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203 lines
5.4 KiB
Markdown
203 lines
5.4 KiB
Markdown
---
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title: Ensemble Empirical Mode Decomposition, EEMD
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tags:
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- advanced
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- EMD
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- HTT
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- signal-processing
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- signal
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date: 2024-04-23
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---
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# Intro
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集合经验模态分解（Ensemble Empirical Mode Decomposition, EEMD）是一种改进的[EMD](signal_processing/algorithm/EMD/basic.md)方法，它通过引入白噪声来解决EMD中的**模态混叠问题**。
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模态混叠是指在分解过程中，不同时间尺度的信号成分错误地混合在一起，导致分解结果不准确。
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上述figure就是一个很好的例子，连续低频正弦信号上叠加了间歇性高频震动的调制信号，因为间歇性高频震动的调制信号干扰了Maximum点的选择，使得局部极值在很短的时间间隔发生多次跳变，进而使得我们的IMF并不准确，不同时间尺度的信号成分错误地混合在一起。
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以下我们也通过我们写的EMD做了示范：
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很明显，在IMF1，发生了混叠
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# Step by Step
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为了解决这种模态混叠的现象，EEMD方法应运而生。其步骤为，
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* 设定总体平均次数$M$
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* 将一个具有标准正态分布的白噪声$n_i(t)$加到原始信号$x(t)$上，产生一个新的信号，$x_i(t) = x(t) + n_i(t)$
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* 对$x_i(t)$进行EMD
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$$
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x_i(t) = \sum_{j=1}^{J}c_{i,j}(t) + r_{i,j}(t)
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$$
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* 重复上述两步骤吗，得到IMF集合
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$$
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c_{1,j}(t), c_{2,j}(t), \cdots,c_{M,j}(t), j=1,2,\cdots,J
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$$
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* 对上述的IMF集合进行集合平局，得到最终的IMF，即：
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$$
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c_j(t) = \frac{1}{M}\sum_{i=1}^{M}c_{i,j}(t)
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$$
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## Algorithm Detail
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* 添加噪声对于我们的信号有什么影响
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* 每次添加的信号有什么要求，加多大的噪声；（提出EEMD的Wu文章中说，标准差为0.1的噪声）
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# Demo Code
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```python
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def EEMD(signal, max_imf = 10, tolerance = 0.01, iterations = 10):
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def __extrema(signal):
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max_peaks = []
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min_peaks = []
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for i in range(1, len(signal) - 1):
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if signal[i] > signal[i-1] and signal[i] > signal[i+1]:
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max_peaks.append(i)
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if signal[i] < signal[i-1] and signal[i] < signal[i+1]:
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min_peaks.append(i)
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return max_peaks, min_peaks
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def __mean_env(signal):
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max_peaks, min_peaks = __extrema(signal)
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max_env = np.interp(range(len(signal)), max_peaks, [signal[i] for i in max_peaks])
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min_env = np.interp(range(len(signal)), min_peaks, [signal[i] for i in min_peaks])
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return (max_env + min_env) / 2
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def __IMF(signal):
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resdiual = signal
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while True:
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imf = signal - __mean_env(resdiual)
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resdiual = signal - imf
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if np.mean(imf) < tolerance:
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break
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return imf, resdiual
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mean = np.mean(signal)
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std = np.std(signal)
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signal = (signal - mean) / std
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imfss = []
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signals = []
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min_imf = max_imf
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tmp_imf = max_imf
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for index in range(iterations):
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imfs = []
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noise = np.random.randn(len(signal))
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noise = noise * 0.1
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new_signal = signal + noise
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signals.append(new_signal)
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tmp_imf = max_imf
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while True:
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imf, residual = __IMF(new_signal)
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imfs.append(imf)
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max_peaks, min_peaks = __extrema(residual)
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if len(max_peaks) < 2 or len(min_peaks) < 2:
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tmp_min = len(imfs)
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if tmp_min < min_imf:
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min_imf = tmp_min
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break
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if np.abs(np.std(residual)) < tolerance or tmp_imf == 0:
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tmp_min = len(imfs)
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if tmp_min < min_imf:
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min_imf = tmp_min
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break
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new_signal = residual
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tmp_imf -= 1
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imfss.append(imfs)
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imf_ans = []
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for i in range(min_imf):
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imf = 0
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for j in range(iterations):
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imf += imfss[j][i]
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imf = imf / iterations
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imf_ans.append(imf)
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signal_ans = 0
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for signal in signals:
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signal_ans += signal
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signal_ans = signal_ans / iterations
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residual_ans = signal_ans
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for imf in imf_ans:
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residual_ans = residual_ans - imf
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imf_ans = [imf * std + mean for imf in imf_ans]
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residual_ans = residual_ans * std + mean
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signal_ans = signal_ans * std + mean
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return imf_ans, residual_ans, signal_ans
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```
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## Results
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通过EEMD，结果如下：
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EEMD的前几个IMF将高频噪声和白噪声过滤，在IMF7显示了信号原有的模态
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# Reference
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* https://zhuanlan.zhihu.com/p/205345681
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* https://blog.csdn.net/weixin_45317919/article/details/109850346 |