weight, 4) +
  poly(Viscera_weight, 4) + poly(Shucked_weight,4) + poly(Shell_weight, 4) + Infant + Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight,  4):Infant, data=abalone_train)

RMSE Score

kable(log_rmse(abalone_int_infant,"Interaction Infant Model"), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Interaction Infant Model	2.0475	2.2021

Interaction Model Assumptions

model_assumptions(abalone_int_infant,"abalone_int_infant")

Anova F Test

anova(abalone_add_poly3_int,abalone_int_infant)

## Analysis of Variance Table
## 
## Model 1: log(Rings) ~ Diameter + Length + Height + poly(Whole_weight, 
##     3) + poly(Viscera_weight, 3) + poly(Shucked_weight, 3) + 
##     poly(Shell_weight, 3) + Infant
## Model 2: log(Rings) ~ Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 
##     4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Infant + 
##     Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Infant
##   Res.Df     RSS Df Sum of Sq     F    Pr(>F)    
## 1   2907 108.739                                 
## 2   2896  98.741 11    9.9988 26.66 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Here we are selecting the best performing interaction model as one of our candidate models. This model has less complexity as well as the lowest rmse.
What we see here is that this model has the lowest rmse on unseen data (test data) compare to previous two full models. In addition to this, this model is also less complex than previous two, These characteristic makes this model as one of the candidate model of our choice.
Now since we have a few candidate model that we like, lets compare these against each other and looks for best model.

Candidate models

From the analysis above we have selected interaction model with polynomial degree of 4 as we have seen it has one of the lowest test rmse with good model assumptions. We will consider this as the first choice of one of the candidate model. This model contains some of the predictors and not the ones with high multicollinearity. This model is abalone_model_int_bic.
Another selected model is the interaction model with all predictors with similar lower rmse. Though this model will be have more parameters hence is a more complex model than earlier model but since our goal is prediction, we do not care about complexity of the model. This model is abalone_int_full
One another candidate model will be the interaction model with categorical variable with 2 factor levels. The the beginning or our analysis we saw that distribution of this predictors was similar between level male and female so we decided to introduced a new variable which essentially had 2 levels, infant and non-infant where non-infant = male and female. This model is also an interaction model with the polynomial degree of 4. This model is abalone_int_infant. Note: (The Additive model and related analysis using the infant predictor have been added in the appendix section). The entire process of analysis with categorical predictor and factor level with 3 values and 2 values are same.

RMSE comparison for candidate models

Candidate model 1 (abalone_model_int_bic)

kable(log_rmse(abalone_model_int_bic,paste("Candidate model 1")), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Candidate model 1	2.0452	2.2041

Candidate model 2 (abalone_int__candidate_full)

kable(log_rmse(abalone_int_full,paste("Candidate model 2")), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Candidate model 2	2.0394	2.2073

Candidate model 3 (abalone_int_infant)

kable(log_rmse(abalone_int_infant,paste("Candidate model 3")), digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Candidate model 3	2.0475	2.2021

As we can see from above that the Candidate model 3 (abalone_int_infant) is performing best among all three models.

Model assumptions for candidate models

Candidate model 1 (abalone_model_int_bic)

model_assumptions(abalone_model_int_bic,"Candidate model 1")

Candidate model 2 (abalone_int_full)

model_assumptions(abalone_int_full,"Candidate model 2")

Candidate model 3 (abalone_int_infant)

model_assumptions(abalone_int_infant,"Candidate model 3")

Based on these plots we can see that all of our candidate models are satisfying both constant variance and normality assumptions nicely.
Next we will compare all of these candidates models for its prediction and confidence interval and observe any influential points in our data. lets look at it.

Unusual observations

We will take a look the unusual observations present in the train data set for each candidate models. First we will look for the high leverage points, outliers and influential points individually and then get the data points with high leverage, outliers and influence by taking the common data points.

####Candidate model 1 (abalone_model_int_bic)

leverage <- find_leverage(abalone_model_int_bic)
outliers <- find_outliers(abalone_model_int_bic)
influence <- find_influence(abalone_model_int_bic)
common_leverage_influence <- intersect(influence,leverage)
common_model_1 <- intersect(outliers,common_leverage_influence)

length(common_model_1)

## [1] 16

We see that there are 16 observations that are high leverage, outlier and influential points.

####Candidate model 2 (abalone_int_full)

leverage <- find_leverage(abalone_int_full)
outliers <- find_outliers(abalone_int_full)
influence <- find_influence(abalone_int_full)
common_leverage_influence <- intersect(influence,leverage)
common_model_2 <- intersect(outliers,common_leverage_influence)

length(common_model_2)

## [1] 15

In this model, we see that there are 15 observations that are high leverage, outlier and influential points.

####Candidate model 3 (abalone_int_infant)

leverage <- find_leverage(abalone_int_infant)
outliers <- find_outliers(abalone_int_infant)
influence <- find_influence(abalone_int_infant)
common_leverage_influence <- intersect(influence,leverage)
common_model_3 <- intersect(outliers,common_leverage_influence)

length(common_model_3)

## [1] 19

This model has 19 such observations that are high leverage, outlier and influential points.

####Common Unusual Observations between all Candidate Models

We have now compiled the list of observations that are common across our candidate models. Lets take a look at these and see if there are any similarities.

common_data <- intersect(common_model_3,intersect(common_model_1,common_model_2))

length(common_data)

## [1] 13

kable(abalone_train[common_data,], digits = 4,format = 'markdown')

Sex	Length	Diameter	Height	Whole_weight	Shucked_weight	Viscera_weight	Shell_weight	Rings	Infant	weight.diff
M	124	102	35	323.0	102.1	38.4	135.0	12	NI	47.5
I	15	11	2	0.4	0.2	0.1	0.3	1	I	-0.2
F	129	105	38	361.7	140.7	77.7	79.0	18	NI	64.3
M	42	30	10	7.7	3.1	1.7	2.0	3	NI	0.9
I	30	20	5	3.0	0.9	0.8	1.0	2	I	0.3
I	62	45	14	21.1	87.0	3.0	8.0	5	I	-76.9
I	91	66	20	74.4	71.6	15.5	22.0	8	I	-34.7
F	91	71	226	118.8	66.4	23.2	26.7	8	NI	2.5
M	133	107	45	436.7	150.7	78.2	177.0	27	NI	30.8
F	160	126	39	505.2	186.6	118.0	124.0	23	NI	76.6
I	55	41	14	21.1	99.0	3.8	6.3	5	I	-88.0
I	76	55	19	48.5	21.2	9.7	42.0	6	I	-24.4
I	63	46	0	26.8	11.5	5.7	70.1	6	I	-60.5

The first thing that we notice right away is there are few observations that we initially determined as illogical or data entry error in our dataset. These observations have negative weight difference which is not possible. This make sense as there could be some error while recording this data.
Another pattern is some of these observations have whole_weight of magnitude of more than 2 from the average whole_weight. Also these observations are having high number of rings i.e. these are older abalones.
Since there are other observations present which qualifies as unusual observations, overall there is no specific pattern among them. Therefore we cannot rule these out from the model and analysis we are considering for prediction.
We will use Parallel Coordinates Plot in order to visualize any obvious pattern in the result section.
Lets see how our candidate models behaves once we remove these datasets.

####RMSE Score after removing the Unusual Observations

Next we will perform a quick experiment by removing the Unusual Observations from the Train Data Set and calculating the RMSE score for the Test Data Set.

abalone_train_updated <- abalone_train[-common_data,]

abalone_model_int_bic_obs_rmse <- get_log_rmse_obs(lm(log(Rings) ~ Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex +     Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Sex,data =abalone_train_updated),abalone_train_updated)

abalone_int_full_obs_rmse <- get_log_rmse_obs(lm(log(Rings) ~ Length + Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Sex + Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Sex, data = abalone_train_updated),abalone_train_updated)
                        
abalone_int_infant_obs_rmse <- get_log_rmse_obs(lm(log(Rings) ~ Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Infant + Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Infant, data = abalone_train_updated),abalone_train_updated)     

model_names <- c("Candidate model 1 (abalone_model_int_bic)",
                 "Candidate model 2 (abalone_int_full)",
                 "Candidate model 3 (abalone_int_infant)"
                 )

result <- data.frame( "Model" = model_names,
                      "RMSE Train" = c(
														abalone_model_int_bic_obs_rmse$train,
														abalone_int_full_obs_rmse$train,
														abalone_int_infant_obs_rmse$train
													 ),
          					  "RMSE Test" = c(
          										abalone_model_int_bic_obs_rmse$test,
          										abalone_int_full_obs_rmse$test,
          										abalone_int_infant_obs_rmse$test
          									 ), 
          					  "RMSE Train all Data" = c( 
          												 get_log_rmse(abalone_model_int_bic)$train,
          												 get_log_rmse(abalone_int_full)$train,
          												 get_log_rmse(abalone_int_infant)$train
          												),
          					  "RMSE Test all data" = c(
          										get_log_rmse(abalone_model_int_bic)$test,
          										get_log_rmse(abalone_int_full)$test,
          										get_log_rmse(abalone_int_infant)$test)
									  )

kable(result, digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test	RMSE.Train.all.Data	RMSE.Test.all.data
Candidate model 1 (abalone_model_int_bic)	2.0094	2.1996	2.0452	2.2041
Candidate model 2 (abalone_int_full)	2.0050	2.2016	2.0394	2.2073
Candidate model 3 (abalone_int_infant)	2.0118	2.1976	2.0475	2.2021

Interesting though, all the models trained without the unusual observations produced better rmse on the test data. In the above table we can see the comparison of the different RMSE values.
Since these unusual observations do not reveal any specific pattern or any unusual behavior we can not remove these from our dataset and will continue our analysis using the candidate models.

Prediction and Confidence intervals

#Candidate model 1
new_data <- abalone_test[1:10,c("Sex","Height","Diameter","Whole_weight","Viscera_weight","Shucked_weight","Shell_weight")]

faraway::vif(abalone_model_int_bic)

#Candidate model 2
new_data_multicol <- abalone_test[1:10,c("Sex","Length","Height","Diameter","Whole_weight", "Viscera_weight","Shucked_weight","Shell_weight")]

#Candidate model 3
new_data_infant <- abalone_test[1:10,c("Length","Height","Diameter","Whole_weight","Viscera_weight",
                                       "Shucked_weight","Shell_weight","Infant")]

Lets start with confidence intervals. The confidence intervals for all of our candidate models are of similar range. i.e. on average at any point the range of interval seems to be the same for all the models. Note that 2 out of 3 candidate models have all the predictors used hence have very high collinearity. Since our models are fitting data nicely, it reduces the variance range at any point on average. This is good.
Moreover, similar to confidence intervals the prediction intervals are also in same range between the models. Again model fitting had effect on this narrower interval range. Note that prediction intervals are much wider than confidence interval which we would expect. Since prediction intervals for all of the models are in similar range even after high collinearity, we would say that these are good. We have used default 95% formula to calculate both confidence and prediction intervals for all of our candidate models.
Based on our analysis on all these candidate models, we think that candidate model 3 abalone_int_infant would be the best choice for predicting the age of abalone. This model is smaller and contains less complexity compare to other two candidate models and yet has lowest rmse. Also the prediction and confidence intervals are very similar among the models. Because of these reasons we would select model abalone_int_infant as best model for our analysis.

Predicting the age of abalone

Now lets predict age of abalone using our selected model. We will select 5 random observations from test data and try to predict the age. Note that model actually predicts the number of rings in abalone. We need to add 1.5 in that ring count to determine the true age of abalone. So essentially we will be comparing the number of rings predicted by our model to actual number of rings in test dataset.

#Take 5 random observations from test data
dropcol <- c("weight.diff","Infant")
test_data <- abalone_test[, !(abalone_test %in% dropcol)]
sample <- sample_n(test_data, 5)

predicted <- round(exp(predict(abalone_int_infant, newdata=sample)))

new_df <- data.frame("Actual no of Rings" = c(sample$Rings), 
                     "Predicted no of Rings" = c(predicted),
                     "Actual age of abalone" = c(round(sample$Rings + 1.5)), 
                     "Predicted age of abalone" = round(predicted + 1.5))

kable(new_df, digits = 4,format = 'markdown')

Actual.no.of.Rings	Predicted.no.of.Rings	Actual.age.of.abalone	Predicted.age.of.abalone
10	9	12	10
13	13	14	14
11	11	12	12
19	11	20	12
11	11	12	12

We can see from above example that we calculated the number of rings as well as the age from both test and model for few observations. We see that our model is predicting the age correctly or very close to real age of abalone. These were 5 sample observations from test data that model has not seen before. We could take some more and perform the same process and see how well our model predicts the age of abalone.
In this analysis report, we have applied various multi regression techniques and methodologies, however that does not mean that this problem can not be solved using other more advanced techniques. We tried to apply some of the advanced methods beyond the scope of this report just to evaluate that if such techniques can further improve our prediction.

Using advanced techniques

In this section we went beyond tradition common regression techniques and tried to apply some advance techniques to see if we can create a model with lower test RMSE.

####Elastic Net The elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods.

We will try to use elastic net to further optimize our prediction.
Like lasso and ridge, the intercept is not penalized and glment takes care of standardization internally.
We will use the same predictors as in the Candidate model 3 (abalone_int_infant).

set.seed(42)
cv_10 <- trainControl(method = "cv", number = 10)

enet <- train(
  Rings ~ Height + Diameter + poly(Whole_weight, 4) + poly(Viscera_weight, 4) + poly(Shucked_weight, 4) + poly(Shell_weight, 4) + Infant + Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight, 4):Infant, data=abalone_train,
  method = "glmnet",
  trControl = cv_10,
  tuneLength = 10
)

get_best_result(enet)

##   alpha       lambda     RMSE  Rsquared      MAE    RMSESD RsquaredSD
## 1   0.4 0.0009173591 2.084348 0.5712596 1.487684 0.1250824 0.02329004
##       MAESD
## 1 0.0657003

result <- data.frame('Model'="Elastic Net","RMSE Train"=c(sqrt(mean((abalone_train$Rings-predict(enet,abalone_train))^2))),"RMSE Test"=c(sqrt(mean((abalone_test$Rings-predict(enet,abalone_test))^2))))

kable(result, digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Elastic Net	2.0501	2.1756

We see that both train and test rmse results are lower than our candidate models.

####Random Forest Regression

Random forest or random decision forests are an ensemble learning method for classification, regression and tasks, that operate by constructing a multitude of decision tress at training time and outputting the class that so the model os the classes (in case of classification) or mean prediction (in case of regression) of the individual trees. Random decision forests correct doe decision tree's habit of overfitting the training dataset.
We have implements the Random Forest regression model for prediction here to see if we can make any further improvement.The RMSE from the Random Forest Regression model turned out to be better than the RMSE Score of the candidate models.

booston_boost <- gbm(Rings ~ ., data = abalone_train, distribution = "gaussian", n.trees = 5000, interaction.depth = 3, shrinkage = 0.01)

boston_boost_tst_pred <- predict(booston_boost, newdata = abalone_test, n.trees = 5000)

boston_boost_train_pred <- predict(booston_boost, newdata = abalone_train, n.trees = 5000)

result <- data.frame('Model'="Random Forest","RMSE Train"=c(rmse(boston_boost_train_pred, abalone_train$Rings)),"RMSE Test"=c(rmse(boston_boost_tst_pred, abalone_test$Rings)))

kable(result, digits = 4,format = 'markdown')

Model	RMSE.Train	RMSE.Test
Random Forest	1.6416	2.1789

Below are some of the pros and cons of Random forest method
- It does not make any assumptions on distribution on data.
- Automatically fits highly non-linear interactions.
- Handles missing values pretty well through proximity.
- Quick to fit even for large datasets.
- Random forests have been observed to overfit for some datasets with noisy classification/regression tasks.
- Not easy to interpret if tree is not small
Since these advanced techniques are beyond the scope of this report, for selection procedure we will not consider these model.

Results

The task to predict the age of abalone was systematically approached in this report. First dataset was explored, unusual observations were discovered and the variables were appropriately standardized.
We started with an additive model that had all the predictors named abalone_add. We continued with checking various assumptions and tests to further improve the accuracy of the model and lower the rmse.
We then performed several transformations for response and predictors variables. We saw improvement in rmse as we performed various variable selection procedures.
After our analysis we have found a couple of candidate models that are good and we have selected model abalone_int_infant as our best model.
Below are some of the key points that are important that illustrates our choice of this conclusion.
- After applying above techniques, we arrived at 3 possible candidate models.
- Candidate model 1 was abalone_model_int_bic which was the result of various selection procedure over interaction model. we saw that this model had lowest AIC score as well as 2nd lowest rmse. Model contains some high colinear predictors but when checked the rmse, prediction and confidence intervals, it was an appropriate choice.
- Candidate model 2 was abalone_int_full. Recall that exhaustive seach procedure indicated that when we all of the predictors, we get the best model. So we decided to add all the predicors in the model with interaction terms and polynomial degree of 4. This model has the highest multicollinearity but from the rmse and prediction and confidence intervals came out to be pretty reasonable.
- Candidate model 3 was abalone_int_infant. As our initial data analysis showed that categorical variable Sex had 3 factor levels. female, male and infant. Levels female and male we extremely similar in terms of their distribution and therefore we combined those two levels in one, making total levels of predictor Sex to be two. Because of this reduction of level this model probably is the simplest model compare to previous two models. we performed thorough analysis for this and the final model had similar rmse and prediction and confidence intervals.This is why we selected this model as one of our candidate model.
Out of above 3 candidate model candidate model 3 abalone_int_infant turned out to be slightly better and at the same time less complex and therefore it is our model of choice for this analysis. There were number of reason for this. Lets discuss about it.
Below is the formula for model selected for this analysis

abalone_int_infant <- lm(log(Rings) ~  Height + Diameter + poly(Whole_weight, 4) +
  poly(Viscera_weight, 4) + poly(Shucked_weight,4) + poly(Shell_weight, 4) + Infant + Diameter:poly(Shucked_weight, 4) + poly(Shucked_weight,  4):Infant, data=abalone_train)

Here is the quick comparison of the Test/Train RMSE of the all 3 Candidate models.

Models	RMSE.Train	RMSE.Test
Candidate model 1 (abalone_model_int_bic)	2.0452	2.2041
Candidate model 2 (abalone_int_full)	2.0394	2.2073
Candidate model 3 (abalone_int_infant)	2.0475	2.2021

The chart below shows the rmse of the selected model (marked in Red), candidate models & rest of the model throughout this analysis report and their test rmse comparison. We can clearly see that our selected model abalone_int_infant also renamed int infant in plot below has lowest test rmse.

A clear pattern here is when model is smaller and simpler then test rmse is high but as we moved forward with better model and as it started to get more flexible (hence larger in size), the test rmse went down.
Below are the normality assumptions for chosen model. We can clearly see that as we fitted model with log transformation and interaction terms the constant variation and normality assumptions got better and better.

model_assumptions(abalone_int_infant,"Candidate model 3")

One key points to note here is that the presence of multicollinearity does not really affect the prediction (which is our goal for this analysis) we were able to reduce its affect upto certain level. We verify that by fitted vs residual plots and it can be seen that the model that we have selected has the residuals centered around 0 and roughly the constant variation.
If we plot the Unusual Observations using Parallel Coordinates Plot, we can see that the Height for these observations is shorter compare to an average height of abalone. The same is true in regards to Whole_weight and Viscera_weight which as lower. Also the weight.diff of some of these observations are negative which is what we found during the initial data analysis.
Below parallel coordinate plot depicts the characteristics of these unusual observations where Infant = 1(orange lines) and Non-Infant [Male/Female] = 2(blue lines).

Now if we just take a sample from our train data and plot using Parallel Coordinates, we can see that generally Height, Whole_weight and Viscera_weight are evenly distributed.

We also saw that removing unusual observations from the dataset lowers the test rmse further, however we did not remove those from our dataset as they appears to be valid assumptions and not an extreme cases.
We can see that the RMSE score of the Elastic Net and Random Forest are lower than our candidate models, however the difference is not very significant.

Model	RMSE.Train	RMSE.Test
Elastic Net	2.0501	2.1756
Random Forest	1.6416	2.1789

Discussion

Lets start the discussions with dataset that we are analyzing. From the initial summary statistics we see that there was very high collinearity present in the data. almost all of the variables had high vif values.
Another key important observations that we made was data quality. We found that there were some instances where the observations were not recorded properly. for example for 2 observations the height was 0 which is practically impossible. but when we look at the entire observations and other variables for the same, we did not find any unusual pattern. We realized that it could be a data entry error therefore we decided to keep those in our dataset for analysis. Below are the those 2 observations.

Sex	Length	Diameter	Height	Whole_weight	Shucked_weight	Viscera_weight	Shell_weight	Rings	Infant	weight.diff
I	86	68	0	85.6	41.3	17.2	23.0	8	I	4.1
I	63	46	0	26.8	11.5	5.7	70.1	6	I	-60.5

We also see that there were some observations that did not stand up to the logic. Predictor Whole_weight is supposed to be a linear function of predictors Shucked_weight, Viscera_weight, Shell_weight and addition unknown mass of blood/water lost during the shucking process. In other words sum of all other weight predictors should be equal or less than Whole_weight. But we found 153 such observations that violated this assumptions. When looked closely, we identified that there were no similarities or any patterns among the observations, so we concluded that this might simply be a data recording error and therefore these observations were not removed from the dataset. Below are those 153 observations.

Sex	Length	Diameter	Height	Whole_weight	Shucked_weight	Viscera_weight	Shell_weight	Rings	Infant	weight.diff
I	48	35	9	14.0	6.3	4.7	4.0	5	I	-1.0
I	41	30	11	8.4	5.1	3.0	2.4	5	I	-2.1
I	42	30	10	8.4	3.5	2.5	3.0	4	I	-0.6
I	78	59	19	40.6	17.5	9.0	15.0	7	I	-0.9
M	94	74	24	115.9	58.6	45.4	28.0	9	NI	-16.1
M	90	69	21	82.3	36.0	22.5	27.0	7	NI	-3.2

Next interesting trend that we identified during exploratory analysis was distribution of predictor Sex factor levels. Factor levels female and male were extremely similar with respect to all other predictors. Also overall data was also roughly evenly distributed among level value of female, male and infant. Below are the distribution between the levels and their distribution overall with respect to other predictors and response variable.

Sex	Number of observations
M	1527
F	1307
I	1342

In the above boxplots we can see that shape of distribution for female and male is extremely similar for all predictors.
Now lets take a look on its frequencies.

Again it is very similar. These outcome pointed us to another direction where we created a new variables in dataset by combining the female and male into one level names as non-infant. So now the new categorical variable had two levels infant and non-infant. In one of our candidate model we used this new categorical instead of original to see its effect. At the end this was our model of choice.
Next important discovery during the initial data analysis was presence of very high multicollinearity among predictors. And this makes sense as we previously discovered that Whole_weight predictor was somewhat dependent on other weight predictors. Length of abalone was extremely correlated to its Diameter. Below were the Variation inflation factor values for model coefficients from additive model.

##           SexI           SexM         Length       Diameter         Height 
##       1.982824       1.400405      42.874862      44.907429       3.114493 
##   Whole_weight Shucked_weight Viscera_weight   Shell_weight 
##     118.819537      29.579686      18.591058      22.182789

The constant variation and normality assumptions were violated for the initial model. Using various variance stabilizing techniques and transformations we were able to come up a good model. These selected candidate models had interaction terms used, polynomial transformations for predictors and log transformation for response variable.
Interestingly even after the presence of high multicollinearity, during the model selection procedure we ended up selecting the model with all 8 parameters. i.e. all the selection procedures were suggesting that lowest AIC comes when used all the predictors from the model. Though after the model selection when we used various response and predictors transformations, we were able to reduce the size of the model a bit while maintaining similar test rmse.
We saw that we had few unusual observations that does have slight impact on lowering the test rmse but since these observations do not represent significant change or shift the model coefficients, we didn't remove them from the dataset.
Talking about this analysis in the context of the data, we could see from the model that Length variable is probably not going to be a important factor for predicting the age of abalone. we can see that we did not select this in our model. This sort of make sense as Length will be more or less correlated with Diameter (i.e. more the length more the diameter) and since we have already taken diameter into our model, we don't need Length from the abalone dataset.
Another very interesting fact while predicting is that whole weight alone can not be used to predict the age. we know that shucking process dries up the blood and water from abalone which is weighted again, and combination of weight at various stages (peeling, shucking, drying) are also important factor to determine the age of Abalone.

Actual.no.of.Rings	Predicted.no.of.Rings	Actual.age.of.abalone	Predicted.age.of.abalone
11	10	12	12
10	11	12	12
12	11	14	12
20	14	22	16
7	7	8	8
9	8	10	10
8	10	10	12
6	6	8	8
10	11	12	12
10	11	12	12

We took 10 sample observations from test data and predicted the age using our selected model (abalone_int_infant) and on average it correctly predicted the age 3 out of 10 times. From the test rmse value of selected model (2.2021) considering range of out response variable rings (1 to 29), We think that our model is fitting really well for predicting the age of Abalone.

Appendix

Below are the custom functions that are used throughout our analysis for creating plots and various other calculations. This code is also mentioned in Methods section with echo set to FALSE.

#Function to calculate RMSE
rmse <-  function(actual, predicted) {
  sqrt(mean((actual - predicted) ^ 2))
}

#high leverage points
calculate_leverage <- function(model){
  (hatvalues(model) > 2 * mean(hatvalues(model)))
}

#large residuals
calculated_residuales <- function(model){
  (abs(rstandard(model)) > 2)
}

#influential points
calculate_influence <- function(model){
  sum(cooks.distance(model) > 4 / length(cooks.distance(model)))
}

#Function for boxplot
create_boxplot <- function(xvar,yvar,xtitle,ytitle){ 
  ggplot(abalone, aes(x=xvar,y=yvar, fill=Sex))+
  geom_boxplot() + 
  xlab(paste(xtitle)) +  
  ylab(paste(ytitle))
}

#function for density plots
create_den_plots <- function(xvar, title){
  ggplot(abalone, aes(x = xvar, fill = Sex)) + 
    geom_density(alpha = 0.8) + 
    ylab("Number of abalone") + 
    ggtitle(paste(title)) + 
    xlab("Sex") 
    #facet_wrap(~Sex, ncol = 3)
}

#variable added plot
create_varplot <- function(big_model, small_model){
plot(resid(big_model) ~ resid(small_model), 
     col = "dodgerblue", pch = 20,
     xlab = "Residuals, Added Predictor", 
     ylab = "Residuals, Original Model")
abline(h = 0, lty = 2)
abline(v = 0, lty = 2)
abline(lm(resid(big_model) ~ resid(small_model)),
       col = "darkorange", lwd = 2)
}

#Scatter plots 
create_scatter_plot <- function(xvar,yvar,xtitle,ytitle) {
  ggplot(abalone_train,aes(x=xvar,y=yvar, col=Sex)) + 
    geom_point() + 
    geom_jitter() + 
    xlab(paste(xtitle)) +  
    ylab(paste(ytitle))
}

#fitted vs residuals plots
create_fitted_residuals <- function(model,name){
plot(fitted(model), resid(model), col = "grey", pch = 20,
     xlab = "Fitted", ylab = "Residuals", main = paste("Fitted vs Residuals (",name,")"))
abline(h = 0, col = "darkorange", lwd = 2)
}

#normal q-q plot
create_qq_normal <- function(model,name){
  qqnorm(resid(model), main = paste("Normal Q-Q Plot (",name,")"), col = "darkgrey")
  qqline(resid(model), col = "dodgerblue", lwd = 2)
}


compare_rmse <- function(model,name){
  abalone_add_train_rmse <- rmse(abalone_train$Rings, predict(model,abalone_train))
  abalone_add_test_rmse <- rmse(abalone_test$Rings, predict(model,abalone_test))
  result <- data.frame('Model'=c(name),"RMSE Train"=c(abalone_add_train_rmse),"RMSE Test"=c(abalone_add_test_rmse))
}


log_rmse <- function(model,name){
  abalone_add_train_rmse  <- sqrt(mean((abalone_train$Rings - exp(fitted(model))) ^2))
  abalone_add_test_rmse <- sqrt(mean((abalone_test$Rings - exp(predict(model, newdata=abalone_test))) ^ 2))
  result <- data.frame('Model'=c(name),"RMSE Train"=c(abalone_add_train_rmse),"RMSE Test"=c(abalone_add_test_rmse))
}


get_log_rmse <- function(model){
  abalone_add_train_rmse <- sqrt(mean((abalone_train$Rings - exp(fitted(model))) ^2))
  abalone_add_test_rmse <- sqrt(mean((abalone_test$Rings - exp(predict(model, newdata=abalone_test))) ^ 2))
  data.frame(train=abalone_add_train_rmse,test=abalone_add_test_rmse)
}

get_log_rmse_obs <- function(model,train_data){
  abalone_add_train_rmse <- sqrt(mean((train_data$Rings - exp(fitted(model))) ^2))
  abalone_add_test_rmse <- sqrt(mean((abalone_test$Rings - exp(predict(model, newdata=abalone_test))) ^ 2))
  data.frame(train=abalone_add_train_rmse,test=abalone_add_test_rmse)
}

model_assumptions <- function(model,name){
  par(mfrow = c(1, 2),oma=c(0,0,0,0))
  create_fitted_residuals(model,name)
  create_qq_normal(model,name)
}

test_model <- function(degree){
    model <- lm(log(Rings) ~  Diameter+ Length + Height + poly(Whole_weight,degree) + poly(Viscera_weight,degree) + poly(Shucked_weight,degree) + poly(Shell_weight,degree) + Sex, data=abalone_train)
    model
}

test_int_model <- function(d){
  model <- lm(log(Rings) ~ Length + Height + Diameter + poly(Whole_weight, d) +
  poly(Viscera_weight, d) + poly(Shucked_weight,d) + poly(Shell_weight, d) + Sex + Diameter:poly(Shucked_weight, d) + poly(Shucked_weight,  d):Sex, data=abalone_train)
  model
}


find_leverage <- function(model){
  return (which(hatvalues(model) > 2 * mean(hatvalues(model))))
}

find_outliers <- function(model){
  which(abs(rstandard(model)) > 2)
}

find_influence <- function(model){
 which(cooks.distance(model)> 4 / length(cooks.distance(model)))
}

get_best_result <- function(caret_fit) {
  best_result <- caret_fit$results[as.numeric(rownames(caret_fit$bestTune)), ]
  rownames(best_result) <- NULL
  best_result
}

get_rmse <- function(model){
  abalone_add_train_rmse <- sqrt(mean((abalone_train$Rings - (fitted(model))) ^2))
  abalone_add_test_rmse <- sqrt(mean((abalone_test$Rings - (predict(model, newdata=abalone_test))) ^ 2))
  data.frame(train=abalone_add_train_rmse,test=abalone_add_test_rmse)
}

Outlier plots from result section

unusual_observations <- abalone_train[common_data,]

data <- data.frame(Infant=as.numeric(unusual_observations$Infant),weight.diff=unusual_observations$weight.diff,Length=unusual_observations$Length,Diameter=unusual_observations$Diameter,Whole_weight=unusual_observations$Whole_weight,Viscera_weight=unusual_observations$Viscera_weight,Shucked_weight=unusual_observations$Shucked_weight,Shell_weight=unusual_observations$Shell_weight,Height=unusual_observations$Height,Rings=unusual_observations$Rings)
plot_colors <- c("Darkorange", "Dodgerblue")

MASS::parcoord(data, col = plot_colors[data$Infant],lwd=2,var.label=TRUE)

set.seed(42)
indexes_sample <- sample(1:nrow(abalone_train), size=0.2*nrow(abalone_train))
abalone_sample <- abalone_train[indexes_sample,]

data_sample <- data.frame(Infant=as.numeric(abalone_sample$Infant),weight.diff=abalone_sample$weight.diff,Length=abalone_sample$Length,Diameter=abalone_sample$Diameter,Whole_weight=abalone_sample$Whole_weight,Viscera_weight=abalone_sample$Viscera_weight,Shucked_weight=abalone_sample$Shucked_weight,Shell_weight=abalone_sample$Shell_weight,Height=abalone_sample$Height,Rings=abalone_sample$Rings)

plot_colors <- c("Darkorange", "Dodgerblue")

MASS::parcoord(data_sample, col = plot_colors[data_sample$Infant],lwd=.5,var.label=TRUE)

Below code is used to create Model Size vs Test rmse

model_names <- c(
					"Candidate model 1 (abalone_model_int_bic)",
					"Candidate model 2 (abalone_int_full)",
					"Candidate model 3 (abalone_int_infant)"
				)

result <- data.frame(
          					  "Models" = model_names,
          					  "RMSE Train" = c( 
                      											 get_log_rmse(abalone_model_int_bic)$train,
                      											 get_log_rmse(abalone_int_full)$train,
                      											 get_log_rmse(abalone_int_infant)$train
          										             ),
          					   "RMSE Test" = c(
                    										get_log_rmse(abalone_model_int_bic)$test,
                    										get_log_rmse(abalone_int_full)$test,
                    										get_log_rmse(abalone_int_infant)$test
                    									)
									   )
					

kable(result, digits = 4,format = 'markdown')

rmse_abalone_add_small <- get_rmse(abalone_add_small)
rmse_abalone_add <- get_rmse(abalone_add)
rmse_abalone_add_log <- get_log_rmse(abalone_add_log)

rmse_abalone_model_int_bic <- get_log_rmse(abalone_model_int_bic)
rmse_abalone_int_full <- get_log_rmse(abalone_int_full)
rmse_abalone_int_infant <- get_log_rmse(abalone_int_infant)

train_rmse_plot <- c(rmse_abalone_add_small$train,rmse_abalone_add$train,rmse_abalone_add_log$train,rmse_abalone_model_int_bic$train,rmse_abalone_int_full$train,rmse_abalone_int_infant$train)

test_rmse_plot <- c(rmse_abalone_add_small$test,rmse_abalone_add$test,rmse_abalone_add_log$test,rmse_abalone_model_int_bic$test,rmse_abalone_int_full$test,rmse_abalone_int_infant$test)


model_length <- c(length(coef(abalone_add_small)),length(coef(abalone_add)),length(coef(abalone_add_log)),length(coef(abalone_model_int_bic)),length(coef(abalone_int_full)),length(coef(abalone_int_infant)))

namebank <- c("add small","add","add log","int bic","int full","int infant")

plot_frame <- data.frame(namebank,model_length,test_rmse_plot)

ggplot(plot_frame, aes(x = model_length, y = test_rmse_plot, label = namebank)) +
  geom_point() + 
  geom_point(data = plot_frame[6,], aes(x = model_length, y = test_rmse_plot), colour ="Red", size = 5 ) + 
  geom_line() + 
  scale_y_continuous(limits = c(2.19,2.37)) + 
  geom_text((aes(label=namebank, hjust = 0, vjust = 2))) + 
  theme(axis.title.x = element_text(colour = "Black"),axis.title.y = element_text(colour = "Black")) + 
  labs(title =  "Model Size vs Test RMSE",x = "Model Length", y = "Test RMSE" )

Name		Name	Last commit message	Last commit date
Latest commit History 60 Commits
Abalone_files/figure-html		Abalone_files/figure-html
Abalone.Rmd		Abalone.Rmd
Abalone.md		Abalone.md
Abalone.pdf		Abalone.pdf
Abalone_data.csv		Abalone_data.csv
README.md		README.md
_config.yml		_config.yml

Actual.no.of.Rings	Predicted.no.of.Rings	Actual.age.of.abalone	Predicted.age.of.abalone
11	10	12	12
10	11	12	12
12	11	14	12
20	14	22	16
7	7	8	8
9	8	10	10
8	10	10	12
6	6	8	8
10	11	12	12
10	11	12	12

Actual.no.of.Rings	Predicted.no.of.Rings	Actual.age.of.abalone	Predicted.age.of.abalone
11	10	12	12
10	11	12	12
12	11	14	12
20	14	22	16
7	7	8	8
9	8	10	10
8	10	10	12
6	6	8	8
10	11	12	12
10	11	12	12

nishitpatel01/predicting-age-of-abalone-using-regression

Folders and files

Latest commit

History

Repository files navigation

predicting-age-of-abalone-using-regression

Introduction

Dataset

Background

Description

Variable List

Interest

"Predicting the age of abalone using regression"

Methods

Data summary and statistics

Categorial Variables

Numeric Variables

Response variable

Data quality

Variance inflation factor of the additive model without the Whole_weight & Diameter

RMSE Score of abalone_add_small

Anova F Test

Additive Model Assumptions

Exhaustive search

Model Assumptions

RMSE Score

Polynomials

RMSE Score

RMSE Score

ANOVA F Test

RMSE Score

ANOVA F Test

RMSE Score

ANOVA F Test

RMSE Score

ANOVA F Test

AIC and BIC on poly additive model:

RMSE Score - AIC

RMSE Score - BIC

ANOVA F Test

AIC and BIC on Interaction model

RMSE Score - BIC/AIC

Infant Model Analysis

Additive Infant Models

RMSE Score

Additive Model Assumptions

Box-Cox Transformations

Additive Infant Model with log response transformations

RMSE Score

Additive Model Assumptions

Polynomial Model

RMSE Score

Polynomial Model Assumptions

Anova F Test

Interaction Model

RMSE Score

Interaction Model Assumptions

Anova F Test

Candidate models

RMSE comparison for candidate models

Model assumptions for candidate models

Unusual observations

Prediction and Confidence intervals

Predicting the age of abalone

Using advanced techniques

Results

Discussion

Appendix

About

Topics

Resources

Stars

Watchers

Forks

Releases

Packages 0

Packages

Actual.no.of.Rings	Predicted.no.of.Rings	Actual.age.of.abalone	Predicted.age.of.abalone
11	10	12	12
10	11	12	12
12	11	14	12
20	14	22	16
7	7	8	8
9	8	10	10
8	10	10	12
6	6	8	8
10	11	12	12
10	11	12	12