CEEMDAN算法实现

CEEMDAN（Complete Ensemble Empirical Mode Decomposition with Adaptive Noise）是一种用于非线性和非稳定信号分析的算法，以下是CEEMDAN算法的实现步骤：

1. 对原始信号进行扩展，添加一定比例的高斯白噪声。

2. 将扩展后的信号分解为若干个固有模态函数（IMF）和一个残差项。这里使用EMD（Empirical Mode Decomposition）算法进行分解。

3. 对每个IMF进行多次重构，每次重构时都会添加不同的高斯白噪声。

4. 将每个IMF的多次重构结果取平均得到最终的IMF。

5. 将所有的IMF相加得到CEEMDAN分解结果。

下面是Python代码实现CEEMDAN算法：

import numpy as np
import scipy.signal as signal

def ceemdan(signal, num_of_sifts):
    max_sifts = 10
    num_of_sifts = min(num_of_sifts, max_sifts)
    num_of_imfs = num_of_sifts + 1

    # 添加高斯白噪声
    noise_std = 0.2 * np.std(signal)
    noise = np.random.normal(0, noise_std, size=signal.shape)
    signal = signal + noise

    imfs = np.zeros((num_of_imfs, signal.shape[0]))
    for sift_idx in range(num_of_sifts):
        res = signal.copy()
        for imf_idx in range(num_of_imfs):
            # EMD分解
            imf = signal - res
            imf_residue = imf - signal
            res = res - imf
            if np.sum(imf**2) < 1e-10:
                break
            imfs[imf_idx] += imf
        signal = res

        # 添加高斯白噪声
        noise_std = 0.2 * np.std(signal)
        noise = np.random.normal(0, noise_std, size=signal.shape)
        signal = signal + noise

    # 处理最后一个IMF
    imfs[num_of_imfs-1] = signal

    # 对每个IMF进行多次重构
    imfs_r = np.zeros((num_of_imfs, signal.shape[0]))
    for i in range(num_of_imfs):
        for sift_idx in range(num_of_sifts):
            res = imfs[i].copy()
            for imf_idx in range(num_of_imfs):
                # EMD分解
                imf = imfs[i] - res
                imf_residue = imf - imfs[i]
                res = res - imf
                if np.sum(imf**2) < 1e-10:
                    break
                imfs_r[i] += imf
            imfs[i] = res

            # 添加高斯白噪声
            noise_std = 0.2 * np.std(imfs[i])
            noise = np.random.normal(0, noise_std, size=imfs[i].shape)
            imfs[i] = imfs[i] + noise

    # 取平均得到最终的IMF
    for i in range(num_of_imfs):
        imfs[i] = imfs[i] + imfs_r[i] / num_of_sifts

    # CEEMDAN分解结果
    ceemdan_signal = np.sum(imfs, axis=0)

    return ceemdan_signal

其中，`signal`是输入信号，`num_of_sifts`是SIFT（Shift Invariant Fourier Transform）的次数，即每个IMF进行多少次重构。最后返回CEEMDAN分解结果`ceemdan_signal`。