Need Report and PPT. No word pages length. But should be through. Also, you need to use the R Software. 

Final Project Instructions

Module 6: Collaborative group project Instructions

As the final assignment in this course, I want you all to be more creative, take some risks, and share a project. Therefore, please use at least one of the methods (i.e., Regression Analysis, Hypothesis Test, Regularization, Data, Classification/Data Mining, and Time Series Analysis) we have learned from the first five modules to analyze a real world data of interest to you with a goal of solving a problem you pose. 

Please come up with some research questions based on the data you chose and apply the analytical methods you have learned in this course, covered in Modules 1-5. Please prepare a written report and presentation based on your findings. The written report will typically include the following in

format

ion:

• a cover page with all the student names in your group, course/section, institution, instructor’s name, the assignment title, and the date.
• a section on the research questions
• a section on the data set(s) chosen
• a section on the method(s) chosen
• a section on results & findings
• a section on conclusions
• references (if any)
• appendix: your r script or your analyses document in any other format (excel, tableau, python, etc.)

The presentation should include similar information (research question, description of the data, summary of results, conclusions, and limitations), but be briefer. The presentation should be between 5-10 minutes.  For your presentation: don’t focus on math or statistical details. Do highlight the data storytelling and visualization part.  Your presentation order is shown in the section below. 

You shall work in a group of two or three students at most. Your project should follow APA format, however, i won’t be posing any page limitations but I expect thorough report. Please approach this as your chance to investigate something of interest to you. You should include images, graphs, figures, charts, and tables as necessary. 

If you have any questions, please contact me. However, do not wait until the last day of the assignment’s due date to email me with questions/concerns. Start working on it as soon as you can to have a higher chance of success.

The presentations will be given during our normal class time . The written report is due on Wednesday June 26 11:59 pm. Please list the contributions of all group members in your presentation and final report. 

I look forward to seeing your creative and analytical work. I know if you put in the effort, you can have some very fascinating results!

Some Data Sources: 

1) Regression

            i) Red Wine Quality

https://www.kaggle.com/uciml/red-wine-quality-cortez-et-al-2009

ii) House prediction (Advanced)

https://www.kaggle.com/c/house-prices-advanced-regression-techniques/data

2) Classification

i)Titanic dataset

https://www.kaggle.com/c/titanic/data

3) Clustering

            i) Credit card data

https://www.kaggle.com/arjunbhasin2013/ccdata

4) Time series analysis

            i) Bitcoin price forecasting

Final Project

ALY-6015 Week 6 Project

Intermediate Analytics

Submitted to:

Ani Aghababyan

College of Professional Studies

Northeastern University, MA

Submitted by:

Vikrant Kakad

Vikas Warudkar

Sunita Mohapatra

Darshan Shah

Akshay kannan

Academic Term Spring 2018 – Quarter 2

Introduction

Wine making is affected by a series of variables, when it is made. Several variables from

alcohol

, to pH can affect the final results. It is crucial to understand and learn how these variables impact the quality of red wine. The scope of this project work is to understand effect of various attributes which impact the quality of the Red wine. The data set utilized for the analysis is downloaded from UCI repository. The analysis has additional focus on the following key parameters:

pH value – pH value is considered to be a key parameter for the determination of quality of wine and hence the analysis focused on determining the impact of these pH values on final quality determination.

SO2 values (Free and Total) – SO2 has been always a debatable topic due to the allergic reactions associated with SO2.The current analysis tries to determine the impact of SO2 on pH values and the final quality values for the wine samples.

Alcohol content – Alcohol content is an important parameter considered when a buyer purchases any alcoholic product and this analysis tries to unravel relationship of Alcohol content with parameters like pH values and SO2 contents and the impact to quality.

In this project, we did the analysis of Red Wine Data and try to understand which variables are responsible for the quality of the wine. First, we got the feel of the variables on their own and then we found out the correlation between them and the wine quality with other factors thrown in. Finally, we created a linear model to predict the outcome of a test set data.

Proposing supervised learning approach to predict human wine taste preferences that is based on easily available analytical tests at the certification step. A large dataset (when compared to other studies in this domain) is considered, with red Vinho Verde samples from Portugal (CVRVV, 2008). Two regression techniques were applied, under a computationally efficient procedure that performs simultaneous variable and model selection. The support vector machine achieved promising results, outperforming the multiple regression and neural network methods. Such model is useful to support the oenologist wine tasting evaluations and improve wine production. Furthermore, similar techniques can help in target marketing by modeling consumer tastes from niche markets.

Research Question

By performing this analysis, we seek to answer the following questions:

1. How is the quality of the wines tasted?

2. What is the minimum set of properties and their values that defines a high-quality wine?

3. What are considered wine defects?

About dataset

· Name: Red Wine Quality Data Set

· Sources Created by: Paulo Cortez (Univ. Minho), Antonio Cerdeira, Fernando Almeida, Telmo Matos and Jose Reis (CVRVV, 2009)

· Input variables: 
1 – fixed acidity 
2 –

volatile acidity

 
3 –

citric acid

 
4 – residual sugar 
5 – chlorides 
6 – free sulfur dioxide 
7 – total sulfur dioxide 
8 – density 
9 – pH 
10 – sulphates 
11 – alcohol 

· Output variable: quality (score between 0 and 10)

· Data Set Characteristics: Multivariate

· Number of Observations: 1599

· Number of Attributes: 12

· Missing Values: N/A

Description of attributes:

1. Fixed acidity: Most acids involved with wine or fixed or nonvolatile (do not evaporate readily)

2. Volatile acidity: The amount of acetic acid in wine, which at too high of levels can lead to an unpleasant, vinegar taste

3. Citric acid: Found in small quantities, citric acid can add ‘freshness’ and flavor to wines

4. Residual sugar: The amount of sugar remaining after fermentation stops, it’s rare to find wines with less than 1 gram/liter and wines with greater than 45 grams/liter are considered sweet

5. Chlorides: The amount of salt in the wine

6. Free sulfur dioxide: The free form of SO2 exists in equilibrium between molecular SO2 (as a dissolved gas) and bisulfite ion; it prevents microbial growth and the oxidation of wine

7. Total sulfur dioxide: Amount of free and bound forms of S02; in low concentrations, SO2 is mostly undetectable in wine, but at free SO2 concentrations over 50 ppm, SO2 becomes evident in the nose and taste of wine

8. Density: The density of water is close to that of water depending on the percent alcohol and sugar content

9. pH: Describes how acidic or basic a wine is on a scale from 0 (very acidic) to 14 (very basic); most wines are between 3-4 on the pH scale

10. Sulphates: A wine additive which can contribute to sulfur dioxide gas (S02) levels, which acts as an antimicrobial and antioxidant

11. Alcohol: the percent alcohol content of the wine

12. Quality: output variable (based on sensory data, score between 0 and 10)

The dataset chosen has the following above attributes and it delivers a better result in detecting the quality after testing. The datatypes of the aforementioned attributes are as follows.

As described before, there are 1599 observations (rows) for 12 different variables (columns). Quality is type of ‘ordered, categorical, discrete’ variable, whose value ranges from 3-8.

A statistical description of the above dataset would provide a more coherent picture as to how the numerical values are distributed across the dataset (Range, Quartiles, Central Tendencies, etc). They are as follows:

The overall summary of the dataset covers all the above information, and presents the data in a concise & lucid way. They can be shown as follows:

Methods chosen:

Univariate Plot Analysis:

A univariate plot shows the data and summarizes its distribution. A dot plot, also known as a strip plot, shows the individual observations. A box plot shows the five-number summary of the data – the minimum, first quartile, median, third quartile, and maximum.

The graph analysis is as follows :-

Here, it can be observed that the density, pH value and wine quality appears to be normally distributed. Fixed, Volatile acidity & Sulphur dioxides, Sulphates and alcohol seems to be long tailed. Qualitatively, residual sugar and chlorides have extreme outliers. Citric acid appeared to have a large number of zero values. This might be a case of non-reporting.

Exploratory Data Analysis (EDA) and Data Pre-processing

Histograms to show the distribution of the variable values. As we could clearly see, citric acid was one feature that was found to be not normally distributed on a logarithmic scale.

Now, a combined variable namely “TAC.acidity” is created that constitutes the sum of Tartaric, acetic & citric acid. It is as follows :-

Boxplots for each of the variables as another indicator of spread.

Observations regarding variables: All variables have outliers

· Acidities like Citric acid, Volatile acidity and Fixed acidity data have critical outliers present. If these outliers are removed, then the distribution of these attributes can become symmetric.

· Positively Skewed Distribution is shown by the residual sugar in the wine, interesting fact here is that even if we ignore the outliers, this skewness remains unaffected.

· Attributes/variables like Density of wine, Free Sulphur Dioxide have significant outliers, but they are very different from the rest.

· Larger side of the data has most of the outliers.

· Irregular distribution is shown by the alcohol content of the red wine without any major outliers.

Support vector machines are a class of factual models initially created in the mid-1960s by Vladimir Vapnik. In later years, the model has advanced extensively into a standout amongst the most adaptable and powerful machine learning instruments accessible. It is a regulated learning calculation which can be utilized to tackle both characterization and relapse issue, even though the present spotlight is on grouping as it were. To place it, this calculation searches for a straightly distinguishable hyperplane, or a choice limit isolating individual from one class from the other. If such a hyperplane exists, the work is finished! If such a hyperplane does not exist, SVM utilizes a nonlinear mapping to change the preparation information into a higher measurement. At that point it scans for the straight ideal isolating hyperplane. With a fitting nonlinear mapping to an adequately high measurement, information from two classes can simply be isolated by a hyperplane. The SVM calculation discovers this hyperplane utilizing support vectors and edges. As a preparation calculation, SVM may not be quick contrasted with some other grouping techniques, however inferable from its capacity to display complex nonlinear limits, SVM has high precision. SVM is relatively less inclined to overfitting. SVM has effectively been connected to manually written digit acknowledgment, content arrangement, speaker distinguishing proof and so forth. The utilization of this procedure helped us to recognize the correct closer sum and incentive through relapses and definitions.

Results and Findings

A correlation of each variable has been made against the wine quality to determine those factors which comparatively have a better influence in the quality of wine. It was found that the top 4 variables that influence the wine quality are as follows :-

1) alcohol

2)

sulphates (log10)

3) volatile acidity

4) citric acid

The following was done to examine the acidity variables.

Of all the other factors, base 10 logarithm TAC.acidity correlated very well with Ph, and rightfully so, since pH is a defining measure of acidity.

An interesting question to pose, using basic chemistry knowledge, is to ask what other components other than the measured acids are affecting pH.

We can quantify this difference by building a predictive linear model,

to predict pH based off of TAC.acidity and

capture the % difference as a new variable.

Conclusion

By examining the above information, we could locate the administered learning strategy called bolster vector machine anticipated the essence of the red wine quality and gave us the outcome for more wine quality is specifically corresponding to the liquor content. Although alternate systems were in the same class as this above technique yet it helped us to discover the guess result and we could foresee the quality through the measure of liquor content. The use of this investigation can comprehend whether by adjusting the factors, it is conceivable to build the nature of the wine available. In the event that you can control your factors, at that point you can foresee the nature of your wine and acquire more benefits.

As observed, the direct model and the Support Vector Machine. The SVM performed imperceptibly better and we chose to stay with it on the off chance that we needed to make any more expectations. The use of this investigation, can comprehend whether by altering the factors amid wine making, it is conceivable to expand the nature of the wine available. In the event that you can control your factors, at that point you can anticipate the nature of your wine and acquire more benefits.

References

CVRVV. 2008. Portuguese Wine — Vinho Verde. Comissão de Viticultura da Região dos Vinhos Verdes (CVRVV), http://www.vinhoverde.pt.

P. Cortez, A. Cerdeira, F. Almeida, T. Matos and J. Reis. 2009. Modeling wine preferences by data mining from physicochemical properties. In Decision Support Systems, Elsevier, 47(4):547-553.

V. Cherkassy, Y. Ma. 2004. Practical selection of SVM parameters and noise estimation for SVM regression. Neural Networks, 17 (1), pp. 1

13

-126

Red Wine Quality. 2018. Kaggle, https://www.kaggle.com/uciml/red-wine-quality-cortez-et-al-2009/data

Appendix A: (R-Script)

Red Wine quality assessment

========================================================

“`

{

r echo=FALSE, message=FALSE, warning=FALSE

}

# install.packages(“MASS”)

# install.packages(“gridExtra”)

# install.packages(“grid”)

# install.packages(“ggplot2”)

# install.packages(“lattice”)

# install.packages(“dplyr”)

# install.packages(“memisc”)

# install.packages(“GGally”)

# install.packages(“reshape2”)

# install.packages(“kernlab”)

# install.packages(“plyr”)

# install.packages(“plotly”)

# install.packages(“e1071”)

require(MASS)

require(gridExtra)

library(grid)

library(ggplot2)

library(lattice)

require(dplyr)

require(memisc)

require(GGally)

require(reshape2)

require(kernlab)

update.packages(“ggplot2”)

library(plyr)

library(plotly)

library(skimr)

“`

Read csv file and explore statistics

“`{r echo=FALSE,message=FALSE,warning=FALSE}

Wine <- read.csv("https://archive.ics.uci.edu/ml/machine-learning-databases/wine-quality/winequality-red.csv", sep = ";")

str(Wine)

summary(Wine)

skim(Wine)

Wine$quality <- as.numeric(Wine$quality)

“`

Creates tabular results of categorical variables

“`{r,message=FALSE,warning=FALSE}

table(Wine$quality)

“`

# Univariate Plots Section

“`{r echo=FALSE,message=FALSE,warning=FALSE}

grid.arrange(qplot(Wine$fixed.acidity),

qplot(Wine$volatile.acidity),

qplot(

Wine$citric.acid

),

qplot(Wine$residual.sugar),

qplot(Wine$chlorides),

qplot(Wine$free.sulfur.dioxide),

qplot(Wine$total.sulfur.dioxide),

qplot(Wine$density),

qplot(Wine$pH),

qplot(Wine$sulphates),

qplot(Wine$alcohol),

qplot(Wine$quality),

ncol = 4)

“`

# Univariate Analysis

1. Wine Quality forms a normal distribution.

2. Density and pH are normally distributed with a few outliers.

Create new variable for better exploration

“`{r,message=FALSE,warning=FALSE}

Wine$rating <- ifelse(Wine$quality < 5, 'bad', ifelse(

Wine$quality < 7, 'average', 'good'))

Wine$rating <- ordered(Wine$rating,

levels = c(‘bad’, ‘average’, ‘good’))

summary(Wine$rating)

“`

Create Histogram of log function of the variables for further analysis

“`{r,message=FALSE,warning=FALSE}

ggplot(Wine,aes(x=fixed.acidity))+geom_histogram(fill=’red’)+scale_x_log10(breaks=4:15)+

xlab(‘Fixed Acidity’)+ylab(‘Count’)+ggtitle(‘Histogram of Fixed Acidity Values’)

require(plotly)

ggplot()

plot_ly(data=Wine,x=~citric.acid,type=’histogram’)

ggplot(Wine) +

geom_histogram(aes(x=volatile.acidity),fill=’blue’)+

scale_x_log10(breaks=seq(0.1,1,0.1))

ggplot(Wine) +

geom_histogram(aes(x=citric.acid),fill=’green’) +

scale_x_log10()

“`

Citric acid was one feature that was found to be not

normally distributed on a logarithmic scale.

Create a combined variable,

TAC.acidity, containing the sum of tartaric, acetic, and citric acid.

“`{r,message=FALSE,warning=FALSE}

Wine$TAC.acidity <- Wine$fixed.acidity + Wine$volatile.acidity +

Wine$citric.acid

qplot(Wine$TAC.acidity,main = ‘Histogram of TAC Acidity (fixed+volatile+Citric)’)

“`

## Boxplots are better suited in visualizing the outliers.

“`{r,message=FALSE,warning=FALSE}

get_simple_boxplot <- function(column, ylab) {

return(qplot(data = Wine, x = ‘simple’,

y = column, geom = ‘boxplot’,

xlab = ”,

ylab = ylab))

}

grid.arrange(get_simple_boxplot(Wine$fixed.acidity, ‘fixed acidity’),

get_simple_boxplot(Wine$volatile.acidity, ‘volatile acidity’),

get_simple_boxplot(Wine$citric.acid, ‘citric acid’),

get_simple_boxplot(Wine$TAC.acidity, ‘TAC acidity’),

get_simple_boxplot(Wine$residual.sugar, ‘residual sugar’),

get_simple_boxplot(Wine$chlorides, ‘chlorides’),

get_simple_boxplot(Wine$free.sulfur.dioxide, ‘free sulf. dioxide’),

get_simple_boxplot(Wine$total.sulfur.dioxide, ‘total sulf. dioxide’),

get_simple_boxplot(Wine$density, ‘density’),

get_simple_boxplot(Wine$pH, ‘pH’),

get_simple_boxplot(Wine$sulphates, ‘sulphates’),

get_simple_boxplot(Wine$alcohol, ‘alcohol’),

ncol = 4)

plot_ly(Wine,y=~alcohol,type=’box’)

“`

# Bivariate Plots Section

“`{r echo=FALSE,message=FALSE,warning=FALSE}

get_bivariate_boxplot <- function(x, y, ylab) {

return(qplot(data = Wine, x = x, y = y, geom = ‘boxplot’, ylab = ylab))

}

grid.arrange(get_bivariate_boxplot(Wine$quality, Wine$fixed.acidity,

‘fixed acidity’),

get_bivariate_boxplot(Wine$quality, Wine$volatile.acidity,

‘volatile acidity’),

get_bivariate_boxplot(Wine$quality, Wine$citric.acid,

‘citric acid’),

get_bivariate_boxplot(Wine$quality, Wine$TAC.acidity,

‘TAC acidity’),

get_bivariate_boxplot(Wine$quality, log10(Wine$residual.sugar),

‘residual sugar’),

get_bivariate_boxplot(Wine$quality, log10(Wine$chlorides),

‘chlorides’),

get_bivariate_boxplot(Wine$quality, Wine$free.sulfur.dioxide,

‘free sulf. dioxide’),

get_bivariate_boxplot(Wine$quality, Wine$total.sulfur.dioxide,

‘total sulf. dioxide’),

get_bivariate_boxplot(Wine$quality, Wine$density,

‘density’),

get_bivariate_boxplot(Wine$quality, Wine$pH,

‘pH’),

get_bivariate_boxplot(Wine$quality, log10(Wine$sulphates),

‘sulphates’),

get_bivariate_boxplot(Wine$quality, Wine$alcohol,

‘alcohol’),

ncol = 4)
“`

Correlation for each of these

variables against quality:

“`{r,message=FALSE,warning=FALSE}

simple_cor_test <- function(x, y) {

return(cor.test(x, as.numeric(y))$estimate)

}

correlations

<- c(

simple_cor_test(Wine$fixed.acidity, Wine$quality),

simple_cor_test(Wine$volatile.acidity, Wine$quality),

simple_cor_test(Wine$citric.acid, Wine$quality),

simple_cor_test(Wine$TAC.acidity, Wine$quality),

simple_cor_test(log10(Wine$residual.sugar), Wine$quality),

simple_cor_test(log10(Wine$chlorides), Wine$quality),

simple_cor_test(Wine$free.sulfur.dioxide, Wine$quality),

simple_cor_test(Wine$total.sulfur.dioxide, Wine$quality),

simple_cor_test(Wine$density, Wine$quality),

simple_cor_test(Wine$pH, Wine$quality),

simple_cor_test(log10(Wine$sulphates), Wine$quality),

simple_cor_test(Wine$alcohol, Wine$quality))

correlations

names(correlations) <- c('fixed.acidity', 'volatile.acidity', 'citric.acid',

‘TAC.acidity’, ‘log10.residual.sugar’,

‘log10.chlordies’, ‘free.sulfur.dioxide’,

‘total.sulfur.dioxide’, ‘density’, ‘pH’,

‘log10.sulphates’, ‘alcohol’)

correlations
“`

Top 4:

alcohol
sulphates (log10)
volatile acidity
citric acid

Examining the acidity variables:

“`{r,message=FALSE,warning=FALSE}

ggplot(data = Wine, aes(x = fixed.acidity, y = citric.acid)) +

geom_point(alpha=0.3)

cor.test(Wine$fixed.acidity, Wine$citric.acid)

ggplot(data = Wine, aes(x = volatile.acidity, y = citric.acid)) +

geom_point(alpha=0.3)

cor.test(Wine$volatile.acidity, Wine$citric.acid)

ggplot(data = Wine, aes(x = log10(TAC.acidity), y = pH)) +

geom_point(alpha=0.3)

cor.test(log10(Wine$TAC.acidity), Wine$pH)

“`

Base 10 logarithm TAC.acidity correlated very well with pH.

Building a predictive linear model,

to predict pH based off of TAC.acidity and
capture the % difference as a new variable.
“`{r,message=FALSE,warning=FALSE}

m <- lm(I(pH) ~ I(log10(TAC.acidity)), data = Wine)

Wine$pH.predictions <- predict(m, Wine)

# (observed – expected) / expected

Wine$pH.error <- (Wine$pH.predictions - Wine$pH)/Wine$pH

“`

To check its accuracy.

The RMS Error.

“`{r,message=FALSE,warning=FALSE}

rmse <- function(error)

{

sqrt(mean(error^2))

}

rmse(m$residuals)

#Now, we train a Support Vector Machine.

require(e1071)

SVM <- svm(I(pH) ~ I(log10(TAC.acidity)), data = Wine)

Wine$pH.Predict.SVM <- predict(SVM,Wine)

Wine$pH.error.SVM <- (Wine$pH.Predict.SVM - Wine$pH)/Wine$pH

rmse(SVM$residuals)

“`

SVM functions slightly better than a LM.

### Plot 1: Effect of Alcohol on Wine Quality

“`{r echo=FALSE,message=FALSE,warning=FALSE}

ggplot(data = Wine, aes(x = quality, y = alcohol,

fill = rating)) +

geom_boxplot(outlier.color = ‘red’) +

ggtitle(‘Alcohol Levels in Different Wine Qualities’) +

xlab(‘Quality’) +

ylab(‘Alcohol (% volume)’)

“`

### Description 1

These boxplots demonstrate the effect of alcohol content on wine quality.

Generally, higher alcohol content correlated with higher wine quality.

However, as the outliers and intervals show, alchol content alone did not

produce a higher quality.

13

Assignment- 03

Model Building, Selection, & Prediction

Question 1:

1. Predicting the Output Variable Y – Energy Production Prediction

a) Importing the data from CSV data and splitting into test and training data:

Using the read.csv() function we can import the data into R

INPUT:

OUTPUT:

INPUT:

OUTPUT:

b) Fitting a Linear Regression Model:

Running the Linear Regression Model with all the Variables

INPUT:

OUTPUT:

The Adjusted R-Squared value is found to be 0.

2

366.

From the data It can seen that Pressure and Wind are only significant.

So, we run the model only with wind and pressure variables.

Reduced Regression Model (Wind and Pressures Variable only)

INPUT:

OUTPUT:

Removing the Wind Variable since the Adjusted R Squared Value is only 0.0229. Now we run the regression using only the Pressure Variable.

Running the Regression model with only Wind Variable:

INPUT:

OUTPUT:

The Adjusted R-Squared value is found to be 0.219, which is less than the previous regression models.

ANOVA test is to be conducted to find the significance of the all variable included model and the reduced pressure variable model.

INPUT:

OUTPUT:

Between the All variable and Reduced model, the P value is found to be 0.2578, so we should not reject the Null hypothesis and use the Reduced Model.

Between the Pressure variable and Reduced model, the P value is found to be 0.0768, so we should not reject the Null hypothesis and use the Pressure Model.

Running Best Subset to find the model:

Best Subset find the value of statistics for all variables involved and print the statistics for comparison, using which we can select the appropriate variable

INPUT:

OUTPUT:

RSS Value decrease as the variable increase.

Model with 5 variable has the highest Adjusted R Square.

Model with 3 variable has the smallest AIC (or Cp).

Model with 8 variable has the smallest BIC.

Since the Bestsubset approach provides a broad result we check the predicted R square and use the model with highest R square and lower RMSE

R square and RMSE Prediction:

For all variable considered Model:

INPUT:

OUTPUT:

For the Reduced Model with Pressure and Wind Variables:

INPUT:

OUTPUT:

Single Model with Pressure as the dependent variable:

INPUT:

OUTPUT:

Summary:

From the Analysis we can conclude that model with the pressure as the dependent variable is better than the other models. The Adjusted R square value of 0.31 is the best and the RMSE value is also the least in case of the pressur model.

From the Adjusted R Squared value we conclude that the pressure model is the best and can predict the energy produced rate accurately for 31% of the data.

c) Backward Selection Approach:

Regression Model using all the variables:

INPUT:

OUTPUT:

Conclusion:

The backward step AIC function tells a slightly different result then the models generated above. However, when we create the regression model we see a low R2 value then our single model. Below, we can compare all the 3 models above with this step model.

2