House Price Prediction

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Overview

This notebook investigates the use of machine learning model, namely linear regression, to predict house price using the following dataset.

Download Dataset (.csv)

The analysis covers these main steps:

1. Data preprocessing and Exploratory Data Analysis (EDA)
2. Statistical analysis for feature selection
3. Model training
4. Model evaluation

The primary objective is to understand how to perform end-to-end machine learning process.

Install and Import Packages

%pip install --upgrade -q matplotlib numpy pandas scikit-learn scipy seaborn

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import seaborn as sns

Load Dataset

df = pd.read_csv('data/house_price.csv')
df.head()

	Id	MSSubClass	MSZoning	LotFrontage	LotArea	Street	Alley	LotShape	LandContour	Utilities	...	PoolArea	PoolQC	Fence	MiscFeature	MiscVal	MoSold	YrSold	SaleType	SaleCondition	SalePrice
0	1	60	RL	65.0	8450	Pave	NaN	Reg	Lvl	AllPub	...	0	NaN	NaN	NaN	0	2	2008	WD	Normal	208500
1	2	20	RL	80.0	9600	Pave	NaN	Reg	Lvl	AllPub	...	0	NaN	NaN	NaN	0	5	2007	WD	Normal	181500
2	3	60	RL	68.0	11250	Pave	NaN	IR1	Lvl	AllPub	...	0	NaN	NaN	NaN	0	9	2008	WD	Normal	223500
3	4	70	RL	60.0	9550	Pave	NaN	IR1	Lvl	AllPub	...	0	NaN	NaN	NaN	0	2	2006	WD	Abnorml	140000
4	5	60	RL	84.0	14260	Pave	NaN	IR1	Lvl	AllPub	...	0	NaN	NaN	NaN	0	12	2008	WD	Normal	250000

5 rows × 81 columns

Data Preprocessing and EDA

Data Preprocessing Phase 1

First, let’s take a look at the dataset overview.

df.info()

<class 'pandas.DataFrame'>
RangeIndex: 1460 entries, 0 to 1459
Data columns (total 81 columns):
 #   Column         Non-Null Count  Dtype  
---  ------         --------------  -----  
 0   Id             1460 non-null   int64  
 1   MSSubClass     1460 non-null   int64  
 2   MSZoning       1460 non-null   str    
 3   LotFrontage    1201 non-null   float64
 4   LotArea        1460 non-null   int64  
 5   Street         1460 non-null   str    
 6   Alley          91 non-null     str    
 7   LotShape       1460 non-null   str    
 8   LandContour    1460 non-null   str    
 9   Utilities      1460 non-null   str    
 10  LotConfig      1460 non-null   str    
 11  LandSlope      1460 non-null   str    
 12  Neighborhood   1460 non-null   str    
 13  Condition1     1460 non-null   str    
 14  Condition2     1460 non-null   str    
 15  BldgType       1460 non-null   str    
 16  HouseStyle     1460 non-null   str    
 17  OverallQual    1460 non-null   int64  
 18  OverallCond    1460 non-null   int64  
 19  YearBuilt      1460 non-null   int64  
 20  YearRemodAdd   1460 non-null   int64  
 21  RoofStyle      1460 non-null   str    
 22  RoofMatl       1460 non-null   str    
 23  Exterior1st    1460 non-null   str    
 24  Exterior2nd    1460 non-null   str    
 25  MasVnrType     588 non-null    str    
 26  MasVnrArea     1452 non-null   float64
 27  ExterQual      1460 non-null   str    
 28  ExterCond      1460 non-null   str    
 29  Foundation     1460 non-null   str    
 30  BsmtQual       1423 non-null   str    
 31  BsmtCond       1423 non-null   str    
 32  BsmtExposure   1422 non-null   str    
 33  BsmtFinType1   1423 non-null   str    
 34  BsmtFinSF1     1460 non-null   int64  
 35  BsmtFinType2   1422 non-null   str    
 36  BsmtFinSF2     1460 non-null   int64  
 37  BsmtUnfSF      1460 non-null   int64  
 38  TotalBsmtSF    1460 non-null   int64  
 39  Heating        1460 non-null   str    
 40  HeatingQC      1460 non-null   str    
 41  CentralAir     1460 non-null   str    
 42  Electrical     1459 non-null   str    
 43  1stFlrSF       1460 non-null   int64  
 44  2ndFlrSF       1460 non-null   int64  
 45  LowQualFinSF   1460 non-null   int64  
 46  GrLivArea      1460 non-null   int64  
 47  BsmtFullBath   1460 non-null   int64  
 48  BsmtHalfBath   1460 non-null   int64  
 49  FullBath       1460 non-null   int64  
 50  HalfBath       1460 non-null   int64  
 51  BedroomAbvGr   1460 non-null   int64  
 52  KitchenAbvGr   1460 non-null   int64  
 53  KitchenQual    1460 non-null   str    
 54  TotRmsAbvGrd   1460 non-null   int64  
 55  Functional     1460 non-null   str    
 56  Fireplaces     1460 non-null   int64  
 57  FireplaceQu    770 non-null    str    
 58  GarageType     1379 non-null   str    
 59  GarageYrBlt    1379 non-null   float64
 60  GarageFinish   1379 non-null   str    
 61  GarageCars     1460 non-null   int64  
 62  GarageArea     1460 non-null   int64  
 63  GarageQual     1379 non-null   str    
 64  GarageCond     1379 non-null   str    
 65  PavedDrive     1460 non-null   str    
 66  WoodDeckSF     1460 non-null   int64  
 67  OpenPorchSF    1460 non-null   int64  
 68  EnclosedPorch  1460 non-null   int64  
 69  3SsnPorch      1460 non-null   int64  
 70  ScreenPorch    1460 non-null   int64  
 71  PoolArea       1460 non-null   int64  
 72  PoolQC         7 non-null      str    
 73  Fence          281 non-null    str    
 74  MiscFeature    54 non-null     str    
 75  MiscVal        1460 non-null   int64  
 76  MoSold         1460 non-null   int64  
 77  YrSold         1460 non-null   int64  
 78  SaleType       1460 non-null   str    
 79  SaleCondition  1460 non-null   str    
 80  SalePrice      1460 non-null   int64  
dtypes: float64(3), int64(35), str(43)
memory usage: 924.0 KB

Based on the output above, here are some informations about the dataset. - The dataset has 1460 rows and 81 columns. - The target variable is ‘SalePrice’. - The features consist of both numerical and categorical features. - Notice that there are some features with missing values.

Let’s also see the distribution of the target variable (SalePrice).

df['SalePrice'].plot(kind='hist')

Dropping Unnecessary Columns

# drop column 'Id' because it is same as an index
try:
    df = df.drop(columns=['Id'])
    print('Successfully Removed Column Id.')
except KeyError:
    print('Column Id Not Found or Has Been Removed.')

Successfully Removed Column Id.

Checking dan Dropping Duplicated Rows

# check and drop duplicated rows (if exist)
total_duplicates = df.duplicated().sum()
print(f'Total duplicated rows: {total_duplicates}')
if total_duplicates:
    df = df.drop_duplicates()
    print('All duplicated rows has been removed.')
else:
    print('No duplicated rows need to be dropped.')

Total duplicated rows: 0
No duplicated rows need to be dropped.

Now, let’s focus on identifying which features have missing values and each missing values counts.

Checking Missing Values

check_missing_values_series = df.isna().sum().sort_values()
missing_values_series = check_missing_values_series[check_missing_values_series > 0]

missing_values_columns = missing_values_series.index.tolist()

missing_dtypes = [str(df[column].dtypes) for column in missing_values_columns]
missing_dtypes_series = pd.Series(data=missing_dtypes, index=missing_values_columns)

missing_unique = [df[column].unique().tolist() for column in missing_values_columns]
missing_unique_series = pd.Series(data=missing_unique, index=missing_values_columns)

missing_values_df = pd.concat([missing_values_series, missing_dtypes_series, missing_unique_series], axis=1)
missing_values_df = missing_values_df.rename(columns={0: 'Missing Values Counts', 1: 'Dtype', 2: 'Unique Values'})
print(missing_values_df)
print()
print(f'Total Columns with Missing Values: {len(missing_values_df)}')

              Missing Values Counts    Dtype  \
Electrical                        1      str   
MasVnrArea                        8  float64   
BsmtFinType1                     37      str   
BsmtCond                         37      str   
BsmtQual                         37      str   
BsmtFinType2                     38      str   
BsmtExposure                     38      str   
GarageType                       81      str   
GarageCond                       81      str   
GarageFinish                     81      str   
GarageQual                       81      str   
GarageYrBlt                      81  float64   
LotFrontage                     259  float64   
FireplaceQu                     690      str   
MasVnrType                      872      str   
Fence                          1179      str   
Alley                          1369      str   
MiscFeature                    1406      str   
PoolQC                         1453      str   

                                                  Unique Values  
Electrical               [SBrkr, FuseF, FuseA, FuseP, Mix, nan]  
MasVnrArea    [196.0, 0.0, 162.0, 350.0, 186.0, 240.0, 286.0...  
BsmtFinType1                [GLQ, ALQ, Unf, Rec, BLQ, nan, LwQ]  
BsmtCond                                  [TA, Gd, nan, Fa, Po]  
BsmtQual                                  [Gd, TA, Ex, nan, Fa]  
BsmtFinType2                [Unf, BLQ, nan, ALQ, Rec, LwQ, GLQ]  
BsmtExposure                              [No, Gd, Mn, Av, nan]  
GarageType    [Attchd, Detchd, BuiltIn, CarPort, nan, Basmen...  
GarageCond                            [TA, Fa, nan, Gd, Po, Ex]  
GarageFinish                               [RFn, Unf, Fin, nan]  
GarageQual                            [TA, Fa, Gd, nan, Ex, Po]  
GarageYrBlt   [2003.0, 1976.0, 2001.0, 1998.0, 2000.0, 1993....  
LotFrontage   [65.0, 80.0, 68.0, 60.0, 84.0, 85.0, 75.0, nan...  
FireplaceQu                           [nan, TA, Gd, Fa, Ex, Po]  
MasVnrType                        [BrkFace, nan, Stone, BrkCmn]  
Fence                           [nan, MnPrv, GdWo, GdPrv, MnWw]  
Alley                                         [nan, Grvl, Pave]  
MiscFeature                       [nan, Shed, Gar2, Othr, TenC]  
PoolQC                                        [nan, Ex, Fa, Gd]  

Total Columns with Missing Values: 19

Based on the output above, there are 19 columns with missing values.

Here are the strategies for handling those missing values.

• Drop the GarageYrBlt column. From a logical and temporal standpoint, the construction year of a garage functions as a close linear approximation of the primary structure’s construction year (YearBuilt), as residential garages are typically built concurrently with the main house. Imputing the missing values (which mathematically represent the absence of a garage) with a continuous year would introduce noise. Therefore, treating GarageYrBlt as a redundant temporal proxy and dropping it is a mathematically sound decision, as the variance of the property’s age is already rigorously accounted for by YearBuilt.

• Impute the numerical features (MasVnrArea, LotFrontage) using the median to mitigate the influence of extreme values.

• Impute the categorical features using a constant value (‘None’) since the missing values systematically represent the absence of a specific house facility (e.g., no fence, no alley), making ‘None’ a structurally meaningful category.

• The only exception is the Electrical feature, where the single missing value is imputed using the mode (most frequent value), assuming the missing entry is a data entry omission rather than the complete absence of an electrical system.

Handling Missing Values

Train-Test Split

Before handling the missing values, split the the data into train and test set. Here, we are using 80:20 ratio for the train and test set.

We must split our data first, to avoid data leakage

from sklearn.model_selection import train_test_split

# drop GarageYrBlt
try:
    df = df.drop(columns=['GarageYrBlt'])
    try:
        missing_values_columns.remove('GarageYrBlt')
    except:
        pass
    print('Successfully Removed Column GarageYrBlt.')
except KeyError:
    print('Column GarageYrBlt Not Found or Has Been Removed.')

# separate the target variable
target = 'SalePrice'
y = df[target].values

X_df = df.drop(columns=[target])

# split the data into train and test set
X_train_df, X_test_df, y_train, y_test = train_test_split(X_df, y, test_size=0.2, random_state=42)
print(f'X_train_df shape: {X_train_df.shape}')
print(f'X_test_df shape: {X_test_df.shape}')
print(f'y_train shape: {y_train.shape}')
print(f'y_test shape: {y_test.shape}')

Successfully Removed Column GarageYrBlt.
X_train_df shape: (1168, 78)
X_test_df shape: (292, 78)
y_train shape: (1168,)
y_test shape: (292,)

Divide the data into the column without missing values, column with numerical missing values, column with categorical missing values, and column with special treatment categorical missing values.

missing_num_columns = ['MasVnrArea', 'LotFrontage']
missing_special_cat_columns = ['Electrical']
missing_cat_columns = set(missing_values_columns.copy())
remove_from_missing_cat = set(missing_num_columns + missing_special_cat_columns)
missing_cat_columns = list(missing_cat_columns - remove_from_missing_cat)

not_missing_columns = df.drop(columns=missing_values_columns+[target]).columns.tolist()

X_train_not_missing = X_train_df[not_missing_columns].values
X_test_not_missing = X_test_df[not_missing_columns].values

X_train_missing_num = X_train_df[missing_num_columns].values
X_test_missing_num = X_test_df[missing_num_columns].values

X_train_missing_cat = X_train_df[missing_cat_columns].values
X_test_missing_cat = X_test_df[missing_cat_columns].values

X_train_missing_special_cat = X_train_df[missing_special_cat_columns].values
X_test_missing_special_cat = X_test_df[missing_special_cat_columns].values

print(f'X_train_not_missing shape: {X_train_not_missing.shape}')
print(f'X_test_not_missing shape: {X_test_not_missing.shape}')
print(f'X_train_missing_num shape: {X_train_missing_num.shape}')
print(f'X_test_missing_num shape: {X_test_missing_num.shape}')
print(f'X_train_missing_cat shape: {X_train_missing_cat.shape}')
print(f'X_test_missing_cat shape: {X_test_missing_cat.shape}')
print(f'X_train_missing_special_cat shape: {X_train_missing_special_cat.shape}')
print(f'X_test_missing_special_cat shape: {X_test_missing_special_cat.shape}')

X_train_not_missing shape: (1168, 60)
X_test_not_missing shape: (292, 60)
X_train_missing_num shape: (1168, 2)
X_test_missing_num shape: (292, 2)
X_train_missing_cat shape: (1168, 15)
X_test_missing_cat shape: (292, 15)
X_train_missing_special_cat shape: (1168, 1)
X_test_missing_special_cat shape: (292, 1)

Using SimpleImputer

Then, we can do the imputation using SimpleImputer from the scikit-learn.

from sklearn.impute import SimpleImputer

# create each imputer instance
num_imputer = SimpleImputer(strategy='median')
cat_imputer = SimpleImputer(strategy='constant', fill_value='None')
special_cat_imputer = SimpleImputer(strategy='most_frequent')

# fit-transform to the missing train set and only transform to the missing test set to avoid data leakage
X_train_imputed_num = num_imputer.fit_transform(X_train_missing_num)
X_test_imputed_num = num_imputer.transform(X_test_missing_num)

X_train_imputed_cat = cat_imputer.fit_transform(X_train_missing_cat)
X_test_imputed_cat = cat_imputer.transform(X_test_missing_cat)

X_train_imputed_special_cat = special_cat_imputer.fit_transform(X_train_missing_special_cat)
X_test_imputed_special_cat = special_cat_imputer.transform(X_test_missing_special_cat)

# concatenate all train and test set
X_train_imputed = np.concatenate((X_train_not_missing, X_train_imputed_num, X_train_imputed_cat, X_train_imputed_special_cat), axis=1)
X_test_imputed = np.concatenate((X_test_not_missing, X_test_imputed_num, X_test_imputed_cat, X_test_imputed_special_cat), axis=1)

print(f'X_train_imputed shape: {X_train_imputed.shape}')
print(f'X_test_imputed shape: {X_test_imputed.shape}')

X_train_imputed shape: (1168, 78)
X_test_imputed shape: (292, 78)

# convert X_train_imputed and X_test_imputed to DataFrame
current_all_columns = not_missing_columns + missing_num_columns + missing_cat_columns + missing_special_cat_columns
X_train_imputed_df = pd.DataFrame(data=X_train_imputed, columns=current_all_columns)
X_test_imputed_df = pd.DataFrame(data=X_test_imputed, columns=current_all_columns)

X_train_imputed_df.info()

<class 'pandas.DataFrame'>
RangeIndex: 1168 entries, 0 to 1167
Data columns (total 78 columns):
 #   Column         Non-Null Count  Dtype 
---  ------         --------------  ----- 
 0   MSSubClass     1168 non-null   object
 1   MSZoning       1168 non-null   str   
 2   LotArea        1168 non-null   object
 3   Street         1168 non-null   str   
 4   LotShape       1168 non-null   str   
 5   LandContour    1168 non-null   str   
 6   Utilities      1168 non-null   str   
 7   LotConfig      1168 non-null   str   
 8   LandSlope      1168 non-null   str   
 9   Neighborhood   1168 non-null   str   
 10  Condition1     1168 non-null   str   
 11  Condition2     1168 non-null   str   
 12  BldgType       1168 non-null   str   
 13  HouseStyle     1168 non-null   str   
 14  OverallQual    1168 non-null   object
 15  OverallCond    1168 non-null   object
 16  YearBuilt      1168 non-null   object
 17  YearRemodAdd   1168 non-null   object
 18  RoofStyle      1168 non-null   str   
 19  RoofMatl       1168 non-null   str   
 20  Exterior1st    1168 non-null   str   
 21  Exterior2nd    1168 non-null   str   
 22  ExterQual      1168 non-null   str   
 23  ExterCond      1168 non-null   str   
 24  Foundation     1168 non-null   str   
 25  BsmtFinSF1     1168 non-null   object
 26  BsmtFinSF2     1168 non-null   object
 27  BsmtUnfSF      1168 non-null   object
 28  TotalBsmtSF    1168 non-null   object
 29  Heating        1168 non-null   str   
 30  HeatingQC      1168 non-null   str   
 31  CentralAir     1168 non-null   str   
 32  1stFlrSF       1168 non-null   object
 33  2ndFlrSF       1168 non-null   object
 34  LowQualFinSF   1168 non-null   object
 35  GrLivArea      1168 non-null   object
 36  BsmtFullBath   1168 non-null   object
 37  BsmtHalfBath   1168 non-null   object
 38  FullBath       1168 non-null   object
 39  HalfBath       1168 non-null   object
 40  BedroomAbvGr   1168 non-null   object
 41  KitchenAbvGr   1168 non-null   object
 42  KitchenQual    1168 non-null   str   
 43  TotRmsAbvGrd   1168 non-null   object
 44  Functional     1168 non-null   str   
 45  Fireplaces     1168 non-null   object
 46  GarageCars     1168 non-null   object
 47  GarageArea     1168 non-null   object
 48  PavedDrive     1168 non-null   str   
 49  WoodDeckSF     1168 non-null   object
 50  OpenPorchSF    1168 non-null   object
 51  EnclosedPorch  1168 non-null   object
 52  3SsnPorch      1168 non-null   object
 53  ScreenPorch    1168 non-null   object
 54  PoolArea       1168 non-null   object
 55  MiscVal        1168 non-null   object
 56  MoSold         1168 non-null   object
 57  YrSold         1168 non-null   object
 58  SaleType       1168 non-null   str   
 59  SaleCondition  1168 non-null   str   
 60  MasVnrArea     1168 non-null   object
 61  LotFrontage    1168 non-null   object
 62  BsmtCond       1168 non-null   str   
 63  MasVnrType     1168 non-null   str   
 64  GarageType     1168 non-null   str   
 65  GarageQual     1168 non-null   str   
 66  Fence          1168 non-null   str   
 67  BsmtFinType1   1168 non-null   str   
 68  BsmtExposure   1168 non-null   str   
 69  FireplaceQu    1168 non-null   str   
 70  GarageCond     1168 non-null   str   
 71  GarageFinish   1168 non-null   str   
 72  Alley          1168 non-null   str   
 73  BsmtFinType2   1168 non-null   str   
 74  BsmtQual       1168 non-null   str   
 75  PoolQC         1168 non-null   str   
 76  MiscFeature    1168 non-null   str   
 77  Electrical     1168 non-null   str   
dtypes: object(35), str(43)
memory usage: 711.9+ KB

X_test_imputed_df.info()

<class 'pandas.DataFrame'>
RangeIndex: 292 entries, 0 to 291
Data columns (total 78 columns):
 #   Column         Non-Null Count  Dtype 
---  ------         --------------  ----- 
 0   MSSubClass     292 non-null    object
 1   MSZoning       292 non-null    str   
 2   LotArea        292 non-null    object
 3   Street         292 non-null    str   
 4   LotShape       292 non-null    str   
 5   LandContour    292 non-null    str   
 6   Utilities      292 non-null    str   
 7   LotConfig      292 non-null    str   
 8   LandSlope      292 non-null    str   
 9   Neighborhood   292 non-null    str   
 10  Condition1     292 non-null    str   
 11  Condition2     292 non-null    str   
 12  BldgType       292 non-null    str   
 13  HouseStyle     292 non-null    str   
 14  OverallQual    292 non-null    object
 15  OverallCond    292 non-null    object
 16  YearBuilt      292 non-null    object
 17  YearRemodAdd   292 non-null    object
 18  RoofStyle      292 non-null    str   
 19  RoofMatl       292 non-null    str   
 20  Exterior1st    292 non-null    str   
 21  Exterior2nd    292 non-null    str   
 22  ExterQual      292 non-null    str   
 23  ExterCond      292 non-null    str   
 24  Foundation     292 non-null    str   
 25  BsmtFinSF1     292 non-null    object
 26  BsmtFinSF2     292 non-null    object
 27  BsmtUnfSF      292 non-null    object
 28  TotalBsmtSF    292 non-null    object
 29  Heating        292 non-null    str   
 30  HeatingQC      292 non-null    str   
 31  CentralAir     292 non-null    str   
 32  1stFlrSF       292 non-null    object
 33  2ndFlrSF       292 non-null    object
 34  LowQualFinSF   292 non-null    object
 35  GrLivArea      292 non-null    object
 36  BsmtFullBath   292 non-null    object
 37  BsmtHalfBath   292 non-null    object
 38  FullBath       292 non-null    object
 39  HalfBath       292 non-null    object
 40  BedroomAbvGr   292 non-null    object
 41  KitchenAbvGr   292 non-null    object
 42  KitchenQual    292 non-null    str   
 43  TotRmsAbvGrd   292 non-null    object
 44  Functional     292 non-null    str   
 45  Fireplaces     292 non-null    object
 46  GarageCars     292 non-null    object
 47  GarageArea     292 non-null    object
 48  PavedDrive     292 non-null    str   
 49  WoodDeckSF     292 non-null    object
 50  OpenPorchSF    292 non-null    object
 51  EnclosedPorch  292 non-null    object
 52  3SsnPorch      292 non-null    object
 53  ScreenPorch    292 non-null    object
 54  PoolArea       292 non-null    object
 55  MiscVal        292 non-null    object
 56  MoSold         292 non-null    object
 57  YrSold         292 non-null    object
 58  SaleType       292 non-null    str   
 59  SaleCondition  292 non-null    str   
 60  MasVnrArea     292 non-null    object
 61  LotFrontage    292 non-null    object
 62  BsmtCond       292 non-null    str   
 63  MasVnrType     292 non-null    str   
 64  GarageType     292 non-null    str   
 65  GarageQual     292 non-null    str   
 66  Fence          292 non-null    str   
 67  BsmtFinType1   292 non-null    str   
 68  BsmtExposure   292 non-null    str   
 69  FireplaceQu    292 non-null    str   
 70  GarageCond     292 non-null    str   
 71  GarageFinish   292 non-null    str   
 72  Alley          292 non-null    str   
 73  BsmtFinType2   292 non-null    str   
 74  BsmtQual       292 non-null    str   
 75  PoolQC         292 non-null    str   
 76  MiscFeature    292 non-null    str   
 77  Electrical     292 non-null    str   
dtypes: object(35), str(43)
memory usage: 178.1+ KB

For the next exploratory data analysis, it is conducted exclusively on the Training Set. The Test Set remains completely untouched and unseen, acting as pure out-of-sample data to ensure an unbiased evaluation of the final regression model.

Feature Engineering

Now, we are going to do some feature engineering. Before that, let’s take a look at the unique values and the mode of current each column.

print('-----Unique Values and Mode in Each Column-----\n\n')
for column in current_all_columns:
    print(f'Unique Values in {column}:')
    print(X_train_imputed_df[column].unique().tolist())
    print()
    print(f'Mode: {X_train_imputed_df[column].mode()[0]}')
    print('='*50)

-----Unique Values and Mode in Each Column-----


Unique Values in MSSubClass:
[20, 60, 30, 50, 85, 80, 45, 90, 160, 40, 190, 75, 120, 70, 180]

Mode: 20
==================================================
Unique Values in MSZoning:
['RL', 'RM', 'FV', 'RH', 'C (all)']

Mode: RL
==================================================
Unique Values in LotArea:
[8400, 7837, 8777, 7200, 5000, 9800, 11911, 5436, 12090, 9158, 13400, 11880, 13600, 9600, 8068, 21000, 8235, 7950, 12416, 10594, 11643, 6240, 10237, 10083, 4251, 17500, 8544, 9541, 12144, 5604, 10240, 9991, 7252, 12160, 9353, 7018, 13284, 6600, 7060, 3604, 8892, 5362, 4456, 11639, 8000, 7082, 14054, 11003, 11846, 8520, 11029, 10246, 13500, 13693, 3842, 11200, 10475, 12772, 10205, 6342, 7500, 10437, 12046, 10762, 9375, 29959, 9204, 3182, 10140, 9060, 12342, 9750, 11767, 8121, 7134, 9056, 9790, 7800, 16560, 11751, 1680, 10603, 9187, 32463, 11207, 14112, 13710, 9554, 8712, 8100, 11143, 9786, 6000, 21750, 4928, 11694, 11409, 14948, 2500, 11249, 12358, 8029, 46589, 8525, 8250, 10500, 25000, 10800, 5001, 10186, 5330, 9591, 7931, 20896, 9560, 11308, 12461, 9505, 9549, 5063, 11700, 13125, 5687, 6371, 9967, 8281, 8973, 16647, 8176, 5820, 8125, 12354, 13518, 10400, 3675, 8640, 11670, 10667, 9020, 11218, 13825, 13031, 12552, 7390, 9120, 15863, 11250, 8814, 11198, 11050, 9473, 13515, 11425, 4280, 12640, 10264, 11341, 15623, 11952, 9571, 11923, 11841, 7990, 7227, 11214, 14684, 11625, 21286, 8405, 20544, 15523, 9638, 8800, 8402, 8688, 10289, 10010, 9965, 5389, 10768, 14331, 24682, 10142, 9084, 4130, 17043, 17423, 8780, 7763, 11600, 13175, 10440, 10084, 9100, 8773, 8470, 8393, 11146, 9000, 11194, 7340, 15611, 13101, 2448, 17104, 6292, 12384, 10480, 5100, 8385, 8198, 4426, 11065, 11844, 10316, 11888, 9900, 7407, 8885, 12968, 31770, 19296, 8816, 12257, 9066, 10182, 5119, 6768, 15426, 12936, 13472, 14000, 10625, 5600, 13132, 10192, 14859, 2544, 15602, 9246, 9550, 14774, 9480, 12099, 14115, 9037, 9230, 6040, 10355, 13517, 10463, 11839, 10000, 8910, 9236, 14720, 8846, 12508, 10769, 11500, 11075, 9738, 3600, 1869, 10357, 8877, 8487, 16545, 19900, 6762, 12150, 7711, 10420, 5700, 8963, 8924, 16059, 8190, 11414, 5900, 4590, 25339, 7415, 9360, 6305, 7301, 7064, 14175, 9240, 11584, 10011, 7560, 11478, 11361, 11216, 3635, 9808, 21384, 8791, 14230, 14157, 13159, 11096, 20431, 10175, 8854, 13568, 21453, 8900, 14892, 10141, 12886, 8935, 7658, 18890, 40094, 6130, 6970, 11235, 11616, 3072, 8944, 11435, 11040, 6173, 13072, 13360, 18450, 7314, 7180, 11287, 12011, 10918, 7917, 8012, 10159, 9525, 9180, 2628, 3500, 8472, 13837, 8820, 7128, 10994, 10991, 11426, 11070, 11999, 13000, 3180, 11000, 115149, 16033, 10106, 11317, 7920, 16492, 4671, 9042, 10921, 13286, 9920, 6993, 1890, 2217, 13173, 7875, 9986, 13704, 7910, 2645, 9382, 9340, 10386, 8536, 8335, 14442, 159000, 9500, 7599, 9337, 11526, 10552, 10852, 8600, 9855, 12665, 7100, 11717, 8635, 8530, 10206, 53227, 8760, 8993, 4400, 9672, 12537, 10364, 23595, 8755, 14375, 7758, 7943, 3964, 15865, 6442, 11316, 12030, 2522, 12513, 10634, 12376, 6882, 5350, 8445, 7420, 9018, 8738, 4435, 12474, 13728, 63887, 9503, 8004, 8172, 20781, 9803, 10152, 9313, 5250, 9590, 11423, 13650, 7740, 6420, 9439, 7750, 12546, 17217, 10382, 7226, 4224, 9101, 10880, 6402, 8500, 9764, 10574, 12435, 12180, 7175, 9636, 23257, 9650, 2160, 15431, 6155, 35760, 9950, 8521, 10215, 8248, 9760, 9938, 6629, 9125, 6380, 6853, 8775, 14154, 7862, 8280, 2308, 5310, 17600, 8410, 8450, 7540, 28698, 9735, 8480, 12328, 11645, 11512, 9300, 12589, 5306, 5684, 6780, 27650, 10200, 9873, 7588, 17755, 11228, 4750, 15593, 13891, 12224, 15750, 9450, 10846, 10944, 7535, 5925, 9548, 18800, 7879, 16737, 14145, 16635, 5400, 16321, 10448, 11160, 4270, 3636, 16669, 10562, 8750, 10197, 9531, 5587, 6120, 9490, 8740, 7851, 11404, 10900, 12182, 7388, 11475, 9572, 22950, 16905, 26178, 7922, 12327, 14536, 15256, 4500, 11428, 8064, 8246, 4608, 7150, 7728, 7596, 8049, 8978, 9297, 5232, 1596, 3000, 10395, 10029, 13695, 17871, 3696, 7000, 11302, 8872, 11764, 6951, 14450, 11796, 10859, 13560, 10226, 4274, 7244, 14778, 8089, 7472, 4571, 14762, 7446, 2289, 10020, 4403, 5790, 9179, 10304, 14803, 2280, 9937, 7861, 7259, 22420, 12803, 8070, 45600, 11924, 8850, 2665, 7577, 3196, 19378, 7056, 164660, 1477, 12220, 6911, 8556, 11988, 9742, 9248, 9262, 14191, 5586, 14977, 9906, 9286, 4017, 9825, 12118, 10659, 8072, 9717, 12205, 14585, 7630, 10482, 12243, 4800, 9250, 10120, 11900, 8461, 10007, 10896, 9830, 6900, 5825, 12394, 10656, 12822, 36500, 17671, 25286, 10150, 14260, 14100, 5500, 9842, 2368, 3922, 11957, 15870, 3363, 18386, 12134, 215245, 13869, 7350, 21695, 10004, 12085, 6820, 2268, 10320, 6792, 16259, 16770, 13418, 8529, 7153, 15576, 8462, 12735, 14572, 6300, 1526, 11340, 12227, 12095, 5062, 7032, 6171, 10791, 4388, 12122, 10570, 8731, 10721, 12464, 2117, 16226, 12692, 9587, 10496, 10208, 12003, 11241, 7585, 8737, 17503, 3013, 10680, 2572, 11400, 9612, 7677, 16381, 8532, 13680, 9291, 17400, 5105, 7558, 3982, 11932, 12798, 11120, 1300, 10615, 12393, 8834, 6270, 13450, 6180, 3136, 10402, 10267, 8212, 50271, 12493, 12925, 8926, 11664, 12000, 4920, 9303, 9350, 8765, 1953, 7024, 10998, 6435, 17542, 8471, 10356, 12677, 26142, 4043, 11838, 6627, 9400, 10410, 9947, 15384, 12168, 10335, 12203, 25419, 6204, 10005, 16659, 14963, 9464, 9135, 9142, 16692, 5890, 53107, 5664, 17920, 3010, 7050, 14781, 12108, 14303, 10012, 10261, 4060, 8700, 7700, 7162, 7892, 19690, 6931, 16196, 10530, 10592, 14850, 11777, 11362, 11787, 11275, 14601, 9364, 10593, 9247, 9765, 21780, 11184, 1533, 13673, 8339, 10832, 7590, 10421, 9849, 9492, 13860, 11885, 7207, 6960, 11248, 9170, 7052, 13006, 8546, 11175, 8658, 5814, 11394, 9819, 8160, 12209, 8724, 1974, 10452, 10920, 11170, 15578, 8123, 9430, 11333, 11782, 7937, 11883, 11345, 12137, 4712, 10125, 13688, 8767, 12438, 6858, 7406, 2016, 11851, 7313, 8499, 13346, 10635, 11210, 3880, 10652, 7795, 10927, 10708, 12155, 14215, 13700, 13214, 13907, 10207, 4923, 2651, 4282, 14559, 8749, 10970, 8366, 7449, 4118, 7015, 35133, 9196, 7136, 7094, 12274, 12444, 8849, 53504, 14200, 9405, 57200, 3951, 10624, 10628, 6060, 9317, 7804, 7642, 3684]

Mode: 9600
==================================================
Unique Values in Street:
['Pave', 'Grvl']

Mode: Pave
==================================================
Unique Values in LotShape:
['Reg', 'IR1', 'IR2', 'IR3']

Mode: Reg
==================================================
Unique Values in LandContour:
['Lvl', 'Bnk', 'HLS', 'Low']

Mode: Lvl
==================================================
Unique Values in Utilities:
['AllPub', 'NoSeWa']

Mode: AllPub
==================================================
Unique Values in LotConfig:
['Inside', 'Corner', 'CulDSac', 'FR2', 'FR3']

Mode: Inside
==================================================
Unique Values in LandSlope:
['Gtl', 'Mod', 'Sev']

Mode: Gtl
==================================================
Unique Values in Neighborhood:
['NAmes', 'Gilbert', 'Edwards', 'SWISU', 'CollgCr', 'IDOTRR', 'NWAmes', 'Somerst', 'Sawyer', 'Crawfor', 'OldTown', 'Timber', 'BrkSide', 'SawyerW', 'NridgHt', 'Mitchel', 'NoRidge', 'Blmngtn', 'BrDale', 'NPkVill', 'Veenker', 'ClearCr', 'StoneBr', 'MeadowV', 'Blueste']

Mode: NAmes
==================================================
Unique Values in Condition1:
['Norm', 'Feedr', 'RRAn', 'Artery', 'PosA', 'PosN', 'RRAe', 'RRNn', 'RRNe']

Mode: Norm
==================================================
Unique Values in Condition2:
['Norm', 'RRNn', 'PosA', 'Feedr', 'RRAe', 'PosN', 'Artery', 'RRAn']

Mode: Norm
==================================================
Unique Values in BldgType:
['1Fam', 'Duplex', 'TwnhsE', '2fmCon', 'Twnhs']

Mode: 1Fam
==================================================
Unique Values in HouseStyle:
['1Story', '2Story', '1.5Fin', 'SFoyer', 'SLvl', '1.5Unf', '2.5Unf', '2.5Fin']

Mode: 1Story
==================================================
Unique Values in OverallQual:
[5, 6, 4, 8, 7, 3, 10, 9, 1, 2]

Mode: 6
==================================================
Unique Values in OverallCond:
[6, 7, 5, 8, 4, 9, 3, 2, 1]

Mode: 5
==================================================
Unique Values in YearBuilt:
[1957, 1993, 1910, 1937, 1924, 1972, 2005, 1922, 1981, 2007, 1966, 1978, 1965, 1969, 2002, 1953, 1955, 1977, 1986, 1926, 1928, 2003, 2006, 1959, 1949, 2009, 1925, 1980, 1976, 1982, 1970, 1979, 1954, 1997, 1960, 1920, 1995, 1950, 1916, 1962, 2008, 1923, 1968, 1893, 2004, 1985, 1875, 2000, 1994, 1963, 1974, 1939, 1958, 1946, 1967, 1932, 1971, 1983, 1961, 1964, 1898, 1929, 1931, 1915, 1941, 1911, 1999, 1992, 1940, 2001, 1991, 1912, 1975, 1996, 1919, 1913, 1990, 1998, 1948, 1900, 1936, 1935, 1904, 1880, 1930, 1914, 1973, 1942, 1956, 1951, 1918, 1938, 1885, 1921, 1989, 1947, 1988, 1890, 1984, 1872, 1952, 1987, 1892, 1945, 1908, 1882, 1906, 1934, 1905, 1917, 2010]

Mode: 2005
==================================================
Unique Values in YearRemodAdd:
[1957, 1994, 1950, 1972, 2005, 2007, 1981, 1966, 1978, 1990, 1969, 2002, 1953, 1995, 1977, 1986, 2003, 2009, 2006, 1980, 1993, 1982, 1970, 1979, 1954, 2008, 1998, 1960, 1996, 2004, 1984, 2001, 2000, 1985, 1962, 1976, 1963, 1999, 1958, 1955, 1967, 1971, 1983, 1975, 1997, 1964, 1965, 1992, 1968, 1991, 2010, 1987, 1989, 1988, 1974, 1973, 1959, 1956, 1951, 1961, 1952]

Mode: 1950
==================================================
Unique Values in RoofStyle:
['Gable', 'Hip', 'Mansard', 'Gambrel', 'Flat', 'Shed']

Mode: Gable
==================================================
Unique Values in RoofMatl:
['CompShg', 'WdShngl', 'WdShake', 'Metal', 'Tar&Grv', 'ClyTile', 'Roll']

Mode: CompShg
==================================================
Unique Values in Exterior1st:
['MetalSd', 'VinylSd', 'Wd Sdng', 'BrkFace', 'CemntBd', 'Plywood', 'HdBoard', 'Stucco', 'WdShing', 'AsbShng', 'ImStucc', 'BrkComm', 'CBlock', 'AsphShn', 'Stone']

Mode: VinylSd
==================================================
Unique Values in Exterior2nd:
['MetalSd', 'VinylSd', 'Wd Sdng', 'CmentBd', 'Plywood', 'HdBoard', 'Stucco', 'Wd Shng', 'AsbShng', 'ImStucc', 'BrkFace', 'AsphShn', 'Brk Cmn', 'Stone', 'Other', 'CBlock']

Mode: VinylSd
==================================================
Unique Values in ExterQual:
['TA', 'Gd', 'Ex', 'Fa']

Mode: TA
==================================================
Unique Values in ExterCond:
['Gd', 'TA', 'Ex', 'Fa', 'Po']

Mode: TA
==================================================
Unique Values in Foundation:
['CBlock', 'PConc', 'BrkTil', 'Slab', 'Wood', 'Stone']

Mode: PConc
==================================================
Unique Values in BsmtFinSF1:
[922, 0, 569, 218, 894, 735, 588, 187, 516, 704, 454, 35, 180, 535, 1398, 500, 833, 1406, 375, 468, 625, 1116, 685, 1024, 1086, 1064, 641, 1309, 219, 398, 765, 1567, 619, 648, 1373, 1258, 498, 784, 533, 1696, 156, 595, 25, 16, 663, 204, 608, 352, 384, 251, 507, 503, 705, 387, 1200, 336, 1159, 714, 1014, 420, 380, 860, 600, 120, 48, 292, 1330, 425, 299, 334, 360, 1361, 787, 607, 432, 956, 904, 674, 712, 280, 1088, 374, 1721, 936, 1456, 906, 437, 210, 733, 410, 553, 567, 1390, 846, 256, 547, 24, 385, 624, 592, 222, 902, 602, 308, 329, 523, 247, 925, 804, 486, 365, 1162, 1302, 2096, 816, 485, 841, 241, 460, 763, 206, 836, 1071, 262, 350, 1150, 1180, 1157, 1274, 656, 236, 1904, 614, 504, 1285, 565, 790, 338, 528, 322, 231, 573, 554, 403, 720, 662, 987, 301, 737, 639, 672, 56, 668, 1220, 1238, 832, 821, 593, 824, 397, 789, 116, 885, 1247, 216, 386, 970, 732, 428, 661, 144, 695, 1085, 655, 298, 1036, 660, 20, 616, 562, 392, 321, 738, 781, 912, 686, 1440, 36, 575, 1016, 759, 982, 620, 560, 1369, 680, 988, 315, 369, 1218, 644, 788, 1249, 1410, 490, 716, 938, 1056, 1320, 444, 456, 2260, 932, 438, 599, 941, 713, 724, 390, 421, 165, 1646, 747, 312, 104, 1002, 364, 976, 448, 1219, 1261, 351, 767, 828, 520, 1234, 354, 465, 495, 273, 1572, 572, 666, 358, 344, 378, 1464, 1186, 697, 1201, 442, 1018, 786, 708, 862, 33, 464, 1121, 734, 772, 111, 224, 903, 837, 905, 524, 1470, 851, 331, 1280, 626, 637, 5644, 457, 544, 297, 1332, 400, 586, 259, 1358, 57, 324, 353, 678, 859, 1138, 874, 1097, 1040, 702, 361, 348, 623, 510, 168, 1767, 1387, 842, 492, 822, 750, 551, 1005, 27, 257, 1270, 706, 773, 249, 630, 539, 719, 783, 64, 176, 50, 1573, 603, 552, 1619, 1000, 483, 1447, 213, 1246, 370, 1277, 1065, 1288, 288, 1480, 489, 320, 252, 405, 550, 482, 967, 965, 731, 1441, 1300, 929, 897, 188, 556, 290, 803, 609, 659, 1053, 1084, 694, 1606, 462, 294, 642, 739, 605, 831, 646, 317, 1422, 658, 540, 250, 1106, 728, 945, 266, 311, 612, 775, 578, 633, 381, 356, 1636, 566, 830, 774, 1128, 513, 415, 450, 998, 777, 430, 548, 1334, 509, 1436, 643, 1350, 196, 404, 1046, 915, 429, 746, 49, 568, 594, 197, 521, 480, 740, 170, 806, 72, 1351, 812, 192, 234, 75, 330, 427, 1236, 349, 182, 852, 808, 1004, 368, 814, 549, 515, 896, 564, 943, 264, 867, 958, 645, 536, 436, 281, 1231, 220, 994, 300, 883, 756, 604, 1282, 570, 1110, 1213, 426, 239, 1154, 493, 285, 440, 1170, 284, 416, 700, 991, 203, 1810, 1400, 419, 865, 742, 270, 309, 980, 408, 125, 506, 459, 559, 664, 209, 326, 611, 275, 679, 854, 795, 340, 617, 975, 538, 985, 1158, 306, 978, 984, 133, 1314, 920, 776, 532, 319, 1443, 282, 895, 328, 1039, 1260, 371, 919, 310, 1148, 226, 394, 990, 696, 1059, 108, 698, 1445, 690, 296, 1216, 1163, 946, 899, 810, 819, 916, 751, 152, 684, 138, 1153, 866, 141, 546, 631, 379, 417, 650, 1196, 505, 185, 484, 1336, 28, 1416, 445, 40, 778, 1104, 622, 167]

Mode: 0
==================================================
Unique Values in BsmtFinSF2:
[0, 228, 627, 128, 869, 645, 208, 1061, 149, 872, 377, 630, 544, 180, 182, 532, 1057, 175, 634, 311, 28, 344, 336, 438, 465, 396, 177, 391, 123, 210, 380, 108, 163, 466, 324, 690, 64, 168, 1127, 193, 80, 81, 276, 620, 110, 211, 723, 841, 93, 712, 202, 181, 713, 290, 273, 41, 354, 823, 374, 68, 441, 551, 32, 121, 546, 539, 106, 495, 764, 215, 531, 279, 165, 362, 668, 105, 411, 150, 173, 147, 1080, 1029, 219, 174, 468, 670, 258, 125, 692, 791, 820, 345, 1120, 117, 486, 606, 306, 608, 580, 420, 1031, 184, 469, 1085, 144, 132, 35, 557, 500, 239, 96, 547, 230, 972, 768, 435, 334, 264]

Mode: 0
==================================================
Unique Values in BsmtUnfSF:
[392, 799, 796, 162, 808, 0, 684, 61, 606, 1496, 380, 567, 314, 1164, 1010, 905, 155, 768, 748, 1042, 783, 343, 625, 496, 1502, 1228, 457, 396, 165, 173, 1293, 481, 864, 319, 175, 35, 689, 1065, 661, 736, 572, 1428, 554, 317, 686, 1319, 879, 252, 225, 968, 435, 1237, 2153, 221, 40, 1694, 460, 141, 780, 281, 413, 692, 626, 1284, 378, 247, 1330, 356, 978, 834, 416, 953, 707, 491, 286, 449, 1139, 96, 410, 90, 88, 138, 490, 397, 132, 849, 1580, 312, 862, 958, 1774, 476, 122, 459, 611, 322, 360, 728, 940, 305, 506, 432, 994, 1632, 384, 76, 761, 140, 363, 504, 168, 134, 884, 143, 422, 570, 811, 912, 210, 906, 813, 1926, 1304, 732, 1905, 70, 288, 1055, 533, 99, 769, 196, 14, 620, 697, 301, 1104, 310, 349, 1122, 536, 324, 764, 522, 75, 426, 300, 1276, 1800, 924, 832, 930, 198, 720, 229, 791, 404, 804, 914, 1616, 237, 195, 599, 316, 415, 280, 1630, 526, 528, 841, 270, 1362, 244, 208, 131, 818, 589, 650, 1528, 121, 1390, 1013, 1098, 1709, 636, 698, 1406, 1079, 596, 1497, 191, 100, 793, 400, 588, 390, 638, 186, 1085, 2046, 257, 844, 239, 441, 1198, 336, 440, 80, 107, 130, 216, 2336, 706, 245, 604, 1092, 747, 1670, 600, 254, 1656, 540, 1393, 342, 166, 64, 1048, 203, 1064, 379, 519, 192, 893, 224, 1588, 279, 1217, 712, 1078, 323, 846, 580, 29, 172, 1480, 503, 1035, 501, 1050, 1176, 487, 880, 678, 36, 1288, 1530, 176, 816, 179, 920, 420, 434, 292, 200, 114, 710, 671, 486, 549, 1489, 398, 1212, 785, 485, 15, 361, 1566, 673, 641, 438, 935, 952, 274, 831, 1296, 652, 878, 108, 234, 1375, 1584, 792, 602, 277, 217, 163, 111, 508, 474, 386, 598, 284, 550, 408, 1128, 868, 1571, 1907, 1181, 448, 1053, 969, 424, 112, 840, 557, 461, 115, 464, 412, 447, 135, 1795, 1541, 409, 776, 1468, 125, 328, 1125, 291, 105, 374, 1436, 718, 876, 1090, 1442, 295, 255, 248, 1095, 927, 1302, 648, 645, 702, 344, 806, 74, 220, 354, 126, 750, 1043, 1346, 357, 193, 160, 970, 715, 318, 975, 402, 466, 941, 1203, 462, 425, 82, 223, 622, 219, 1286, 55, 762, 1140, 798, 268, 124, 1422, 1082, 403, 1257, 664, 788, 543, 756, 372, 78, 993, 1372, 262, 495, 778, 1063, 1266, 1191, 932, 103, 658, 150, 340, 473, 860, 571, 742, 30, 171, 1120, 1290, 499, 701, 1734, 297, 458, 84, 995, 779, 207, 1045, 1686, 294, 700, 120, 26, 238, 672, 576, 276, 235, 901, 311, 971, 1107, 383, 326, 188, 556, 1626, 204, 303, 960, 189, 256, 784, 948, 92, 680, 483, 273, 405, 427, 320, 468, 1152, 355, 721, 1074, 218, 530, 628, 2121, 847, 1097, 1392, 1622, 149, 568, 624, 544, 333, 892, 240, 399, 442, 289, 136, 104, 242, 560, 907, 418, 1094, 1030, 414, 177, 117, 1145, 151, 815, 1342, 1273, 1335, 161, 167, 1836, 740, 1240, 467, 1777, 1573, 967, 1072, 304, 1710, 93, 264, 331, 81, 1374, 999, 169, 1129, 352, 1184, 733, 212, 525, 1638, 83, 812, 431, 456, 1680, 612, 1258, 1550, 230, 153, 976, 1470, 896, 23, 72, 730, 197, 1368, 1249, 1195, 1117, 1434, 662, 916, 1200, 563, 1116, 180, 77, 1252, 308, 1969, 201, 1264, 484, 774, 810, 765, 341, 1380, 1351, 630, 734, 1357, 406, 585, 190, 366, 546, 364, 1017, 154, 538, 936, 717, 50, 1405, 1226, 32, 1141, 777, 375, 746, 722, 452, 666, 174, 110, 156, 972, 253, 595, 1026, 510, 1753, 290, 389, 577, 429, 537, 660, 371, 795, 1020, 133, 278, 1232, 482, 187, 1307, 752, 38, 1088, 918, 350, 564, 39, 285, 797, 925, 299, 578, 801, 539, 411, 370, 1163, 1349, 894, 586, 1689, 359, 542, 309, 250, 705, 964, 1121, 891, 1316, 445, 184, 1194, 444, 594, 520, 982, 89, 170, 726, 814, 708, 877, 1649, 1234, 515, 1424, 338, 548, 635, 616, 367, 315, 1594, 1494, 512, 470, 516, 1158, 2002, 874, 640, 178, 1459, 298, 637, 693, 524, 1560, 1417, 479, 60, 1420, 500, 1373]

Mode: 0
==================================================
Unique Values in TotalBsmtSF:
[1314, 799, 796, 731, 1026, 894, 684, 1422, 1496, 814, 1024, 1271, 768, 1164, 1010, 1809, 825, 690, 1606, 1248, 1042, 783, 1176, 625, 1902, 0, 1502, 1228, 832, 864, 1686, 1281, 858, 1293, 1505, 1086, 1383, 816, 1344, 689, 1065, 661, 736, 572, 1428, 773, 686, 1319, 879, 1017, 1792, 968, 1054, 648, 1237, 2153, 1594, 1298, 1694, 958, 925, 780, 2109, 848, 626, 1284, 973, 1144, 1346, 1040, 560, 978, 1442, 953, 384, 707, 1372, 793, 952, 1844, 483, 1610, 1084, 1249, 802, 1152, 910, 777, 992, 849, 1580, 912, 862, 1078, 1822, 1452, 884, 1200, 720, 728, 1629, 940, 1092, 1113, 994, 1632, 1522, 750, 1473, 420, 1713, 1269, 2077, 1104, 1624, 1494, 702, 1358, 1221, 1008, 1390, 1056, 1162, 813, 1926, 1304, 547, 756, 1905, 1055, 533, 691, 991, 1098, 928, 824, 1022, 1274, 1122, 1440, 1128, 764, 440, 1728, 1588, 1392, 2396, 808, 1276, 1800, 924, 930, 2158, 1039, 861, 791, 804, 936, 1120, 1616, 1073, 1389, 1466, 1595, 1437, 1630, 528, 841, 956, 616, 270, 1362, 2216, 833, 931, 1416, 1542, 650, 1552, 1414, 1013, 1626, 1709, 636, 698, 1406, 1079, 596, 654, 803, 588, 985, 1085, 2046, 844, 1347, 1080, 1318, 1004, 1660, 1698, 723, 2392, 1103, 1034, 1053, 747, 570, 1670, 600, 1501, 1656, 1393, 1136, 1476, 1064, 1650, 540, 1214, 725, 893, 771, 1475, 655, 1479, 2033, 870, 983, 866, 1196, 784, 836, 1500, 1947, 1187, 1050, 1212, 1142, 880, 1328, 1052, 1288, 948, 1554, 839, 1161, 1240, 920, 980, 972, 1188, 539, 1070, 1704, 1193, 1489, 1386, 1573, 732, 1324, 1566, 1922, 846, 641, 1848, 1425, 990, 938, 1746, 520, 831, 1752, 1361, 3138, 1051, 672, 1375, 1584, 792, 1258, 876, 1158, 1021, 1232, 807, 1604, 1930, 840, 1204, 1088, 918, 1571, 1907, 1181, 896, 969, 1643, 1833, 644, 319, 1517, 943, 960, 630, 546, 1652, 1795, 1541, 1075, 776, 1468, 2000, 1125, 1477, 1444, 1306, 845, 1398, 959, 1436, 718, 1090, 1117, 1563, 1364, 1095, 1391, 1302, 1766, 806, 1332, 819, 1029, 942, 2217, 1370, 1262, 817, 970, 715, 608, 1673, 649, 975, 1682, 1127, 637, 6110, 941, 1568, 1048, 868, 1581, 622, 1286, 408, 1140, 1107, 798, 1408, 735, 1097, 1082, 963, 744, 800, 1257, 1350, 714, 3094, 611, 1203, 900, 993, 1267, 778, 1063, 1266, 1218, 932, 855, 485, 1478, 658, 856, 888, 1012, 860, 811, 1019, 1268, 742, 585, 1461, 1184, 1734, 1165, 1719, 1223, 1251, 1980, 1208, 1602, 1149, 1484, 2110, 544, 1582, 794, 1600, 576, 1153, 713, 1453, 1210, 1057, 1485, 1216, 1060, 290, 892, 1106, 1394, 1309, 680, 462, 612, 915, 1032, 1151, 1114, 1680, 721, 1074, 1814, 530, 105, 2121, 847, 927, 1622, 1241, 1296, 1419, 624, 788, 683, 1470, 873, 2078, 1256, 1028, 1272, 907, 1094, 1030, 1175, 1575, 815, 757, 1273, 1499, 2006, 810, 1860, 1145, 1244, 1777, 967, 901, 1976, 708, 1710, 360, 1374, 264, 740, 525, 1638, 1434, 916, 926, 765, 1574, 1096, 976, 559, 2136, 372, 1516, 1488, 1368, 1195, 1454, 1278, 662, 1845, 1530, 1330, 1116, 1190, 944, 853, 822, 1252, 3200, 1264, 774, 1224, 696, 2633, 1373, 1297, 1126, 551, 1520, 1300, 1510, 561, 1462, 1001, 1041, 1405, 1226, 1006, 1842, 1141, 1100, 1205, 1338, 872, 951, 984, 1192, 506, 1518, 1069, 1753, 954, 1261, 945, 1486, 704, 965, 1490, 660, 1225, 1360, 795, 1160, 935, 1260, 617, 1067, 1020, 1108, 1763, 482, 828, 1986, 1138, 1156, 697, 746, 1482, 981, 602, 1836, 1379, 854, 1838, 663, 1720, 999, 1132, 1163, 1349, 1148, 712, 1689, 1952, 1299, 458, 727, 964, 1121, 1316, 1220, 1194, 1856, 882, 1253, 1154, 982, 1109, 1536, 971, 1003, 1504, 1242, 1649, 1143, 1572, 1310, 1234, 1199, 1424, 1561, 1525, 1235, 1657, 1614, 716, 1058, 1617, 2002, 996, 874, 1593, 673, 1459, 693, 709, 1560, 979, 1417, 1932, 1684, 1277, 2524]

Mode: 0
==================================================
Unique Values in Heating:
['GasA', 'Grav', 'Wall', 'GasW', 'OthW', 'Floor']

Mode: GasA
==================================================
Unique Values in HeatingQC:
['TA', 'Gd', 'Ex', 'Fa', 'Po']

Mode: Ex
==================================================
Unique Values in CentralAir:
['Y', 'N']

Mode: Y
==================================================
Unique Values in 1stFlrSF:
[1314, 799, 796, 981, 1026, 894, 684, 1422, 1496, 913, 1086, 1601, 1186, 1164, 1010, 2259, 825, 698, 1651, 789, 1338, 1042, 783, 1200, 625, 1902, 1040, 1502, 1228, 1036, 864, 1686, 1620, 858, 1301, 1505, 1224, 1383, 816, 1344, 703, 1065, 661, 736, 572, 1428, 773, 948, 1537, 879, 1792, 968, 1512, 960, 1521, 2069, 1646, 1298, 1694, 958, 999, 780, 814, 2113, 1118, 626, 1284, 979, 1144, 1504, 1309, 1442, 768, 953, 1360, 707, 1342, 793, 1170, 1844, 483, 1610, 1080, 1622, 802, 1152, 910, 1306, 1075, 1580, 1085, 950, 1771, 1828, 1148, 1476, 959, 916, 854, 728, 1024, 1092, 1113, 1028, 1632, 1548, 1061, 1484, 708, 1713, 1269, 2097, 1130, 1624, 884, 1494, 1041, 912, 936, 1358, 774, 1221, 1053, 1701, 1056, 1163, 822, 1966, 1682, 1072, 764, 1905, 827, 1350, 1055, 1021, 691, 991, 1098, 1054, 928, 1133, 1687, 1104, 1022, 1274, 1134, 1128, 1060, 1008, 694, 1728, 1588, 1392, 2411, 1161, 1276, 1800, 924, 832, 956, 2196, 1039, 720, 961, 1236, 1699, 1120, 1616, 1073, 1389, 861, 575, 1466, 1437, 1630, 1094, 892, 616, 729, 1586, 2234, 833, 1283, 1644, 2073, 1552, 925, 1414, 1013, 1712, 1717, 1089, 786, 1454, 1079, 998, 664, 790, 1142, 985, 848, 2046, 992, 1445, 1372, 902, 1656, 1382, 2020, 1052, 1328, 1004, 1660, 1709, 735, 2392, 1103, 1050, 803, 1173, 372, 1222, 1670, 520, 1801, 888, 1136, 1680, 889, 1214, 725, 901, 753, 1532, 1194, 2053, 914, 1472, 983, 866, 1196, 1126, 845, 1168, 949, 1220, 1500, 1310, 2207, 1208, 1440, 1212, 1175, 880, 869, 1554, 1416, 1381, 954, 495, 980, 920, 972, 1704, 1523, 1489, 1569, 1573, 1012, 1600, 1922, 846, 1047, 1848, 1425, 952, 990, 988, 1746, 885, 1464, 831, 1752, 1361, 3138, 784, 672, 334, 600, 1584, 792, 1258, 1158, 964, 1465, 1232, 1617, 1340, 1940, 1216, 1149, 1377, 1188, 918, 1571, 1362, 1959, 1234, 1182, 997, 1824, 1850, 808, 840, 1035, 1888, 943, 1698, 1063, 1535, 630, 546, 1652, 1795, 1541, 1507, 1479, 2000, 1125, 1124, 1839, 1444, 1127, 905, 1423, 1700, 1689, 1576, 876, 1370, 804, 1563, 1663, 1095, 1391, 1302, 734, 1766, 1253, 1097, 1078, 806, 1332, 1226, 1029, 1291, 750, 2217, 1262, 824, 970, 1281, 1319, 1077, 679, 1005, 1742, 897, 4692, 2156, 1048, 938, 1146, 1252, 741, 1294, 779, 1140, 1107, 798, 1679, 1110, 596, 1082, 963, 752, 1266, 887, 1334, 624, 1664, 2402, 751, 1831, 944, 756, 932, 682, 686, 1062, 697, 812, 1296, 1071, 1218, 855, 1001, 1478, 658, 856, 1160, 754, 1034, 860, 811, 1357, 1264, 742, 1064, 813, 1509, 702, 1184, 966, 1625, 1304, 1734, 1336, 1719, 1223, 1251, 941, 1980, 1621, 1602, 1378, 2110, 1707, 909, 882, 1015, 851, 1165, 1153, 689, 1327, 757, 1453, 1518, 1238, 1057, 1629, 1634, 438, 747, 1190, 1394, 1339, 1178, 526, 612, 933, 1032, 1114, 1724, 841, 1088, 1826, 581, 923, 2121, 847, 1067, 1268, 1112, 1241, 1640, 1419, 1547, 788, 1776, 904, 2084, 1256, 807, 1436, 1272, 1307, 1038, 899, 1575, 815, 1456, 842, 1604, 1619, 1661, 810, 1836, 1240, 1244, 1167, 1578, 993, 1976, 1260, 1710, 976, 1429, 1138, 1557, 1203, 1145, 1287, 778, 525, 1582, 1968, 926, 765, 1574, 1096, 1675, 2036, 896, 576, 794, 1516, 1368, 1249, 1195, 1132, 1434, 1872, 1530, 1542, 1116, 1549, 1020, 769, 1210, 3228, 1166, 1074, 1277, 893, 1493, 1002, 915, 1506, 2136, 696, 2633, 1468, 900, 1555, 1297, 1733, 551, 1172, 1567, 561, 1014, 1462, 986, 1405, 838, 1006, 1842, 1246, 1100, 2117, 1122, 951, 1090, 984, 969, 1069, 1788, 1261, 875, 1486, 849, 1217, 1501, 660, 1156, 955, 1288, 865, 908, 1108, 1659, 1779, 1248, 1282, 965, 1787, 1987, 1181, 962, 1051, 1520, 1411, 760, 1320, 1838, 663, 1720, 680, 1337, 1349, 1696, 1299, 1668, 835, 829, 1364, 1121, 1721, 1316, 1324, 1856, 1811, 1154, 1007, 1155, 1536, 971, 1003, 1242, 1279, 1572, 1561, 1204, 1525, 1235, 1614, 716, 1058, 1867, 2018, 996, 874, 1593, 673, 1363, 1459, 693, 1560, 1932, 1684, 1690, 1141, 772, 2524]

Mode: 864
==================================================
Unique Values in 2ndFlrSF:
[0, 772, 787, 665, 876, 358, 800, 1257, 728, 1296, 534, 701, 625, 689, 589, 716, 524, 885, 980, 984, 981, 1142, 1254, 574, 240, 860, 912, 591, 871, 560, 432, 711, 707, 325, 1175, 504, 709, 648, 846, 1063, 886, 649, 208, 568, 1237, 408, 910, 762, 940, 776, 1066, 862, 1054, 1151, 702, 1215, 780, 656, 1182, 843, 512, 1174, 783, 834, 790, 580, 807, 956, 687, 998, 684, 1370, 677, 903, 764, 2065, 808, 930, 551, 406, 857, 748, 1362, 761, 1128, 495, 581, 661, 390, 482, 840, 832, 793, 866, 833, 850, 873, 1274, 1203, 848, 836, 660, 1028, 720, 892, 698, 623, 756, 1177, 1332, 566, 754, 900, 741, 797, 1185, 809, 445, 767, 902, 272, 1140, 745, 539, 1540, 844, 1116, 883, 1020, 729, 1427, 1208, 685, 778, 864, 880, 829, 1259, 1538, 714, 600, 725, 915, 838, 804, 694, 639, 564, 1141, 806, 561, 960, 939, 1288, 547, 828, 371, 695, 546, 700, 954, 881, 712, 596, 978, 540, 213, 670, 472, 679, 571, 384, 766, 192, 1038, 592, 1230, 750, 1070, 739, 457, 1523, 582, 1286, 590, 464, 872, 439, 950, 1103, 473, 668, 622, 336, 640, 983, 842, 349, 252, 785, 475, 1134, 628, 611, 1796, 1306, 920, 548, 813, 887, 672, 545, 1101, 467, 634, 526, 854, 966, 576, 742, 703, 779, 901, 1426, 686, 1017, 1088, 1032, 904, 368, 888, 784, 914, 650, 620, 348, 651, 1100, 878, 567, 1357, 1518, 1281, 520, 636, 744, 1000, 462, 612, 975, 896, 530, 1106, 1112, 556, 587, 586, 765, 908, 601, 884, 595, 928, 550, 1051, 1349, 1060, 875, 630, 167, 855, 994, 110, 518, 708, 1031, 1111, 1323, 664, 1440, 631, 826, 1053, 1479, 1611, 732, 1818, 448, 533, 676, 713, 438, 1347, 644, 1320, 1330, 1178, 831, 1242, 678, 1067, 1194, 454, 441, 1072, 1001, 941, 1098, 1044, 752, 868, 605, 932, 811, 704, 1336, 1360, 795, 1392, 989, 874, 1414, 691, 220, 830, 788, 924, 626, 602, 925, 1015, 895, 858, 511, 936, 734, 573, 727, 1243, 378, 1157, 977, 1216, 1097, 370, 992, 1152, 456, 1096, 1312, 796, 1120, 1129, 1218, 918, 224, 979, 1589, 351, 653, 514]

Mode: 0
==================================================
Unique Values in LowQualFinSF:
[0, 232, 528, 397, 513, 80, 120, 420, 384, 205, 234, 572, 53, 515, 390, 392, 481, 473, 144, 360]

Mode: 0
==================================================
Unique Values in GrLivArea:
[1314, 1571, 796, 1768, 1691, 894, 1560, 1154, 1422, 1496, 913, 1086, 1601, 1986, 1164, 2267, 2259, 825, 1426, 1651, 789, 2634, 1576, 1484, 1200, 1250, 1902, 1040, 1502, 1228, 1268, 864, 1686, 1620, 858, 1301, 1505, 1224, 1383, 816, 1344, 1392, 1065, 1452, 1096, 1428, 1658, 1928, 1537, 1863, 2007, 1792, 968, 2654, 960, 2775, 2643, 1646, 1298, 1694, 958, 999, 1020, 1674, 2113, 2030, 1217, 2169, 1850, 1144, 1504, 1309, 1258, 1442, 1664, 1360, 1414, 1342, 1118, 2345, 1844, 987, 1610, 1080, 1622, 1511, 1152, 1558, 1911, 1306, 2138, 2466, 1734, 1158, 1771, 1828, 1716, 2713, 1367, 1826, 1382, 1456, 2448, 1964, 1092, 1113, 1804, 1632, 2614, 1923, 708, 1713, 1269, 2097, 2184, 1624, 2035, 1494, 1743, 2127, 1358, 1430, 1221, 2094, 1053, 1701, 1056, 1163, 1665, 1196, 3140, 1682, 1072, 1547, 1905, 1661, 1350, 1845, 1498, 1947, 1098, 1054, 928, 1820, 3082, 1788, 1022, 1274, 2504, 2153, 2031, 1824, 1008, 694, 1728, 1588, 4476, 1969, 1276, 1800, 924, 832, 1886, 2196, 1039, 1271, 2093, 2447, 1120, 1616, 1073, 1389, 861, 1135, 2828, 1437, 1630, 1855, 1675, 2084, 1111, 729, 1586, 2234, 833, 1283, 1644, 2073, 1539, 1552, 925, 1526, 1712, 1717, 1750, 1176, 1936, 1919, 1762, 1626, 790, 1935, 1666, 985, 848, 1970, 2046, 1865, 2134, 2646, 1236, 902, 912, 1656, 2020, 1052, 2531, 1852, 1660, 1709, 1764, 1395, 2392, 1103, 2078, 803, 1173, 1784, 1920, 1670, 1223, 1801, 2599, 2468, 1362, 1680, 1440, 1214, 1479, 2329, 1194, 3238, 914, 1641, 1573, 1472, 1126, 845, 1968, 1368, 1220, 1500, 2450, 2207, 1208, 1795, 1212, 2715, 1724, 1328, 2417, 948, 2019, 869, 1554, 2112, 1416, 1593, 1381, 954, 1922, 980, 1786, 972, 2263, 1725, 1704, 1523, 1489, 1569, 1790, 1600, 1692, 1047, 2728, 1425, 952, 990, 988, 1746, 1718, 1464, 1752, 2620, 4676, 784, 1386, 334, 1584, 1517, 964, 2380, 1232, 1982, 1924, 2080, 1647, 1617, 1340, 1940, 884, 1284, 2290, 2183, 1869, 2082, 1959, 2374, 2142, 1992, 2285, 1355, 1668, 1406, 1888, 1638, 1698, 1063, 1535, 630, 1652, 1541, 1507, 1441, 672, 2192, 2000, 1125, 1124, 1839, 2144, 2260, 1977, 2555, 1700, 1671, 1376, 1689, 2554, 1370, 1285, 1474, 2013, 1563, 1663, 1192, 1774, 1962, 1302, 1766, 1253, 1097, 1078, 1572, 1524, 2060, 1818, 1029, 2521, 2217, 1262, 1894, 1738, 1319, 3222, 768, 1077, 1183, 2291, 2332, 1348, 2108, 1336, 5642, 1470, 2156, 1048, 2358, 1146, 1363, 1148, 1294, 1248, 1419, 1140, 2090, 1640, 1679, 1218, 1142, 1110, 1953, 752, 1740, 1385, 1266, 2021, 1334, 1252, 2526, 2402, 3627, 2520, 944, 1553, 1230, 1611, 1949, 1369, 1482, 1372, 1296, 2172, 932, 1322, 1635, 1478, 1184, 1710, 2126, 1394, 970, 1034, 1720, 1387, 1357, 1264, 1767, 813, 754, 2069, 1509, 1481, 1802, 2610, 1466, 1625, 2287, 2822, 2054, 1719, 1251, 1829, 1574, 720, 1980, 1621, 1602, 2119, 2110, 774, 1707, 2161, 1796, 1015, 1578, 860, 1165, 1960, 1327, 1324, 2810, 3608, 2519, 1057, 2132, 1629, 1188, 1216, 1634, 1060, 1548, 438, 747, 892, 1190, 1339, 1178, 1983, 764, 1908, 1032, 1114, 841, 1088, 2270, 923, 910, 1374, 2121, 1412, 1241, 2320, 788, 1776, 904, 1605, 1868, 1256, 2624, 1375, 1095, 2200, 1346, 2482, 2098, 899, 2324, 1575, 1690, 1475, 2462, 1604, 1836, 2372, 1240, 1244, 1167, 1198, 1976, 616, 1050, 2087, 1429, 1812, 1138, 1557, 1145, 1432, 3395, 1287, 778, 1733, 2149, 1582, 1742, 2198, 3447, 1316, 2601, 1365, 1708, 3493, 2036, 1109, 1516, 1469, 1550, 2596, 2452, 1434, 1721, 1501, 2337, 1872, 1530, 2872, 1445, 1116, 943, 1608, 840, 1978, 2727, 1851, 1525, 3228, 1166, 1846, 1074, 2344, 893, 1493, 1004, 1458, 1506, 2136, 2633, 1468, 773, 900, 1555, 1297, 1529, 1102, 1913, 1567, 1580, 2792, 1688, 1229, 1699, 936, 1948, 2224, 2002, 1993, 1861, 1041, 986, 1405, 1226, 838, 1006, 1842, 1100, 2117, 1082, 1122, 1338, 1756, 951, 1090, 1512, 1577, 2167, 969, 1518, 1958, 1069, 2414, 1261, 875, 1486, 1564, 1490, 3194, 1320, 2612, 3112, 2230, 2022, 1288, 1310, 1067, 2784, 1659, 1779, 1929, 2696, 1912, 965, 1787, 1987, 1181, 1128, 918, 1520, 1411, 2058, 2201, 981, 1889, 1130, 2398, 1991, 1838, 1352, 1132, 1337, 1349, 1683, 1696, 882, 1895, 1714, 2704, 835, 1639, 1556, 1364, 2236, 1121, 1856, 1134, 1961, 1094, 1811, 2034, 2514, 2365, 1436, 1062, 1155, 1536, 971, 1003, 1242, 2340, 1279, 2668, 2576, 1561, 1204, 1378, 1235, 2240, 2295, 1614, 866, 1904, 1867, 2376, 2018, 996, 1761, 1391, 1459, 1026, 2640, 1108, 693, 1203, 1932, 1684, 3279, 2157, 698, 1687, 1141, 1277, 1123, 2524, 1981]

Mode: 864
==================================================
Unique Values in BsmtFullBath:
[1, 0, 2, 3]

Mode: 0
==================================================
Unique Values in BsmtHalfBath:
[0, 1, 2]

Mode: 0
==================================================
Unique Values in FullBath:
[1, 2, 0, 3]

Mode: 2
==================================================
Unique Values in HalfBath:
[0, 1, 2]

Mode: 0
==================================================
Unique Values in BedroomAbvGr:
[3, 2, 4, 6, 5, 0, 1, 8]

Mode: 3
==================================================
Unique Values in KitchenAbvGr:
[1, 2, 3, 0]

Mode: 1
==================================================
Unique Values in KitchenQual:
['TA', 'Gd', 'Ex', 'Fa']

Mode: TA
==================================================
Unique Values in TotRmsAbvGrd:
[5, 7, 4, 6, 8, 12, 10, 9, 3, 11, 2, 14]

Mode: 6
==================================================
Unique Values in Functional:
['Typ', 'Min2', 'Min1', 'Maj2', 'Mod', 'Maj1', 'Sev']

Mode: Typ
==================================================
Unique Values in Fireplaces:
[0, 1, 2, 3]

Mode: 0
==================================================
Unique Values in GarageCars:
[1, 2, 0, 4, 3]

Mode: 2
==================================================
Unique Values in GarageArea:
[294, 380, 0, 240, 308, 552, 400, 576, 474, 484, 478, 486, 528, 390, 450, 720, 440, 616, 200, 968, 225, 393, 555, 567, 644, 271, 288, 612, 490, 572, 505, 280, 354, 264, 784, 540, 461, 480, 431, 462, 660, 812, 874, 619, 364, 880, 694, 482, 403, 776, 301, 300, 663, 839, 551, 647, 467, 336, 437, 286, 460, 457, 410, 360, 546, 1356, 413, 282, 471, 756, 610, 208, 774, 281, 858, 560, 164, 521, 711, 504, 442, 624, 626, 564, 606, 856, 1134, 836, 757, 384, 434, 481, 539, 833, 615, 750, 611, 220, 562, 820, 530, 525, 614, 788, 550, 520, 409, 678, 260, 672, 304, 508, 656, 736, 577, 275, 352, 574, 472, 813, 534, 350, 702, 625, 701, 312, 542, 338, 834, 515, 418, 1052, 608, 451, 765, 512, 502, 618, 435, 1166, 506, 292, 500, 686, 782, 429, 494, 588, 908, 684, 685, 426, 160, 483, 228, 328, 420, 753, 441, 923, 884, 630, 752, 470, 497, 180, 487, 690, 642, 779, 872, 583, 318, 475, 800, 213, 514, 825, 666, 423, 578, 297, 290, 392, 216, 719, 396, 570, 1069, 632, 831, 527, 532, 627, 398, 676, 636, 905, 447, 772, 573, 544, 673, 890, 650, 826, 273, 706, 192, 758, 427, 493, 600, 459, 792, 432, 596, 575, 592, 533, 252, 325, 516, 786, 456, 620, 722, 766, 509, 648, 739, 768, 840, 895, 843, 404, 492, 234, 888, 430, 416, 389, 198, 704, 479, 455, 529, 868, 436, 402, 373, 586, 261, 621, 510, 818, 368, 270, 594, 205, 846, 1418, 548, 603, 438, 670, 662, 595, 602, 189, 1043, 453, 422, 807, 721, 248, 299, 947, 388, 306, 255, 498, 538, 565, 256, 343, 667, 1020, 375, 473, 613, 566, 770, 319, 522, 511, 789, 324, 541, 628, 246, 808, 531, 754, 866, 439, 671, 414, 732, 244, 320, 860, 569, 924, 601, 513, 379, 1220, 276, 864, 326, 878, 250, 377, 900, 210, 444, 844, 397, 458, 433, 870, 424, 1014, 824, 928, 691, 496, 604, 645, 495, 683, 315, 889, 680, 330, 804, 898, 452, 477, 342, 830, 738, 871, 740, 894, 468, 582, 309, 327, 518, 936, 287, 795, 501, 499, 454, 748, 1025, 408, 476, 862, 349, 954, 366, 816, 693, 463, 598, 305, 726, 556, 412, 590, 865, 303, 853, 746, 682, 995, 554, 841, 523, 526]

Mode: 0
==================================================
Unique Values in PavedDrive:
['Y', 'P', 'N']

Mode: Y
==================================================
Unique Values in WoodDeckSF:
[250, 0, 328, 256, 100, 276, 168, 120, 166, 140, 252, 192, 384, 216, 511, 74, 242, 224, 194, 206, 88, 105, 414, 128, 165, 160, 150, 236, 156, 265, 439, 95, 227, 289, 182, 40, 159, 126, 237, 517, 38, 172, 240, 164, 144, 72, 142, 98, 169, 158, 176, 635, 280, 315, 180, 136, 264, 253, 171, 288, 89, 84, 364, 272, 208, 125, 270, 113, 48, 110, 349, 70, 132, 210, 188, 36, 177, 322, 340, 222, 97, 294, 248, 203, 668, 184, 78, 283, 233, 245, 186, 305, 301, 321, 356, 127, 509, 255, 304, 143, 148, 178, 201, 302, 196, 155, 335, 55, 116, 106, 104, 380, 519, 327, 238, 404, 468, 262, 80, 296, 431, 418, 371, 195, 220, 268, 64, 728, 81, 66, 367, 263, 58, 214, 316, 187, 235, 536, 303, 353, 342, 361, 207, 147, 260, 394, 486, 441, 53, 123, 52, 24, 122, 12, 306, 108, 574, 162, 209, 500, 320, 406, 44, 362, 124, 112, 392, 45, 28, 448, 87, 736, 215, 149, 239, 670, 135, 173, 409, 366, 133, 344, 390, 309, 314, 183, 325, 200, 35, 292, 57, 290, 300, 197, 550, 219, 42, 212, 92, 295, 312, 403, 146, 857, 355, 90, 333, 185, 261, 154, 202, 298, 416, 421, 198, 586, 402, 96, 229, 379, 232, 243, 228, 269, 117, 257, 63, 576, 139, 436, 115, 370, 400, 324, 49, 179, 476, 60, 153, 181, 138, 495, 503, 474]

Mode: 0
==================================================
Unique Values in OpenPorchSF:
[0, 40, 38, 96, 130, 42, 46, 120, 50, 72, 41, 54, 207, 114, 65, 28, 131, 78, 121, 116, 102, 8, 84, 36, 17, 52, 49, 502, 53, 26, 33, 224, 87, 98, 88, 20, 35, 122, 68, 75, 81, 156, 214, 205, 108, 66, 24, 76, 192, 80, 60, 243, 62, 39, 267, 48, 30, 184, 113, 182, 45, 191, 61, 104, 34, 44, 136, 100, 63, 142, 211, 29, 32, 70, 59, 144, 152, 21, 234, 103, 105, 124, 51, 27, 141, 69, 74, 82, 158, 106, 99, 166, 154, 64, 73, 111, 86, 189, 244, 126, 22, 204, 155, 135, 57, 140, 312, 56, 146, 94, 406, 172, 265, 95, 238, 112, 25, 162, 137, 90, 118, 228, 4, 176, 134, 47, 292, 150, 55, 16, 15, 128, 151, 288, 364, 187, 170, 132, 285, 523, 258, 160, 129, 208, 260, 252, 125, 18, 58, 101, 133, 240, 341, 85, 110, 287, 93, 159, 169, 319, 168, 43, 198, 12, 291, 282, 304, 547, 91, 123, 210, 188, 117, 250, 23, 231, 418, 235, 236, 185, 138, 213, 83, 247, 262, 263, 77, 200]

Mode: 0
==================================================
Unique Values in EnclosedPorch:
[0, 164, 264, 242, 252, 192, 112, 162, 96, 81, 102, 128, 228, 184, 176, 36, 275, 158, 120, 224, 236, 226, 126, 34, 108, 42, 68, 156, 293, 221, 268, 144, 220, 244, 272, 77, 234, 177, 216, 145, 116, 44, 190, 143, 194, 90, 39, 115, 208, 20, 99, 137, 84, 214, 40, 196, 114, 183, 286, 100, 32, 91, 24, 150, 198, 168, 240, 202, 52, 330, 60, 129, 301, 239, 160, 134, 291, 105, 185, 205, 170, 552, 212, 180, 123, 189, 259, 56, 19, 154, 280, 210, 136, 254, 386, 248, 50, 140]

Mode: 0
==================================================
Unique Values in 3SsnPorch:
[0, 162, 144, 180, 320, 407, 130, 216, 196, 168, 290, 245, 153, 182, 140, 508, 304]

Mode: 0
==================================================
Unique Values in ScreenPorch:
[0, 189, 222, 155, 197, 271, 128, 260, 160, 120, 170, 216, 80, 95, 276, 116, 180, 176, 161, 224, 147, 40, 291, 100, 374, 145, 126, 252, 200, 233, 184, 90, 156, 192, 168, 225, 163, 143, 153, 410, 185, 142, 190, 175, 178, 119, 99, 182, 198, 152, 140, 385, 480, 288, 130, 440, 312, 204, 53, 122, 144, 287, 63, 273, 263, 210]

Mode: 0
==================================================
Unique Values in PoolArea:
[0, 648, 555, 480, 512, 519, 738]

Mode: 0
==================================================
Unique Values in MiscVal:
[0, 15500, 600, 500, 800, 400, 3500, 700, 1300, 450, 8300, 2000, 350, 1200, 480, 620, 1150, 1400, 2500]

Mode: 0
==================================================
Unique Values in MoSold:
[6, 5, 7, 3, 4, 10, 12, 11, 8, 9, 2, 1]

Mode: 6
==================================================
Unique Values in YrSold:
[2010, 2009, 2008, 2007, 2006]

Mode: 2009
==================================================
Unique Values in SaleType:
['WD', 'New', 'COD', 'ConLI', 'Oth', 'ConLw', 'Con', 'ConLD', 'CWD']

Mode: WD
==================================================
Unique Values in SaleCondition:
['Normal', 'Abnorml', 'Partial', 'Family', 'Alloca', 'AdjLand']

Mode: Normal
==================================================
Unique Values in MasVnrArea:
[0.0, 252.0, 210.0, 140.0, 1047.0, 206.0, 176.0, 168.0, 184.0, 132.0, 368.0, 340.0, 268.0, 157.0, 90.0, 275.0, 200.0, 66.0, 237.0, 160.0, 562.0, 220.0, 772.0, 174.0, 85.0, 72.0, 660.0, 298.0, 149.0, 14.0, 99.0, 243.0, 480.0, 232.0, 28.0, 86.0, 125.0, 75.0, 452.0, 528.0, 260.0, 166.0, 109.0, 262.0, 154.0, 180.0, 365.0, 128.0, 860.0, 288.0, 80.0, 302.0, 259.0, 151.0, 245.0, 204.0, 183.0, 16.0, 104.0, 234.0, 123.0, 156.0, 228.0, 171.0, 630.0, 360.0, 748.0, 238.0, 175.0, 119.0, 196.0, 98.0, 122.0, 250.0, 40.0, 108.0, 106.0, 233.0, 146.0, 169.0, 788.0, 464.0, 256.0, 112.0, 399.0, 320.0, 513.0, 420.0, 60.0, 922.0, 100.0, 81.0, 143.0, 27.0, 564.0, 165.0, 224.0, 388.0, 32.0, 289.0, 410.0, 579.0, 225.0, 74.0, 127.0, 731.0, 287.0, 24.0, 335.0, 451.0, 284.0, 500.0, 153.0, 162.0, 186.0, 396.0, 96.0, 64.0, 130.0, 110.0, 640.0, 870.0, 272.0, 95.0, 412.0, 1.0, 762.0, 51.0, 328.0, 18.0, 144.0, 113.0, 82.0, 530.0, 450.0, 178.0, 366.0, 70.0, 576.0, 768.0, 351.0, 378.0, 101.0, 48.0, 300.0, 136.0, 428.0, 150.0, 456.0, 604.0, 246.0, 472.0, 189.0, 571.0, 22.0, 468.0, 38.0, 541.0, 600.0, 192.0, 68.0, 44.0, 254.0, 50.0, 170.0, 796.0, 336.0, 120.0, 442.0, 479.0, 310.0, 240.0, 258.0, 172.0, 212.0, 1378.0, 290.0, 350.0, 30.0, 84.0, 324.0, 219.0, 481.0, 424.0, 56.0, 448.0, 425.0, 223.0, 126.0, 333.0, 653.0, 491.0, 236.0, 202.0, 226.0, 293.0, 116.0, 248.0, 247.0, 54.0, 344.0, 76.0, 297.0, 359.0, 281.0, 348.0, 315.0, 216.0, 270.0, 138.0, 432.0, 158.0, 285.0, 816.0, 105.0, 89.0, 286.0, 147.0, 415.0, 305.0, 194.0, 304.0, 63.0, 391.0, 11.0, 387.0, 381.0, 274.0, 621.0, 164.0, 632.0, 312.0, 53.0, 117.0, 94.0, 370.0, 88.0, 102.0, 115.0, 261.0, 342.0, 182.0, 921.0, 215.0, 145.0, 705.0, 142.0, 650.0, 163.0, 436.0, 42.0, 408.0, 46.0, 975.0, 299.0, 34.0, 135.0, 554.0, 318.0, 148.0, 41.0, 209.0, 92.0, 97.0, 594.0, 584.0, 338.0, 31.0, 573.0, 673.0, 266.0, 567.0, 664.0, 466.0, 244.0, 203.0, 306.0, 280.0, 255.0, 380.0, 161.0, 423.0, 426.0, 616.0, 603.0, 309.0]

Mode: 0.0
==================================================
Unique Values in LotFrontage:
[70.0, 59.0, 67.0, 60.0, 50.0, 122.0, 40.0, 78.0, 76.0, 85.0, 82.0, 80.0, 49.0, 75.0, 47.0, 55.0, 91.0, 52.0, 65.0, 35.0, 100.0, 79.0, 128.0, 71.0, 64.0, 34.0, 53.0, 33.0, 95.0, 58.0, 72.0, 105.0, 44.0, 69.0, 54.0, 43.0, 74.0, 21.0, 90.0, 102.0, 77.0, 51.0, 107.0, 120.0, 96.0, 89.0, 30.0, 103.0, 62.0, 104.0, 84.0, 68.0, 160.0, 92.0, 81.0, 88.0, 115.0, 46.0, 36.0, 73.0, 39.0, 56.0, 48.0, 63.0, 87.0, 106.0, 86.0, 141.0, 24.0, 108.0, 99.0, 57.0, 114.0, 42.0, 83.0, 121.0, 98.0, 144.0, 110.0, 61.0, 130.0, 93.0, 137.0, 118.0, 66.0, 313.0, 168.0, 45.0, 109.0, 124.0, 149.0, 134.0, 32.0, 129.0, 97.0, 37.0, 112.0, 41.0, 94.0, 152.0, 150.0, 111.0, 182.0, 174.0, 38.0, 101.0, 153.0]

Mode: 70.0
==================================================
Unique Values in BsmtCond:
['TA', 'Fa', 'None', 'Gd', 'Po']

Mode: TA
==================================================
Unique Values in MasVnrType:
['None', 'BrkFace', 'Stone', 'BrkCmn']

Mode: None
==================================================
Unique Values in GarageType:
['Attchd', 'None', 'Detchd', 'BuiltIn', 'Basment', '2Types', 'CarPort']

Mode: Attchd
==================================================
Unique Values in GarageQual:
['TA', 'None', 'Po', 'Fa', 'Gd', 'Ex']

Mode: TA
==================================================
Unique Values in Fence:
['None', 'MnPrv', 'GdWo', 'GdPrv', 'MnWw']

Mode: None
==================================================
Unique Values in BsmtFinType1:
['Rec', 'Unf', 'ALQ', 'LwQ', 'BLQ', 'GLQ', 'None']

Mode: Unf
==================================================
Unique Values in BsmtExposure:
['No', 'Av', 'Gd', 'Mn', 'None']

Mode: No
==================================================
Unique Values in FireplaceQu:
['None', 'TA', 'Gd', 'Fa', 'Ex', 'Po']

Mode: None
==================================================
Unique Values in GarageCond:
['TA', 'None', 'Po', 'Fa', 'Gd', 'Ex']

Mode: TA
==================================================
Unique Values in GarageFinish:
['RFn', 'None', 'Unf', 'Fin']

Mode: Unf
==================================================
Unique Values in Alley:
['None', 'Pave', 'Grvl']

Mode: None
==================================================
Unique Values in BsmtFinType2:
['Unf', 'LwQ', 'Rec', 'BLQ', 'None', 'GLQ', 'ALQ']

Mode: Unf
==================================================
Unique Values in BsmtQual:
['TA', 'Gd', 'Fa', 'None', 'Ex']

Mode: TA
==================================================
Unique Values in PoolQC:
['None', 'Fa', 'Ex', 'Gd']

Mode: None
==================================================
Unique Values in MiscFeature:
['None', 'Othr', 'Shed', 'Gar2', 'TenC']

Mode: None
==================================================
Unique Values in Electrical:
['SBrkr', 'FuseA', 'FuseF', 'FuseP']

Mode: SBrkr
==================================================

From the output above, we can do some feature engineering as follows.

• Create new numerical feature HouseAge and AgeAftRemod where HouseAge = YrSold - YearBuilt and AgeAftRemod = YrSold - YearRemodAdd. Then, drop YrSold, YearBuilt, and YearRemodAdd.

• Create new numerical feature TotalBath where TotalBath = BsmtFullBath + 0.5*BsmtHalfBath + FullBath + 0.5*HalfBath. Then, drop BsmtFullBath, BsmtHalfBath, FullBath and HalfBath. Half baths are assigned a constant utility weight of \(0.5\) to preserve the exact functional utility and economic weight of the property’s sanitation facilities.

• Create new numerical feature TotalPorchSF where TotalPorchSF = WoodDeckSF + OpenPorchSF + EnclosedPorch + 3SsnPorch + ScreenPorch. Then, drop WoodDeckSF, OpenPorchSF, EnclosedPorch, 3SsnPorch, and ScreenPorch. This is based on the inspections that reveal high sparsity (excessive zero values) across these individual features because properties rarely possess all five porch types simultaneously.

Also, we can make some hypotheses.

• The feature TotalBsmtSF is hypothesized to be a perfect linear combination of its sub-components: TotalBsmtSF = BsmtFinSF1 + BsmtFinSF2 + BsmtUnfSF. A program will be executed to verify if the equation holds true across 100% of the rows. If this deterministic relationship is validated, the three sub-components will be systematically dropped to eliminate redundancy and prevent multicollinearity.

• The feature GrLivArea is hypothesized to be a perfect linear combination of its sub-components: GrLivArea = 1stFlrSF + 2ndFlrSF + LowQualFinSF. A program will be executed to verify if the equation holds true across 100% of the rows. If this deterministic relationship is validated, the three sub-components will be systematically dropped to eliminate redundancy and prevent multicollinearity.

# execute the plan

# HouseAge and AgeAftRemod
X_train_imputed_df['HouseAge'] = X_train_imputed_df['YrSold'] - X_train_imputed_df['YearBuilt']
X_train_imputed_df['AgeAftRemod'] = X_train_imputed_df['YrSold'] - X_train_imputed_df['YearRemodAdd']

X_test_imputed_df['HouseAge'] = X_test_imputed_df['YrSold'] - X_test_imputed_df['YearBuilt']
X_test_imputed_df['AgeAftRemod'] = X_test_imputed_df['YrSold'] - X_test_imputed_df['YearRemodAdd']


# TotalBath
X_train_imputed_df['TotalBath'] = (X_train_imputed_df['BsmtFullBath'] + 0.5*X_train_imputed_df['BsmtHalfBath'] + X_train_imputed_df['FullBath'] 
                                   + 0.5*X_train_imputed_df['HalfBath'])
X_test_imputed_df['TotalBath'] = (X_test_imputed_df['BsmtFullBath'] + 0.5*X_test_imputed_df['BsmtHalfBath'] + X_test_imputed_df['FullBath'] 
                                  + 0.5*X_test_imputed_df['HalfBath'])


# TotalPorchSF
X_train_imputed_df['TotalPorchSF'] = (X_train_imputed_df['WoodDeckSF'] + X_train_imputed_df['OpenPorchSF'] + X_train_imputed_df['EnclosedPorch'] 
                                      + X_train_imputed_df['3SsnPorch'] + X_train_imputed_df['ScreenPorch'])
X_test_imputed_df['TotalPorchSF'] = (X_test_imputed_df['WoodDeckSF'] + X_test_imputed_df['OpenPorchSF'] + X_test_imputed_df['EnclosedPorch'] 
                                      + X_test_imputed_df['3SsnPorch'] + X_test_imputed_df['ScreenPorch'])


X_train_imputed_df = X_train_imputed_df.drop(columns=['YrSold', 'YearBuilt', 'YearRemodAdd', 'BsmtFullBath', 'BsmtHalfBath',
                                                       'FullBath', 'HalfBath', 'WoodDeckSF', 'OpenPorchSF', 'EnclosedPorch', 
                                                       '3SsnPorch', 'ScreenPorch'])
X_test_imputed_df = X_test_imputed_df.drop(columns=['YrSold', 'YearBuilt', 'YearRemodAdd', 'BsmtFullBath', 'BsmtHalfBath',
                                                     'FullBath', 'HalfBath', 'WoodDeckSF', 'OpenPorchSF', 'EnclosedPorch', 
                                                     '3SsnPorch', 'ScreenPorch'])


# TotalBsmtSF
basement_check = (X_train_imputed_df['TotalBsmtSF'] == (X_train_imputed_df['BsmtFinSF1'] + X_train_imputed_df['BsmtFinSF2'] + X_train_imputed_df['BsmtUnfSF'])).all()
if basement_check:
    X_train_imputed_df = X_train_imputed_df.drop(columns=['BsmtFinSF1', 'BsmtFinSF2', 'BsmtUnfSF'])
    X_test_imputed_df = X_test_imputed_df.drop(columns=['BsmtFinSF1', 'BsmtFinSF2', 'BsmtUnfSF'])


# GrLivArea
grliv_check = (X_train_imputed_df['GrLivArea'] == (X_train_imputed_df['1stFlrSF'] + X_train_imputed_df['2ndFlrSF'] + X_train_imputed_df['LowQualFinSF'])).all()
if grliv_check:
    X_train_imputed_df = X_train_imputed_df.drop(columns=['1stFlrSF', '2ndFlrSF', 'LowQualFinSF'])
    X_test_imputed_df = X_test_imputed_df.drop(columns=['1stFlrSF', '2ndFlrSF', 'LowQualFinSF'])


print(f'X_train_imputed_df shape: {X_train_imputed_df.shape}')
print(f'X_test_imputed_df shape: {X_test_imputed_df.shape}')

X_train_imputed_df shape: (1168, 64)
X_test_imputed_df shape: (292, 64)

X_train_imputed_df.head()

	MSSubClass	MSZoning	LotArea	Street	LotShape	LandContour	Utilities	LotConfig	LandSlope	Neighborhood	...	Alley	BsmtFinType2	BsmtQual	PoolQC	MiscFeature	Electrical	HouseAge	AgeAftRemod	TotalBath	TotalPorchSF
0	20	RL	8400	Pave	Reg	Lvl	AllPub	Inside	Gtl	NAmes	...	None	Unf	TA	None	None	SBrkr	53	53	2.0	250
1	60	RL	7837	Pave	IR1	Lvl	AllPub	Inside	Gtl	Gilbert	...	None	Unf	Gd	None	None	SBrkr	16	15	2.5	40
2	30	RL	8777	Pave	Reg	Lvl	AllPub	Inside	Gtl	Edwards	...	None	Unf	Fa	None	None	FuseA	98	58	1.0	492
3	50	RL	7200	Pave	Reg	Lvl	AllPub	Corner	Gtl	SWISU	...	None	Unf	Gd	None	None	SBrkr	70	57	2.5	264
4	50	RL	5000	Pave	Reg	Lvl	AllPub	Inside	Gtl	SWISU	...	Pave	Unf	TA	None	None	SBrkr	86	60	2.0	242

5 rows × 64 columns

X_test_imputed_df.head()

	MSSubClass	MSZoning	LotArea	Street	LotShape	LandContour	Utilities	LotConfig	LandSlope	Neighborhood	...	Alley	BsmtFinType2	BsmtQual	PoolQC	MiscFeature	Electrical	HouseAge	AgeAftRemod	TotalBath	TotalPorchSF
0	20	RL	8414	Pave	Reg	Lvl	AllPub	Inside	Gtl	Sawyer	...	None	Unf	TA	None	None	SBrkr	43	3	1.5	192
1	60	RL	12256	Pave	IR1	Lvl	AllPub	Corner	Gtl	NoRidge	...	None	Unf	Ex	None	None	SBrkr	16	15	3.5	218
2	30	RM	8960	Pave	Reg	Lvl	AllPub	Inside	Gtl	OldTown	...	Grvl	Unf	TA	None	None	FuseA	83	60	1.0	130
3	50	RM	5000	Pave	Reg	Lvl	AllPub	Corner	Gtl	BrkSide	...	None	Unf	TA	None	None	SBrkr	59	56	2.0	60
4	20	RL	12898	Pave	IR1	HLS	AllPub	Inside	Gtl	Timber	...	None	Unf	Ex	None	None	SBrkr	2	1	3.0	228

5 rows × 64 columns

Data Visualization and Feature Selection

First, divide the training dataset into 2 parts: numerical features only and categorical features only.

current_all_columns = X_train_imputed_df.columns.tolist() # update
num_columns = ['LotArea', 'TotalBsmtSF', 'GrLivArea', 'GarageCars', 'GarageArea', 
               'MasVnrArea', 'LotFrontage', 'HouseAge', 'AgeAftRemod', 'TotalBath', 
               'TotalPorchSF']
cat_columns = list(set(current_all_columns.copy()) - set(num_columns))

X_train_imputed_num_df = X_train_imputed_df[num_columns]
X_train_imputed_cat_df = X_train_imputed_df[cat_columns]

X_test_imputed_num_df = X_test_imputed_df[num_columns]
X_test_imputed_cat_df = X_test_imputed_df[cat_columns]

print(f'X_train_imputed_num_df shape: {X_train_imputed_num_df.shape}')
print(f'X_train_imputed_cat_df shape: {X_train_imputed_cat_df.shape}')
print(f'X_test_imputed_num_df shape: {X_test_imputed_num_df.shape}')
print(f'X_test_imputed_cat_df shape: {X_test_imputed_cat_df.shape}')

X_train_imputed_num_df shape: (1168, 11)
X_train_imputed_cat_df shape: (1168, 53)
X_test_imputed_num_df shape: (292, 11)
X_test_imputed_cat_df shape: (292, 53)

Numerical Features

Let’s explore which numerical features have a strong relationship with the target variable by creating scatter plots and examining their correlation values.

corr_train_df = X_train_imputed_num_df.copy()
corr_train_df[target] = y_train

corr_matrix = corr_train_df.corr()
target_corr = corr_matrix[target]

n_columns = 3
n_rows = (len(num_columns) + n_columns - 1) // n_columns

fig, axes = plt.subplots(n_rows, n_columns, figsize=(15, 5 * n_rows))
axes = axes.flatten()

for i, column in enumerate(num_columns):
    ax = axes[i]
    ax.scatter(X_train_imputed_df[column].values, y_train)
    ax.set_xlabel(column, fontsize=10)
    ax.set_ylabel(target, fontsize=10)

    ax.text(0.05, 0.95, f'Pearson Correlation Coefficient (r) = {target_corr[column]:.3f}', fontsize=10, color='black', transform=ax.transAxes)

for j in range(i + 1, len(axes)):
    fig.delaxes(axes[j])

plt.show()

Let’s also see the correlation between numerical features using heatmap visualization.

plt.subplots(figsize=(15,15))
sns.heatmap(corr_matrix.iloc[:-1, :-1], vmin=-1, center=0, vmax=1, annot=True)
plt.show()

To select the most predictive features for the linear regression model, a threshold of \(|r|> 0.5\) is strictly applied to filter the continuous numerical features. We will get [‘TotalBsmtSF’, ‘GrLivArea’, ‘GarageCars’, ‘GarageArea’, ‘HouseAge’, ‘AgeAftRemod’, ‘TotalBath’]

However, notice that the heatmap above reveals a severe multicollinearity between GarageCars and GarageArea (\(r \approx 0.88\)). This strong linear dependency occurs because both features encapsulate the exact same underlying proxy: the physical capacity of the garage. To maintain the full-rank assumption for the linear regression model and avoid variance inflation, one of these features must be eliminated. GarageArea is intentionally preserved because it is a pure continuous numerical variable with higher granularity, whereas GarageCars is dropped to eliminate redundancy.

abs_target_corr = target_corr.abs().drop(target)
selected_num_series = abs_target_corr[abs_target_corr > 0.5]
selected_num = selected_num_series.index.tolist()

# remove GarageCars
selected_num.remove('GarageCars')

print(f'Selected Numerical Features: {selected_num}')
print(f'Number of Selected Numerical Features: {len(selected_num)} Out of {len(num_columns)} Numerical Features.\n')
print('Pearson Correlation Coefficient (r):')
print(selected_num_series.drop('GarageCars'))

Selected Numerical Features: ['TotalBsmtSF', 'GrLivArea', 'GarageArea', 'HouseAge', 'AgeAftRemod', 'TotalBath']
Number of Selected Numerical Features: 6 Out of 11 Numerical Features.

Pearson Correlation Coefficient (r):
TotalBsmtSF    0.597766
GrLivArea      0.695652
GarageArea     0.624139
HouseAge       0.516428
AgeAftRemod    0.509649
TotalBath      0.621280
Name: SalePrice, dtype: float64

Categorical Features

Let’s also create box plots to visualize the categorical features.

boxplot_train_df = X_train_imputed_cat_df.copy()
boxplot_train_df[target] = y_train

n_columns = 3
n_rows = (len(cat_columns) + n_columns - 1) // n_columns

fig, axes = plt.subplots(n_rows, n_columns, figsize=(15, 5 * n_rows))
axes = axes.flatten()

for i, column in enumerate(cat_columns):
    sns.boxplot(boxplot_train_df, x=column, y=target, ax=axes[i])

for j in range(i + 1, len(axes)):
    fig.delaxes(axes[j])

plt.show()

To determine which categorical and ordinal features hold significant predictive power over the continuous target variable (SalePrice), an Analysis of Variance (ANOVA) F-test is employed. Mathematically, ANOVA evaluates whether the mean of the target variable differs significantly across the distinct groups or levels defined by each categorical feature. The statistical hypotheses for this evaluation are formulated as follows:

• Null Hypothesis (\(H_0\)): The mean SalePrice is identical across all categories of the feature.

• Alternative Hypothesis (\(H_1\)): At least one category mean is significantly different from the others.

A stringent significance threshold of \(\alpha = 0.05\) is applied. Categorical features that yield a \(p\text{-value} < 0.05\) allow us to rigorously reject the null hypothesis, confirming that the variance between the category means is statistically significant and not due to random chance. These verified features are selected for downstream dummy encoding and modeling, while non-significant features are discarded to keep the model parsimonious.

A common conceptual question is how a simple comparison of group means can dictate whether a categorical feature is important for a regression model. The mathematical logic lies in the concept of variance explanation.

If we group the house prices (SalePrice) based on a specific categorical feature (for instance, Neighborhood), and we discover that the mean price of houses in Neighborhood A is structurally different from Neighborhood B and Neighborhood C, it demonstrates that knowing the category provides immediate information about the target’s magnitude. Mathematically, this implies that the variance between the groups is much larger than the variance within the groups.

from scipy import stats

anova_df = X_train_imputed_cat_df[cat_columns].copy()
anova_df[target] = y_train

anova_results = []

print("----- Running ANOVA Test -----\n")

for column in cat_columns:

    unique_cats = anova_df[column].unique()
    
    grouped_data = [anova_df[anova_df[column] == cat][target].values for cat in unique_cats]

    f_stat, p_val = stats.f_oneway(*grouped_data)
    

    anova_results.append({
        'Feature': column,
        'F_Statistic': f_stat,
        'P_Value': p_val
    })

# convert to DataFrame and sort by F_Statistics in descending order
anova_summary_df = pd.DataFrame(anova_results).sort_values(by='F_Statistic', ascending=False).reset_index(drop=True)

print("Results:\n")
print(anova_summary_df)

# select features with p-value < 0.05
selected_cat = anova_summary_df[anova_summary_df['P_Value'] < 0.05]['Feature'].tolist()
print(f"\nSelected Categorical Features: {selected_cat}\n")
print(f'Number of Selected Categorical Features: {len(selected_cat)} Out of {len(cat_columns)} Categorical Features.')

----- Running ANOVA Test -----

Results:

          Feature  F_Statistic        P_Value
0     KitchenQual   332.403111  7.223574e-156
1       ExterQual   318.566812  5.695797e-151
2     OverallQual   262.806466  1.627739e-272
3        BsmtQual   248.162401  3.875122e-154
4    GarageFinish   171.170540   6.430908e-92
5      Fireplaces   114.707812   4.466366e-65
6     FireplaceQu    92.211841   7.717475e-82
7      CentralAir    89.825144   1.399942e-20
8      MasVnrType    79.691818   6.858075e-47
9       HeatingQC    77.812050   1.597372e-58
10     Foundation    77.154994   6.121416e-70
11     GarageType    64.091698   7.896260e-69
12   Neighborhood    54.116418  2.317899e-169
13   BsmtFinType1    50.857581   1.097677e-55
14   BsmtExposure    46.909362   1.377122e-36
15   TotRmsAbvGrd    40.899515   5.324178e-75
16  SaleCondition    34.303105   9.045336e-33
17     PavedDrive    32.925986   1.230150e-14
18       MSZoning    31.026797   1.412720e-24
19       LotShape    29.314662   2.841706e-18
20     MSSubClass    27.468999   1.149867e-62
21     Electrical    23.587138   8.009490e-15
22     GarageQual    21.757164   7.135050e-21
23     GarageCond    21.579494   1.059399e-20
24       SaleType    21.368142   1.708349e-30
25    OverallCond    20.112204   1.183219e-28
26     HouseStyle    16.446771   9.178892e-21
27       BsmtCond    16.432600   4.189772e-13
28    Exterior1st    16.242190   9.562360e-37
29    Exterior2nd    15.631165   2.255237e-37
30      RoofStyle    14.648311   5.999028e-14
31          Alley    13.271476   1.999632e-06
32    LandContour    12.952648   2.514628e-08
33   BedroomAbvGr    11.973405   8.225818e-15
34         PoolQC    11.221984   2.918959e-07
35       BldgType    10.928077   1.061013e-08
36          Fence    10.539999   2.169467e-08
37       PoolArea     9.770752   1.595538e-10
38   KitchenAbvGr     8.830580   8.635620e-06
39      ExterCond     7.466410   6.146250e-06
40      LotConfig     6.971157   1.517764e-05
41   BsmtFinType2     6.392812   1.228230e-06
42     Condition1     5.414498   1.028592e-06
43        Heating     4.021229   1.271593e-03
44     Functional     3.526161   1.832685e-03
45      LandSlope     2.918212   5.442526e-02
46       RoofMatl     2.654246   1.461373e-02
47    MiscFeature     1.710591   1.452517e-01
48     Condition2     1.619626   1.257925e-01
49         MoSold     0.846926   5.928499e-01
50        MiscVal     0.748021   7.623634e-01
51      Utilities     0.323535   5.696006e-01
52         Street     0.279746   5.969682e-01

Selected Categorical Features: ['KitchenQual', 'ExterQual', 'OverallQual', 'BsmtQual', 'GarageFinish', 'Fireplaces', 'FireplaceQu', 'CentralAir', 'MasVnrType', 'HeatingQC', 'Foundation', 'GarageType', 'Neighborhood', 'BsmtFinType1', 'BsmtExposure', 'TotRmsAbvGrd', 'SaleCondition', 'PavedDrive', 'MSZoning', 'LotShape', 'MSSubClass', 'Electrical', 'GarageQual', 'GarageCond', 'SaleType', 'OverallCond', 'HouseStyle', 'BsmtCond', 'Exterior1st', 'Exterior2nd', 'RoofStyle', 'Alley', 'LandContour', 'BedroomAbvGr', 'PoolQC', 'BldgType', 'Fence', 'PoolArea', 'KitchenAbvGr', 'ExterCond', 'LotConfig', 'BsmtFinType2', 'Condition1', 'Heating', 'Functional', 'RoofMatl']

Number of Selected Categorical Features: 46 Out of 53 Categorical Features.

Based on the results above, while applying the standard significance threshold (\(\text{p-value} < \alpha\)) successfully filters out statistically irrelevant features, it still retains an excessively large number of categorical predictors due to the sample size’s high sensitivity. Introducing too many dummy variables post-encoding risks inflating the model’s feature space, which leads to structural overfitting and severely degrades the interpretability of a linear estimator.

To enforce a strict parsimonious model constraints, a Select K-Best approach is adopted by ranking the features based on their F-Statistic magnitude. Mathematically, the F-statistic represents the ratio of variance between the groups to the variance within the groups (\(F = \frac{\text{MS}_{\text{between}}}{\text{MS}_{\text{within}}}\)). A larger F-value indicates a clean and distinct separation of target distributions among the categories, yielding a significantly higher signal-to-noise ratio. Therefore, filtering the top 10 categorical features with the highest F-scores guarantees that we capture the absolute strongest predictors while maintaining a lean and highly stable design matrix.

# reselect features with top 10 F-statistics
selected_cat = anova_summary_df['Feature'].head(10).tolist()
print(f"Selected Top-10 Categorical Features: {selected_cat}\n")

Selected Top-10 Categorical Features: ['KitchenQual', 'ExterQual', 'OverallQual', 'BsmtQual', 'GarageFinish', 'Fireplaces', 'FireplaceQu', 'CentralAir', 'MasVnrType', 'HeatingQC']

Data Preprocessing Phase 2

Let’s apply the selected numerical and categorical features.

X_train_selected_num_df = X_train_imputed_num_df[selected_num]
X_train_selected_cat_df = X_train_imputed_cat_df[selected_cat]

X_test_selected_num_df = X_test_imputed_num_df[selected_num]
X_test_selected_cat_df = X_test_imputed_cat_df[selected_cat]

print(f'X_train_selected_num_df shape: {X_train_selected_num_df.shape}')
print(f'X_train_selected_cat_df shape: {X_train_selected_cat_df.shape}')
print(f'X_test_selected_num_df shape: {X_test_selected_num_df.shape}')
print(f'X_test_selected_cat_df shape: {X_test_selected_cat_df.shape}')

X_train_selected_num_df shape: (1168, 6)
X_train_selected_cat_df shape: (1168, 10)
X_test_selected_num_df shape: (292, 6)
X_test_selected_cat_df shape: (292, 10)

Numerical Features Standardization

As observed in the scatter plots for numerical features, the numerical features exhibit vastly different scales and measurement ranges (e.g., GrLivArea ranges in thousands of square feet, whereas TotalBath values range from 1 to 5). Linear regression models are highly sensitive to these scale discrepancies because the optimization algorithm might inadvertently penalize or over-allocate weight to features with larger absolute magnitudes.

To mitigate this, StandardScaler is applied to transform all numerical features to possess a mean of 0 and a standard deviation of 1 (\(\mu = 0, \sigma = 1\)).

from sklearn.preprocessing import StandardScaler

scaler = StandardScaler()

X_train_scaled_num = scaler.fit_transform(X_train_selected_num_df)
X_test_scaled_num = scaler.transform(X_test_selected_num_df)

X_train_scaled_num_df = pd.DataFrame(data=X_train_scaled_num, columns=selected_num)
X_test_scaled_num_df = pd.DataFrame(data=X_test_scaled_num, columns=selected_num)

X_train_scaled_num_df.head()

	TotalBsmtSF	GrLivArea	GarageArea	HouseAge	AgeAftRemod	TotalBath
0	0.572612	-0.407093	-0.863837	0.526141	1.452679	-0.285144
1	-0.596547	0.083170	-0.456264	-0.679464	-0.382563	0.356567
2	-0.603357	-1.395250	-2.257169	1.992419	1.694159	-1.568564
3	-0.750921	0.458975	-1.119755	1.080068	1.645863	0.356567
4	-0.081209	0.312087	-0.797488	1.601411	1.790750	-0.285144

X_test_scaled_num_df.head()

	TotalBsmtSF	GrLivArea	GarageArea	HouseAge	AgeAftRemod	TotalBath
0	-0.006292	-0.876372	-1.006014	0.200302	-0.962113	-0.926854
1	0.910874	2.088099	1.117159	-0.679464	-0.382563	1.639987
2	-0.122072	-0.952678	-0.551048	1.503660	1.790750	-1.568564
3	-0.131153	0.260581	-0.266695	0.721645	1.597567	-0.285144
4	1.267297	0.176644	2.065003	-1.135640	-1.058704	0.998277

Categorical Features Encoding

Note that machine learning algorithms can only process number. This means we need to encode the categorical features.

Before, that divide each X_train_selected_cat_df and X_test_selected_cat_df into 2 parts: ordinal categorical features only and nominal categorical features only.

ordinal_cat_columns = ['KitchenQual', 'ExterQual', 'OverallQual', 'BsmtQual', 'FireplaceQu', 'HeatingQC']
nominal_cat_columns = list(set(selected_cat.copy()) - set(ordinal_cat_columns))

X_train_ordinal_cat_df = X_train_selected_cat_df[ordinal_cat_columns]
X_train_nominal_cat_df = X_train_selected_cat_df[nominal_cat_columns]

X_test_ordinal_cat_df = X_test_selected_cat_df[ordinal_cat_columns]
X_test_nominal_cat_df = X_test_selected_cat_df[nominal_cat_columns]

print(f'X_train_ordinal_cat_df shape: {X_train_ordinal_cat_df.shape}')
print(f'X_train_nominal_cat_df shape: {X_train_nominal_cat_df.shape}')
print(f'X_test_ordinal_cat_df shape: {X_test_ordinal_cat_df.shape}')
print(f'X_test_nominal_cat_df shape: {X_test_nominal_cat_df.shape}')

X_train_ordinal_cat_df shape: (1168, 6)
X_train_nominal_cat_df shape: (1168, 4)
X_test_ordinal_cat_df shape: (292, 6)
X_test_nominal_cat_df shape: (292, 4)

Ordinal Encoding

print('Value Counts of Ordinal Categorical Features:\n')
for column in X_train_ordinal_cat_df.columns.tolist():
    print(X_train_ordinal_cat_df[column].value_counts())
    print('='*25)

Value Counts of Ordinal Categorical Features:

KitchenQual
TA    589
Gd    470
Ex     77
Fa     32
Name: count, dtype: int64
=========================
ExterQual
TA    727
Gd    388
Ex     42
Fa     11
Name: count, dtype: int64
=========================
OverallQual
6     311
5     308
7     258
8     135
4      88
9      36
3      16
10     13
1       2
2       1
Name: count, dtype: int64
=========================
BsmtQual
TA      521
Gd      493
Ex       97
Fa       29
None     28
Name: count, dtype: int64
=========================
FireplaceQu
None    547
Gd      305
TA      252
Fa       27
Ex       21
Po       16
Name: count, dtype: int64
=========================
HeatingQC
Ex    583
TA    347
Gd    195
Fa     42
Po      1
Name: count, dtype: int64
=========================

Features capturing quality and condition (e.g., ExterQual, KitchenQual) inherently possess a natural mathematical hierarchy. Rather than treating them as nominal text labels, these variables are processed using Ordinal Encoding to translate their qualitative values into a structurally ordered numerical scale (e.g., None \(\rightarrow\) 0, Po \(\rightarrow\) 1, Fa \(\rightarrow\) 2, TA \(\rightarrow\) 3, Gd \(\rightarrow\) 4, Ex \(\rightarrow\) 5).

This encoding scheme preserves the explicit sequence of quality tiers, allowing the linear model to interpret the monotonic progression from poor to excellent directly as a continuous predictive signal.

Ordinal mapping

def ordinal_mapping(x):
    if x == 'None':
        return 0
    elif x == 'Po':
        return 1
    elif x == 'Fa':
        return 2
    elif x == 'TA':
        return 3
    elif x == 'Gd':
        return 4
    elif x == 'Ex':
        return 5
    else:
        return x

X_train_encoded_ordinal_cat_df = X_train_ordinal_cat_df.map(ordinal_mapping)
X_test_encoded_ordinal_cat_df = X_test_ordinal_cat_df.map(ordinal_mapping)

print(f'X_train_encoded_ordinal_cat_df shape: {X_train_encoded_ordinal_cat_df.shape}')
print(f'X_test_encoded_ordinal_cat_df shape: {X_test_encoded_ordinal_cat_df.shape}')

X_train_encoded_ordinal_cat_df shape: (1168, 6)
X_test_encoded_ordinal_cat_df shape: (292, 6)

X_train_encoded_ordinal_cat_df.head()

	KitchenQual	ExterQual	OverallQual	BsmtQual	FireplaceQu	HeatingQC
0	3	3	5	3	0	3
1	3	4	6	4	3	4
2	3	3	5	2	0	4
3	4	3	5	4	3	5
4	4	3	5	3	4	3

X_test_encoded_ordinal_cat_df.head()

	KitchenQual	ExterQual	OverallQual	BsmtQual	FireplaceQu	HeatingQC
0	3	3	6	3	0	3
1	4	4	8	5	3	5
2	3	3	5	3	4	4
3	3	3	6	3	4	5
4	5	4	9	5	5	5

One-Hot Encoding

For categorical features that lack any inherent logical ordering or mathematical ranking (such as Foundation or Neighborhood), One-Hot Encoding is implemented. This process converts each nominal category into a set of distinct binary dummy columns (\(0\) or \(1\)).

To prevent the ‘dummy variable trap’—which causes perfect multicollinearity within linear estimators—the first category of each feature is dropped (drop=‘first’). This approach mathematically guarantees a full-rank design matrix while enabling the model to compute isolated intercept shifts for each architectural category.

X_train_encoded_nominal_cat_df = pd.get_dummies(X_train_nominal_cat_df, prefix=nominal_cat_columns, drop_first=True, dtype=int)
X_test_encoded_nominal_cat_df = pd.get_dummies(X_test_nominal_cat_df, prefix=nominal_cat_columns, drop_first=True, dtype=int)

print(f'X_train_encoded_nominal_cat_df shape: {X_train_encoded_nominal_cat_df.shape}')
print(f'X_test_encoded_nominal_cat_df shape: {X_test_encoded_nominal_cat_df.shape}')

X_train_encoded_nominal_cat_df shape: (1168, 10)
X_test_encoded_nominal_cat_df shape: (292, 10)

X_train_encoded_nominal_cat_df.head()

	CentralAir_Y	MasVnrType_BrkFace	MasVnrType_None	MasVnrType_Stone	Fireplaces_1	Fireplaces_2	Fireplaces_3	GarageFinish_None	GarageFinish_RFn	GarageFinish_Unf
0	1	0	1	0	0	0	0	0	1	0
1	1	0	1	0	1	0	0	0	1	0
2	1	0	1	0	0	0	0	1	0	0
3	1	1	0	0	0	1	0	0	0	1
4	1	0	1	0	1	0	0	0	0	1

X_test_encoded_nominal_cat_df.head()

	CentralAir_Y	MasVnrType_BrkFace	MasVnrType_None	MasVnrType_Stone	Fireplaces_1	Fireplaces_2	Fireplaces_3	GarageFinish_None	GarageFinish_RFn	GarageFinish_Unf
0	1	0	1	0	0	0	0	0	1	0
1	1	1	0	0	0	1	0	0	1	0
2	1	0	1	0	1	0	0	0	0	1
3	1	0	1	0	0	1	0	0	0	1
4	1	0	0	1	1	0	0	0	0	0

Concatenate scaled numeric dataframe, encoded ordinal categorical dataframe, and encoded nominal categorical dataframe for each train and test set.

X_train_preprocessed_df = pd.concat([X_train_scaled_num_df, X_train_encoded_ordinal_cat_df, X_train_encoded_nominal_cat_df], axis=1)
X_test_preprocessed_df = pd.concat([X_test_scaled_num_df, X_test_encoded_ordinal_cat_df, X_test_encoded_nominal_cat_df], axis=1)

print(f'X_train_preprocessed_df shape: {X_train_preprocessed_df.shape}')
print(f'X_test_preprocessed_df shape: {X_test_preprocessed_df.shape}')

X_train_preprocessed_df shape: (1168, 22)
X_test_preprocessed_df shape: (292, 22)

X_train_preprocessed_df.head()

	TotalBsmtSF	GrLivArea	GarageArea	HouseAge	AgeAftRemod	TotalBath	KitchenQual	ExterQual	OverallQual	BsmtQual	...	CentralAir_Y	MasVnrType_BrkFace	MasVnrType_None	MasVnrType_Stone	Fireplaces_1	Fireplaces_2	Fireplaces_3	GarageFinish_None	GarageFinish_RFn	GarageFinish_Unf
0	0.572612	-0.407093	-0.863837	0.526141	1.452679	-0.285144	3	3	5	3	...	1	0	1	0	0	0	0	0	1	0
1	-0.596547	0.083170	-0.456264	-0.679464	-0.382563	0.356567	3	4	6	4	...	1	0	1	0	1	0	0	0	1	0
2	-0.603357	-1.395250	-2.257169	1.992419	1.694159	-1.568564	3	3	5	2	...	1	0	1	0	0	0	0	1	0	0
3	-0.750921	0.458975	-1.119755	1.080068	1.645863	0.356567	4	3	5	4	...	1	1	0	0	0	1	0	0	0	1
4	-0.081209	0.312087	-0.797488	1.601411	1.790750	-0.285144	4	3	5	3	...	1	0	1	0	1	0	0	0	0	1

5 rows × 22 columns

X_test_preprocessed_df.head()

	TotalBsmtSF	GrLivArea	GarageArea	HouseAge	AgeAftRemod	TotalBath	KitchenQual	ExterQual	OverallQual	BsmtQual	...	CentralAir_Y	MasVnrType_BrkFace	MasVnrType_None	MasVnrType_Stone	Fireplaces_1	Fireplaces_2	Fireplaces_3	GarageFinish_None	GarageFinish_RFn	GarageFinish_Unf
0	-0.006292	-0.876372	-1.006014	0.200302	-0.962113	-0.926854	3	3	6	3	...	1	0	1	0	0	0	0	0	1	0
1	0.910874	2.088099	1.117159	-0.679464	-0.382563	1.639987	4	4	8	5	...	1	1	0	0	0	1	0	0	1	0
2	-0.122072	-0.952678	-0.551048	1.503660	1.790750	-1.568564	3	3	5	3	...	1	0	1	0	1	0	0	0	0	1
3	-0.131153	0.260581	-0.266695	0.721645	1.597567	-0.285144	3	3	6	3	...	1	0	1	0	0	1	0	0	0	1
4	1.267297	0.176644	2.065003	-1.135640	-1.058704	0.998277	5	4	9	5	...	1	0	0	1	1	0	0	0	0	0

5 rows × 22 columns

X_train_preprocessed = X_train_preprocessed_df.values
X_test_preprocessed = X_test_preprocessed_df.values

Train Model

Now, let’s train a linear regression model with X_train_preprocessed and y_train.

from sklearn.linear_model import LinearRegression

lin_reg = LinearRegression()
lin_reg.fit(X_train_preprocessed, y_train)

LinearRegression()

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predictors = X_train_preprocessed_df.columns.tolist()
coefficients = lin_reg.coef_
intercept = lin_reg.intercept_

eq = f'{target} = {intercept:.2f}'
i = 1
for coef, pred in zip(coefficients, predictors):
    eq += f' + {coef:.2f}*{pred}'
    if i % 5 == 0:
        eq += '\n'
    i += 1
print(eq)

SalePrice = 396.19 + 8260.04*TotalBsmtSF + 20038.39*GrLivArea + 11274.64*GarageArea + -3033.16*HouseAge + -1192.06*AgeAftRemod
 + 6646.29*TotalBath + 11092.08*KitchenQual + 11627.78*ExterQual + 12488.63*OverallQual + 3959.27*BsmtQual
 + 5189.51*FireplaceQu + -91.17*HeatingQC + 5715.66*CentralAir_Y + 4663.61*MasVnrType_BrkFace + 7223.31*MasVnrType_None
 + 10034.29*MasVnrType_Stone + -9060.35*Fireplaces_1 + 14493.01*Fireplaces_2 + -109233.73*Fireplaces_3 + 8994.24*GarageFinish_None
 + -8216.61*GarageFinish_RFn + -8537.54*GarageFinish_Unf

Model Evaluation

y_pred = lin_reg.predict(X_test_preprocessed)
print(y_pred)

[125237.87756637 318127.65169004 115179.15202396 189704.9374924
 315374.19539881  85098.6060117  227095.14398652 163833.63012701
  85346.1822527  118351.72027517 159919.19188096 105234.08314025
 108580.53643215 210172.68585276 179258.86775992 122189.91364991
 226841.74088901 106301.46772854 124221.21115974 217063.95368565
 165138.83201738 226019.55234336 205132.46547194 118557.07936313
 230425.23107108 179922.17960739 203847.3542917   86499.17616534
 184031.53759274 215612.9116712  111283.58605479 262794.31860959
 247423.34181674  86356.40304748 272683.44370795 149352.9280821
 151372.71164388 234504.58450411 291831.68282501  67866.46205121
 140110.20560223 256919.10900556 101255.58220467 341213.68029865
 123429.44169886 146224.29300267  98579.4576827  114769.99993277
 354461.47630698 123314.34862556  97822.3059975  231249.7438844
 134001.04355544 309889.8961586  173120.28271276 248776.6149112
 216477.45581803 165217.66596539 157906.09354524  68814.91137739
  39698.48448559 143151.49908794 289051.35939729 274206.57232382
 284137.25501883 219964.34294772  93933.58142521 323180.01351578
 112338.4953307  157369.45719433 109805.93285315 119214.4775887
  87635.67623133  56033.47446782 386625.48077802 180150.05681447
 298320.51213087 282114.73564832 120667.97215548  95234.15055406
 120111.24342838  99115.78639867 113567.98824221  77998.12207205
 152284.36443612 117627.49389872 277581.86885191 200795.9645377
 165216.47553862 209773.14536714 170662.41799854 151681.6349822
  94486.03823084 279194.55460452 139821.42316314 201643.53838289
 208088.84089612 202439.14359711 219541.09435954 253255.69857093
 187942.88001435 220748.45657698 273754.67408817 133560.44298716
 179093.22429218 198258.08978158 147922.61360524 251937.31925261
 136020.84034955 190958.8569759   10566.30000323 119665.69487297
 118814.17743873 118169.3228029  222673.28339336 145859.52905396
  85679.14869779  94746.75120775 143437.89340885 296806.69203113
 142137.08659447 164903.66775089 183346.78710467 174418.97176847
 196474.76932251 106016.6353928  249394.54623408  93436.08314302
 136957.06938751 189086.63593395 204457.85162687 315263.30436226
 202455.2266898  154013.89280248  29563.57977052 332658.43214886
 324584.84528571 115314.29892025 250846.53890568 478717.17453828
 322029.07966606 102652.30743494 203871.20602696 176739.4200901
 131903.51748454 102669.92367369 213168.70280127 210413.82909887
 115382.75990707  69764.22440754  81943.41244433 135483.24660796
 271261.05236305 169240.62024613  78639.40441424 156175.44073093
 132153.07011163 150770.28296477  75328.06968828 120770.29295011
 210584.66968558 144977.30662262 269931.40704915 147008.36233701
 108465.82989417 153325.38771634 228429.05611818 325215.07868292
 358815.01993843 220666.94553616 218893.87869159  84934.5651789
 129626.59064407 183292.86076145 295405.94474932 131568.08617999
 138748.72847806 212417.46810774 128531.45284759 183268.96460604
 166880.31688435  77253.10059053 117340.59413078 173120.14099709
 267447.47440548 160776.94422982 290651.25534416 241396.64229031
 215803.73460295  88579.87827967 132252.46720149 105581.53839646
 159211.49134684 141050.45347732 215800.63144223 185274.2032171
 235071.98850554  83427.04010841 237867.08904435 116465.91086813
 238222.10884708 178937.64320176 117201.42466119 299385.79686092
 210577.64008267 125029.10039732 242461.44969723 128651.38815319
 169906.03603578  93393.65844562 251082.420071   143653.69094223
 112389.72242115 156234.92196556 201162.51307601 250939.86747882
 231143.24472829 150959.71338102 129888.30479017 135619.37719165
 116525.80887498 240851.77530373 200630.57029793  98357.65742991
 234637.08257696 127148.85582399  97699.33318144  86776.16348221
 165892.09011131  92867.53476235  69791.04879435 181439.12669687
 124431.13443628 100575.82739673 253206.19576717 127749.88534179
 206476.3866852  156629.33036045 239040.80124831 131607.64902129
 106771.53750921 272605.00085453 210064.6087068  354589.72619491
 202212.27275461 129588.94424753 151807.85016635 177038.14776955
 142530.03150575  86658.23068558 189937.21425107 188929.96775401
 124929.2115678   94871.65325933 116842.16803943 143671.94145885
 115142.16557399 117934.42596739 189115.57292436 258012.09136909
 287426.34740529 178963.68859872 108403.73243496 241756.38429776
 296397.55076322 238814.38882567 175751.94739934 161167.36299473
 105447.33104861 176063.75469809 358679.47802879 224258.29039712
 246487.58801236 111214.17630735  86395.80798058 142954.8718669
 204120.2065942  298375.18824033 215116.34537545 139598.74387171
 216477.08054162 120084.20273816 209876.81893061 109260.28390159
 279132.19171997 184947.67241146 230937.55980842 102951.98849324
 242862.21164168 221576.84193895 123013.86185747  92993.53691855]

r2 = lin_reg.score(X_test_preprocessed, y_test)

print(f'R^2 = {r2}')

R^2 = 0.8138148685027489

The trained linear regression model achieved a Coefficient of Determination (\(R^2\) score) of 0.8138 on the independent test dataset. Mathematically, this signifies that approximately 81.38% of the total variance in house prices (SalePrice) can be successfully explained and predicted by the features selected and engineered within this framework.

Remaining below 20% of unexplained variance is likely attributed to unmeasured stochastic factors or non-linear dynamics. An \(R^2\) score above 0.80 indicates a highly robust, generalisable, and statistically stable linear estimator.

To visually inspect the prediction alignment, an overlapping histogram comparing the ground-truth distribution (truth in green) against the model’s predictions (prediction in blue) is constructed.

plt.hist(y_test, color='green', alpha=.5)
plt.hist(y_pred, color='blue', alpha=.5)
plt.legend(['truth', 'prediction'], loc='upper right')
plt.show()

The visualization demonstrates that the model captures the central tendency, skewness, and peak density of the dataset exceptionally well, particularly within the mid-tier market segment between $100,000 and $250,000. However, minor distribution variance is visible at the extreme right tail (above $500,000), where the model slightly underestimates high-leverage outliers. This is a characteristic constraint of linear estimators when modeling premium, ultra-luxury property prices that follow exponential pricing dynamics rather than strictly linear traits.