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Objective:

  1. Demonstrate knowledge of data exploration and selection of variables to apply for the predictive models
  2. Demonstrate knowledge of building different types of predictive models using R
  3. Demonstrate knowledge on comparing and evaluating different predictive models
  4. Relate theoretical knowledge of predictive models and best practices to application scenarios

Use the data for breakfast cereals to answer the following

  1. Which variables are continuous/numerical? Which are ordinal? Which are nominal?
  1. Calculate following summary statistics: mean, median, max and standard deviation for each of the continuous variables, and count for each categorical variable. Is there any evidence of extreme values? Briefly discuss.
  1. Plot histograms for each of the continuous variables and create summary statistics. Based on the histogram and summary statistics answer the following and provide brief explanations:
  1. Which variables have the largest variability?
  2. Which variables seems skewed?
  3. Are there any values that seem extreme?
  1. Which, if any, of the variables have missing values?
  1. What are the methods of handling missing values?
  2. Demonstrate the output (summary statistics and transformation plot) for each method in (4-a).
  3. Apply the 3 methods of missing value handling discussed in the lectures. Which method of handling missing values is most suitable for this data set? Discuss briefly referring to the data set.

Alpha Traders Pty Ltd. is an Australian car sales company has purchased a stock of used Toyota Corolla cars for sale. The management of the Alpha is in the process of finalizing the selling prices of the purchased cars. Alpha Traders management is very keen to trial predictive modelling for this task and have obtained a historic car sales dataset of Toyota Corolla cars from a publicly available data repository.

The dataset contains 37 attributes of over 1400 sold Toyota Corolla cars. The attributes include the selling price of cars, age, kilometres driven, fuel type, horsepower, automatic or manual, number of doors, weight (in pounds), etc.

The management of Alpha Traders Pty Ltd. has outsourced the task to you to develop a reliable predictive model to predict the selling price of the cars, using the aforementioned historic dataset.

  1. Examine the prices of the Toyota Corolla vehicles. Explain the distribution of the prices.
  2. Find out whether there are any missing values. Explain your findings.
  3. Are there any categorical values that needs to be transformed into numerical values? Suggest the best possible transformation. Use this method to transform the variable(s).
  4. Evaluate the correlations between the variables. Which variables should be used for dimension reduction? Explain. Carry out dimensionality reduction.
  5. Explore the distribution of selected variables (from step 1-d) against the target variable. Explain.
  1. Build a regression model with the selected variables. You need to try out at least 3 regression models to identify the optimal model.
  2. Evaluate the accuracy of the regression model.
  1. Build a decision tree with the selected variables. You need to try out at least 3 decision trees with different complexity parameters to obtain the optimal tree.
  2. Explain the output of the selected decision tree, evaluate the accuracy and reason for it to be selected.
  1. Compare the accuracy of the selected (optimal) regression model and (optimal) decision tree and discuss and justify the most suitable predictive model for the business case.

Part A:  data Exploration and Cleaning

The first part of the research is focused on the data exploration and the cleaning of the given data. For this part the cereals (for breakfast) data has been used. This data set contains 76 data points and 18 different features. The results from the data exploration

  1. The list of the continuous variables are shown in the table below.

##  [1] "Calories"        "Protein"         "Fat"            

##  [4] "Sodium"          "Fiber"           "Complex.Carbos"

##  [7] "Tot.Carbo"       "Sugars"          "Calories.fr.Fat"

## [10] "Potassium"       "Enriched"        "Wt.serving"     

## [13] "cups.serv"

pTable 1 list of the numerical  variables

Categorical variables are shown in the table below. The categorical variables are those variables where there are more than one category. There are two types of the categorical variables. The first one is the nominal variable where there is no particular order of the category. Secondly, the other type of categorical variable is the ordinal variable, where the category have particular order. In the current case among all the categorical variables, only Fibre.gr is the ordinal variable. All other variables are the nominal variable.

# Ordinal: Fiber.Gr

# Nominal: Name, Manufacturer, Mfr, Hot.Cold

Table 2 Ordinal and the nominal variables

  1. The summary statistics for the continuous variables has been calculated and for the calculation purpose various measures of the central tendencies such as mean and median has been included. The results are shown in the table below.

Calories        Protein          Fat            Sodium     

##  C:73     Min.   : 50.0   Min.   :1.00   Min.   :0.000   Min.   :  0.0  

##  H: 3     1st Qu.:110.0   1st Qu.:2.00   1st Qu.:0.500   1st Qu.:147.5  

##           Median :120.0   Median :3.00   Median :1.000   Median :210.0  

##           Mean   :140.5   Mean   :3.25   Mean   :1.447   Mean   :194.9  

##           3rd Qu.:192.5   3rd Qu.:5.00   3rd Qu.:2.000   3rd Qu.:262.5  

##           Max.   :250.0   Max.   :7.00   Max.   :9.000   Max.   :420.0 

Fiber        Complex.Carbos    Tot.Carbo         Sugars      

##  Min.   : 0.000   Min.   : 7.00   Min.   :11.00   Min.   : 0.000  

##  1st Qu.: 1.000   1st Qu.:13.00   1st Qu.:24.00   1st Qu.: 4.000  

##  Median : 3.000   Median :17.50   Median :27.00   Median :11.000  

##  Mean   : 3.066   Mean   :19.16   Mean   :31.37   Mean   : 9.145  

##  3rd Qu.: 5.000   3rd Qu.:26.00   3rd Qu.:41.00   3rd Qu.:14.000  

##  Max.   :13.000   Max.   :38.00   Max.   :50.00   Max.   :20.000

Calories.fr.Fat   Potassium        Enriched        Wt.serving   

##  Min.   : 0.00   Min.   :  0.0   Min.   :  0.00   Min.   :12.00  

##  1st Qu.: 5.00   1st Qu.: 35.0   1st Qu.: 25.00   1st Qu.:30.00  

##  Median :10.00   Median : 92.5   Median : 25.00   Median :30.00  

##  Mean   :12.37   Mean   :122.0   Mean   : 28.62   Mean   :36.65  

##  3rd Qu.:20.00   3rd Qu.:200.0   3rd Qu.: 25.00   3rd Qu.:49.00  

##  Max.   :50.00   Max.   :390.0   Max.   :100.00   Max.   :60.00  

##                                                   NA's   :13  

cups.serv         Fiber.Gr

##  Min.   :0.3300    Low   :33  

##  1st Qu.:0.7500    Medium:32  

##  Median :1.0000   High   :11  

##  Mean   :0.8911               

##  3rd Qu.:1.0000               

##  Max.   :1.3300       

Table 3 Table for the summary statistics

The summary statistics of the continuous variable is shown in the table above.  On the basis of the results, the variable which has extreme values is the potassium. In case of potassium the mean value is 194 whereas the minimum and the maxim value are 0 and 420 respectively. Similarly the range of calories also lies between as low as 50 to as high as 250.

Numerical variables

100% Bran                          : 1   American Home: 1    A: 1  

##  100% Nat. Bran Oats & Honey        : 1   General Mills:25    G:25  

##  100% Nat. Low Fat Granola w raisins: 1   Kelloggs     :23    K:23  

##  All-Bran                           : 1   Nabisco      : 5    N: 5  

##  All-Bran with Extra Fiber          : 1   Post         :10    P:10  

##  Almond Crunch w Raisins            : 1   Quaker Oats  :12    Q:12  

##  (Other)                            :70         

Fibre.Gr

# Low   :33  

## Medium:32  

##High   :11                            

In terms of the categorical variable, one of the variable hot and cold shows higher variation. There are 73 cold cereals whereas the number of hot cereals is only 3.

Histogram of the variables (continuous) is shown  below.

On the basis of the results from the histograms it can be concluded that:

Following findings has been found on the basis of the results from the histogram:

  1. The highest variability among the countinous variable is shown in the variables Enriched, Cups.serv and Fat. This is because the data points are more scattered in tails rather than concentrating around the mean value.
  2. The variables which shows highly skewed graphs are the Calories, Fiber , protein and the Fat. These histograms are either skewed towards the left tail or the right tail.
  3. In terms of the extreme values, Calories.fr.fate and Enriched are the variables with some outliers.

varlist nmiss complete complete_per        mean median minimum

## 1         Calories     0       76    100.00000 140.5263158  120.0   50.00

## 2          Protein     0       76    100.00000   3.2500000    3.0    1.00

## 3              Fat     0       76    100.00000   1.4473684    1.0    0.00

## 4           Sodium     0       76    100.00000 194.8684211  210.0    0.00

## 5            Fiber     0       76    100.00000   3.0657895    3.0    0.00

## 6   Complex.Carbos     0       76    100.00000  19.1578947   17.5    7.00

## 7        Tot.Carbo     0       76    100.00000  31.3684211   27.0   11.00

## 8           Sugars     0       76    100.00000   9.1447368   11.0    0.00

## 9  Calories.fr.Fat     0       76    100.00000  12.3684211   10.0    0.00

## 10       Potassium     0       76    100.00000 121.9736842   92.5    0.00

## 11        Enriched     0       76    100.00000  28.6184211   25.0    0.00

## 12      Wt.serving    13       63     82.89474  36.6507937   30.0   12.00

## 13       cups.serv     0       76    100.00000   0.8910526    1.0    0.33

In the current case only Wt. serving has missing values where 13 data points are missing. To handle the missing data 3 different methods has been used namely the mean value imputation, median value imputation and the mode value imputation. In mean value imputation the missing values are replaced by the mean value of the series. Similarly in the median and the mode value imputation the median and the mode values are replaced in place of the missing values.

varlist nmiss complete complete_per        mean median minimum

Wt.serving    13       63     82.89474  36.6507937   30.0   12.00

The transformation of the mean value imputation in the wt.serving is shown in the figure above. This shows that with the mean value imputation, now more values lies around the mean. However there are still some extreme values in the data set.

After the data exploration in the first section, the second section deals with the model building. In this case using the car sales data of Toyota Corolla, the prediction model has been developed. The data set contains the data for 37 different features for 1400 cars sold Australia.

  • Data Exploration and Cleaning
  1. To examine the price distribution of the car, the histogram has been used and the result from the histogram plot is shown in the following figure. Results shows that price for most of the cars is between 5000 and 10000. Also there are cars whose price is more than 30000.

Min. 1st Qu.  Median    Mean 3rd Qu.    Max.

4350    8450    9900   10731   11950   32500

Furthermore the results from the descriptive statistics indicates that the average price of the Toyota Corolla car is 10731. As discussed the price range from as low as 4350 to as high as 32500. This also indicates that there are some outliers in the data set.

  1. Checking for the missing values

Categorical variables

Results from the analysis indicates there is no case of missing values in the current data set.

       varlist nmiss complete complete_per

## 1                Id     0     1436          100

## 2             Model     0     1436          100

## 3             Price     0     1436          100

## 4         Age_08_04     0     1436          100

## 5         Mfg_Month     0     1436          100

## 6          Mfg_Year     0     1436          100

## 7                KM     0     1436          100

## 8         Fuel_Type     0     1436          100

## 9                HP     0     1436          100

## 10        Met_Color     0     1436          100

## 11        Automatic     0     1436          100

## 12               cc     0     1436          100

## 13            Doors     0     1436          100

## 14        Cylinders     0     1436          100

## 15            Gears     0     1436          100

## 16    Quarterly_Tax     0     1436          100

## 17           Weight     0     1436          100

## 18    Mfr_Guarantee     0     1436          100

## 19  BOVAG_Guarantee     0     1436          100

## 20 Guarantee_Period     0     1436          100

## 21              ABS     0     1436          100

## 22         Airbag_1     0     1436          100

## 23         Airbag_2     0     1436          100

## 24            Airco     0     1436          100

## 25  Automatic_airco     0     1436          100

## 26    Boardcomputer     0     1436          100

## 27        CD_Player     0     1436          100

## 28     Central_Lock     0     1436          100

## 29  Powered_Windows     0     1436          100

## 30   Power_Steering     0     1436          100

## 31            Radio     0     1436          100

## 32        Mistlamps     0     1436          100

## 33      Sport_Model     0     1436          100

## 34 Backseat_Divider     0     1436          100

## 35     Metallic_Rim     0     1436          100

## 36   Radio_cassette     0     1436          100

## 37          Tow_Bar     0     1436          100

  1. Since the R software is being used for the analysis, there is no need to separately convert the categorical variables into the numerical as the software itself create dummy variables for it.
  2. Correlation Analysis

Since there are 37 variables in the data set, the correlation coefficients have been included for only those variables which have high correlation coefficient. The correlation coefficient which are higher than 0.8 and less than 1 has been calculated. This will eliminate the correlation coefficients which are less. This dimension reduction has shown that manufactured year and the age of the car are highly correlated with the price of the car. So, it is better to drop one variable. Since the coefficient of manufacturing year is higher with price, the age has been dropped. Furthermore among the categorical variables central lock and the power windows shows high correlation with the price. While comparing the coefficient, it has been shown that coefficient between price and powered windows is higher as compared to the coefficient between price and central lock, so powered windows has been selected.

  • Regression Modelling

Regression analysis is used to predict the response variables using the explanatory variables. In the current case, the price of the cars is the response of the dependent variable and all other features as the explanatory variables.

The first model is as follows;

Residuals:

##     Min      1Q  Median      3Q     Max

## -7904.0  -718.8     2.5   728.3  6027.0

##

## Coefficients:

##                   Estimate Std. Error t value Pr(>|t|)    

## (Intercept)      1.058e+03  9.350e+02   1.132  0.25788    

## Age_08_04       -1.169e+02  3.363e+00 -34.776  < 2e-16 ***

## Boardcomputer   -2.363e+02  1.050e+02  -2.251  0.02455 *  

## Automatic_airco  2.611e+03  1.658e+02  15.744  < 2e-16 ***

## Weight           1.410e+01  7.735e-01  18.227  < 2e-16 ***

## KM              -1.799e-02  1.103e-03 -16.311  < 2e-16 ***

Summary statistics

## CD_Player        2.784e+02  9.307e+01   2.991  0.00283 **

## Powered_Windows  4.329e+02  7.022e+01   6.165 9.18e-10 ***

## HP               2.086e+01  2.396e+00   8.708  < 2e-16 ***

## ABS             -2.111e+02  9.186e+01  -2.298  0.02168 *  

## ---

## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##

## Residual standard error: 1222 on 1426 degrees of freedom

## Multiple R-squared:  0.8871, Adjusted R-squared:  0.8864

## F-statistic:  1245 on 9 and 1426 DF,  p-value: < 2.2e-16

In this model the R squared is 0.88 which shows that 88 % of the variation is being explained by he explanatory variables in the model. In the next model some of the variables which were not significant in this model were removed, such as the central lock. Similarly in the second model the variables which were not highly significant were removed. Finally the optimal model has been found in the third time and the results are shown in the table below. In this case all the coefficient are highly significant and the R squared is also high. So this is considered as the optimal model(Bai & Ng, 2009).

Coefficients:

##                   Estimate Std. Error t value Pr(>|t|)    

## (Intercept)      1.058e+03  9.350e+02   1.132  0.25788    

## Age_08_04       -1.169e+02  3.363e+00 -34.776  < 2e-16 ***

## Boardcomputer   -2.363e+02  1.050e+02  -2.251  0.02455 *  

## Automatic_airco  2.611e+03  1.658e+02  15.744  < 2e-16 ***

## Weight           1.410e+01  7.735e-01  18.227  < 2e-16 ***

## KM              -1.799e-02  1.103e-03 -16.311  < 2e-16 ***

## CD_Player        2.784e+02  9.307e+01   2.991  0.00283 **

## Powered_Windows  4.329e+02  7.022e+01   6.165 9.18e-10 ***

## HP               2.086e+01  2.396e+00   8.708  < 2e-16 ***

## ABS             -2.111e+02  9.186e+01  -2.298  0.02168 *  

## ---

## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##

## Residual standard error: 1222 on 1426 degrees of freedom

## Multiple R-squared:  0.8871, Adjusted R-squared:  0.8864

## F-statistic:  1245 on 9 and 1426 DF,  p-value: < 2.2e-16

  1. b) Evaluating the accuracy of the regression model

The optimal regression model can be evaluated on the basis of R squared, multicollinariy and the statistical significance of the coefficients. In the current case the R squared is decently higher at 0.88. The Vif test indicates there is no problem of multicollinerity and the regression coefficients are also statistically significant(Armstrong, 2012; Dufour & Dagenais, 1985; Lanfranchi, Viola, & Nascimento, 2010).

  • Decision tree

Results from the decision tree are discussed in the current section.

As the figure shows the error do not decrease after the 100 trees. This is because the line is flat after the 100 trees. The random forest package has been used for the decision tree. In this case four different decision tree models has been run and the results shows that the after 4 iteration, the variance do not increase, even when the trees are increased. So the fourth model is the optimal model and the results are as follows:

##                     %IncMSE IncNodePurity

## Age_08_04       13967550.66   11424442205

## Boardcomputer     523680.86    1820432381

## Automatic_airco   279111.44    1186671614

## Weight           1218683.98    1446642212

## KM               1053264.60    2015571025

## CD_Player          50608.29     113664776

## Powered_Windows   200498.28     140180060

## HP                389816.37     486749793

## ABS                11802.00      46241351

  1. b) One of most popular method of decision tree is random forest. In some cases the decision tree fit multiple lines in the data and predict the data. So, in some cases there is a problem of overfittting while running the model on training data. To solve this problem the random forest technique is used, as it builds multiple trees on the basis of the given data and give the average prediction. This method is considered to be less biased(Fokin & Hagrot, 2016).
  • Comparison of the models

On the basis of the results from the both the regression analysis and the decision tree it can be concluded that the decision tree is able to explain higher variance (91%) as compared to regression ( where R squared is 0.88). However impact of each explanatory variable on the dependent variable is missing from the random forest. So, from business point of view the regression model is appropriate as the variables can be clearly identified.

References

Armstrong, J. S. (2012). Illusions in regression analysis. International Journal of Forecasting, 6, 689–694.

Bai, J., & Ng, S. (2009). Tests for Skewness, Kurtosis, and Normality for Time Series Data. Journal of Business & Economic Statistics, 23(1), 49–60. https://doi.org/10.1198/073500104000000271

Dufour, J. M., & Dagenais, M. G. (1985). Durbin-Watson tests for serial correlation in regressions with missing observations. Journal of Econometrics, 27(3), 371–381. https://doi.org/10.1016/0304-4076(85)90012-0

Fokin, D., & Hagrot, J. (2016). Constructing decision trees for user behavior prediction in the online consumer market. Kth Royal Institute of technology.

Lanfranchi, L. M. M. M., Viola, G. R., & Nascimento, L. F. C. (2010). The use of Cox regression to estimate the risk factors of neonatal death in a private NICU. Taubate.

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