做网站用哪些软件搜索引擎原理
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；