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Predicting the ratings of restaurants on Zomato's dataset using regression. (MAE = 0.186)

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Restaurant Rating Prediction

Predicting the aggregate rating of Zomato restaurants using Machine Learning.

Tools used: Python (Numpy, Pandas, Matplotlib, Seaborn, Scikit-learn, Dython)

Sections:

Summary:

Visualizing the data using Seaborn and Dython, engineering new features like cuisines and restaurant names, using Random Forest Regressor from Scikit-learn for prediction, visualizing and evaluating model performance, and finally tuning it using RandomizedSearchCV to further improve performance.

Final model with a Mean Absolute Error of around 0.183. We will also evaluate other metrics like MSE, RMSE, MDAE and MAPE.

Value: This model can help restaurant aggregators and food delivery companies like Zomato in predicting the aggregate rating of new restaurants, or existing unrated ones. That can in turn increase customer satisfaction.

png

Data Analysis and Cleaning

First, we import the libraries that we will be using:

import numpy as np
import pandas as pd
import seaborn as sns
from dython import nominal
import matplotlib.pyplot as plt
%matplotlib inline

from IPython.core.display import HTML    # To centralize the plots
HTML("""
<style>
.output_png {
    display: table-cell;
    text-align: center;
    vertical-align: middle;
}
</style>
""")

Importing the data:

df = pd.read_csv('zomato.csv', encoding='ISO-8859-1') # Specifying the encoding is important or it will raise UTF error

Let's get to know our data:

df.shape
(9551, 21)

So we have 9551 rows and 21 columns.

Let's see the columns:

df.columns
Index(['Restaurant ID', 'Restaurant Name', 'Country Code', 'City', 'Address',
       'Locality', 'Locality Verbose', 'Longitude', 'Latitude', 'Cuisines',
       'Average Cost for two', 'Currency', 'Has Table booking',
       'Has Online delivery', 'Is delivering now', 'Switch to order menu',
       'Price range', 'Aggregate rating', 'Rating color', 'Rating text',
       'Votes'],
      dtype='object')

Let's take a look at the first 5 rows of the dataset, to get an idea of the data:

pd.set_option('display.max_columns',21)
df.head()
Restaurant ID Restaurant Name Country Code City Address Locality Locality Verbose Longitude Latitude Cuisines Average Cost for two Currency Has Table booking Has Online delivery Is delivering now Switch to order menu Price range Aggregate rating Rating color Rating text Votes
0 6317637 Le Petit Souffle 162 Makati City Third Floor, Century City Mall, Kalayaan Avenu... Century City Mall, Poblacion, Makati City Century City Mall, Poblacion, Makati City, Mak... 121.027535 14.565443 French, Japanese, Desserts 1100 Botswana Pula(P) Yes No No No 3 4.8 Dark Green Excellent 314
1 6304287 Izakaya Kikufuji 162 Makati City Little Tokyo, 2277 Chino Roces Avenue, Legaspi... Little Tokyo, Legaspi Village, Makati City Little Tokyo, Legaspi Village, Makati City, Ma... 121.014101 14.553708 Japanese 1200 Botswana Pula(P) Yes No No No 3 4.5 Dark Green Excellent 591
2 6300002 Heat - Edsa Shangri-La 162 Mandaluyong City Edsa Shangri-La, 1 Garden Way, Ortigas, Mandal... Edsa Shangri-La, Ortigas, Mandaluyong City Edsa Shangri-La, Ortigas, Mandaluyong City, Ma... 121.056831 14.581404 Seafood, Asian, Filipino, Indian 4000 Botswana Pula(P) Yes No No No 4 4.4 Green Very Good 270
3 6318506 Ooma 162 Mandaluyong City Third Floor, Mega Fashion Hall, SM Megamall, O... SM Megamall, Ortigas, Mandaluyong City SM Megamall, Ortigas, Mandaluyong City, Mandal... 121.056475 14.585318 Japanese, Sushi 1500 Botswana Pula(P) No No No No 4 4.9 Dark Green Excellent 365
4 6314302 Sambo Kojin 162 Mandaluyong City Third Floor, Mega Atrium, SM Megamall, Ortigas... SM Megamall, Ortigas, Mandaluyong City SM Megamall, Ortigas, Mandaluyong City, Mandal... 121.057508 14.584450 Japanese, Korean 1500 Botswana Pula(P) Yes No No No 4 4.8 Dark Green Excellent 229

Let's take a closer look at the columns:

df.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 9551 entries, 0 to 9550
Data columns (total 21 columns):
 #   Column                Non-Null Count  Dtype  
---  ------                --------------  -----  
 0   Restaurant ID         9551 non-null   int64  
 1   Restaurant Name       9551 non-null   object 
 2   Country Code          9551 non-null   int64  
 3   City                  9551 non-null   object 
 4   Address               9551 non-null   object 
 5   Locality              9551 non-null   object 
 6   Locality Verbose      9551 non-null   object 
 7   Longitude             9551 non-null   float64
 8   Latitude              9551 non-null   float64
 9   Cuisines              9542 non-null   object 
 10  Average Cost for two  9551 non-null   int64  
 11  Currency              9551 non-null   object 
 12  Has Table booking     9551 non-null   object 
 13  Has Online delivery   9551 non-null   object 
 14  Is delivering now     9551 non-null   object 
 15  Switch to order menu  9551 non-null   object 
 16  Price range           9551 non-null   int64  
 17  Aggregate rating      9551 non-null   float64
 18  Rating color          9551 non-null   object 
 19  Rating text           9551 non-null   object 
 20  Votes                 9551 non-null   int64  
dtypes: float64(3), int64(5), object(13)
memory usage: 1.5+ MB

Observation: It seems Cuisines has some null values. We'll take a look at those.

df.describe() # Looking at just the numerical columns
Restaurant ID Country Code Longitude Latitude Average Cost for two Price range Aggregate rating Votes
count 9.551000e+03 9551.000000 9551.000000 9551.000000 9551.000000 9551.000000 9551.000000 9551.000000
mean 9.051128e+06 18.365616 64.126574 25.854381 1199.210763 1.804837 2.666370 156.909748
std 8.791521e+06 56.750546 41.467058 11.007935 16121.183073 0.905609 1.516378 430.169145
min 5.300000e+01 1.000000 -157.948486 -41.330428 0.000000 1.000000 0.000000 0.000000
25% 3.019625e+05 1.000000 77.081343 28.478713 250.000000 1.000000 2.500000 5.000000
50% 6.004089e+06 1.000000 77.191964 28.570469 400.000000 2.000000 3.200000 31.000000
75% 1.835229e+07 1.000000 77.282006 28.642758 700.000000 2.000000 3.700000 131.000000
max 1.850065e+07 216.000000 174.832089 55.976980 800000.000000 4.000000 4.900000 10934.000000

Looks like no restaurant has a full 5 stars rating. Interesting.

Now let's take a look at the null values of our columns:

sns.heatmap(df.isnull().sum().values.reshape(-1,1), \
            annot=True, cmap=plt.cm.Blues, yticklabels=df.columns)
plt.xlabel('Null Values')
plt.show()

png

Observation: Cuisines has 9 null values.

Since we can't determine what cuisines a restaurant has from the other features in the dataset, we will just drop these null values.

df.dropna(inplace=True)

There. Let's take a look at the null counts again, just to check:

sns.heatmap(df.isnull().sum().values.reshape(-1,1), \
            annot=True, cmap=plt.cm.Blues, yticklabels=df.columns)
plt.xlabel('Null Values')
plt.show()

png

Perfect.

There is something interesting about the Switch to order menu column:

df['Switch to order menu']
0       No
1       No
2       No
3       No
4       No
        ..
9546    No
9547    No
9548    No
9549    No
9550    No
Name: Switch to order menu, Length: 9542, dtype: object
df['Switch to order menu'].value_counts()
No    9542
Name: Switch to order menu, dtype: int64

Observation: Switch to order menu has no other value than 'No'.

Since that is not much use for us, we are going to drop it.

df.drop('Switch to order menu', axis=1, inplace = True)

Since once of the categorical columns turned out to be useless for us, it makes sense to also take a look at the rest of them:

df.columns
Index(['Restaurant ID', 'Restaurant Name', 'Country Code', 'City', 'Address',
       'Locality', 'Locality Verbose', 'Longitude', 'Latitude', 'Cuisines',
       'Average Cost for two', 'Currency', 'Has Table booking',
       'Has Online delivery', 'Is delivering now', 'Price range',
       'Aggregate rating', 'Rating color', 'Rating text', 'Votes'],
      dtype='object')
df['Restaurant Name'].value_counts()
Cafe Coffee Day             83
Domino's Pizza              79
Subway                      63
Green Chick Chop            51
McDonald's                  48
                            ..
The Town House Cafe          1
The G.T. Road                1
The Darzi Bar & Kitchen      1
Smoke On Water               1
Walter's Coffee Roastery     1
Name: Restaurant Name, Length: 7437, dtype: int64
df.Locality.value_counts().value_counts() # Remember, we can specify a column both as df['column'] and df.column
1      550
2      172
3      103
4       51
5       42
      ... 
44       1
45       1
50       1
51       1
122      1
Name: Locality, Length: 82, dtype: int64
df['Has Table booking'].value_counts()
No     8384
Yes    1158
Name: Has Table booking, dtype: int64
df['Has Online delivery'].value_counts()
No     7091
Yes    2451
Name: Has Online delivery, dtype: int64
df['Is delivering now'].value_counts()
No     9508
Yes      34
Name: Is delivering now, dtype: int64
df.City.value_counts()
New Delhi         5473
Gurgaon           1118
Noida             1080
Faridabad          251
Ghaziabad           25
                  ... 
Lincoln              1
Lakeview             1
Lakes Entrance       1
Inverloch            1
Panchkula            1
Name: City, Length: 140, dtype: int64

Observation: So, all of these columns do have more than one value. That means they could actually be useful.

Now we are going to use the Dython library to make a correlation plot of all the features. What I like about this library is that it lets you easily plot the correlation between both categorical and continuous features, something that is not easy to do with Pandas.

nominal.associations(df,figsize=(20,10),mark_columns=True,title="Correlation Matrix") # correlation matrix
plt.show()

png

Feature Engineering and Preprocessing

If we look at the Aggregate rating (con) row, we can see how correlated it is with the rest of the features.

The first highly correlated feature is the Restaurant name (nom) column, with 95%. Let's take a look at this column and see what we can do.

print( f"Total number of restaurants:    {df['Restaurant Name'].value_counts().shape[0]}")
print(f"Restaurants with 1 value count: {(df['Restaurant Name'].value_counts() == 1).sum()}")
Total number of restaurants:    7437
Restaurants with 1 value count: 6703

That's a lot of restaurants. and a lot of them also value count of just 1.

We won't be able to include all of these in a model. So let's just pick the top 10.

df['Restaurant Name'].value_counts().head(10)
Cafe Coffee Day     83
Domino's Pizza      79
Subway              63
Green Chick Chop    51
McDonald's          48
Keventers           34
Pizza Hut           30
Giani               29
Baskin Robbins      28
Barbeque Nation     26
Name: Restaurant Name, dtype: int64

Now we are going to define a function to get dummies just for these 10 restaurants. Dummies are columns with values 0 and 1; 0 meaning false and 1 meaning true.

So, for example, if we make a dummy column for "Cafe Coffee Day", the rows in the dummy column will have 1 as the value if the restaurant's name is 'Cafe Coffee Day', and 0 if not.

def dummy(rest_name,column):
    df[column] = df['Restaurant Name'].apply(lambda x: 1 if str(x).strip()==rest_name\
                                             else 0)
dummy('Cafe Coffee Day','cafe_coffee_day')

Here is a visual example to see how the columns look:

df.loc[df['cafe_coffee_day']==1].head(3)
Restaurant ID Restaurant Name Country Code City Address Locality Locality Verbose Longitude Latitude Cuisines Average Cost for two Currency Has Table booking Has Online delivery Is delivering now Price range Aggregate rating Rating color Rating text Votes cafe_coffee_day
932 9650 Cafe Coffee Day 1 Faridabad SCF 42, Shopping Centre, Main Huda Market, Sec... Sector 15 Sector 15, Faridabad 77.323611 28.395267 Cafe 450 Indian Rupees(Rs.) No No No 1 3.3 Orange Average 67 1
1126 8590 Cafe Coffee Day 1 Ghaziabad 1st Floor, Shipra Mall, Gulmohar Road, Indirap... Shipra Mall, Indirapuram Shipra Mall, Indirapuram, Ghaziabad 77.370208 28.634047 Cafe 450 Indian Rupees(Rs.) No No No 1 3.2 Orange Average 63 1
1283 631 Cafe Coffee Day 1 Gurgaon Upper Ground Floor, DLF Mega Mall, DLF Phase 1... DLF Mega Mall, DLF Phase 1 DLF Mega Mall, DLF Phase 1, Gurgaon 77.093595 28.475489 Cafe 450 Indian Rupees(Rs.) No No No 1 2.6 Orange Average 27 1

Wherever the Restaurant Name column's value is "Cafe Coffee Day", the value of the cafe_coffee_day column is 1.

We will apply this function for all of the 10 most frequent restaurants:

def dum_col(x):
    return x.strip().lower().replace(' ','_')

def dummy(lst,column):
    for i in lst.index:
        df[dum_col(i)] = df[column].apply(lambda x: i in x)
restaurants = df['Restaurant Name'].value_counts().head(10)
dummy(restaurants,'Restaurant Name')
df.head()
Restaurant ID Restaurant Name Country Code City Address Locality Locality Verbose Longitude Latitude Cuisines ... cafe_coffee_day domino's_pizza subway green_chick_chop mcdonald's keventers pizza_hut giani baskin_robbins barbeque_nation
0 6317637 Le Petit Souffle 162 Makati City Third Floor, Century City Mall, Kalayaan Avenu... Century City Mall, Poblacion, Makati City Century City Mall, Poblacion, Makati City, Mak... 121.027535 14.565443 French, Japanese, Desserts ... False False False False False False False False False False
1 6304287 Izakaya Kikufuji 162 Makati City Little Tokyo, 2277 Chino Roces Avenue, Legaspi... Little Tokyo, Legaspi Village, Makati City Little Tokyo, Legaspi Village, Makati City, Ma... 121.014101 14.553708 Japanese ... False False False False False False False False False False
2 6300002 Heat - Edsa Shangri-La 162 Mandaluyong City Edsa Shangri-La, 1 Garden Way, Ortigas, Mandal... Edsa Shangri-La, Ortigas, Mandaluyong City Edsa Shangri-La, Ortigas, Mandaluyong City, Ma... 121.056831 14.581404 Seafood, Asian, Filipino, Indian ... False False False False False False False False False False
3 6318506 Ooma 162 Mandaluyong City Third Floor, Mega Fashion Hall, SM Megamall, O... SM Megamall, Ortigas, Mandaluyong City SM Megamall, Ortigas, Mandaluyong City, Mandal... 121.056475 14.585318 Japanese, Sushi ... False False False False False False False False False False
4 6314302 Sambo Kojin 162 Mandaluyong City Third Floor, Mega Atrium, SM Megamall, Ortigas... SM Megamall, Ortigas, Mandaluyong City SM Megamall, Ortigas, Mandaluyong City, Mandal... 121.057508 14.584450 Japanese, Korean ... False False False False False False False False False False

5 rows × 30 columns

Now we have True or False values for each of the top 10 restaurants. In python, True and False can also be written as 1 and 0.

Let's take a look at how many restaurants are named 'Cafe Coffee Day', using our new column:

print(f"Number of Cafe Coffee Day's: {df.loc[df['cafe_coffee_day']==1].size}")
Number of Cafe Coffee Day's: 2730
df.shape
(9542, 30)

Observation: So out of our 9542 different restaurants, 2730 are Cafe Coffee Day's.

Now let's take a look at the correlation between the Aggregate rating and the new columns that we have created.

features = ['Price range','Votes','Country Code','Restaurant ID','Longitude',
            'Has Table booking','Has Online delivery','cafe_coffee_day',
            "domino's_pizza",'subway','green_chick_chop',"mcdonald's",'keventers',
            'pizza_hut','giani','baskin_robbins','barbeque_nation',
            'Aggregate rating']# --> Only added to see correlation, must be removed later
nominal.associations(df[features],figsize=(20,10),mark_columns=True,\
                     title="Correlation Matrix (features)")
plt.show()

png

Observation: Except for barbeque_nation, the rest of the created features seem to have extremely low correlations.

Since the best practice is to keep the model simplistic and use only the best features, we are going to drop all the features except for this one.

features = ['Price range','Votes','Country Code','Restaurant ID','Longitude',
            'Has Table booking','Has Online delivery','barbeque_nation']

This is going to be our final list of features for training and testing our model.

Important Note: We are not going to include the features Rating color and Rating text in this list. Their inclusion will not result in an actually useful model.

Model Building and Tuning

Now, using these features, we are going to build models to predict our target variable.

Building

We know that predicting the Aggregate rating feature is a regression problem. Since its correlation with other features is not high enough, a linear model like Linear Regression will not be optimal.

Instead, we are going to use a Random Forest Regressor model for this problem.

First, we are going to split the data into independent variables (Features) and a dependent variable (Target).

So, our features (the columns we will use to predict):

X = pd.get_dummies(df[features])
X
Price range Votes Country Code Restaurant ID Longitude barbeque_nation Has Table booking_No Has Table booking_Yes Has Online delivery_No Has Online delivery_Yes
0 3 314 162 6317637 121.027535 False 0 1 1 0
1 3 591 162 6304287 121.014101 False 0 1 1 0
2 4 270 162 6300002 121.056831 False 0 1 1 0
3 4 365 162 6318506 121.056475 False 1 0 1 0
4 4 229 162 6314302 121.057508 False 0 1 1 0
... ... ... ... ... ... ... ... ... ... ...
9546 3 788 208 5915730 28.977392 False 1 0 1 0
9547 3 1034 208 5908749 29.041297 False 1 0 1 0
9548 4 661 208 5915807 29.034640 False 1 0 1 0
9549 4 901 208 5916112 29.036019 False 1 0 1 0
9550 2 591 208 5927402 29.026016 False 1 0 1 0

9542 rows × 10 columns

Our target (the column we want to predict):

y = df['Aggregate rating']

Now, we want to split them into train and test sets.

We will use the train set to train the model, and the test set to test the performance of the model.

from sklearn.model_selection import train_test_split

X_train,X_test,y_train,y_test = train_test_split(X,y,test_size=0.2,random_state=1)

Next, we will import the model that we want to use, i.e, RandomForestRegressor:

from sklearn.ensemble import RandomForestRegressor
rf = RandomForestRegressor(random_state = 2)

Now we fit the train sets into the model, and use it to predict the test set:

rf.fit(X_train,y_train)
y_pred = rf.predict(X_test)

And check its performance using the test and the prediction sets:

from sklearn import metrics
mse = metrics.mean_squared_error(y_test, y_pred)
rmse = metrics.mean_squared_error(y_test,y_pred,squared=False)
mae = metrics.mean_absolute_error(y_test, y_pred)
medae = metrics.median_absolute_error(y_test, y_pred)


print(f"Mean Squared Error (MSE): {mse}")
print(f"Root Mean Squared Error (RMSE): {rmse}")
print(f"Mean Absolute Error (MAE): {mae}")
print(f"Median Absolute Error (MEDAE): {medae}")
print(f'Test variance: {np.var(y_test)}')
Mean Squared Error (MSE): 0.08397072865374543
Root Mean Squared Error (RMSE): 0.28977703265397936
Mean Absolute Error (MAE): 0.18649607124148773
Median Absolute Error (MEDAE): 0.11999999999999966
Test variance: 2.2502005690560023

Let's plot the residuals:

residuals = y_test - y_pred
# plot the residuals
plt.scatter(np.linspace(0,5,1909), residuals,c=residuals,cmap='magma', edgecolors='black', linewidths=.1)
plt.colorbar(label="Quality", orientation="vertical")
# plot a horizontal line at y = 0
plt.hlines(y = 0,
xmin = 0, xmax=5,
linestyle='--',colors='black')
# set xlim
plt.xlim((0, 5))
plt.xlabel('Aggregate Rating'); plt.ylabel('Residuals')
plt.show()

png

A residual is the difference between the observed value of the target and the predicted value. The closer the residual is to 0, the better job our model is doing.

print(f"Error range: {residuals.max()-residuals.min()}")
Error range: 2.7820000000000076

So our prediction's error range is around 2.782.

Tuning

Now we are going to run RandomizedSearchCV to tune the model by improving the hyperparameters.

# from sklearn.model_selection import RandomizedSearchCV

# n_estimators = [int(x) for x in np.linspace(start = 200, stop = 2000, num = 10)]
# max_features = ['auto', 'sqrt']
# max_depth = [int(x) for x in np.linspace(10, 110, num = 11)]
# max_depth.append(None)
# min_samples_split = [2, 5, 10]
# min_samples_leaf = [1, 2, 4]
# bootstrap = [True, False]

# random_grid = {'n_estimators': n_estimators,
#                'max_features': max_features,
#                'max_depth': max_depth,
#                'min_samples_split': min_samples_split,
#                'min_samples_leaf': min_samples_leaf,
#                'bootstrap': bootstrap}

# rf2 = RandomForestRegressor(random_state=2)

# rf_rscv = RandomizedSearchCV(estimator=rf2, param_distributions=random_grid,\
#                              n_iter = 100, cv = 3, verbose=2, random_state=2, n_jobs = -1)
# rf_rscv.fit(X_train,y_train)
# print(rf_rscv.best_params_)

# Output:
#      n_estimators= 1200,
#      min_samples_split= 10,
#      min_samples_leaf= 1,
#      max_depth = 30,
#      bootstrap= True,
#      random_state=2

A hyperparameter is a machine learning parameter whose value is chosen before a learning algorithm is trained. It has an impact on the model's performance.

Now we are going to use these hyperparameters to make a new Random Forests model, fit the data into it and then score it:

rf_random = RandomForestRegressor(
      n_estimators= 1200,
      min_samples_split= 10,
      min_samples_leaf= 1,
      max_depth = 30,
      max_features='sqrt',
      bootstrap= True,
      random_state=2) # Best RandomizedSearch parameters

rf_random.fit(X_train,y_train)
random_pred = rf_random.predict(X_test)
random_mse = metrics.mean_squared_error(y_test, random_pred)
random_rmse = metrics.mean_squared_error(y_test, random_pred, squared=False)
random_mae = metrics.mean_absolute_error(y_test, random_pred)
random_medae = metrics.median_absolute_error(y_test, random_pred)

print(f"Mean Squared Error (MSE): {random_mse}")
print(f"Root Mean Squared Error (RMSE): {random_rmse}")
print(f"Mean Absolute Error (MAE): {random_mae}")
print(f"Median Absolute Error (MEDAE): {random_medae}")
print(f'Test variance: {np.var(y_test)}')
Mean Squared Error (MSE): 0.07950896506171087
Root Mean Squared Error (RMSE): 0.2819733410478921
Mean Absolute Error (MAE): 0.18367410616812943
Median Absolute Error (MEDAE): 0.1146495007615349
Test variance: 2.2502005690560023
print('Improvements:')
print(f"Mean Squared Error (MSE):       {mse} => {random_mse}")
print(f"Root Mean Squared Error (RMSE): {rmse} => {random_rmse}")
print(f"Mean Absolute Error (MAE):      {mae} => {random_mae}")
print(f"Median Absolute Error (MEDAE):  {mae} => {random_medae}")
print(f'Test variance: {np.var(y_test)}')
Improvements:
Mean Squared Error (MSE):       0.08397072865374543 => 0.07950896506171087
Root Mean Squared Error (RMSE): 0.28977703265397936 => 0.2819733410478921
Mean Absolute Error (MAE):      0.18649607124148773 => 0.18367410616812943
Median Absolute Error (MEDAE):  0.18649607124148773 => 0.1146495007615349
Test variance: 2.2502005690560023

There is decrease in the model's errors.

We can also run GridSearchCV on the parameters around these to maybe tune the model further. But we are done with model tuning for this project.

Results

Let's plot the residuals for this final model:

f_residuals = y_test - random_pred
# plot the residuals
plt.scatter(np.linspace(0,5,1909), f_residuals, c = f_residuals, cmap='magma', edgecolors='black', linewidths=.1)
plt.colorbar(label = "Quality", orientation = "vertical")
# plot a horizontal line at y = 0
plt.hlines(y = 0, xmin = 0, xmax = 5, linestyle = '--', colors = 'black')
# set xlim
plt.xlim((0, 5))
plt.xlabel('Aggregate Rating'); plt.ylabel('Residuals')
plt.show()

png

print(f"Errors range of first model: {residuals.max() - residuals.min()}")
print(f"Errors range of second model: {f_residuals.max() - f_residuals.min()}")
print(f"Error difference of models: {(residuals.max() - residuals.min()) - (f_residuals.max() - f_residuals.min())}")
Errors range of first model: 2.7820000000000076
Errors range of second model: 2.554883167428941
Error difference of models: 0.2271168325710664

When compared to the previous model (with default hyperparameters), our final model has a .227 reduction in range of error.

FIN

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