diff --git a/content/signal_processing/algorithm/EMD/attachments/Figure_1 1.png b/content/signal_processing/algorithm/EMD/attachments/Figure_1 1.png new file mode 100644 index 000000000..de136f6df Binary files /dev/null and b/content/signal_processing/algorithm/EMD/attachments/Figure_1 1.png differ diff --git a/content/signal_processing/algorithm/EMD/attachments/Figure_1.png b/content/signal_processing/algorithm/EMD/attachments/Figure_1.png new file mode 100644 index 000000000..1bee32edd Binary files /dev/null and b/content/signal_processing/algorithm/EMD/attachments/Figure_1.png differ diff --git a/content/signal_processing/algorithm/EMD/attachments/Figure_2 1.png b/content/signal_processing/algorithm/EMD/attachments/Figure_2 1.png new file mode 100644 index 000000000..ff941bbd1 Binary files /dev/null and b/content/signal_processing/algorithm/EMD/attachments/Figure_2 1.png differ diff --git a/content/signal_processing/algorithm/EMD/attachments/Figure_2.png b/content/signal_processing/algorithm/EMD/attachments/Figure_2.png new file mode 100644 index 000000000..318db89f2 Binary files /dev/null and b/content/signal_processing/algorithm/EMD/attachments/Figure_2.png differ diff --git a/content/signal_processing/algorithm/EMD/attachments/Figure_3 1.png b/content/signal_processing/algorithm/EMD/attachments/Figure_3 1.png new file mode 100644 index 000000000..9a1e21ab1 Binary files /dev/null and b/content/signal_processing/algorithm/EMD/attachments/Figure_3 1.png differ diff --git a/content/signal_processing/algorithm/EMD/attachments/Figure_3.png b/content/signal_processing/algorithm/EMD/attachments/Figure_3.png new file mode 100644 index 000000000..cfc6f75f7 Binary files /dev/null and b/content/signal_processing/algorithm/EMD/attachments/Figure_3.png differ diff --git a/content/signal_processing/algorithm/EMD/attachments/Pasted image 20240423162631.png b/content/signal_processing/algorithm/EMD/attachments/Pasted image 20240423162631.png new file mode 100644 index 000000000..5eec539bc Binary files /dev/null and b/content/signal_processing/algorithm/EMD/attachments/Pasted image 20240423162631.png differ diff --git a/content/signal_processing/algorithm/EMD/basic.md b/content/signal_processing/algorithm/EMD/basic.md index 6a6d6b98b..34e93d667 100644 --- a/content/signal_processing/algorithm/EMD/basic.md +++ b/content/signal_processing/algorithm/EMD/basic.md @@ -7,7 +7,7 @@ tags: - signal - algorithm - math -date: 2024-04-24 +date: 2024-04-23 --- # Introduction @@ -104,7 +104,7 @@ $$ import numpy as np import numpy as np -def EMD(signal, max_imf = 10, tolerance = 0.1): +def EMD(signal, max_imf = 10, tolerance = 0.01): def __extrema(signal): @@ -131,12 +131,18 @@ def EMD(signal, max_imf = 10, tolerance = 0.1): def __IMF(signal): - imf = signal - __mean_env(signal) - resdiual = signal - imf + resdiual = signal + + while True: + + imf = signal - __mean_env(resdiual) + resdiual = signal - imf + + if np.mean(imf) < tolerance: + break return imf, resdiual - # standardize the signal mean = np.mean(signal) std = np.std(signal) @@ -151,6 +157,11 @@ def EMD(signal, max_imf = 10, tolerance = 0.1): imfs.append(imf) + max_peaks, min_peaks = __extrema(residual) + + if len(max_peaks) < 2 or len(min_peaks) < 2: + break + if np.abs(np.std(residual)) < tolerance or max_imf == 0: break @@ -161,7 +172,6 @@ def EMD(signal, max_imf = 10, tolerance = 0.1): residual = residual * std + mean return imfs, residual - ``` # Reference diff --git a/content/signal_processing/algorithm/EMD/eemd.md b/content/signal_processing/algorithm/EMD/eemd.md new file mode 100644 index 000000000..5e3e799c5 --- /dev/null +++ b/content/signal_processing/algorithm/EMD/eemd.md @@ -0,0 +1,203 @@ +--- +title: Ensemble Empirical Mode Decomposition, EEMD +tags: + - advanced + - EMD + - HTT + - signal-processing + - signal +date: 2024-04-23 +--- + +# Intro + +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**。 + + +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。 + +![](signal_processing/algorithm/EMD/attachments/Pasted%20image%2020240423162631.png) + +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。 + +以下我们也通过我们写的EMD做了示范： + + +![](signal_processing/algorithm/EMD/attachments/Figure_1.png) + + +![](signal_processing/algorithm/EMD/attachments/Figure_3.png) + + +![](signal_processing/algorithm/EMD/attachments/Figure_2.png) + +很明显，在IMF1，发生了混叠 + +# Step by Step + +为了解决这种模态混叠的现象，EEMD方法应运而生。其步骤为， + +* 设定总体平均次数$M$ +* 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)$ +* 对$x_i(t)$进行EMD + $$ + x_i(t) = \sum_{j=1}^{J}c_{i,j}(t) + r_{i,j}(t) + $$ +* 重复上述两步骤吗，得到IMF集合 + $$ + c_{1,j}(t), c_{2,j}(t), \cdots,c_{M,j}(t), j=1,2,\cdots,J + $$ +* 对上述的IMF集合进行集合平局，得到最终的IMF，即： + $$ + c_j(t) = \frac{1}{M}\sum_{i=1}^{M}c_{i,j}(t) + $$ + +## Algorithm Detail + +* 添加噪声对于我们的信号有什么影响 +* 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） + + +# Demo Code + +```python +def EEMD(signal, max_imf = 10, tolerance = 0.01, iterations = 10): + + + def __extrema(signal): + + max_peaks = [] + min_peaks = [] + + for i in range(1, len(signal) - 1): + if signal[i] > signal[i-1] and signal[i] > signal[i+1]: + max_peaks.append(i) + if signal[i] < signal[i-1] and signal[i] < signal[i+1]: + min_peaks.append(i) + + return max_peaks, min_peaks + + + def __mean_env(signal): + + max_peaks, min_peaks = __extrema(signal) + + max_env = np.interp(range(len(signal)), max_peaks, [signal[i] for i in max_peaks]) + min_env = np.interp(range(len(signal)), min_peaks, [signal[i] for i in min_peaks]) + + return (max_env + min_env) / 2 + + def __IMF(signal): + + resdiual = signal + + while True: + + imf = signal - __mean_env(resdiual) + resdiual = signal - imf + + if np.mean(imf) < tolerance: + break + + return imf, resdiual + + + mean = np.mean(signal) + std = np.std(signal) + + signal = (signal - mean) / std + + imfss = [] + signals = [] + min_imf = max_imf + tmp_imf = max_imf + + for index in range(iterations): + + imfs = [] + + noise = np.random.randn(len(signal)) + noise = noise * 0.1 + + new_signal = signal + noise + signals.append(new_signal) + + tmp_imf = max_imf + + while True: + + imf, residual = __IMF(new_signal) + + imfs.append(imf) + + max_peaks, min_peaks = __extrema(residual) + + if len(max_peaks) < 2 or len(min_peaks) < 2: + + tmp_min = len(imfs) + if tmp_min < min_imf: + min_imf = tmp_min + + break + + if np.abs(np.std(residual)) < tolerance or tmp_imf == 0: + + tmp_min = len(imfs) + if tmp_min < min_imf: + min_imf = tmp_min + + break + + new_signal = residual + tmp_imf -= 1 + + imfss.append(imfs) + + imf_ans = [] + + for i in range(min_imf): + + imf = 0 + + for j in range(iterations): + imf += imfss[j][i] + + imf = imf / iterations + imf_ans.append(imf) + + signal_ans = 0 + + for signal in signals: + signal_ans += signal + + signal_ans = signal_ans / iterations + + residual_ans = signal_ans + + for imf in imf_ans: + + residual_ans = residual_ans - imf + + imf_ans = [imf * std + mean for imf in imf_ans] + residual_ans = residual_ans * std + mean + signal_ans = signal_ans * std + mean + + return imf_ans, residual_ans, signal_ans +``` + + +## Results + +通过EEMD，结果如下： + +![](signal_processing/algorithm/EMD/attachments/Figure_1%201.png) + +![](signal_processing/algorithm/EMD/attachments/Figure_2%201.png) + +![](signal_processing/algorithm/EMD/attachments/Figure_3%201.png) + +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 +# Reference + +* https://zhuanlan.zhihu.com/p/205345681 +* https://blog.csdn.net/weixin_45317919/article/details/109850346 \ No newline at end of file diff --git a/content/signal_processing/signal_processing_MOC.md b/content/signal_processing/signal_processing_MOC.md index 93a843503..00bfac9b2 100644 --- a/content/signal_processing/signal_processing_MOC.md +++ b/content/signal_processing/signal_processing_MOC.md @@ -53,6 +53,7 @@ date: 2024-03-18 ## Empirical Mode Decomposition * [EMD Basic](signal_processing/algorithm/EMD/basic.md) +* [EEMD](signal_processing/algorithm/EMD/eemd.md) # Software