BMW Pricing Challenge
Adetya Aggarwal
19/08/2020
Assignment Description
Select a small data set from the available public data sets (you can find a list of public data sets here http://www.teymourian.de/public-data-sets-for-data-analytic-projects/ ).
Describe a research scenario and specify a research question based on data analytic methods that we learned in our class, for example methods like, one and two sample means, t-test, correlation tests, simple and multiple linear regression, ANOVA and ANCOVA, one and two-Sample Tests for Proportions and logistic regression.
Clean up your data and reduce it to no more than 500 observations if your data set is large.
Response - The Dataset selected for this Project is ‘BMW Pricing Challenge’. it is available on Kaggle.com and has been sampled to select 500 rows from roughly from 5000.
Research Scenario Description (no more than 200 words)
Describe your research scenario in no more than 200 words. This is a general description of the use case. Similar to our class examples, we first describe the overall scenario and then we specify a specific research question based on it.
Response - In 2019 alone, about 41 million used cars were sold in the US. Evaluation of the value of a used car is one of the primary challenges faced by the automotive companies and dealers around the world. Apart from the physical aspects like the color, model, transmission type etc., it is also a derivation of other factors like prestige value and perception of the quality of the maker.
In this project, we try to model the price, using the data analytic method of Multiple Linear Regression. We try to find out which aspects of a BMW car affect its price and which do not and we try to gauge how influential each factor is. In the end, we formulate and test the accuracy of model that can predict the price of the next car based on the relevant factors discovered during analysis.
Describe the data set (no more than 200 words)
Describe briefly the data set. Describe each columns of the data set if you use the column in your analysis. Clean up your data before usage, for example you can remove the outliers. Remove unused columns. If possible provide a Link to the main data set source.
Response - The dataset consists of the used sale prices of certain BMW car. It has roughly 5000 rows and 18 columns.
The first column is the maker of the car, as this data relates to only BMW; all the cars are of make BMW.
The second column relates to the model name of the car assigned by BMW. There are 75 unique models in in the dataset
Column 3 and 4 relate to the total mileage and Engine power of each car respectively.
Column 5 denotes the registration date of the car when the first owner bought it.
Column 6 provides the information regarding the fuel used in each car. There are four different types of fuels used in the cars of this dataset, namely - Diesel, Petrol, Hybrid Petrol and Electro. For this analysis, we are going to only focus on Diesel and Petrol cars as the no. of cars who use Hybrid Petrol and Electro are minute in comparison with Petrol and Diesel cars and hence treated like outliers and removed from the dataset.
Column 7 states the color of the car and Column 8 the type. There are a total of ten different colored cars in the dataset and of eight different types namely - Convertible, Coupe, Estate, Hatchback, Sedan, Subcompact, SUV and Van.
Columns nine to sixteen provide the information regarding the presence of (or lack of) certain features in each car. The data provider does not provide the information regarding description of those features.
Column 17 is the final sale price of the car. This is the most important variable for us in the data. We will try to model this variable based on other variables.
Column 18 is the final date on which the car was sold to the second owner.
The original dataset can be found at - https://www.kaggle.com/danielkyrka/bmw-pricing-challenge
Research Question (no more than 100 words)
Describe briefly in one or two sentences the main research question. This is similar to the last sentence of our class examples.
Can we quantify the physical and other features of pre-owned vehicles to generate a model of the price of a car which can we used to predict the price of the other similar natured cars?
Your solution R code
## 'data.frame': 4843 obs. of 18 variables:
## $ maker_key : chr "BMW" "BMW" "BMW" "BMW" ...
## $ model_key : chr "118" "M4" "320" "420" ...
## $ mileage : int 140411 13929 183297 128035 97097 152352 205219 115560 123886 139541 ...
## $ engine_power : int 100 317 120 135 160 225 145 105 125 135 ...
## $ registration_date: chr "2012-02-01" "2016-04-01" "2012-04-01" "2014-07-01" ...
## $ fuel : chr "diesel" "petrol" "diesel" "diesel" ...
## $ paint_color : chr "black" "grey" "white" "red" ...
## $ car_type : chr "convertible" "convertible" "convertible" "convertible" ...
## $ feature_1 : logi TRUE TRUE FALSE TRUE TRUE TRUE ...
## $ feature_2 : logi TRUE TRUE FALSE TRUE TRUE TRUE ...
## $ feature_3 : logi FALSE FALSE FALSE FALSE FALSE FALSE ...
## $ feature_4 : logi FALSE FALSE FALSE FALSE FALSE FALSE ...
## $ feature_5 : logi TRUE FALSE TRUE TRUE FALSE TRUE ...
## $ feature_6 : logi TRUE TRUE FALSE TRUE TRUE TRUE ...
## $ feature_7 : logi TRUE TRUE TRUE TRUE TRUE TRUE ...
## $ feature_8 : logi FALSE TRUE FALSE TRUE TRUE TRUE ...
## $ price : int 11300 69700 10200 25100 33400 17100 12400 6100 6200 17300 ...
## $ sold_at : chr "2018-01-01" "2018-02-01" "2018-02-01" "2018-02-01" ...## maker_key model_key mileage engine_power
## Length:4843 Length:4843 Min. : -64 Min. : 0
## Class :character Class :character 1st Qu.: 102914 1st Qu.:100
## Mode :character Mode :character Median : 141080 Median :120
## Mean : 140963 Mean :129
## 3rd Qu.: 175196 3rd Qu.:135
## Max. :1000376 Max. :423
## registration_date fuel paint_color car_type
## Length:4843 Length:4843 Length:4843 Length:4843
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
## feature_1 feature_2 feature_3 feature_4
## Mode :logical Mode :logical Mode :logical Mode :logical
## FALSE:2181 FALSE:1004 FALSE:3865 FALSE:3881
## TRUE :2662 TRUE :3839 TRUE :978 TRUE :962
##
##
##
## feature_5 feature_6 feature_7 feature_8
## Mode :logical Mode :logical Mode :logical Mode :logical
## FALSE:2613 FALSE:3674 FALSE:329 FALSE:2223
## TRUE :2230 TRUE :1169 TRUE :4514 TRUE :2620
##
##
##
## price sold_at
## Min. : 100 Length:4843
## 1st Qu.: 10800 Class :character
## Median : 14200 Mode :character
## Mean : 15828
## 3rd Qu.: 18600
## Max. :178500After loading the dataset and checking the basic structure, we can see that there is one car with negative mileage and another one, which has zero engine power. After close examination, it was concluded that the negative mileage car was probably a new car as it was the newest car in our dataset but must have been bought by a dealer from another dealer and hence part of the used car dataset. The mileage is thus converted to zero.
The other car with zero engine power also has all the features listed as false. A car with zero engine power and no features is only possible if it is a totalled/salvage titled car and hence is excluded from the data as we are only interested in working, street legal cars for this analysis.
| Model_Name | Mileage | Power | Fuel | Color | Type | feature_1 | feature_2 | feature_3 | feature_4 | feature_5 | feature_6 | feature_7 | feature_8 | Price | Age | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 118 | 140411 | 100 | diesel | black | convertible | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 11300 | 2160 |
| 3 | 320 | 183297 | 120 | diesel | white | convertible | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 10200 | 2131 |
| 4 | 420 | 128035 | 135 | diesel | red | convertible | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 25100 | 1310 |
| 5 | 425 | 97097 | 160 | diesel | silver | convertible | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 33400 | 1216 |
| 7 | 325 | 205219 | 145 | diesel | grey | convertible | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 12400 | 3197 |
| 10 | 320 | 139541 | 135 | diesel | white | convertible | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 17300 | 1733 |
A new age column is created to gauge the age of the car at the time of sale to the second owner. It is calculated by extracting the number of days between the registration date and the date of sale and is represented in number of days.
The feature columns are also converted from logical to binary for ease of use when adding to the multiple linear regression model (MLR).
Column maker_key, Registration Date and Sold date are dropped as maker is same for all the cars ‘BMW’ and does not provide any new information. The date columns are dropped because we already extracted the age of the car from them and hence do not provide other information required for this analysis.
As mentioned earlier, cars of fuel type hybrid and electro are removed. (11 rows removed out of 4842) as they will not be considered for the model. Some of the column names are hence changed for better aesthetics.
The dataset is then organised by type and fuel (type) which will form the basis of sampling of the dataset. First six rows of the modified dataset (before sampling) is shown above.
## Stratum 1
##
## Population total and number of selected units: 28 3
## Stratum 2
##
## Population total and number of selected units: 19 2
## Stratum 3
##
## Population total and number of selected units: 82 9
## Stratum 4
##
## Population total and number of selected units: 21 3
## Stratum 5
##
## Population total and number of selected units: 1574 160
## Stratum 6
##
## Population total and number of selected units: 32 4
## Stratum 7
##
## Population total and number of selected units: 659 67
## Stratum 8
##
## Population total and number of selected units: 35 4
## Stratum 9
##
## Population total and number of selected units: 1105 113
## Stratum 10
##
## Population total and number of selected units: 62 7
## Stratum 11
##
## Population total and number of selected units: 109 12
## Stratum 12
##
## Population total and number of selected units: 7 1
## Stratum 13
##
## Population total and number of selected units: 1041 106
## Stratum 14
##
## Population total and number of selected units: 13 2
## Stratum 15
##
## Population total and number of selected units: 42 5
## Stratum 16
##
## Population total and number of selected units: 2 1
## Number of strata 16
## Total number of selected units 499## Model_Name Mileage Power Color
## Length:499 Min. : 476 Min. : 70.0 Length:499
## Class :character 1st Qu.:101510 1st Qu.:100.0 Class :character
## Mode :character Median :138024 Median :120.0 Mode :character
## Mean :138302 Mean :130.2
## 3rd Qu.:174801 3rd Qu.:140.0
## Max. :370014 Max. :317.0
## feature_1 feature_2 feature_3 feature_4
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:1.0000 1st Qu.:0.0000 1st Qu.:0.0000
## Median :1.0000 Median :1.0000 Median :0.0000 Median :0.0000
## Mean :0.5671 Mean :0.8016 Mean :0.2305 Mean :0.2004
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:0.0000
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## feature_5 feature_6 feature_7 feature_8
## Min. :0.000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.000 1st Qu.:0.0000 1st Qu.:1.0000 1st Qu.:0.0000
## Median :0.000 Median :0.0000 Median :1.0000 Median :1.0000
## Mean :0.493 Mean :0.2866 Mean :0.9399 Mean :0.5371
## 3rd Qu.:1.000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. :1.000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## Price Age Type Fuel
## Min. : 800 Min. : 272 Length:499 Length:499
## 1st Qu.:11000 1st Qu.:1490 Class :character Class :character
## Median :14400 Median :1764 Mode :character Mode :character
## Mean :16146 Mean :1948
## 3rd Qu.:18350 3rd Qu.:2114
## Max. :73100 Max. :6451Five-hundred rows are then sampled from the dataset to form the basis of the analysis. The sampled strata details and summary of the sampled dataset is displayed above.
Formal Test to test (using the 5-step procedure) whether the set of these predictors are associated with price at the alpha = 0.05
1. Set up the hypotheses and select the alpha level
\(H_0:\beta\)Predictors \(= 0\)Â (None of the Independent variables are not Predictors of Price)
\(H_1: H_0:\) \(\beta\)Predictors \(\neq\) 0 (at least one of the slope coefficients is different than 0 and is a predictor of Price)
\(\alpha\) = 0.05
2. Select the appropriate test statistic
we are going to use a F-test statistic.
\[\mathbf{F} = \frac{MS\ Reg}{MS\ Res}\ where\ df = k,n - k - 1\]
3.State the decision rule
Reject \(H_0\) if \(p-value <= \alpha\). Otherwise, do not reject \(H_0\)
4. Compute the test statistic and the associated p-value.
##
## Call:
## lm(formula = Price ~ ., data = MLR.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18407 -1450 0 1508 17637
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value
## (Intercept) 13676.616016 4063.065027 3.366
## Model_Name116 -379.560267 2229.276462 -0.170
## Model_Name118 531.352980 2422.531795 0.219
## Model_Name120 -395.491108 2950.480959 -0.134
## Model_Name123 -1068.143897 4352.905662 -0.245
## Model_Name125 -6504.031412 4347.085177 -1.496
## Model_Name218 -12113.538373 4953.317595 -2.446
## Model_Name218 Active Tourer -1195.020942 4187.128295 -0.285
## Model_Name218 Gran Tourer -3320.852907 4206.887647 -0.789
## Model_Name225 Active Tourer -96.913365 5280.544036 -0.018
## Model_Name316 1831.775809 2537.969392 0.722
## Model_Name318 2116.095544 2531.953979 0.836
## Model_Name318 Gran Turismo 2922.631634 2484.880404 1.176
## Model_Name320 1686.453728 2598.470699 0.649
## Model_Name320 Gran Turismo 1949.318613 2702.184494 0.721
## Model_Name325 -884.685405 3333.693333 -0.265
## Model_Name325 Gran Turismo 5001.972148 4336.413513 1.153
## Model_Name330 1420.635758 3283.247451 0.433
## Model_Name335 -73.754789 4212.202166 -0.018
## Model_Name420 5874.055062 3440.204922 1.707
## Model_Name420 Gran Coupé 7166.720041 2911.406278 2.462
## Model_Name425 12108.084749 5100.481103 2.374
## Model_Name435 12551.299567 5324.603997 2.357
## Model_Name435 Gran Coupé 15375.973168 4635.786986 3.317
## Model_Name518 4329.622606 2888.657707 1.499
## Model_Name520 3998.306819 2659.453626 1.503
## Model_Name520 Gran Turismo 7478.695245 3015.981514 2.480
## Model_Name525 3956.657582 2872.259282 1.378
## Model_Name528 4888.992518 4496.819057 1.087
## Model_Name530 5764.482747 3103.828658 1.857
## Model_Name530 Gran Turismo 5112.734364 4887.218875 1.046
## Model_Name535 3432.881447 4192.245172 0.819
## Model_Name630 621.902293 5088.656228 0.122
## Model_Name640 12770.272193 4523.796025 2.823
## Model_Name640 Gran Coupé 15802.078861 4823.890951 3.276
## Model_Name730 7911.491925 3330.493936 2.375
## Model_Name740 31426.110262 3952.578911 7.951
## Model_NameM235 6519.471476 5275.934266 1.236
## Model_NameM4 36608.459248 5857.396835 6.250
## Model_NameM550 15846.823109 5255.191593 3.015
## Model_NameX1 2251.070155 4373.701368 0.515
## Model_NameX3 4810.391539 4396.348133 1.094
## Model_NameX4 10842.518936 4867.135296 2.228
## Model_NameX5 12600.106877 4643.694958 2.713
## Model_NameX5 M 29475.414959 5122.129532 5.755
## Model_NameX6 14720.371011 5200.074801 2.831
## Model_NameX6 M 36306.894625 6238.419819 5.820
## Mileage -0.034984 0.003729 -9.382
## Power 52.816340 14.816859 3.565
## Colorblack 707.422298 1888.266846 0.375
## Colorblue 86.498705 1907.858782 0.045
## Colorbrown -952.105353 1958.594961 -0.486
## Colorgreen -3900.705528 3417.949930 -1.141
## Colorgrey 1395.139998 1907.634961 0.731
## Colorred -831.586541 2769.003013 -0.300
## Colorsilver 1737.519752 1981.997441 0.877
## Colorwhite 1464.588481 1949.263072 0.751
## feature_1 441.633037 388.200998 1.138
## feature_2 267.510039 514.683506 0.520
## feature_3 393.254775 426.486754 0.922
## feature_4 892.308269 568.454201 1.570
## feature_5 282.763118 418.827865 0.675
## feature_6 508.168520 409.589546 1.241
## feature_7 2063.617048 845.491628 2.441
## feature_8 1247.267092 420.546714 2.966
## Age -3.046208 0.257066 -11.850
## Typecoupe -1174.550498 3251.465753 -0.361
## Typeestate -2459.506841 2339.743446 -1.051
## Typehatchback -1649.577686 2645.902806 -0.623
## Typesedan -1213.847079 2346.422686 -0.517
## Typesubcompact -1816.297268 2801.542256 -0.648
## Typesuv -1948.995019 4272.969309 -0.456
## Typevan NA NA NA
## Fuelpetrol -722.939314 943.282219 -0.766
## Pr(>|t|)
## (Intercept) 0.000832 ***
## Model_Name116 0.864885
## Model_Name118 0.826492
## Model_Name120 0.893432
## Model_Name123 0.806275
## Model_Name125 0.135347
## Model_Name218 0.014867 *
## Model_Name218 Active Tourer 0.775474
## Model_Name218 Gran Tourer 0.430326
## Model_Name225 Active Tourer 0.985366
## Model_Name316 0.470845
## Model_Name318 0.403761
## Model_Name318 Gran Turismo 0.240185
## Model_Name320 0.516677
## Model_Name320 Gran Turismo 0.471068
## Model_Name325 0.790847
## Model_Name325 Gran Turismo 0.249360
## Model_Name330 0.665457
## Model_Name335 0.986038
## Model_Name420 0.088463 .
## Model_Name420 Gran Coupé 0.014227 *
## Model_Name425 0.018043 *
## Model_Name435 0.018864 *
## Model_Name435 Gran Coupé 0.000989 ***
## Model_Name518 0.134657
## Model_Name520 0.133469
## Model_Name520 Gran Turismo 0.013535 *
## Model_Name525 0.169068
## Model_Name528 0.277558
## Model_Name530 0.063970 .
## Model_Name530 Gran Turismo 0.296088
## Model_Name535 0.413321
## Model_Name630 0.902788
## Model_Name640 0.004982 **
## Model_Name640 Gran Coupé 0.001140 **
## Model_Name730 0.017968 *
## Model_Name740 0.0000000000000168 ***
## Model_NameM235 0.217251
## Model_NameM4 0.0000000009973957 ***
## Model_NameM550 0.002719 **
## Model_NameX1 0.607041
## Model_NameX3 0.274495
## Model_NameX4 0.026422 *
## Model_NameX5 0.006930 **
## Model_NameX5 M 0.0000000166262998 ***
## Model_NameX6 0.004863 **
## Model_NameX6 M 0.0000000115930715 ***
## Mileage < 0.0000000000000002 ***
## Power 0.000406 ***
## Colorblack 0.708114
## Colorblue 0.963859
## Colorbrown 0.627135
## Colorgreen 0.254411
## Colorgrey 0.464970
## Colorred 0.764080
## Colorsilver 0.381170
## Colorwhite 0.452854
## feature_1 0.255910
## feature_2 0.603503
## feature_3 0.357009
## feature_4 0.117225
## feature_5 0.499960
## feature_6 0.215408
## feature_7 0.015064 *
## feature_8 0.003189 **
## Age < 0.0000000000000002 ***
## Typecoupe 0.718101
## Typeestate 0.293769
## Typehatchback 0.533325
## Typesedan 0.605203
## Typesubcompact 0.517127
## Typesuv 0.648535
## Typevan NA
## Fuelpetrol 0.443858
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3439 on 426 degrees of freedom
## Multiple R-squared: 0.8819, Adjusted R-squared: 0.8619
## F-statistic: 44.18 on 72 and 426 DF, p-value: < 0.00000000000000022
\(F-Statistic =\) 44.18 on 72 and 426 DF
\(p-value =\) less than < 0.00000000000000022
\(Adjusted\ R^2 =\) 0.86
5.Conclusion
Reject \(H_0\) since \(p <= \alpha\). We have significant evidence at the \(\alpha=0.05\) level that independent variables when taken together are predictive of price. That is, there is evidence of a linear association between Price and all other independent variables.
From the above test we can conclude that some but not all variables are predictors of price. we now create a new model after removing the redundent variables and check the effect on R-squared value. we do this so as to keep the model simple and also avoid the possibility of overfitting.
##
## Call:
## lm(formula = Price ~ . - Color - Type - feature_1 - feature_2 -
## feature_3 - feature_4 - feature_5 - feature_6 - Fuel, data = MLR.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19311 -1568 0 1594 17856
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) 11035.533036 2457.585768 4.490
## Model_Name116 -367.148781 2144.222770 -0.171
## Model_Name118 365.628899 2343.901188 0.156
## Model_Name120 -121.484247 2838.921068 -0.043
## Model_Name123 -156.627857 4249.557013 -0.037
## Model_Name125 -5964.664909 4295.126228 -1.389
## Model_Name218 -9205.343724 4169.884660 -2.208
## Model_Name218 Active Tourer -78.448695 3232.448513 -0.024
## Model_Name218 Gran Tourer -2086.675230 3248.685052 -0.642
## Model_Name225 Active Tourer 339.737970 4320.107547 0.079
## Model_Name316 1673.407342 2170.758919 0.771
## Model_Name318 1885.799624 2147.686961 0.878
## Model_Name318 Gran Turismo 3192.691023 2350.432214 1.358
## Model_Name320 1485.878699 2240.415309 0.663
## Model_Name320 Gran Turismo 3143.603252 2547.820386 1.234
## Model_Name325 -1496.859392 2983.573744 -0.502
## Model_Name325 Gran Turismo 5318.374518 4247.894528 1.252
## Model_Name330 438.037486 2902.284516 0.151
## Model_Name335 861.360304 3836.631219 0.225
## Model_Name420 6622.952058 2711.828377 2.442
## Model_Name420 Gran Coupé 7195.868539 2793.662380 2.576
## Model_Name425 14669.951862 4239.324330 3.460
## Model_Name435 12486.467196 4726.634404 2.642
## Model_Name435 Gran Coupé 14204.081694 4554.170257 3.119
## Model_Name518 5221.792856 2564.436953 2.036
## Model_Name520 3890.926161 2268.184091 1.715
## Model_Name520 Gran Turismo 8539.710306 2870.077660 2.975
## Model_Name525 3559.275249 2493.758610 1.427
## Model_Name528 4427.094292 4357.974928 1.016
## Model_Name530 5371.769142 2750.173068 1.953
## Model_Name530 Gran Turismo 5106.691172 4421.419217 1.155
## Model_Name535 3545.947162 3961.757557 0.895
## Model_Name630 536.703208 4462.547160 0.120
## Model_Name640 12319.849904 3634.666830 3.390
## Model_Name640 Gran Coupé 17055.529137 4598.127713 3.709
## Model_Name730 8672.975605 2933.738990 2.956
## Model_Name740 31513.122136 3612.619228 8.723
## Model_NameM235 6710.995356 4678.554941 1.434
## Model_NameM4 36158.837660 5245.327005 6.894
## Model_NameM550 15086.087913 4985.580422 3.026
## Model_NameX1 2366.960650 2204.327190 1.074
## Model_NameX3 5206.299910 2272.938057 2.291
## Model_NameX4 11127.363522 3054.682786 3.643
## Model_NameX5 12825.357418 2696.690526 4.756
## Model_NameX5 M 30362.034646 3397.816503 8.936
## Model_NameX6 13973.470681 3429.537546 4.074
## Model_NameX6 M 35328.977005 4962.621333 7.119
## Mileage -0.036270 0.003633 -9.984
## Power 64.659908 14.029388 4.609
## feature_7 3104.130639 743.244574 4.176
## feature_8 1425.665215 408.435591 3.491
## Age -3.009185 0.233225 -12.902
## Pr(>|t|)
## (Intercept) 0.00000905948056 ***
## Model_Name116 0.864123
## Model_Name118 0.876110
## Model_Name120 0.965886
## Model_Name123 0.970615
## Model_Name125 0.165614
## Model_Name218 0.027781 *
## Model_Name218 Active Tourer 0.980649
## Model_Name218 Gran Tourer 0.520999
## Model_Name225 Active Tourer 0.937353
## Model_Name316 0.441182
## Model_Name318 0.380382
## Model_Name318 Gran Turismo 0.175040
## Model_Name320 0.507534
## Model_Name320 Gran Turismo 0.217911
## Model_Name325 0.616125
## Model_Name325 Gran Turismo 0.211224
## Model_Name330 0.880100
## Model_Name335 0.822463
## Model_Name420 0.014983 *
## Model_Name420 Gran Coupé 0.010321 *
## Model_Name425 0.000591 ***
## Model_Name435 0.008538 **
## Model_Name435 Gran Coupé 0.001932 **
## Model_Name518 0.042316 *
## Model_Name520 0.086958 .
## Model_Name520 Gran Turismo 0.003084 **
## Model_Name525 0.154199
## Model_Name528 0.310245
## Model_Name530 0.051414 .
## Model_Name530 Gran Turismo 0.248712
## Model_Name535 0.371245
## Model_Name630 0.904325
## Model_Name640 0.000762 ***
## Model_Name640 Gran Coupé 0.000234 ***
## Model_Name730 0.003278 **
## Model_Name740 < 0.0000000000000002 ***
## Model_NameM235 0.152153
## Model_NameM4 0.00000000001862 ***
## Model_NameM550 0.002622 **
## Model_NameX1 0.283501
## Model_NameX3 0.022454 *
## Model_NameX4 0.000301 ***
## Model_NameX5 0.00000266899351 ***
## Model_NameX5 M < 0.0000000000000002 ***
## Model_NameX6 0.00005457688954 ***
## Model_NameX6 M 0.00000000000436 ***
## Mileage < 0.0000000000000002 ***
## Power 0.00000528952795 ***
## feature_7 0.00003561163814 ***
## feature_8 0.000530 ***
## Age < 0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3517 on 447 degrees of freedom
## Multiple R-squared: 0.8704, Adjusted R-squared: 0.8556
## F-statistic: 58.86 on 51 and 447 DF, p-value: < 0.00000000000000022\(F-Statistic =\) 58.86 on 51 and 447 DF
\(p-value =\) less than < 0.00000000000000022
\(Adjusted\ R^2 =\) 0.85
As can be seen that removal of some unimportant features has marginal effect on the adjusted R-squared value but the model is much more concise and less likely to exhibit overfitting.
Residuals
As is evident from the plots, there appears to be no apparent association between data points and the model seems to be reasonable. The residuals, even though display a sort of a cluster with some outliers, the model seems reasonable and they do seem to be normally distributed.
| Model_Name | Mileage | Power | Color | feature_1 | feature_2 | feature_3 | feature_4 | feature_5 | feature_6 | feature_7 | feature_8 | Price | Age | Type | Fuel | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 425 | X6 | 46041 | 230 | grey | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 55500 | 942 | suv | diesel |
| 484 | X5 M | 4530 | 230 | silver | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 73100 | 760 | suv | diesel |
| 329 | 740 | 166286 | 230 | brown | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 31300 | 1764 | sedan | diesel |
| 407 | X5 | 64729 | 190 | grey | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 48200 | 1248 | suv | diesel |
| 457 | X5 | 52253 | 190 | white | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 46600 | 851 | suv | diesel |
| 491 | X3 | 97835 | 190 | brown | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 5400 | 1917 | suv | diesel |
| 431 | X5 | 123474 | 190 | black | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 43400 | 1126 | suv | diesel |
| 481 | X5 | 148480 | 155 | silver | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 35700 | 1521 | suv | diesel |
We use the diagnostic plots to identify some of the outliers. The list of such data points is presented above. Interestingly, all of them are SUV’s baring one sedan and all are diesel-fueled cars. Most of them display a similar trend of high price & low mileage the ratio that seems to be not comparable to other cars in the dataset.
##
## Call:
## lm(formula = Price ~ . - Color - Type - feature_1 - feature_2 -
## feature_3 - feature_4 - feature_5 - feature_6 - Fuel, data = MLR.df[-outliers.index,
## ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -8349 -1430 0 1437 7708
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) 10225.418273 1924.367212 5.314
## Model_Name116 -843.925620 1672.746822 -0.505
## Model_Name118 -321.469508 1831.777816 -0.175
## Model_Name120 -993.841835 2222.883343 -0.447
## Model_Name123 -1558.599069 3325.338571 -0.469
## Model_Name125 -6951.303151 3372.196933 -2.061
## Model_Name218 -10157.398055 3259.185805 -3.117
## Model_Name218 Active Tourer -127.593824 2521.661800 -0.051
## Model_Name218 Gran Tourer -2206.344451 2535.347870 -0.870
## Model_Name225 Active Tourer -764.464683 3393.357307 -0.225
## Model_Name316 1085.419362 1693.471308 0.641
## Model_Name318 1119.368746 1677.565473 0.667
## Model_Name318 Gran Turismo 2529.113626 1834.976274 1.378
## Model_Name320 351.457023 1756.777336 0.200
## Model_Name320 Gran Turismo 2188.677023 1994.421534 1.097
## Model_Name325 -2493.264970 2349.604020 -1.061
## Model_Name325 Gran Turismo 4359.108767 3324.103056 1.311
## Model_Name330 -1160.302529 2293.167830 -0.506
## Model_Name335 -1203.458761 3028.938311 -0.397
## Model_Name420 5953.633711 2124.487920 2.802
## Model_Name420 Gran Coupé 6769.274694 2184.846135 3.098
## Model_Name425 13780.408783 3316.165778 4.156
## Model_Name435 11167.254961 3728.186240 2.995
## Model_Name435 Gran Coupé 13251.860546 3583.226372 3.698
## Model_Name518 4384.862112 2001.429814 2.191
## Model_Name520 2766.727673 1778.065564 1.556
## Model_Name520 Gran Turismo 7214.457320 2245.606021 3.213
## Model_Name525 2007.642009 1963.526512 1.022
## Model_Name528 2967.369134 3415.874014 0.869
## Model_Name530 3818.029050 2175.673297 1.755
## Model_Name530 Gran Turismo 2735.921655 3466.749506 0.789
## Model_Name535 1135.962300 3130.566846 0.363
## Model_Name630 -1639.583114 3505.207019 -0.468
## Model_Name640 10421.315195 2879.577755 3.619
## Model_Name640 Gran Coupé 15660.826000 3618.495416 4.328
## Model_Name730 6772.912888 2313.352094 2.928
## Model_Name740 40159.132270 3071.876714 13.073
## Model_NameM235 5489.814717 3687.361434 1.489
## Model_NameM4 34637.430520 4162.489102 8.321
## Model_NameM550 12929.194470 3942.676643 3.279
## Model_NameX1 1624.392216 1722.977598 0.943
## Model_NameX3 4534.304236 1778.853103 2.549
## Model_NameX4 10572.539560 2393.300732 4.418
## Model_NameX5 8758.307380 2166.313730 4.043
## Model_NameX5 M 20199.100943 2838.896335 7.115
## Model_NameX6 6712.199473 2838.006750 2.365
## Model_NameX6 M 34134.768913 3925.008449 8.697
## Mileage -0.031384 0.002858 -10.980
## Power 72.226982 11.419255 6.325
## feature_7 2628.472080 591.784918 4.442
## feature_8 1582.177719 318.870473 4.962
## Age -2.734781 0.183953 -14.867
## Pr(>|t|)
## (Intercept) 0.00000017125673619 ***
## Model_Name116 0.614153
## Model_Name118 0.860771
## Model_Name120 0.655026
## Model_Name123 0.639514
## Model_Name125 0.039857 *
## Model_Name218 0.001950 **
## Model_Name218 Active Tourer 0.959668
## Model_Name218 Gran Tourer 0.384648
## Model_Name225 Active Tourer 0.821864
## Model_Name316 0.521894
## Model_Name318 0.504958
## Model_Name318 Gran Turismo 0.168818
## Model_Name320 0.841528
## Model_Name320 Gran Turismo 0.273068
## Model_Name325 0.289209
## Model_Name325 Gran Turismo 0.190420
## Model_Name330 0.613123
## Model_Name335 0.691324
## Model_Name420 0.005297 **
## Model_Name420 Gran Coupé 0.002071 **
## Model_Name425 0.00003902587448898 ***
## Model_Name435 0.002896 **
## Model_Name435 Gran Coupé 0.000245 ***
## Model_Name518 0.028987 *
## Model_Name520 0.120421
## Model_Name520 Gran Turismo 0.001412 **
## Model_Name525 0.307123
## Model_Name528 0.385486
## Model_Name530 0.079979 .
## Model_Name530 Gran Turismo 0.430427
## Model_Name535 0.716883
## Model_Name630 0.640191
## Model_Name640 0.000330 ***
## Model_Name640 Gran Coupé 0.00001865291906992 ***
## Model_Name730 0.003592 **
## Model_Name740 < 0.0000000000000002 ***
## Model_NameM235 0.137253
## Model_NameM4 0.00000000000000111 ***
## Model_NameM550 0.001124 **
## Model_NameX1 0.346311
## Model_NameX3 0.011142 *
## Model_NameX4 0.00001258887740148 ***
## Model_NameX5 0.00006233629747939 ***
## Model_NameX5 M 0.00000000000458098 ***
## Model_NameX6 0.018459 *
## Model_NameX6 M < 0.0000000000000002 ***
## Mileage < 0.0000000000000002 ***
## Power 0.00000000062447781 ***
## feature_7 0.00001131483790137 ***
## feature_8 0.00000100062147108 ***
## Age < 0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2742 on 439 degrees of freedom
## Multiple R-squared: 0.9039, Adjusted R-squared: 0.8927
## F-statistic: 80.96 on 51 and 439 DF, p-value: < 0.00000000000000022\(F-Statistic =\) 80.96 on 51 and 449 DF
\(p-value =\) less than < 0.00000000000000022
\(Adjusted\ R^2 =\) 0.89
We remove the outliers and construct our final model with only the relevant features. Immediately it is evident that our adjusted R-score has improved by forty basis points due to the removal of the outliers.
The residual plot of the newly created model is presented above. the residuals seem to follow a perfect normal distribution.
Testing and Conclusion.
## [1] "Table: Prdicted vs Actual Prices from the Test Data"
## [2] ""
## [3] "| Predicted | Actual |"
## [4] "|:-----------:|:------:|"
## [5] "| 9441.32180 | 11300 |"
## [6] "| 10292.15859 | 10200 |"
## [7] "| 22539.52007 | 25100 |"
## [8] "| 7232.01080 | 12400 |"
## [9] "| 7998.01506 | 6100 |"
## [10] "| 13837.24816 | 17300 |"\(\ R^2\ from\ test\ data = 0.7\)
We subtracted the cars, which we used to build the model from the original dataset to come up with the test data set. When we originally sampled our dataset, our sample did not include some of the car models as we sampled the data by type (SUV, Compact, etc.) and fuel (Diesel or Petrol) and not by the model. We identified the models not used in our sample and removed them from our test data to avoid extrapolation. We also removed the cars with fuel type electro and hybrid for the same reason.
Some of the predicted values for the price of the car came as zero, which is not possible. Hence, we converted them to zero. Some of actual vs predicted values are presented above. We finally then calculated our test data R2 squared value which came out to be 0.6621321 or 0.7 rounded.
An important observation in hindsight is that fuel type was not a major predictor of price in BMW cars and model type was a major predictor, either of which was not expected.
Even though 0.7 can be an acceptable performance, our model performed poorly on the test data in comparison with the training data. The main reason of which is the size of the sample. Our train data was relatively much smaller than the test data (4258 rows of test data vs 499 of train). Our train-test split of the data was constrained by the project parameter of five-hundred observations, however, a more robust and holistic sampling/splitting method could result in a better performing model.