Problem Statement
- Analyze the features affecting the price of the house and understand the linear equation for the feature. Understand how we can interpret the linear equation and with the help of that equation we can predict the price.
Univariate Analysis
Exploring the univariate distribution of the response variable Sale price.
- As we look at the mean and median,
median < meanso we can say that our variable is right skewed. But it is not much skewed so we can consider it as normally distributed because in real world we never get perfectly distributed data.
- The reason behind this skewness can be outliers present in the response variable..
Find outlier(s) in your original response variable
- We have found the we have some outliers present in the data which can be reason for the skewness.
Bivariate Analysis
- To do the bivariate analysis we need to find the features which are highly correlated to the response variable that is house price.
Heat Map to find highly correlated features.
We can see top 3 numerical feature that are directly correlated to the response variable.
- Gr_Liv_Area Above Ground Living Area in sqft
- Garage_Area Garage Area in sqft
- Total_Bsmt_SF Total basement Size in sqft
Now, We look at each feature one by one and see if we can improve the relation?
Relation between Sale Price and Above ground living area size
Here, we can say that we have some positive relation between sale price and Above Ground Living Area.
- But we can see there are some outliers in Above Ground Living Area, which leads to the less weak relation.
The Correlation coefficient before removing outliers is 0.7067799
The Correlation coefficient after removing outliers is 0.7084774
Relation between Sale Price and Size of garage
Here, we can say that we have some positive relation between sale price and Size of garage.
- But we can see there are some outliers in total basement size, which leads to the less weak relation.
The Correlation coefficient before removing outliers is 0.6401383
The Correlation coefficient after removing outliers is 0.6422716
Relation between Sale Price and Total Basement size
Here, we can say that we have some positive relation between sale price and total basement size.
- But we can see there are some outliers in total basement size, which leads to the less weak relation.
The Correlation coefficient before removing outliers is 0.6325288
The Correlation coefficient after removing outliers is 0.6632371
Conclusion of EDA
- As we have seen so far that our three features have some outliers and by removing them we can get more strong relation which will help to improve performance of the model.
Data Modeling
- In this section we are going to learn how to fit a linear model for the each numerical feature we have seen above and how to interpret there equation.
Model 1 (Sale Price and Above ground living area size)
- When we fit the model, we can see the following values and with the help of that we can form an linear equation.
# A tibble: 2 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 5906. 3352. 1.76 0.0782
2 Gr_Liv_Area 117. 2.16 54.1 0 \(\hat{y}= 5905.812 + 116.874 \times Above-ground-living-area-size\)
Y intercept value means that, For a house with 0 Above ground living area size, the house price will be 5905.812 USD.
Slope of the fitted line means that, If we increase Above ground living area size by 1 Sqft, the price of the house increases with 116.874 USD.
Model 2 (Sale Price and Size of garage)
# A tibble: 2 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 46925. 3341. 14.0 2.40e- 43
2 Garage_Area 277. 6.30 44.1 1.35e-321\(\hat{y}= 46925.28 + 277.37 \times Size-of-garage\)
Y intercept value means that, For a house with 0 Size of garage, the house price will be 46925.28 USD.
Slope of the fitted line means that, If we increase Size of garage size by 1 Sqft, the price of the house increases with 277.37 USD.
Model 3 (Sale Price and Total Basement size)
# A tibble: 2 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 35394. 3309. 10.7 3.36e-26
2 Total_Bsmt_SF 137. 2.90 47.2 0 \(\hat{y}= 35393.94 + 137 \times Total-Basement-Size\)
Y intercept value means that, For a house with 0 total basement size , the house price will be 35393.94 USD.
Slope of the fitted line means that, If we increase size of total basement by 1 Sqft, the price of the house with 137 USD.
Model Assessment
- In this part we select which model is the best model out of the three that we have created.
- How we selected the model and what measures we have take in consideration for selecting the model?
| Above ground living area size | Size of garage | Total Basement size | |
|---|---|---|---|
| (Intercept) | 5905.812 | 46925.278 | 35393.941 |
| (3352.336) | (3340.582) | (3309.416) | |
| Gr_Liv_Area | 116.874 | ||
| (2.161) | |||
| Garage_Area | 277.371 | ||
| (6.296) | |||
| Total_Bsmt_SF | 136.994 | ||
| (2.900) | |||
| Num.Obs. | 2904 | 2766 | 2844 |
| R2 | 0.502 | 0.413 | 0.440 |
| R2 Adj. | 0.502 | 0.412 | 0.440 |
| AIC | 71512.6 | 68800.1 | 70599.3 |
| BIC | 71530.5 | 68817.9 | 70617.1 |
| Log.Lik. | -35753.310 | -34397.051 | -35296.640 |
| F | 2924.609 | 1940.784 | 2231.943 |
As we can see in the table,
Model 1 (Sale price, above ground living area) has 50% variability of sale price based on the above ground living area, which is the highest.
It has highest adjusted R squared value so we can say it is best.
The reason why we considered Adjusted R squared value to select best model is that,
Adjusted R - squared value only consider those independent variable which really helps in explaining the dependent variable.
Here, we are using many independent variable with the single dependent variable and we do not know which one is good one to explain the dependent variable so we can use the adjusted R-squared value.
Model Diagnostics
Now that you have chosen our best model, we need to determine if it is reliable or not:
- There are three things to check.
Linearity
Distribution of Residuals
Constant Variability of residuals
- Creating two new data columns based on our best model: predicted values for our response variable and the corresponding residuals.
# A tibble: 2,904 x 8
Sale_Price Gr_Liv_Area .fitted .resid .hat .sigma .cooksd
<int> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
1 215000 1656 199450. 15550. 0.000394 53815. 0.0000165
2 105000 896 110625. -5625. 0.000896 53816. 0.00000491
3 172000 1329 161232. 10768. 0.000382 53816. 0.00000765
4 244000 2110 252510. -8510. 0.000983 53816. 0.0000123
5 189900 1629 196294. -6394. 0.000380 53816. 0.00000268
6 195500 1604 193372. 2128. 0.000369 53816. 0.000000289
7 213500 1338 162284. 51216. 0.000377 53807. 0.000171
8 191500 1280 155505. 35995. 0.000409 53812. 0.0000917
9 236500 1616 194775. 41725. 0.000374 53810. 0.000112
10 189000 1804 216747. -27747. 0.000513 53813. 0.0000683
# ... with 2,894 more rows, and 1 more variable: .std.resid <dbl>Checking Linearity
- Creating plots to check the assumption of linearity. State whether or not this condition is met and explain your reasoning.
- We can see some linear relation so Linearity condition also met.
Checking Distribution of Residuals
- Creating a plot to check the assumption of nearly normal residuals. State whether or not this condition is met and explaining reasoning.
Checking Constant Variability of residuals
- Checking the assumption of constant variability. State whether or not this condition is met and explaining reasoning.
- Constant variability is not meeting in this case.
- As we see above in the graph between above ground living area size and residuals that as the size of the above ground living area increases the variability of residuals around the 0 line is increases too. So, we can say that constant variability condition is not met.