### 如何使用 `scipy.optimize` 实现双曲线拟合 在 Python 的 SciPy 库中，`scipy.optimize.curve_fit()` 是一种常用的函数用于数据拟合。该方法基于最小二乘法原理来估计模型中的未知参数。对于双曲线拟合的情况，可以通过定义两个独立的曲线并将其组合成一个目标函数来进行优化。 以下是详细的说明以及示例代码： #### 双曲线拟合的核心概念 为了实现两条曲线的同时拟合，可以将每条曲线表示为单独的数学表达式，并通过共享或分离某些参数的方式构建联合的目标函数[^2]。通常情况下，这种拟合涉及多个自变量和因变量之间的关系建模。 假设我们有两组观测数据 `(x_data1, y_data1)` 和 `(x_data2, y_data2)`，分别对应两条不同的曲线。我们可以定义如下形式的目标函数： \[ f_1(x; a, b) \quad \text{and} \quad f_2(x; c, d), \] 其中 \(a\)、\(b\) 表示第一条曲线的相关参数；而 \(c\)、\(d\) 则代表第二条曲线对应的参数。 接着，我们将这两部分结合起来形成一个新的复合函数作为输入给 `curve_fit()` 方法调用的基础结构。 #### 示例代码展示 下面提供了一个具体的例子演示如何应用上述理论框架完成实际操作过程: ```python import numpy as np from scipy.optimize import curve_fit import matplotlib.pyplot as plt # Define the two separate functions to fit. def func1(x, a, b): """First function.""" return a * x + b # Example of linear relationship. def func2(x, c, d): """Second function.""" return c / (x - d) # Example of hyperbolic relationship. # Combined model that includes both curves. def combined_model(X, a, b, c, d): """ X is an array where each row contains one set of independent variables, and we split it into parts corresponding to different datasets. Parameters: X : ndarray Independent variable data points stacked together. a,b,c,d : float Fitting parameters for respective models. Returns: Array containing concatenated results from applying individual funcs on their subsets within 'X'. """ mask = X[:,0]==1 # Assuming first column indicates dataset membership via binary flagging system {1 or 2}. out = np.zeros(len(X)) out[mask] = func1(X[mask][:,-1], a, b) out[~mask] = func2(X[~mask][:,-1], c, d) return out # Generate synthetic noisy test data based off our defined formulas above plus added random noise component. np.random.seed(0) N_points_per_curve=50 noise_level=.3 xs=np.linspace(-4,+8,N_points_per_curve//2)*[-1]+list(range(N_points_per_curve//2)) ys_true_func1=[func1(_, .7,.9 )for _in xs[:len(xs)//2]] errors=noise_level*np.std(ys_true_func1)/abs(np.diff(xs)[0]) noisy_ydata_for_first_part=list(map(lambda yi,yerr:(yi+yerr*(np.random.randn()-eps)), ys_true_func1,[errors]*len(ys_true_func1))) similarly_do_the_second_half_with_respective_parameters_and_noise... all_xvalues=np.vstack(([[1]]*len(noisy_ydata_for_first_part)+ [[2]]*len(secondhalf))) final_dataset_to_pass_into_combinedmodel=all_xvalues.T.reshape((-1,)) initial_guesses=(.6,.8,.5,.3) # Initial guesses close enough but not perfect matches yet! popt,_pcov_=curve_fit(combined_model , final_dataset_to_pass_into_combinedmodel ,[item for sublist in zip(*both_halves_of_noisyydatas)]flattingthemoutcorrectlyherepleasecheckthispartcarefullybeforeusingitdirectlythough! , p0=initial_guesses ) print("Optimized parameter values:", popt) plt.scatter(all_xvalues[:,1][all_xvalues[:,0]==1], [_[i]if i<len(_)else None for i,_ in enumerate(zip(*both_halves_of_noisyydatas)) if all_xvalues[i,0]==1 ], label='Data Set One') plt.plot(sorted([_[i]if i<len(_)else None for i,_ in enumerate(zip(*both_halves_of_noisyydatas))]), list(map(func1, sorted(_), *(popt[:2]))), color="red", linewidth=2., linestyle="--",label="Fitted Curve For First Dataset") plt.scatter(all_xvalues[:,1][all_xvalues[:,0]==2],[_[j]if j<len(_)else None for j,_ in enumerate(zip(*both_halves_of_noisyydatas)) if all_xvalues[j,0]==2 ] ,color="green", marker="s", s=50,label="Dataset Two Points") plt.plot(sorted([_[k]if k<len(_)else None for k,_ in enumerate(zip(*both_halves_of_noisyydatas))]),list(map(func2,sorted(_),(popt[2:]) ),color="blueviolet",linestyle="-.",linewidth=2.,label="Best Fit Line Second Group") plt.legend() plt.show() ``` 注意：以上代码片段仅作示意用途，在复制粘贴前需适当修改以适应具体应用场景下的细节差异。 ---