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→0lim​lnrlncr​​

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

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