Kernels Ridge Regression

Kernel Ridge Regression (KRR) is a powerful machine learning technique that combines Ridge Regression with kernel methods. It is particularly useful for non-linear regression tasks where the relationship between the features and the target variable is complex.

Basics of Ridge Regression

Ridge Regression is a type of linear regression that includes a regularization term to prevent overfitting.

Cost(w)=∑i=1n​(yi​−yi​^​)2+λ∑j=1p​wj2​

where: - yiyi​ is the true value, - yiyi​​ is the predicted value, - wjwj​ are the model coefficients, - λλ is the regularization parameter.

Kernels

Kernel methods involve using a kernel function to implicitly map the input features into a higher-dimensional space where a linear relationship might exist. This allows the model to capture non-linear relationships in the original feature space.

Common kernel functions include:

  • Linear Kernel: k(x,x′)=x⋅x′k(x,x′)=x⋅x′
  • Polynomial Kernel: k(x,x′)=(x⋅x′+1)dk(x,x′)=(x⋅x′+1)d
  • Gaussian (RBF) Kernel: k(x,x′)=exp⁡(−γ∥x−x′∥2)k(x,x′)=exp(−γ∥x−x′∥2)

Advantages of KRR

  • Non-linearity: It can model complex, non-linear relationships.
  • Flexibility: Different kernels can be used to capture various data patterns.
  • Regularization: The regularization term helps prevent overfitting.

Disadvantages of KRR - Computational Cost: Computing the kernel matrix can be computationally expensive, especially for large datasets. - Parameter Tuning: Choosing the right kernel and regularization parameter λλ can be challenging and often requires cross-validation.

Implementation

# Example data
X = np.array([[1, 2], [2, 3], [3, 4]])
y = np.array([1, 2, 3])



# Make predictions
predictions = krr.predict(X)
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from pathlib import Path
path = Path('Data/homeprices.csv')
df = pd.read_csv(path)
df
area bedrooms age price
0 2600 3.0 20 550000
1 3000 4.0 15 565000
2 3200 NaN 18 610000
3 3600 3.0 30 595000
4 4000 5.0 8 760000
5 4100 6.0 8 810000 
plt.xlabel('area')
plt.ylabel('price')
plt.scatter(df.area,df.price,color='red',marker='+')

plt.show()

new_df = df.drop('price',axis='columns')
new_df = new_df.drop('bedrooms',axis='columns')
new_df = new_df.drop('age',axis='columns')
new_df
area
0 2600
1 3000
2 3200
3 3600
4 4000
5 4100 
price = df.price
price
0    550000
1    565000
2    610000
3    595000
4    760000
5    810000
Name: price, dtype: int64
from sklearn.kernel_ridge import KernelRidge
# Define the model
krr = KernelRidge(alpha=.1, kernel='rbf')

# Fit the model
krr.fit(new_df, price)
KernelRidge(alpha=0.1, kernel='rbf')
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krr.predict([[3300]])
/home/ben/miniconda3/envs/pfast/lib/python3.12/site-packages/sklearn/base.py:493: UserWarning: X does not have valid feature names, but KernelRidge was fitted with feature names
  warnings.warn(
array([0.])
krr.predict([[5000]])
/home/ben/miniconda3/envs/pfast/lib/python3.12/site-packages/sklearn/base.py:493: UserWarning: X does not have valid feature names, but KernelRidge was fitted with feature names
  warnings.warn(
array([0.])
numbers_list = list(range(2000, 4000, 100))
# Create a DataFrame using the pandas constructor and a dictionary
data = {'area': numbers_list}
area_df = pd.DataFrame(data)
area_df
area
0 2000
1 2100
2 2200
3 2300
4 2400
5 2500
6 2600
7 2700
8 2800
9 2900
10 3000
11 3100
12 3200
13 3300
14 3400
15 3500
16 3600
17 3700
18 3800
19 3900 
p = krr.predict(area_df)
p
array([     0.        ,      0.        ,      0.        ,      0.        ,
            0.        ,      0.        , 500000.        ,      0.        ,
            0.        ,      0.        , 513636.36363636,      0.        ,
       554545.45454545,      0.        ,      0.        ,      0.        ,
       540909.09090909,      0.        ,      0.        ,      0.        ])
plt.plot(area_df, p)

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