Gaussian Naive Bayes, Defined: A Visible Information with Code Examples for Learners | by Samy Baladram | Oct, 2024

CLASSIFICATION ALGORITHM

⛳️ Extra CLASSIFICATION ALGORITHM, defined:
· Dummy Classifier
· Okay Nearest Neighbor Classifier
· Bernoulli Naive Bayes
▶ Gaussian Naive Bayes
· Determination Tree Classifier
· Logistic Regression
· Assist Vector Classifier
· Multilayer Perceptron (quickly!)

Constructing on our earlier article about Bernoulli Naive Bayes, which handles binary knowledge, we now discover Gaussian Naive Bayes for steady knowledge. Not like the binary strategy, this algorithm assumes every characteristic follows a traditional (Gaussian) distribution.

Right here, we’ll see how Gaussian Naive Bayes handles steady, bell-shaped knowledge — ringing in correct predictions — all with out stepping into the intricate math of Bayes’ Theorem.

Like different Naive Bayes variants, Gaussian Naive Bayes makes the “naive” assumption of characteristic independence. It assumes that the options are conditionally unbiased given the category label.

Nevertheless, whereas Bernoulli Naive Bayes is fitted to datasets with binary options, Gaussian Naive Bayes assumes that the options observe a steady regular (Gaussian) distribution. Though this assumption might not at all times maintain true in actuality, it simplifies the calculations and infrequently results in surprisingly correct outcomes.

All through this text, we’ll use this synthetic golf dataset (made by creator) for instance. This dataset predicts whether or not an individual will play golf primarily based on climate situations.

# IMPORTING DATASET #
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score
import pandas as pd
import numpy as np
dataset_dict = {
'Rainfall': [0.0, 2.0, 7.0, 18.0, 3.0, 3.0, 0.0, 1.0, 0.0, 25.0, 0.0, 18.0, 9.0, 5.0, 0.0, 1.0, 7.0, 0.0, 0.0, 7.0, 5.0, 3.0, 0.0, 2.0, 0.0, 8.0, 4.0, 4.0],
'Temperature': [29.4, 26.7, 28.3, 21.1, 20.0, 18.3, 17.8, 22.2, 20.6, 23.9, 23.9, 22.2, 27.2, 21.7, 27.2, 23.3, 24.4, 25.6, 27.8, 19.4, 29.4, 22.8, 31.1, 25.0, 26.1, 26.7, 18.9, 28.9],
'Humidity': [85.0, 90.0, 78.0, 96.0, 80.0, 70.0, 65.0, 95.0, 70.0, 80.0, 70.0, 90.0, 75.0, 80.0, 88.0, 92.0, 85.0, 75.0, 92.0, 90.0, 85.0, 88.0, 65.0, 70.0, 60.0, 95.0, 70.0, 78.0],
'WindSpeed': [2.1, 21.2, 1.5, 3.3, 2.0, 17.4, 14.9, 6.9, 2.7, 1.6, 30.3, 10.9, 3.0, 7.5, 10.3, 3.0, 3.9, 21.9, 2.6, 17.3, 9.6, 1.9, 16.0, 4.6, 3.2, 8.3, 3.2, 2.2],
'Play': ['No', 'No', 'Yes', 'Yes', 'Yes', 'No', 'Yes', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'No', 'Yes', 'Yes', 'No', 'No', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'Yes']
}
df = pd.DataFrame(dataset_dict)
# Set characteristic matrix X and goal vector y
X, y = df.drop(columns='Play'), df['Play']
# Cut up the information into coaching and testing units
X_train, X_test, y_train, y_test = train_test_split(X, y, train_size=0.5, shuffle=False)
print(pd.concat([X_train, y_train], axis=1), finish='nn')
print(pd.concat([X_test, y_test], axis=1))

Gaussian Naive Bayes works with steady knowledge, assuming every characteristic follows a Gaussian (regular) distribution.

1. Calculate the likelihood of every class within the coaching knowledge.
2. For every characteristic and sophistication, estimate the imply and variance of the characteristic values inside that class.
3. For a brand new occasion:
   a. For every class, calculate the likelihood density perform (PDF) of every characteristic worth beneath the Gaussian distribution of that characteristic inside the class.
   b. Multiply the category likelihood by the product of the PDF values for all options.
4. Predict the category with the very best ensuing likelihood.

Reworking non-Gaussian distributed knowledge

Do not forget that this algorithm naively assume that every one the enter options are having Gaussian/regular distribution?

Since we aren’t actually positive in regards to the distribution of our knowledge, particularly for options that clearly don’t observe a Gaussian distribution, making use of a energy transformation (like Field-Cox) earlier than utilizing Gaussian Naive Bayes may be helpful. This strategy might help make the information extra Gaussian-like, which aligns higher with the assumptions of the algorithm.

from sklearn.preprocessing import PowerTransformer
# Initialize and match the PowerTransformer
pt = PowerTransformer(standardize=True) # Normal Scaling already included
X_train_transformed = pt.fit_transform(X_train)
X_test_transformed = pt.rework(X_test)

Now we’re prepared for the coaching.

1. Class Chance Calculation: For every class, calculate its likelihood: (Variety of situations on this class) / (Complete variety of situations)

from fractions import Fraction
def calc_target_prob(attr):
total_counts = attr.value_counts().sum()
prob_series = attr.value_counts().apply(lambda x: Fraction(x, total_counts).limit_denominator())
return prob_series
print(calc_target_prob(y_train))

2. Function Chance Calculation : For every characteristic and every class, calculate the imply (μ) and customary deviation (σ) of the characteristic values inside that class utilizing the coaching knowledge. Then, calculate the likelihood utilizing Gaussian Chance Density Operate (PDF) formulation.

def calculate_class_probabilities(X_train_transformed, y_train, feature_names):
courses = y_train.distinctive()
equations = pd.DataFrame(index=courses, columns=feature_names)
for cls in courses:
X_class = X_train_transformed[y_train == cls]
imply = X_class.imply(axis=0)
std = X_class.std(axis=0)
k1 = 1 / (std * np.sqrt(2 * np.pi))
k2 = 2 * (std ** 2)
for i, column in enumerate(feature_names):
equation = f"{k1[i]:.3f}·exp(-(x-({imply[i]:.2f}))²/{k2[i]:.3f})"
equations.loc[cls, column] = equation
return equations
# Use the perform with the reworked coaching knowledge
equation_table = calculate_class_probabilities(X_train_transformed, y_train, X.columns)
# Show the equation desk
print(equation_table)

3. Smoothing: Gaussian Naive Bayes makes use of a singular smoothing strategy. Not like Laplace smoothing in different variants, it provides a tiny worth (0.000000001 occasions the most important variance) to all variances. This prevents numerical instability from division by zero or very small numbers.

Given a brand new occasion with steady options:

1. Chance Assortment:
For every attainable class:
· Begin with the likelihood of this class occurring (class likelihood).
· For every characteristic within the new occasion, calculate the likelihood density perform of that characteristic inside the class.

2. Rating Calculation & Prediction:
For every class:
· Multiply all of the collected PDF values collectively.
· The result’s the rating for this class.
· The category with the very best rating is the prediction.

from scipy.stats import norm
def calculate_class_probability_products(X_train_transformed, y_train, X_new, feature_names, target_name):
courses = y_train.distinctive()
n_features = X_train_transformed.form[1]
# Create column names utilizing precise characteristic names
column_names = [target_name] + record(feature_names) + ['Product']
probability_products = pd.DataFrame(index=courses, columns=column_names)
for cls in courses:
X_class = X_train_transformed[y_train == cls]
imply = X_class.imply(axis=0)
std = X_class.std(axis=0)
prior_prob = np.imply(y_train == cls)
probability_products.loc[cls, target_name] = prior_prob
feature_probs = []
for i, characteristic in enumerate(feature_names):
prob = norm.pdf(X_new[0, i], imply[i], std[i])
probability_products.loc[cls, feature] = prob
feature_probs.append(prob)
product = prior_prob * np.prod(feature_probs)
probability_products.loc[cls, 'Product'] = product
return probability_products
# Assuming X_new is your new pattern reshaped to (1, n_features)
X_new = np.array([-1.28, 1.115, 0.84, 0.68]).reshape(1, -1)
# Calculate likelihood merchandise
prob_products = calculate_class_probability_products(X_train_transformed, y_train, X_new, X.columns, y.title)
# Show the likelihood product desk
print(prob_products)

from sklearn.naive_bayes import GaussianNB
from sklearn.metrics import accuracy_score
# Initialize and prepare the Gaussian Naive Bayes mannequin
gnb = GaussianNB()
gnb.match(X_train_transformed, y_train)
# Make predictions on the take a look at set
y_pred = gnb.predict(X_test_transformed)
# Calculate the accuracy
accuracy = accuracy_score(y_test, y_pred)
# Print the accuracy
print(f"Accuracy: {accuracy:.4f}")

GaussianNB is understood for its simplicity and effectiveness. The principle factor to recollect about its parameters is:

1. priors: That is essentially the most notable parameter, just like Bernoulli Naive Bayes. Generally, you don’t must set it manually. By default, it’s calculated out of your coaching knowledge, which frequently works properly.
2. var_smoothing: It is a stability parameter that you just not often want to regulate. (the default is 0.000000001)

The important thing takeaway is that this algoritm is designed to work properly out-of-the-box. In most conditions, you need to use it with out worrying about parameter tuning.

Execs:

1. Simplicity: Maintains the easy-to-implement and perceive trait.
2. Effectivity: Stays swift in coaching and prediction, making it appropriate for large-scale purposes with steady options.
3. Flexibility with Information: Handles each small and enormous datasets properly, adapting to the dimensions of the issue at hand.
4. Steady Function Dealing with: Thrives with steady and real-valued options, making it excellent for duties like predicting real-valued outputs or working with knowledge the place options fluctuate on a continuum.

Cons:

1. Independence Assumption: Nonetheless assumes that options are conditionally unbiased given the category, which could not maintain in all real-world situations.
2. Gaussian Distribution Assumption: Works finest when characteristic values actually observe a traditional distribution. Non-normal distributions might result in suboptimal efficiency (however may be mounted with Energy Transformation we’ve mentioned)
3. Sensitivity to Outliers: Will be considerably affected by outliers within the coaching knowledge, as they skew the imply and variance calculations.

Gaussian Naive Bayes stands as an environment friendly classifier for a variety of purposes involving steady knowledge. Its capability to deal with real-valued options extends its use past binary classification duties, making it a go-to alternative for quite a few purposes.

Whereas it makes some assumptions about knowledge (characteristic independence and regular distribution), when these situations are met, it provides sturdy efficiency, making it a favourite amongst each freshmen and seasoned knowledge scientists for its steadiness of simplicity and energy.

import pandas as pd
from sklearn.naive_bayes import GaussianNB
from sklearn.preprocessing import PowerTransformer
from sklearn.metrics import accuracy_score
from sklearn.model_selection import train_test_split
# Load the dataset
dataset_dict = {
'Rainfall': [0.0, 2.0, 7.0, 18.0, 3.0, 3.0, 0.0, 1.0, 0.0, 25.0, 0.0, 18.0, 9.0, 5.0, 0.0, 1.0, 7.0, 0.0, 0.0, 7.0, 5.0, 3.0, 0.0, 2.0, 0.0, 8.0, 4.0, 4.0],
'Temperature': [29.4, 26.7, 28.3, 21.1, 20.0, 18.3, 17.8, 22.2, 20.6, 23.9, 23.9, 22.2, 27.2, 21.7, 27.2, 23.3, 24.4, 25.6, 27.8, 19.4, 29.4, 22.8, 31.1, 25.0, 26.1, 26.7, 18.9, 28.9],
'Humidity': [85.0, 90.0, 78.0, 96.0, 80.0, 70.0, 65.0, 95.0, 70.0, 80.0, 70.0, 90.0, 75.0, 80.0, 88.0, 92.0, 85.0, 75.0, 92.0, 90.0, 85.0, 88.0, 65.0, 70.0, 60.0, 95.0, 70.0, 78.0],
'WindSpeed': [2.1, 21.2, 1.5, 3.3, 2.0, 17.4, 14.9, 6.9, 2.7, 1.6, 30.3, 10.9, 3.0, 7.5, 10.3, 3.0, 3.9, 21.9, 2.6, 17.3, 9.6, 1.9, 16.0, 4.6, 3.2, 8.3, 3.2, 2.2],
'Play': ['No', 'No', 'Yes', 'Yes', 'Yes', 'No', 'Yes', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'No', 'Yes', 'Yes', 'No', 'No', 'No', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'Yes', 'No', 'Yes']
}
df = pd.DataFrame(dataset_dict)
# Put together knowledge for mannequin
X, y = df.drop('Play', axis=1), (df['Play'] == 'Sure').astype(int)
# Cut up knowledge into coaching and testing units
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.5, shuffle=False)
# Apply PowerTransformer
pt = PowerTransformer(standardize=True)
X_train_transformed = pt.fit_transform(X_train)
X_test_transformed = pt.rework(X_test)
# Practice the mannequin
nb_clf = GaussianNB()
nb_clf.match(X_train_transformed, y_train)
# Make predictions
y_pred = nb_clf.predict(X_test_transformed)
# Verify accuracy
accuracy = accuracy_score(y_test, y_pred)
print(f"Accuracy: {accuracy:.4f}")