Update Note

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
-        resdiual = signal - imf
+        
+        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

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

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