Python实现相空间重构求关联维数——GP算法、自相关法求时间延迟tau、最近邻算法求嵌入维数m
Python实现相空间重构求关联维数——GP算法、自相关法求时间延迟tau、最近邻算法求嵌入维数m
GP算法：
若有一维时间序列为{x1,x2,…,xn},对其进行相空间重构得到高维相空间的一系列向量：
x i ( τ , m ) = ( x i , x i 1 , ⋯ , x i + ( m − 1 ) τ ) {x_i}(\tau ,m) = \left( {{x_i},{x_{i1}}, \cdots ,{x_{i + {{(m - 1)}_\tau }}}} \right) xi(τ,m)=(xi,xi1,⋯,xi+(m−1)τ)
式中： τ \tau τ为时间延迟， τ \tau τ=k Δ t {\rm{\Delta }}t Δt，其中k为整数，为采样时间间隔；m为嵌入维数；i=1，2，⋯，N；N为重构后向量的个数， N = n − ( m − 1 ) τ N = n - (m - 1)\tau N=n−(m−1)τ。
重构相空间关联维数为：
D 2 = lim r → 0 ln c r ln r {D_2} = \mathop {\lim }\limits_{r \to 0} \frac{{\ln {c_r}}}{{\ln r}} D2=r→0limlnrlncr
c r = 1 N 2 {c_r} = \frac{1}{{{N^2}}} cr=N21 ∑ ∑ H \sum\sum H ∑∑H ( r − ∣ ∣ x j − x k ∣ ∣ ) \left( {r - ||{x_j} - {x_k}||} \right) (r−∣∣xj−xk∣∣)
式中：j≠k；r为m维超球半径；H为Heaviside函数。
def GP(imf,tau): #GP算法求关联维数
N=2000
if (len(imf) != N):
print('请输入指定的数据长度！') # N为指定数据长度
return
elif (isinstance(imf, np.ndarray) != True):
print('数据格式错误！')
return
else:
m_max=10 #最大嵌入维数
ss=50 #r的步长
fig=plt.figure()
for m in range(1,m_max+1):
i_num = N - (m - 1) * tau
kj_m = np.zeros((i_num, m)) # m维重构相空间
for i in range(i_num):
for j in range(m):
kj_m[i][j] = imf[i + j * tau]
dist_min, dist_max = np.linalg.norm(kj_m[0] - kj_m[1]), np.linalg.norm(kj_m[0] - kj_m[1])
Dist_m = np.zeros((i_num, i_num)) # 两向量之间的距离
for i in range(i_num):
for k in range(i_num):
D= np.linalg.norm(kj_m[i] - kj_m[k])
if(D>dist_max):
dist_max=D
elif(D>0 and D<dist_min):
dist_min=D
Dist_m[i][k] = D
dr=(dist_max-dist_min)/(ss-1) #r的间距
r_m=[]
Cr_m=[]
for r_index in range(ss):
r=dist_min+r_index*dr
r_m.append(r)
Temp=np.heaviside(r-Dist_m,1)
for i in range(i_num):
Temp[i][i]=0
Cr_m.append(np.sum(Temp))
r_m=np.log(np.array((r_m)))
Cr_m=np.log(np.array((Cr_m))/(i_num*(i_num-1)))
plt.plot(r_m,Cr_m)
plt.show()
自相关法确定 τ \tau τ：
计算时间序列{x1,x2,…,xn}的自相关函数：
R ( j τ ) = 1 N ∑ R(j\tau )= \frac{1}{{{N}}}\sum R(jτ)=N1∑ x ( i ) x ( i + j τ ) x(i)x(i + j\tau ) x(i)x(i+jτ)
当自相关函数值下降到初始函数值的1- e − 1 {{\rm{e}}^{ - 1}} e−1时。所对应的 τ \tau τ即为时间延迟参数。
# 计算GP算法的时间延迟参数（自相关法）
def get_tau(imf):
N=2000
if (len(imf) != N):
print('请输入指定的数据长度！') # N为指定数据长度
return 0
elif (isinstance(imf, np.ndarray) != True):
print('数据格式错误！')
return 0
else:
j = 1 # j为固定值
tau_max = 20
Rall = np.zeros(tau_max)
for tau in range(tau_max):
R = 0
for i in range(N - j * tau):
R += imf[i] * imf[i + j * tau]
Rall[tau] = R / (N - j * tau)
for tau in range(tau_max):
if Rall[tau] < (Rall[0] * 0.6321):
break
return tau
假近邻算法确定m
对m维相空间每一个向量 X i ( m ) = { x i , x i + τ , ⋯ , x i + ( m − 1 ) τ } {X_{i(m)}} = \left\{ {{x_i},{x_{i + \tau }}, \cdots ,{x_{i + (m - 1)\tau }}} \right\} Xi(m)={xi,xi+τ,⋯,xi+(m−1)τ},i=1,2,…,N,N为向量总数，找出它的最近向量 X j ( m ) X_{j(m)} Xj(m)，计算两者欧氏距离 R m ( i ) = ∣ ∣ X i ( m ) − X j ( m ) ∣ ∣ {R_{m }}(i) = ||{X_{i(m)}} - {X_{j(m )}}|| Rm(i)=∣∣Xi(m)−Xj(m)∣∣，它们在m+1维空间的距离为：
R m + 1 ( i ) = ∣ ∣ X i ( m + 1 ) − X j ( m + 1 ) ∣ ∣ {R_{m + 1}}(i) = ||{X_{i(m + 1)}} - {X_{j(m + 1)}}|| Rm+1(i)=∣∣Xi(m+1)−Xj(m+1)∣∣
如果 R m + 1 ( i ) {R_{m + 1}}(i) Rm+1(i)>> R m ( i ) {R_{m}}(i) Rm(i),则为虚假近邻点，定义比值：
R ( i ) = R(i)= R(i)= [ R m + 1 ( i ) ] 2 − [ R m ( i ) ] 2 [ R m ( i ) ] 2 \sqrt {\frac{{{{\left[ {{R_{m + 1}}(i)} \right]}^2} - {{\left[ {{R_m}(i)} \right]}^2}}}{{{{\left[ {{R_m}(i)} \right]}^2}}}} [Rm(i)]2[Rm+1(i)]2−[Rm(i)]2
若 R ( i ) > R 0 R(i)>R_0 R(i)>R0,则称 X j X_j Xj为 X i X_i Xi的假近邻点， R 0 R_0 R0为阈值通常取大于10.计算该m下虚假近邻点占点比例，直到虚假近邻点百分比很小或不随m增大而减少时，此时的m即为所需嵌入维数。
#计算GP算法的嵌入维数(假近邻算法)
def get_m(imf, tau):
N=2000
if (len(imf) != N):
print('请输入指定的数据长度！') # N为指定数据长度
return 0, 0
elif (isinstance(imf, np.ndarray) != True):
print('数据格式错误！')
return 0, 0
else:
m_max = 10
P_m_all = [] # m_max-1个假近邻点百分率
for m in range(1, m_max + 1):
i_num = N - (m - 1) * tau
kj_m = np.zeros((i_num, m)) # m维重构相空间
for i in range(i_num):
for j in range(m):
kj_m[i][j] = imf[i + j * tau]
if (m > 1):
index = np.argsort(Dist_m)
a_m = 0 # 最近邻点数
for i in range(i_num):
temp = 0
for h in range(i_num):
temp = index[i][h]
if (Dist_m[i][temp] > 0):
break
D = np.linalg.norm(kj_m[i] - kj_m[temp])
D = np.sqrt((D * D) / (Dist_m[i][temp] * Dist_m[i][temp]) - 1)
if (D > 10):
a_m += 1
P_m_all.append(a_m / i_num)
i_num_m = i_num - tau
Dist_m = np.zeros((i_num_m, i_num_m)) # 两向量之间的距离
for i in range(i_num_m):
for k in range(i_num_m):
Dist_m[i][k] = np.linalg.norm(kj_m[i] - kj_m[k])
P_m_all = np.array(P_m_all)
m_all = np.arange(1, m_max)
return m_all, P_m_all
三连、三连、三连
完整测试代码如下：
import numpy as np
from scipy.fftpack import fft
from scipy import fftpack
import matplotlib.pyplot as plt
N_ft=2000 #时频域的点数
# 计算GP算法的时间延迟参数（自相关法）
def get_tau(imf):
if (len(imf) != N_ft):
print('请输入指定的数据长度！') # 需要更改，比如弹出对话框
return 0,0,0
elif (isinstance(imf, np.ndarray) != True):
print('数据格式错误！')
return 0,0,0
else:
j = 1 # j为固定值
tau_max = 20
Rall = np.zeros(tau_max)
for tau in range(tau_max):
R = 0
for i in range(N_ft - j * tau):
R += imf[i] * imf[i + j * tau]
Rall[tau] = R / (N_ft - j * tau)
for tau in range(tau_max):
if Rall[tau] < (Rall[0] * 0.6321):
break
tauall=np.arange(tau_max)
return tauall,Rall,tau
# 计算GP算法的嵌入维数(假近邻算法)
def get_m(imf, tau):
if (len(imf) != N_ft):
print('请输入指定的数据长度！') # 需要更改，比如弹出对话框
return 0, 0
elif (isinstance(imf, np.ndarray) != True):
print('数据格式错误！')
return 0, 0
else:
m_max = 10
P_m_all = [] # m_max-1个假近邻点百分率
for m in range(1, m_max + 1):
i_num = N_ft - (m - 1) * tau
kj_m = np.zeros((i_num, m)) # m维重构相空间
for i in range(i_num):
for j in range(m):
kj_m[i][j] = imf[i + j * tau]
if (m > 1):
index = np.argsort(Dist_m)
a_m = 0 # 最近邻点数
for i in range(i_num):
temp = 0
for h in range(i_num):
temp = index[i][h]
if (Dist_m[i][temp] > 0):
break
D = np.linalg.norm(kj_m[i] - kj_m[temp])
D = np.sqrt((D * D) / (Dist_m[i][temp] * Dist_m[i][temp]) - 1)
if (D > 10):
a_m += 1
P_m_all.append(a_m / i_num)
i_num_m = i_num - tau
Dist_m = np.zeros((i_num_m, i_num_m)) # 两向量之间的距离
for i in range(i_num_m):
for k in range(i_num_m):
Dist_m[i][k] = np.linalg.norm(kj_m[i] - kj_m[k])
P_m_all = np.array(P_m_all)
m_all = np.arange(1, m_max)
return m_all, P_m_all
# GP算法求关联维数(时频域特征)
def GP(imf, tau):
if (len(imf) != N_ft):
print('请输入指定的数据长度！') # 需要更改，比如弹出对话框
return
elif (isinstance(imf, np.ndarray) != True):
print('数据格式错误！')
return
else:
m_max = 10 # 最大嵌入维数
ss = 50 # r的步长
fig = plt.figure(1)
for m in range(1, m_max + 1):
i_num = N_ft - (m - 1) * tau
kj_m = np.zeros((i_num, m)) # m维重构相空间
for i in range(i_num):
for j in range(m):
kj_m[i][j] = imf[i + j * tau]
dist_min, dist_max = np.linalg.norm(kj_m[0] - kj_m[1]), np.linalg.norm(kj_m[0] - kj_m[1])
Dist_m = np.zeros((i_num, i_num)) # 两向量之间的距离
for i in range(i_num):
for k in range(i_num):
D = np.linalg.norm(kj_m[i] - kj_m[k])
if (D > dist_max):
dist_max = D
elif (D > 0 and D < dist_min):
dist_min = D
Dist_m[i][k] = D
dr = (dist_max - dist_min) / (ss - 1) # r的间距
r_m = []
Cr_m = []
for r_index in range(ss):
r = dist_min + r_index * dr
r_m.append(r)
Temp = np.heaviside(r - Dist_m, 1)
for i in range(i_num):
Temp[i][i] = 0
Cr_m.append(np.sum(Temp))
r_m = np.log(np.array((r_m)))
Cr_m = np.log(np.array((Cr_m)) / (i_num * (i_num - 1)))
plt.plot(r_m, Cr_m)
plt.xlabel('ln(r)')
plt.ylabel('ln(C)')
plt.show()
if __name__=='__main__':
# 检验关联维数程序
t = []
f1 = 25
f2 = 30
for i in range(N_ft):
t.append(i * 0.001)
t = np.array(t)
# yu = np.ones(M * N)
AEall = np.sin(t * 2 * np.pi * f1) + np.sin(t * 2 * np.pi * f2) #在这里直接改信号
tauall, Rall, tau = get_tau(AEall)
m, P = get_m(AEall, tau)
GP(AEall, 1)
print(tau)
fig2 = plt.figure(2)
yu = np.ones(len(tauall)) * Rall[0] * 0.6321
plt.plot(tauall, Rall)
plt.plot(tauall, yu)
plt.xlabel('tau')
plt.ylabel('R')
plt.show()
fig3 = plt.figure(3)
plt.plot(m, P)
plt.xlabel('m')
plt.ylabel('P')
plt.show()