EMD经验模态分解

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项目上的理论研究看到一种方法EMD+SVD方法识别，看了下EMD算法意义跟傅立叶变换差不多，也是将信号分解为不同的频率，但是区别与傅立叶无线长时间与小波变换选定小波基的问题，EMD给出了自适应分解方法。

反之，如果样本时间序列的本质特征只存在于所发生的当期，并不会延续到未来，亦即样本时间序列的均值、方差、协方差非常数，则这样一个时间序列不足以昭示未来，我们便称这样的样本时间序列是非平稳的。

形象地理解，平稳性就是要求经由样本时间序列所得到的拟合曲线在未来的一段期间内仍能顺着现有的形态“惯性”地延续下去；如果数据非平稳，则说明样本拟合曲线的形态不具有“惯性”延续的特点，也就是基于未来将要获得的样本时间序列所拟合出来的曲线将迥异于当前的样本拟合曲线。

事实上，世界上几乎不存在理想的“平稳”时间序列。欧阳首承教授曾指出：“平稳序列性消除了小概率事件”。即在欧阳教授的溃变论看来，EMD这一方法也是有问题的。但是，该方法确实扩展了平稳化这一传统思想的应用范围，即扩展到了对任何类型的时间序列的处理，也是了不起的新进展。

2：EMD方法：

由原数据减去包络平均后的新数据，若还存在负的局部极大值和正的局部极小值，说明这还不是一个本征模函数，需要继续进行“筛选”。如下图示意：

# -*- coding: utf-8 -*-
"""
Created on Tue Jan 10 15:28:19 2017

@author: admin

EMD算法，经验模态分解方法
ismonotonic 判定当前的时间序列是否是单调序列
findpeaks 寻找当前时间序列的极值点
isImf 判断当前的序列是否为IMF序列
getspline 获取当前样条曲线
emd 经验模态分解方法
"""
import numpy as np 

from scipy.interpolate import interp1d



#判定当前的时间序列是否是单调序列
def ismonotonic(x):
    max_peaks=findpeaks(x)
    min_peaks=findpeaks(-1*x)
    all_num=len(max_peaks)*len(min_peaks)
    if all_num>0:
        return False
    else:
        return True
        
#寻找当前时间序列的极值点
def findpeaks(x):
    
    df_index=np.nonzero(np.diff((np.diff(x)>0)+0)<0)
    
    u_data=np.nonzero((x[df_index[0]+1]>x[df_index[0]]))
    df_index[0][u_data[0]]+=1
    
    return df_index[0]
#判断当前的序列是否为IMF序列
def isImf(x):
    N=np.size(x)
    pass_zero=np.sum(x[0:N-2]*x[1:N-1]<0)
    peaks_num=np.size(findpeaks(x))+np.size(findpeaks(-x))
    if abs(pass_zero-peaks_num)>1:
        return False
    else:
        return True
#获取当前样条曲线
def getspline(x):
    N=np.size(x)
    peaks=findpeaks(x)
    print '当前极值点个数：',len(peaks)
    f=interp1d(np.concatenate(([0,1],peaks,[N+1])),np.concatenate(([0,1],x[peaks],[0])),kind='cubic')
    return f(np.linspace(1,N,N))
    
    
#经验模态分解方法
def emd(x):
    imf=[]
    while not ismonotonic(x):
        x1=x
        sd=np.inf
        while sd>0.1 or  (not isImf(x1)):
            print isImf(x1)
            s1=getspline(x1)
            s2=-getspline(-1*x1)
            x2=x1-(s1+s2)/2
            sd=np.sum((x1-x2)**2)/np.sum(x1**2)
            x1=x2
        
        imf.append(x1)
        x=x-x1
    imf.append(x)
    return imf

if __name__ == '__main__':
    test=np.array([0.109017283,0.923722444,1.651182239,2.020828578,1.799564813,1.422650123,0.662718505,0.330908056,0.147316657,0.396244021,0.453235051,0.916266988,1.053167851,0.63681763,0.532051493,0.08994018,-0.717732329,-0.855063616,-0.680475165,0.071406017,0.913073764,1.855207585,2.682944133,2.667465224,2.479280179,1.992408687,1.348050833,0.994006146,0.822142278,0.591994742,0.886954149,1.260592311,1.230028317,1.217631396,0.644643686,-0.135081116,-0.711964767,-1.064944545,-0.777929926,-0.146634022,0.601942369,1.625442452,2.087659234,2.174841679,1.944062547,1.418676464,0.57214484,0.036795538,-0.000609289,-0.26336984,0.094634086,0.297793896,0.396835119,0.007259407,-0.858480906,-1.394269789,-1.85579278,-2.160075311,-1.824350703,-1.291997551,-0.538101879,0.267661313,0.699446876,1.217420847,0.781731188,0.036382748,-0.437728422,-1.00805078,-1.325132926,-1.159238528,-1.021807838,-0.824617407,-0.79444614,-0.886456075,-1.424007393,-1.955984975,-2.68713261,-2.773011756,-2.671641565,-1.684411326,-0.962753773,0.089646659,0.683509031,0.974626316,0.57909196,0.316738557,-0.261988737,-1.078689298,-1.06546717,-0.981400467,-0.572131872,-0.342081116,-0.090577818,-0.224030397,-0.706924146,-1.320851645,-1.916472444,-1.941698461,-1.716268291,-1.087495455,-0.005622723,0.938624738,1.683385158,2.046688054,1.845156075,1.439595372,0.658465234,0.197282301,0.108990311,0.395379929,0.71815339,0.90361103,0.988130991,0.718943517,0.474426139,-0.188698577,-0.812292702,-0.917224647,-0.816960237,0.075973199,0.93365717,2.02850965,2.385186217,2.84923888,2.551232478,2.008260359,1.432964394,0.969423566,0.511909795,0.821705124,0.830056202,1.101133254,1.200842308,1.09199242,0.374060592,-0.157511494,-0.727140266,-1.015692864,-0.798837593,-0.376662826,0.688543921,1.44398107,2.093600529,2.087501624,1.912169477,1.396776932,0.713375255,0.036906952,-0.423291188,-0.325659809,0.021862763,0.295402323,0.28425045,-0.106604795,-0.38752117,-1.301782234,-1.820414882,-2.227422449,-2.132089253,-1.364103799,-0.484017931,0.44159941,0.801395813,0.86830821,0.930058769,0.21739668,-0.59377128,-1.074263788,-1.348982832,-1.092444315,-0.891845989,-0.665540137,-0.594968238,-0.932225219,-1.26198678,-1.965360528,-2.56298185,-2.770659502,-2.531302519,-1.954089722,-0.876918807,0.106246548,0.881404745,0.934158755,0.612265788,0.211081193,-0.507947232,-0.932325305,-1.113714709,-0.99951518,-0.549967323,-0.368592797,-0.051758275,-0.204206489,-0.627646631,-1.311060114,-1.725636347,-1.767148474,-1.658875671,-0.977381775,0.177248256,0.708712405,1.618516084,2.190492579,1.740386247,1.252984145,0.608749079,0.363034806,-0.092511633,0.177119024,0.644482762,0.916047958,1.110210837,0.883652232,0.475328758,-0.319724597,-0.672705358,-1.040423538,-0.555686523,0.001917248,1.016588957,1.811638672,2.55399336,2.857595409,2.338898565,1.813261182,1.178062425,0.934934391,0.659534713,0.6195693,1.100961118,1.22232423,1.109416176,0.91643165,0.342533287,-0.095494363,-0.828440558,-1.148293964,-0.912005693,-0.343021309,0.673082205,1.48036216,2.05082745,2.227430421,1.911263804,1.214131671,0.666121988,-0.036771033,-0.345128535,-0.323386902,0.086955281,0.31408288,0.062055494,-0.060151388,-0.586766977,-1.370055457,-1.83086834,-2.160409852,-2.03587881,-1.488370129,-0.520083487,0.344308001,0.901225381,1.007126713,0.860427071,0.100294868,-0.466612601,-0.878705299,-1.287068192,-1.292928333,-0.756712711,-0.714507057,-0.574854071,-0.930436328,-1.332537049,-1.953298191,-2.536412336,-2.719579176,-2.400670348,-2.005796991,-0.930500161,-0.232093818,0.63685561,1.076818372,0.788627522,0.19029056,-0.441066144,-0.818181571,-0.94212841,-0.914718186,-0.426309126,-0.338644033,-0.318258323,-0.35855891,-0.751326219,-1.346293002,-1.872268772,-1.813731021,-1.825806728,-1.008774526,-0.096930157,0.912181974,1.59755862,1.995292294,1.683425842,1.226951085,0.741508777,0.392138023,-0.066970314,0.437033816,0.499947822,0.98096552,1.152774873,1.003317487,0.355506082,-0.188551511,-0.71075053,-0.944332939,-0.60684374,0.06883858,0.915825004,1.878621401,2.788487197,2.751431604,2.607898791,1.953746064,1.308510202,0.815496473,0.675471764,0.609660989,0.765761838,1.213866346,1.224456795,1.046346949,0.48951725,-0.1180711,-0.678253216,-1.163668713,-1.006752546,-0.209134924,0.618424395,1.338712995,2.144335655,2.189153994,1.919323153,1.336410932,0.687377732,0.197791543,-0.264135607,-0.389046089,0.10134643,0.123204129,0.362306059,-0.104290662,-0.462812978,-1.389614804,-2.018226227,-2.090158845,-2.156900602,-1.364570545,-0.54400289,0.245051502,0.820141412,1.236885947,0.730875022,0.171962686,-0.574929903,-1.099692438,-1.326460192,-1.037592207,-0.990098498,-0.75882144,-0.663838057,-0.778637544,-1.492871117,-1.93728779,-2.669997981,-2.765273027,-2.703818636,-1.811793161,-0.881770045,-0.101729215,0.665772292,0.900051065,0.810425434,0.203821811,-0.303296165,-0.882685263,-1.112648061,-0.919990368,-0.607717734,-0.16889155,-0.124096817,-0.367863657,-0.7468802,-1.190157527,-1.829022384,-2.013316502,-1.565775935,-0.949470017,-0.036270582,0.885709983,1.516190761,1.9627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    Ts = 0.001;
    #t=np.linspace(0,1,1000)
    #test =np.sin(2*np.pi*10*t) + np.sin(2*np.pi*50*t) + np.sin(2*np.pi*100*t) + 0.1*np.random.random(len(t))
    test_imf=emd(test)
    print test_imf