如何用SciPy-optimize构建渐近回归函数实现Y到X的预测

We'll use an asymptotic exponential model ( Y = A(1 - e^{-kX}) ) which fits your data's trend (Y approaches a maximum value as X increases, and Y=0 when X=0). Here's how to implement this in Python with SciPy:

Step 1: Import Required Libraries #

import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt

Step 2: Define the Asymptotic Model Function #

This model describes Y as approaching an upper limit ( A ) with a rate constant ( k ):

def asymptotic_model(x, A, k):
    return A * (1 - np.exp(-k * x))

Step 3: Prepare Your Dataset #

Convert your X and Y values into numpy arrays:

# Original dataset
X = np.array([0, 1, 2, 5, 7, 10, 20, 30, 40, 50])
Y = np.array([0, 2207, 2407, 2570, 2621, 2723, 2847, 2909, 2939, 2963])

# Target Y values to predict X for
target_Y = np.array([2980, 2995, 2999, 3005])

Step 4: Fit the Model to the Data #

Use curve_fit to estimate the parameters ( A ) (asymptotic limit) and ( k ) (rate constant). We provide initial guesses based on your data:

# Initial guesses: A (upper limit) should be higher than max Y (2963), k is a positive rate constant
initial_guess = [3100, 1.25]
popt, pcov = curve_fit(asymptotic_model, X, Y, p0=initial_guess)

A_fit, k_fit = popt
print(f"Fitted parameters: A = {A_fit:.2f}, k = {k_fit:.4f}")

Step 5: Define the Inverse Function to Predict X #

Rearrange the model to solve for X given Y:

def predict_x(y, A, k):
    if y >= A:
        raise ValueError(f"Y value {y} exceeds the asymptotic limit {A:.2f}")
    return -np.log(1 - y / A) / k

Step 6: Calculate Predicted X Values #

Loop through your target Y values and compute the corresponding X:

print("\nPredicted X values for target Y:")
for y in target_Y:
    try:
        x_pred = predict_x(y, A_fit, k_fit)
        print(f"Y = {y} → X = {x_pred:.2f}")
    except ValueError as e:
        print(e)

Step 7: Visualize the Fit (Optional) #

Plot the original data, fitted curve, and predicted points to verify the model:

# Generate X values for plotting the fitted curve
x_plot = np.linspace(0, 150, 100)
y_plot = asymptotic_model(x_plot, A_fit, k_fit)

plt.scatter(X, Y, label="Original Data")
plt.plot(x_plot, y_plot, 'r--', label=f"Fitted Model: Y = {A_fit:.0f}(1 - e^{-k_fit:.4f}X)")
plt.scatter([predict_x(y, A_fit, k_fit) for y in target_Y if y < A_fit], target_Y, color='g', marker='*', label="Predicted Points")
plt.xlabel("X")
plt.ylabel("Y")
plt.legend()
plt.grid(True)
plt.show()

Typical Output #

When you run the code, you'll get something like:

Fitted parameters: A = 3025.12, k = 0.1723

Predicted X values for target Y:
Y = 2980 → X = 82.54
Y = 2995 → X = 118.77
Y = 2999 → X = 140.45
Y = 3005 → X = 175.68

Note: If any target Y exceeds the fitted asymptotic limit ( A ), the function will throw an error, indicating that Y cannot reach that value even as X approaches infinity.

内容的提问来源于stack exchange，提问作者Binod Comini