```
import numpy as np
def linear_regression(input, output):
x_mean = np.mean(input)
y_mean = np.mean(output)
numerator = np.sum((input - x_mean) * (output - y_mean))
denominator = np.sum((input - x_mean)**2)
alpha_hat = numerator / denominator
beta_hat = y_mean - alpha_hat * x_mean
return alpha_hat, beta_hat
input_data = np.array([1, 2, 3, 4, 5])
output_data = np.array([2, 4, 6, 8, 10])
alpha, beta = linear_regression(input_data, output_data)
print("Slope (alpha):", alpha)
print("Intercept (beta):", beta)
```

we have this code that i put

### Description

The provided code performs a simple linear regression to find the slope (`alpha_hat`

) and intercept (`beta_hat`

) of the best-fit line for the given `input_data`

and `output_data`

. Here’s a step-by-step explanation of the code:

**Calculate Means**: The code first calculates the mean of the input and output arrays.**Calculate Numerator and Denominator**: It then computes the numerator and denominator for the slope calculation. The numerator is the sum of the product of differences between each input/output and their means. The denominator is the sum of squared differences between each input and its mean.**Calculate Slope and Intercept**: Using the numerator and denominator, the code calculates the slope (`alpha_hat`

). The intercept (`beta_hat`

) is then calculated using the mean of the outputs minus the product of the slope and the mean of the inputs.**Return and Print Results**: The slope and intercept are returned and printed.

### Improvements

**Parameter Names**: Use more descriptive parameter names for clarity.**Edge Case Handling**: Add checks for cases where the denominator might be zero.**Documentation**: Add docstrings to describe the function and its parameters.**Vectorization**: Ensure efficient computation using NumPy’s vectorized operations.**Validation**: Validate input lengths to ensure they are the same.**Main Guard**: Use a`main`

guard to allow the script to be imported without executing.

Here is the improved version of the code:

```
import numpy as np
def linear_regression(inputs, outputs):
"""
Perform simple linear regression on the provided data.
Parameters:
inputs (np.ndarray): 1D array of input data (independent variable)
outputs (np.ndarray): 1D array of output data (dependent variable)
Returns:
tuple: slope (alpha_hat) and intercept (beta_hat)
"""
if len(inputs) != len(outputs):
raise ValueError("Input and output arrays must have the same length")
if len(inputs) == 0:
raise ValueError("Input and output arrays must not be empty")
x_mean = np.mean(inputs)
y_mean = np.mean(outputs)
numerator = np.sum((inputs - x_mean) * (outputs - y_mean))
denominator = np.sum((inputs - x_mean)**2)
if denominator == 0:
raise ValueError("Denominator is zero, variance of input data is zero")
alpha_hat = numerator / denominator
beta_hat = y_mean - alpha_hat * x_mean
return alpha_hat, beta_hat
if __name__ == "__main__":
input_data = np.array([1, 2, 3, 4, 5])
output_data = np.array([2, 4, 6, 8, 10])
alpha, beta = linear_regression(input_data, output_data)
print("Slope (alpha):", alpha)
print("Intercept (beta):", beta)
```

### Summary of Improvements

**Descriptive Parameter Names**: Changed`input`

and`output`

to`inputs`

and`outputs`

.**Edge Case Handling**: Added checks for equal lengths, non-empty arrays, and zero variance.**Documentation**: Included a detailed docstring.**Main Guard**: Added a`main`

guard to allow the script to be imported as a module.

To multiply two arrays element-wise (one-to-one) in Python, you can use the `numpy`

library, which is designed for efficient numerical operations on arrays. Here’s how you can do it:

**Using**:`numpy`

- First, you’ll need to install
`numpy`

if you haven’t already:`bash pip install numpy`

- Then, you can use the following code to perform element-wise multiplication:

```
import numpy as np
# Define the two arrays
array1 = np.array([1, 2, 3])
array2 = np.array([4, 5, 6])
# Perform element-wise multiplication
result = np.multiply(array1, array2)
# Print the result
print(result)
```

**Using list comprehension**(if you prefer not to use external libraries):

```
# Define the two arrays
array1 = [1, 2, 3]
array2 = [4, 5, 6]
# Perform element-wise multiplication using list comprehension
result = [a * b for a, b in zip(array1, array2)]
# Print the result
print(result)
```

Both methods will give you the same result. The `numpy`

method is more efficient and is generally recommended for larger arrays due to its performance advantages. The list comprehension method is a simple and quick way to achieve the same for smaller arrays or if you prefer not to use external libraries.

Your code seems to be performing polynomial regression for different features in a dataset and plotting the results. However, there are a few improvements and fixes needed to make it functional and efficient:

**Import necessary libraries:**Ensure all required libraries are imported.**Define the**Make sure this function is defined or imported.`polynomial_regression`

function:**Fix**Initialize`Y_new`

initialization:`Y_new`

with the bias term.**Fix the range for plotting:**Ensure the plotting is consistent with the shapes of`X`

and`Y_new`

.**Calculate**Ensure the`y_new`

for R2 score calculation:`y_new`

is calculated correctly and the R2 score function is defined or imported.**Label corrections in plots:**Make sure the plot labels are clear and distinct.**Proper iteration:**Make sure the loop indices are managed correctly.

Here is the corrected version of your code:

```
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score
# Define polynomial regression function
def polynomial_regression(X, y, degree):
poly_features = PolynomialFeatures(degree=degree, include_bias=False)
X_poly = poly_features.fit_transform(X.reshape(-1, 1))
lin_reg = LinearRegression()
lin_reg.fit(X_poly, y)
return np.append(lin_reg.intercept_, lin_reg.coef_)
# Load data and preprocess
data = pd.read_excel('DataSet.xlsx')
data = data.dropna()
degree = 5
# Loop through features
for i in range(1, 14):
X = data.iloc[:, i:i+1].values.flatten()
y = data['MEDV'].values
# Get polynomial regression coefficients
theta_best = polynomial_regression(X, y, degree)
# Generate predictions
X_new = X.flatten()
Y_new = np.zeros(X_new.shape[0])
Y_new += theta_best[0] # Adding the intercept (bias term)
for j in range(1, degree + 1):
Y_new += theta_best[j] * X_new ** j
# Plot the results
plt.scatter(X, y, color='blue', label='Actual')
plt.plot(X, Y_new, color='red', label='Predictions')
plt.xlabel(f"Feature {i}")
plt.ylabel("MEDV")
plt.legend()
plt.show()
# Calculate and print R2 error
R2_error = r2_score(y, Y_new)
print(f"Feature {i}: R2 error: {R2_error}")
```

### Key Changes and Fixes:

**Initialization of**`Y_new`

:`Y_new`

is initialized with the intercept term`theta_best[0]`

.**Loop Index Correction:**The inner loop uses`j`

to avoid conflict with the outer loop index`i`

.**Fixed R2 score calculation:**The`r2_score`

function from`sklearn.metrics`

is used to calculate the R2 error.**Correct Plotting:**The`plt.scatter`

and`plt.plot`

functions are used appropriately to distinguish between actual and predicted values.**Proper Labeling:**The feature index is dynamically included in the labels for clarity.

Make sure to test the code with your dataset to ensure it runs as expected. If there are any specific errors or issues, please provide the details, and I can assist further.

If you want to implement polynomial regression without using external libraries like `scikit-learn`

, you will need to manually handle the generation of polynomial features and the linear regression. Here’s a step-by-step guide to achieve this:

**Generate Polynomial Features:**

- Create a function to generate polynomial features up to a given degree.

**Perform Linear Regression:**

- Implement the normal equation to compute the best-fit parameters.

Here’s how you can do it:

### Step 1: Generate Polynomial Features

```
import numpy as np
def generate_polynomial_features(X, degree):
n_samples = X.shape[0]
X_poly = np.ones((n_samples, degree + 1))
for d in range(1, degree + 1):
X_poly[:, d] = X ** d
return X_poly
```

### Step 2: Perform Linear Regression

```
def linear_regression(X, y):
# Compute the normal equation
X_transpose = X.T
theta_best = np.linalg.inv(X_transpose.dot(X)).dot(X_transpose).dot(y)
return theta_best
```

### Step 3: Polynomial Regression

Now combine the above steps to perform polynomial regression:

```
def polynomial_regression(X, y, degree):
# Generate polynomial features
X_poly = generate_polynomial_features(X, degree)
# Perform linear regression using normal equation
theta_best = linear_regression(X_poly, y)
return theta_best
# Example usage
import pandas as pd
import matplotlib.pyplot as plt
# Load data and preprocess
data = pd.read_excel('DataSet.xlsx')
data = data.dropna()
degree = 5
for i in range(1, 14):
X = data.iloc[:, i:i+1].values.flatten()
y = data['MEDV'].values
# Get polynomial regression coefficients
theta_best = polynomial_regression(X, y, degree)
# Generate predictions
X_new = X.flatten()
Y_new = np.zeros(X_new.shape[0])
for j in range(degree + 1):
Y_new += theta_best[j] * X_new ** j
# Plot the results
plt.scatter(X, y, color='blue', label='Actual')
plt.plot(X, Y_new, color='red', label='Predictions')
plt.xlabel(f"Feature {i}")
plt.ylabel("MEDV")
plt.legend()
plt.show()
# Calculate and print R2 error
y_new = Y_new # Y_new already contains the predictions
R2_error = 1 - np.sum((y - y_new) ** 2) / np.sum((y - np.mean(y)) ** 2)
print(f"Feature {i}: R2 error: {R2_error}")
```

### Explanation:

`generate_polynomial_features`

function:

- This function takes an input array
`X`

and a degree and returns a matrix containing the polynomial features up to the specified degree.

`linear_regression`

function:

- This function computes the best-fit parameters using the normal equation. It takes the matrix
`X`

and the target vector`y`

, and returns the parameters`theta_best`

.

`polynomial_regression`

function:

- This function combines the generation of polynomial features and the linear regression to return the coefficients for the polynomial regression.

**Example Usage:**

- The example usage section demonstrates how to load data, preprocess it, perform polynomial regression for each feature, generate predictions, and plot the results.

By following these steps, you can perform polynomial regression without relying on external libraries like `scikit-learn`

.

To use the `polynomial_regression_gradient_descent`

function to predict data for all features, you need to follow these steps:

**Normalize the Features:**Normalize the features in your dataset to ensure that gradient descent converges efficiently.**Generate Polynomial Features:**Create polynomial features for each feature in your dataset up to the specified degree.**Perform Polynomial Regression:**Use the gradient descent method to fit the polynomial regression model to each feature and predict the output.**Plot the Results:**Visualize the actual and predicted values for each feature.

Here is the complete code that integrates your functions and performs these steps:

```
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
def polynomial_regression_gradient_descent(feature_matrix, outputs, initial_weights, bias, step_size, tolerance):
weights = np.array(initial_weights)
while True:
predictions = predict_output(feature_matrix, weights, bias)
errors = predictions - outputs
gradients = np.zeros(len(weights))
for i in range(len(weights)):
gradients[i] = feature_derivative(errors, feature_matrix[:, i])
weights -= step_size * gradients
gradient_magnitude = np.sqrt(np.sum(gradients ** 2))
if gradient_magnitude < tolerance:
break
return weights
def normalize_features(data_frame, chosen_features):
for feature in chosen_features:
data_frame[feature] = (data_frame[feature] - data_frame[feature].mean()) / data_frame[feature].std()
return data_frame
def predict_output(feature_matrix, weights, bias):
predictions = np.dot(feature_matrix, weights) + bias
return predictions
def generate_polynomial_features(X, degree):
n_samples = X.shape[0]
X_poly = np.ones((n_samples, degree + 1))
for d in range(1, degree + 1):
X_poly[:, d] = X ** d
return X_poly
def feature_derivative(errors, feature):
return 2 * np.dot(errors, feature)
# Load data and preprocess
data = pd.read_excel('DataSet.xlsx')
data = data.dropna()
chosen_features = data.columns[1:14]
data = normalize_features(data, chosen_features)
degree = 5
step_size = 1e-3
tolerance = 1e-6
for feature in chosen_features:
X = data[feature].values
y = data['MEDV'].values
X_poly = generate_polynomial_features(X, degree)
initial_weights = np.zeros(X_poly.shape[1])
bias = 0
weights = polynomial_regression_gradient_descent(X_poly, y, initial_weights, bias, step_size, tolerance)
X_new = X.flatten()
Y_new = predict_output(X_poly, weights, bias)
# Plot the results
plt.scatter(X, y, color='blue', label='Actual')
plt.plot(X, Y_new, color='red', label='Predictions')
plt.xlabel(f"Feature {feature}")
plt.ylabel("MEDV")
plt.legend()
plt.show()
# Calculate and print R2 error
R2_error = 1 - np.sum((y - Y_new) ** 2) / np.sum((y - np.mean(y)) ** 2)
print(f"Feature {feature}: R2 error: {R2_error}")
```

### Explanation:

**Normalization of Features:**The`normalize_features`

function is used to normalize the features in the dataset. This helps in ensuring that the gradient descent converges efficiently.**Polynomial Features Generation:**The`generate_polynomial_features`

function creates polynomial features up to the specified degree for each feature.**Gradient Descent for Polynomial Regression:**The`polynomial_regression_gradient_descent`

function is used to fit the polynomial regression model using gradient descent.**Prediction and Plotting:**The predicted values are generated using the`predict_output`

function and are then plotted along with the actual values for visualization. The R2 error is also calculated to evaluate the model’s performance.

By following these steps, you can predict the output for all features using polynomial regression with gradient descent.

To improve the efficiency of the gradient descent algorithm and reduce the runtime, you can consider the following approaches:

**Batch Gradient Descent:**Instead of updating the weights for each individual feature in a loop, you can perform batch gradient descent where you update the weights for all features simultaneously.**Vectorized Operations:**Utilize vectorized operations with NumPy to speed up computations.**Stochastic Gradient Descent (SGD):**Use stochastic gradient descent, which updates weights for each training example, to speed up convergence.**Adjust Learning Rate:**Experiment with different learning rates to find an optimal balance between speed and convergence.

Here is an optimized version of your code using vectorized operations and batch gradient descent:

```
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
def polynomial_regression_gradient_descent(feature_matrix, outputs, initial_weights, bias, step_size, tolerance, max_iterations=1000):
weights = np.array(initial_weights)
iteration = 0
while iteration < max_iterations:
predictions = predict_output(feature_matrix, weights, bias)
errors = predictions - outputs
gradients = 2 * feature_matrix.T.dot(errors) / len(outputs)
weights -= step_size * gradients
bias -= step_size * 2 * np.sum(errors) / len(outputs)
gradient_magnitude = np.sqrt(np.sum(gradients ** 2))
if gradient_magnitude < tolerance:
break
iteration += 1
return weights, bias
def normalize_features(data_frame, chosen_features):
for feature in chosen_features:
data_frame[feature] = (data_frame[feature] - data_frame[feature].mean()) / data_frame[feature].std()
return data_frame
def predict_output(feature_matrix, weights, bias):
return feature_matrix.dot(weights) + bias
def generate_polynomial_features(X, degree):
n_samples = X.shape[0]
X_poly = np.ones((n_samples, degree + 1))
for d in range(1, degree + 1):
X_poly[:, d] = X ** d
return X_poly
# Load data and preprocess
data = pd.read_excel('DataSet.xlsx')
data = data.dropna()
chosen_features = data.columns[1:14]
data = normalize_features(data, chosen_features)
degree = 5
step_size = 1e-3
tolerance = 1e-6
for feature in chosen_features:
X = data[feature].values
y = data['MEDV'].values
X_poly = generate_polynomial_features(X, degree)
initial_weights = np.zeros(X_poly.shape[1])
bias = 0
weights, bias = polynomial_regression_gradient_descent(X_poly, y, initial_weights, bias, step_size, tolerance)
Y_new = predict_output(X_poly, weights, bias)
# Plot the results
plt.scatter(X, y, color='blue', label='Actual')
plt.plot(X, Y_new, color='red', label='Predictions')
plt.xlabel(f"Feature {feature}")
plt.ylabel("MEDV")
plt.legend()
plt.show()
# Calculate and print R2 error
R2_error = 1 - np.sum((y - Y_new) ** 2) / np.sum((y - np.mean(y)) ** 2)
print(f"Feature {feature}: R2 error: {R2_error}")
```

### Explanation of Improvements:

**Vectorized Gradient Calculation:**The gradients are calculated in a vectorized manner using`feature_matrix.T.dot(errors)`

, which speeds up the computation.**Bias Update:**The bias term is updated in each iteration using the mean of the errors.**Max Iterations:**A maximum number of iterations is set to prevent the algorithm from running indefinitely.

These optimizations should help reduce the runtime of the gradient descent algorithm. If further speed improvements are necessary, consider using stochastic gradient descent (SGD) or mini-batch gradient descent, which update the weights more frequently and can converge faster in some cases.

The warnings you’re seeing indicate numerical issues during the gradient descent process, likely due to an excessively high learning rate or gradient values becoming too large. Here are a few adjustments you can make to mitigate these issues:

**Reduce the learning rate:**Lower the step size to prevent the weights from changing too drastically.**Feature scaling:**Ensure the features are properly scaled to avoid large gradient values.**Gradient clipping:**Limit the maximum value of gradients to prevent overflow.

Here’s the updated code with these adjustments:

**Reduced learning rate:**Start with a smaller step size.**Gradient clipping:**Limit the gradients to prevent them from becoming too large.

### Updated Code

```
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
def polynomial_regression_gradient_descent(feature_matrix, outputs, initial_weights, bias, step_size, tolerance, max_iterations=1000):
weights = np.array(initial_weights)
iteration = 0
while iteration < max_iterations:
predictions = predict_output(feature_matrix, weights, bias)
errors = predictions - outputs
gradients = 2 * feature_matrix.T.dot(errors) / len(outputs)
# Gradient clipping to prevent overflow
gradients = np.clip(gradients, -1e2, 1e2)
weights -= step_size * gradients
bias -= step_size * 2 * np.sum(errors) / len(outputs)
gradient_magnitude = np.sqrt(np.sum(gradients ** 2))
if gradient_magnitude < tolerance:
break
iteration += 1
return weights, bias
def normalize_features(data_frame, chosen_features):
for feature in chosen_features:
data_frame[feature] = (data_frame[feature] - data_frame[feature].mean()) / data_frame[feature].std()
return data_frame
def predict_output(feature_matrix, weights, bias):
return feature_matrix.dot(weights) + bias
def generate_polynomial_features(X, degree):
n_samples = X.shape[0]
X_poly = np.ones((n_samples, degree + 1))
for d in range(1, degree + 1):
X_poly[:, d] = X ** d
return X_poly
# Load data and preprocess
data = pd.read_excel('DataSet.xlsx')
data = data.dropna()
chosen_features = data.columns[1:14]
data = normalize_features(data, chosen_features)
degree = 5
step_size = 1e-4 # Reduced step size
tolerance = 1e-6
for feature in chosen_features:
X = data[feature].values
y = data['MEDV'].values
X_poly = generate_polynomial_features(X, degree)
initial_weights = np.zeros(X_poly.shape[1])
bias = 0
weights, bias = polynomial_regression_gradient_descent(X_poly, y, initial_weights, bias, step_size, tolerance)
Y_new = predict_output(X_poly, weights, bias)
# Plot the results
plt.scatter(X, y, color='blue', label='Actual')
plt.plot(np.sort(X), Y_new[np.argsort(X)], color='red', label='Predictions')
plt.xlabel(f"Feature {feature}")
plt.ylabel("MEDV")
plt.legend()
plt.show()
# Calculate and print R2 error
R2_error = 1 - np.sum((y - Y_new) ** 2) / np.sum((y - np.mean(y)) ** 2)
print(f"Feature {feature}: R2 error: {R2_error}")
```

### Key Adjustments:

**Reduced Learning Rate:**The`step_size`

has been decreased to`1e-4`

.**Gradient Clipping:**Gradients are clipped to the range`[-1e2, 1e2]`

to prevent them from becoming too large.**Plotting Sorted Data:**The predictions are plotted against the sorted input feature values to ensure a smooth line.

These changes should help in stabilizing the gradient descent process and preventing numerical issues. If you still face issues, you might need to experiment with the learning rate further or consider using a more sophisticated optimization algorithm.