尺度函数 ： scaling function （在一些文档中又称为父函数 father wavelet ）
小波函数 ： wavelet function（在一些文档中又称为母函数 mother wavelet）
连续的小波变换 ：CWT
离散的小波变换 ：DWT
小波变换的基本知识：
不同的小波基函数，是由同一个基本小波函数经缩放和平移生成的。
小波变换是将原始图像与小波基函数以及尺度函数进行内积运算,所以一个尺度函数和一个小波基函数就可以确定一个小波变换
小波变换后低频分量
基本的小波变换函数
二维离散小波变换：
（官网上的例子）
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt def test_pywt(): import numpy as np import matplotlib.pyplot as plt import pywt import pywt.data # Load image original = pywt.data.camera() # Wavelet transform of image, and plot approximation and details titles = ['Approximation', ' Horizontal detail', 'Vertical detail', 'Diagonal detail'] coeffs2 = pywt.dwt2(original, 'bior1.3') LL, (LH, HL, HH) = coeffs2 plt.imshow(original) plt.colorbar(shrink=0.8) fig = plt.figure(figsize=(12, 3)) for i, a in enumerate([LL, LH, HL, HH]): ax = fig.add_subplot(1, 4, i + 1) ax.imshow(a, interpolation="nearest", cmap=plt.cm.gray) ax.set_title(titles[i], fontsize=10) ax.set_xticks([]) ax.set_yticks([]) fig.tight_layout() plt.show() # test_pywt()
#coding=gbk
'''
Created on 2018年10月9日
这个模块是为了测试 pywt 库 的相关用法
@author: Administrator
'''
import pywt
print(pywt.families()) #打印出小波族
# ['haar', 'db', 'sym', 'coif', 'bior', 'rbio', 'dmey', 'gaus', 'mexh', 'morl', 'cgau', 'shan', 'fbsp', 'cmor'] for family in pywt.families(): #打印出每个小波族的每个小波函数 print('%s family: '%(family) + ','.join(pywt.wavelist(family))) # haar family: haar # db family: db1,db2,db3,db4,db5,db6,db7,db8,db9,db10,db11,db12,db13,db14,db15,db16,db17,db18,db19,db20,db21,db22,db23,db24,db25,db26,db27,db28,db29,db30,db31,db32,db33,db34,db35,db36,db37,db38 # sym family: sym2,sym3,sym4,sym5,sym6,sym7,sym8,sym9,sym10,sym11,sym12,sym13,sym14,sym15,sym16,sym17,sym18,sym19,sym20 # coif family: coif1,coif2,coif3,coif4,coif5,coif6,coif7,coif8,coif9,coif10,coif11,coif12,coif13,coif14,coif15,coif16,coif17 # bior family: bior1.1,bior1.3,bior1.5,bior2.2,bior2.4,bior2.6,bior2.8,bior3.1,bior3.3,bior3.5,bior3.7,bior3.9,bior4.4,bior5.5,bior6.8 # rbio family: rbio1.1,rbio1.3,rbio1.5,rbio2.2,rbio2.4,rbio2.6,rbio2.8,rbio3.1,rbio3.3,rbio3.5,rbio3.7,rbio3.9,rbio4.4,rbio5.5,rbio6.8 # dmey family: dmey # gaus family: gaus1,gaus2,gaus3,gaus4,gaus5,gaus6,gaus7,gaus8 # mexh family: mexh # morl family: morl # cgau family: cgau1,cgau2,cgau3,cgau4,cgau5,cgau6,cgau7,cgau8 # shan family: shan # fbsp family: fbsp # cmor family: cmor db3 = pywt.Wavelet('db3') #创建一个小波对象 print(db3) # Filters length: 6 #滤波器长度 # Orthogonal: True #正交 # Biorthogonal: True #双正交 # Symmetry: asymmetric #对称性，不对称 # DWT: True #离散小波变换 # CWT: False #连续小波变换 def print_array(arr): print('[%s]'%','.join(['%.14f'%x for x in arr])) #离散小波变换的小波滤波系数 # dec_lo Decomposition filter values 分解滤波值， rec 重构滤波值 #db3.filter_bank 返回4 个属性 print(db3.filter_bank == (db3.dec_lo, db3.dec_hi, db3.rec_lo, db3.rec_hi)) #True print(db3.dec_len) print(db3.rec_len) #6 # DWT 与 IDWT #使用db2 小波函数做dwt x = [3, 7, 1, 1, -2, 5, 4, 6] cA, cD = pywt.dwt(x, 'db2') #得到近似值和细节系数 print(cA) # [5.65685425 7.39923721 0.22414387 3.33677403 7.77817459] print(cD) # [-2.44948974 -1.60368225 -4.44140056 -0.41361256 1.22474487] #IDWT print(pywt.idwt(cA, cD, 'db2')) # [ 3. 7. 1. 1. -2. 5. 4. 6.] #传入小波对象，设置模式 w = pywt.Wavelet('sym3') cA, cD = pywt.dwt(x, wavelet=w, mode='constant') print(cA) # [ 4.38354585 3.80302657 7.31813271 -0.58565539 4.09727044 7.81994027] print(cD) # [-1.33068221 -2.78795192 -3.16825651 -0.67715519 -0.09722957 -0.07045258] print(pywt.Modes.modes) # ['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect', 'antisymmetric', 'antireflect'] print(pywt.idwt([1,2,0,1], None, 'db3', 'symmetric')) print(pywt.idwt([1,2,0,1], [0,0,0,0], 'db3', 'symmetric')) # [ 0.83431373 -0.23479575 0.16178801 0.87734409] # [ 0.83431373 -0.23479575 0.16178801 0.87734409] #小波包 wavelet packets X = [1, 2, 3, 4, 5, 6, 7, 8] wp = pywt.WaveletPacket(data=X, wavelet='db3', mode='symmetric', maxlevel=3) print(wp) print(wp.data) #[1 2 3 4 5 6 7 8 9] print(repr(wp.path)) print(wp.level) # 0 #分解级别为0 print(wp['ad'].maxlevel) # 3 #访问小波包的子节点 #第一层： print(wp['a'].data) # [ 4.52111203 1.54666942 2.57019338 5.3986205 8.19182134 11.27067814 # 12.65348525] # 当设置分解的 maxlevel 时，分解得到的data #[ 4.52111203 1.54666942 2.57019338 5.3986205 8.20681003 11.18125264] 设置为2 时 print(wp['a'].path) # a #第2 层 print(wp['aa'].data) # [ 3.63890166 6.00349136 2.89780988 6.80941869 15.41549196] print(wp['ad'].data) # [ 1.25531439 -0.60300027 0.36403471 0.59368086 -0.53821027] print(wp['aa'].path) # aa print(wp['ad'].path) # ad #第3 层时： print(wp['aaa'].data) # [ 6.7736584 5.78857317 5.69392399 10.98672847 19.92241106] # print(wp['aaaa'].data) #超过最大层时，会报错 #获取特定层数的所有节点 print([node.path for node in wp.get_level(3, 'natural')]) #第3层有8个 # ['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd'] #依据频带频率进行划分 print([node.path for node in wp.get_level(3, 'freq')]) # ['aaa', 'aad', 'add', 'ada', 'dda', 'ddd', 'dad', 'daa'] #从小波包中 重建数据 X = [1, 2, 3, 4, 5, 6, 7, 8] wp = pywt.WaveletPacket(data=X, wavelet='db1', mode='symmetric', maxlevel=3) print(wp['ad'].data) # [-2,-2] new_wp = pywt.WaveletPacket(data=None, wavelet='db1', mode='symmetric') new_wp['a'] = wp['a'] new_wp['aa'] = wp['aa'].data new_wp['ad'] = [-2,-2] # wp['ad'].data new_wp['d'] = wp['d'] print(new_wp.reconstruct(update=False)) # new_wp['a'] = wp['a'] 直接使用高低频也可进行重构 # new_wp['d'] = wp['d'] print(new_wp) #: None print(new_wp.reconstruct(update=True)) #更新设置为True时。 print(new_wp) # : [1. 2. 3. 4. 5. 6. 7. 8.] #获取叶子结点 print([node.path for node in new_wp.get_leaf_nodes(decompose=False)]) # ['aa', 'ad', 'd'] print([node.path for node in new_wp.get_leaf_nodes(decompose=True)]) # ['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd'] #从小波包树中移除结点 dummy = wp.get_level(2) for i in wp.get_leaf_nodes(False): print(i.path, i.data) # aa [ 5. 13.] # ad [-2. -2.] # da [-1. -1.] # dd [-1.11022302e-16 0.00000000e+00] node = wp['ad'] print(node) #ad: [-2. -2.] del wp['ad'] #删除结点 for i in wp.get_leaf_nodes(False): print(i.path, i.data) # aa [ 5. 13.] # da [-1. -1.] # dd [-1.11022302e-16 0.00000000e+00] print(wp.reconstruct()) #进行重建 # [2. 3. 2. 3. 6. 7. 6. 7.] wp['ad'].data = node.data #还原已删除的结点 print(wp.reconstruct()) # [1. 2. 3. 4. 5. 6. 7. 8.] print(wp['a']) print(wp.a) filename = r'D:\ml_datasets\PHM\c6\c_6_001.csv' data = pd.read_csv(filename) data = data.iloc[100000:110000, 3] cA1, cD1 = pywt.dwt(data, 'db3') #得到近似值和细节系数 wap = pywt.WaveletPacket(data=data, wavelet='db3') dataa = wap['a'].data print(wap['a'].data) print(len(wap['a'].data)) plt.figure(num='ca') plt.plot(dataa) plt.figure(num='data') plt.plot(dataa) plt.show() # plt.figure(num='ca') # plt.plot(cA1) # plt.figure(num='cd') # plt.plot(cD1) # plt.figure(num='data') # plt.plot(data) # plt.show()
pywt 库中 upcoef 函数的使用
pywt.upcoef(part='a'or'd', coeffs=cA, cD, wavelet, level, take)
data = np.array([8,9,10,11,1,2,3,4,5,6,7]).reshape(-1,) print('origin data:',data) (cA, cD) = pywt.dwt(data, 'haar') print('cA para:',cA) # cA para: [12.02081528 14.8492424 2.12132034 4.94974747 7.77817459 9.89949494] print('cD para:',cD) print('take length:len(cA)',len(cA)) # take length:len(cA) 6 print('take length:len(cD)',len(cD)) print(pywt.upcoef('a', cA, 'haar',take=len(cA)) + pywt.upcoef('d', cD, 'haar',take=len(cD))) #截取原始 数据中间的take 个数值 print('take length:len(data)',len(data)) print(pywt.upcoef('a', cA, 'haar',take=len(data)) + pywt.upcoef('d', cD, 'haar',take=len(data))) print(pywt.upcoef('a', cA, 'haar',take=1) + pywt.upcoef('d', cD, 'haar',take=1)) print(pywt.upcoef('a', cA, 'haar',take=2) + pywt.upcoef('d', cD, 'haar',take=2)) print(pywt.upcoef('a', cA, 'haar',take=3)) # [1.5 1.5 3.5] print(pywt.upcoef('d', cD, 'haar',take=3)) # [-0.5 0.5 -0.5] print(pywt.upcoef('a', cA, 'haar',take=3) + pywt.upcoef('d', cD, 'haar',take=3)) # [1. 2. 3.] # 从最中间的位置截取指定长度的数据；如果为1，就是最中间的2；如果为2，就是最中间的两个数，就是2与3；以此类推；
