Data Analysis IOWA House Price Prediction Kaggle Competition
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Real Estate Price Predictions Using Advanced Analytics and Machine Learning
The Competition
The real estate price prediction in Ames, Iowa is a popular competition on Kaggle and provides a great dataset to apply creative feature engineering and advanced regression techniques. In this paper, we analyzed the training and test data sets using uni-variate and bi-variate techniques, handled missing values, dealt with outliers, applied machine learning models and tuned hyper parameters.
Datasets
Competitors are given training and test datasets: Training set has 1460 rows and 81 columns and test has 1459 rows and 80 columns (test set has no target variable). We import our csv files and load the data to Pandas dataframes.
Exploratory Data Analysis (EDA)
Categorical vs Numeric Variables
We start by splitting our data that includes 81 variables into 45 categorical features and 36 numerical predictors. This gave us a good introduction to the dataset and allowed us to come up with a logical approach to our data wrangling. We observed that some of our numerical variables would be better represented as categories, as their numerical values were not intended to be interpreted as a magnitude.
Binning
Binning is widely used to create a better linear relationship between the predictors and the target. We choose to use binning for the below predictors. Note that, original (unbinned) predictors must be dropped after binning.
Dummification
Some numerical variables are not continuous and must be regarded as categorical variables.
- MSSubClass
- OverallQual
- OverallCond
- BsmtFullBath
- BsmtHalfBath
- FullBath
- HalfBath
- BedroomAbvGr
- KitchenAbvGr
- TotRmsAbvGrd
- Fireplaces
- GarageCars
- YrSold
We cast them to category type together with the newly created binned variables and then dummify them with all other categorical variables.
from_num_to_cat = ['MSSubClass','OverallQual','OverallCond','BsmtFullBath','BsmtHalfBath','FullBath','HalfBath','BedroomAbvGr','KitchenAbvGr','TotRmsAbvGrd','Fireplaces','GarageCars','MoSold_Binned','YearBuilt_Binned','YearRemodAdd_Binned','YrSold','GarageYrBlt_Binned','PoolArea_Binned']
df_concat[from_num_to_cat] = df_concat[from_num_to_cat].apply(lambda x: x.astype('category'))
df_concat_dummified = pd.get_dummies(df_concat, drop_first=True)
Missing Values
We continue by detecting missing data and resolving the missing values appropriately.
For many of our variables, missing data indicates that the property does not include such feature. This is the case for features describing pools, alleys, fences, fireplaces, basements, and miscellaneous. For categorical features, we should impute 'None' and for numerical variables describing these features, we should impute 0. For missing values due to lack of information, we choose to impute with the median because it is more robust to outliers than imputing with the mean. For features with very few unique values, we can impute with the mode. We were able to successfully account for all missing values and continued with our EDA.
Uni-variate Analysis
We use histograms to illustrate how symmetric (or asymmetric) our data is. This will give us a sense on what transformations we need to make before doing any modelling. By this point we already suspect that we will be using some multi-linear regression to make our predictions.
Bi-variate Analysis
Regression plots are helpful to see the linear relationship between each predictor and the target.
Outliers
Outlier detection and elimination is a very important before applying any machine learning model on a data set. Even one outlier can alter the model predictions dramatically. Outliers can be detected several ways but the most efficient is to visualize data using scatter plots.
The above scatter plot of GrLivArea to target variable SalePrice shows that there are 2 data points with huge living area but comparably lower SalePrice. We remove those observations so that our predictions are more robust when we input unseen test data, and are not skewed by our outliers.
Feature Engineering and Selection
Another important part of EDA is to check for the correlation. The best way to visualize correlation is to use the heatmap.
Some columns in the data set are redundant because they are simply a part of another column. For example; below equations hold so that the added variables provide no value to the models. Dropping them will reduce multicollinearity, too.
BsmtFinSF1 + BsmtFinSF2 + BsmtUnfSF = TotalBsmtSF
1stFlrSF + 2ndFlrSF + LowQualFinSF = GrLivArea
OpenPorchSF + EnclosedPorch + 3SsnPorch + ScreenPorch + WoodDeckSF = TotalPorchSF
df_concat.drop(['BsmtFinSF1', 'BsmtFinSF2', 'BsmtUnfSF'], axis=1, inplace=True)
df_concat.drop(['1stFlrSF','2ndFlrSF','LowQualFinSF'], axis=1, inplace=True)
df_concat.drop(['OpenPorchSF','EnclosedPorch','3SsnPorch','ScreenPorch','WoodDeckSF'], axis=1, inplace=True)
Data Transformations
Log Transform
Taking logarithm of a continuous variable is often used to solve for the skewness. Below figure reveals that target variable SalePrice is highly skewed.
sns.distplot(train['SalePrice']);
# skewness and kurtosis
print("Skewness: %f" % train['SalePrice'].skew())
print("Kurtosis: %f" % train['SalePrice'].kurt())
Observe the skewness and the histogram after we apply log transformation on the target variable.
Modelling
We use Python's scikit-learn package because it provides all models we need and its object oriented approach makes it easy to fit models.
Preparations
Some preparations are needed before going to the modelling. Since we combine the train and test data sets for east manipulation, we now need to separate them back into train and test. Since test data set must not have target variable, we drop SalePrice_Log even though it consists of all NAs and drop the index.
test = df_concat_dummified.loc[df_concat.SalePrice_Log.isnull(),]
train = df_concat_dummified.loc[~df_concat.SalePrice_Log.isnull(),]
test.drop('SalePrice_Log', inplace=True, axis=1)
test.reset_index(drop=True, inplace=True)
Furthermore, we initialize a Pandas data frame for submitting the predictions with the Id column from the test data set. To fit the models, we specify X (predictors) and Y (target). Be mindful that X must not have Id and SalePrice_Log so that we drop them. There is no need for Id column for test data set as well so we drop Id from test, too.
submission = pd.DataFrame()
submission['Id'] = test.Id
Y=train['SalePrice_Log']
X=train.loc[:, ~train.columns.isin(['Id','SalePrice_Log'])]
test2=test.loc[:, ~test.columns.isin(['Id'])]
Last step before modelling is to check for dimensions of train and test and any nulls, positive or negative infinity.
print(X.shape)
print(test2.shape)
print(set(X.columns) - set(test2.columns))
print(len(set(X.columns) - set(test2.columns)))
print(set(test2.columns) - set(X.columns))
print(X.isin([np.nan, np.inf, -np.inf]).any(axis=0).any())
Multiple Linear Regression
We only use numerical columns as predictors for multiple linear regression. First, we simply add all numerical columns to the regression model and check whether the coefficients are significant or not. However, we need to split the training data as train and test to figure out the train and test errors after fitting. We employ a split of 30% test size and 70% train size.
X_train, X_test, y_train, y_test = train_test_split(X, Y, test_size=0.3, random_state=42)
OLS Regression Results
==============================================================================
Dep. Variable: SalePrice_Log R-squared: 0.902
Model: OLS Adj. R-squared: 0.900
Method: Least Squares F-statistic: 366.4
Date: Mon, 10 Jun 2019 Prob (F-statistic): 0.00
Time: 08:49:30 Log-Likelihood: 670.21
No. Observations: 1020 AIC: -1288.
Df Residuals: 994 BIC: -1160.
Df Model: 25
Covariance Type: nonrobust
================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------
const 14.3594 6.164 2.330 0.020 2.264 26.455
BedroomAbvGr -0.0268 0.008 -3.566 0.000 -0.042 -0.012
BsmtFullBath 0.0732 0.009 8.362 0.000 0.056 0.090
BsmtHalfBath 0.0068 0.017 0.398 0.691 -0.027 0.040
Fireplaces 0.0437 0.007 5.896 0.000 0.029 0.058
FullBath 0.0198 0.012 1.637 0.102 -0.004 0.043
GarageArea 9.434e-05 4.31e-05 2.187 0.029 9.67e-06 0.000
GarageCars 0.0318 0.014 2.340 0.019 0.005 0.058
GarageYrBlt 3.075e-05 1.18e-05 2.607 0.009 7.6e-06 5.39e-05
GrLivArea 0.0003 2.03e-05 12.860 0.000 0.000 0.000
HalfBath 0.0210 0.010 2.030 0.043 0.001 0.041
KitchenAbvGr -0.0648 0.023 -2.815 0.005 -0.110 -0.020
LotArea 1.987e-06 3.92e-07 5.073 0.000 1.22e-06 2.76e-06
LotFrontage 0.0004 0.000 1.890 0.059 -1.44e-05 0.001
MSSubClass -0.0004 0.000 -3.422 0.001 -0.001 -0.000
MasVnrArea 1.588e-05 2.78e-05 0.570 0.568 -3.87e-05 7.05e-05
MiscVal -2.386e-05 1.24e-05 -1.925 0.055 -4.82e-05 4.65e-07
MoSold 0.0011 0.002 0.701 0.484 -0.002 0.004
OverallCond 0.0479 0.004 10.842 0.000 0.039 0.057
OverallQual 0.0677 0.005 13.115 0.000 0.058 0.078
PoolArea 3.677e-05 0.000 0.291 0.771 -0.000 0.000
TotRmsAbvGrd 0.0155 0.005 2.849 0.004 0.005 0.026
TotalBsmtSF 0.0001 1.43e-05 9.789 0.000 0.000 0.000
YearBuilt 0.0029 0.000 11.700 0.000 0.002 0.003
YearRemodAdd 0.0011 0.000 3.593 0.000 0.000 0.002
YrSold -0.0058 0.003 -1.873 0.061 -0.012 0.000
Having all numerical variables included as predictors, we have an adjusted R2 of .9 and test error of .885.
Train error is: 0.902
Test error is: 0.885
To determine the coefficients with 0 lying within confidence interval, we issue the command below:
table[table['p value']>=0.05][['name','coef','p value']]
name coef p value
3 BsmtHalfBath 0.006800 0.691
5 FullBath 0.019800 0.102
13 LotFrontage 0.000400 0.059
15 MasVnrArea 0.000016 0.568
16 MiscVal -0.000024 0.055
17 MoSold 0.001100 0.484
20 PoolArea 0.000037 0.771
25 YrSold -0.005800 0.061
After refitting the multiple linear regression model using only the significan coefficients, we find that train error is a bit lower but the test error is a bit higher. It is not much improvement as we expected but still we decide to go on with the reduced model.
print('Train error: %.5f' %mlr.score(X_train_significants, y_train))
print('Test error: %.5f' %mlr.score(X_test_significants, y_test))
Train error: 0.89998
Test error: 0.88699
Ridge
Ridge is a regularized model so that it needs to be tuned for hyper parameters. Penalty term (alpha) is a hyper parameter to tune. and if it is large, model has less variance but it may suffer from high bias. We use cross-validation to find the best alpha considering different combinations of normalization and intercept parameters.
- Alpha range = linspace(0.01, 5, 200)
- Cross-validation folds = [3, 5, 10, sqrt_n]
- Normalization = [False, True]
- Intercept = [False, True]
200 alpha X 4 CV X 2 Norm X 2 Intercept = 3200 config
Below table shows the training and test R2 values, mean squared error and average cross-validation error for each combination of cross-validation folds, normalization and intercept. Values for CV are experimentally proven that they are the best candidates to find the optimum for the variance-bias tradeoff. P.S. sqrt_n is the square root of number of observations. The strategy here is to choose the alpha giving the maximum R2 for the test data set. Hence, we choose;
alpha=0.41, CV=10, normalization=TRUE, intercept=FALSE
CV |
Norm Type |
Intercept Type |
Alpha |
R2 Training |
R2 Test |
MSE |
Avg. CV Score |
CV_3 |
TRUE |
TRUE |
0.34 |
0.94307 |
0.86306 |
0.01809 |
0.90146 |
CV_3 |
TRUE |
FALSE |
0.39 |
0.94784 |
0.89691 |
0.02134 |
0.85406 |
CV_3 |
FALSE |
TRUE |
5 |
0.94958 |
0.86306 |
0.01611 |
0.90035 |
CV_3 |
FALSE |
FALSE |
0.39 |
0.94784 |
0.88388 |
0.02134 |
0.85406 |
CV_5 |
TRUE |
TRUE |
0.21 |
0.94769 |
0.86303 |
0.01805 |
0.90681 |
CV_5 |
TRUE |
FALSE |
0.29 |
0.95005 |
0.89718 |
0.02135 |
0.86948 |
CV_5 |
FALSE |
TRUE |
4.17 |
0.9507 |
0.86303 |
0.01607 |
0.91139 |
CV_5 |
FALSE |
FALSE |
0.29 |
0.95005 |
0.88419 |
0.02135 |
0.86948 |
CV_10 |
TRUE |
TRUE |
0.16 |
0.94968 |
0.86299 |
0.01808 |
0.90625 |
CV_10 |
TRUE |
FALSE |
0.41 |
0.94736 |
0.89738 |
0.02135 |
0.86789 |
CV_10 |
FALSE |
TRUE |
3.27 |
0.95204 |
0.86299 |
0.01604 |
0.91051 |
CV_10 |
FALSE |
FALSE |
0.41 |
0.94736 |
0.88408 |
0.02135 |
0.86789 |
CV_31 |
TRUE |
TRUE |
0.16 |
0.94968 |
0.86299 |
0.01808 |
0.90048 |
CV_31 |
TRUE |
FALSE |
0.41 |
0.94736 |
0.89736 |
0.02135 |
0.8682 |
CV_31 |
FALSE |
TRUE |
3.5 |
0.95169 |
0.86299 |
0.01604 |
0.90696 |
CV_31 |
FALSE |
FALSE |
0.41 |
0.94736 |
0.88408 |
0.02135 |
0.8682 |
Lasso
Lasso is another regularized model. Only difference from Ridge is Lasso has L1 penalty such that when alpha is large enough, coefficients of the predictors are pushed to become 0. That is why Lasso allows for predictor selection. We use the same approach as we did for Ridge and the best alpha we get is:
alpha=0.16, CV=5, normalization=TRUE, intercept=FALSE
ElasticNet
ElasticNet is a combination of Ridge and Lasso and introduces an additional hyper parameter to tune. Rho specifies how much Ridge (or Lasso) one intend to use in the model.
- Alpha range = np.linspace(0.1, 10, 100)
- Rho range = np.linspace(0.01, 1, 100)
- Cross-validation folds = [3, 5, 10, sqrt_n]
- Normalization = [False, True]
- Intercept = [False, True]
- P.S. Set max_iter = 10,000 to solve convergence problem
100 Alpha X 100 Rho X 4 CV X 2 Norm X 2 Inter = 160K config
Best hyper parameters we get:
alpha=0.01, rho=0.01, CV=10, normalization=TRUE, intercept=FALSE
Random Forest
Random forest does not need categorical columns to be dummified so that we omit them in this model. It also has a lot of parameters to tune. Below are the ones we find important:
- Randomly select samples
- bootstrap = [True, False]
- Max. # of levels in tree
- max_depth = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, None]
- Max. # of features to consider at each split
- max_features = ['auto', 'sqrt’]
- Min. # of samples required to split node
- min_samples_leaf = [1, 2, 4]
- Min. # of samples required to split node
- min_samples_split = [2, 5, 10]
- # of Trees
- n_estimators = [200, 400, 600, 800, 1000, 1200, 1400, 1600, 1800, 2000]
2 BS X 12 Levels X 2 Feat X 3 Leafs X 3 Samples X 10 Estimators = 4320 config
Using grid search to find the best parameters, we end up with the below summary table:
CV |
R2 Training |
R2 Test |
Avg CV Score |
MSE |
CV_3 |
0.94598 |
0.7622 |
0.723 |
0.03715 |
CV_5 |
0.946 |
0.76201 |
0.74271 |
0.03718 |
CV_10 |
0.94571 |
0.76276 |
0.72345 |
0.03706 |
CV_31 |
0.94582 |
0.76377 |
0.70945 |
0.03691 |
Since scores are very close to each other, we employ CV=5 and CV=31 which provides best average CV score and best test R2 score, respectively.
Model Comparison
We compare the models we fit using different metrics such as test R2, MSE and CV scores:
Model Results
Here are the best predictions results submitted for each model:
SubmissionNo |
Model |
Parameters |
Score |
16 |
Ridge |
CV=10, Norm=T, Int=F, alpha=0.41 |
0.12757 |
19 |
ElasticNet |
CV=10, Norm=T, Int=F, alpha=0.01, rho=0.01 |
0.12866 |
11 |
Lasso |
CV=5, Norm=T, Int=F, alpha=0.01 |
0.12946 |
15 |
RandomForest |
CV=5, trees=1000 |
0.14313 |
In general, Ridge performed the best and random forest performed the worst. Our best score is 0.12757 with Ridge.
Future Work
We believe there may be some improvements both in the feature engineering and modelling parts of the project. If we are given more time, we think that new features can be added to the model and then we can check for any improvement happen in terms of test error. Another improvement point is that we employ XGBoost but do not have enough time to find the best parameters. We think, XGBoost can give better results when tuned properly. Lastly, ensembling can be applied to the trained models, so that a combination of each can be implemented as a final model. Ensembling may allow for better predictions since each model learns a different part of the data.