Python作迴歸分析

  假設上述是以輸入-輸出對的形式觀察到的數據。請使用指數函數家族，y=αe^βx，其中β<0，進行迴歸分析，並找出最佳的係數α和β。

程式碼:

import numpy as np

import pandas as pd

from scipy.optimize import curve_fit

from sklearn.linear_model import LinearRegression

from sklearn.metrics import mean_squared_error

# The data is assumed to be the observed input-output pairs.

data = {

    'x': range(11),

    'y': [9735, 4597, 2176, 1024, 483, 229, 108, 52, 24, 11, 6]

}

df = pd.DataFrame(data)

# Defining the exponential function y = α * e^(β * x).

def exponential_func(x, a, b):

    return a * np.exp(b * x)

# Use non-linear least squares to fit the exponential function to data.

params, covariance = curve_fit(exponential_func, df['x'], df['y'])

# Extracting the optimum coefficients.

alpha, beta = params

# Now let's perform a linear regression on the log-transformed data for comparison.

# Linear regression requires a reshape of x to 2D array for sklearn

X_linear = df['x'].values.reshape(-1, 1)

y_linear = np.log(df['y'])  # Transform the y values to a linear space

linear_regressor = LinearRegression()

linear_regressor.fit(X_linear, y_linear)

# Predict the values using both models

y_pred_exponential = exponential_func(df['x'], alpha, beta)

y_pred_linear = np.exp(linear_regressor.predict(X_linear))  # Transform predictions back to the original space

# Calculate the mean squared error for both models

mse_exponential = mean_squared_error(df['y'], y_pred_exponential)

mse_linear = mean_squared_error(df['y'], y_pred_linear)

# Output the parameters and the comparison of errors

print(f"Optimum α: {alpha}")

print(f"Optimum β: {beta}")

print(f"MSE for Exponential Regression: {mse_exponential}")

print(f"MSE for Linear Regression: {mse_linear}")

print(f"Difference in MSE: {mse_linear - mse_exponential}")