Open access peer-reviewed chapter

Symplectic Geometry and Its Applications on Time Series Analysis

Written By

Min Lei

Submitted: November 18th, 2019 Reviewed: November 9th, 2020 Published: December 8th, 2020

DOI: 10.5772/intechopen.94958

Chapter metrics overview

439 Chapter Downloads

View Full Metrics

Abstract

This chapter serves to introduce the symplectic geometry theory in time series analysis and its applications in various fields. The basic concepts and basic elements of mathematics relevant to the symplectic geometry are introduced in the second section. It includes the symplectic space, symplectic transformation, Hamiltonian matrix, symplectic principal component analysis (SPCA), symplectic geometry spectrum analysis (SGSA), symplectic geometry mode decomposition (SGMD), and symplectic entropy (SymEn), etc. In addition, it also briefly reviews the applications of symplectic geometry on time series analysis, such as the embedding dimension estimation, nonlinear testing, noise reduction, as well as fault diagnosis. Readers who are familiar with the mathematical preliminaries may omit the second section, i.e. the theory part, and go directly to the third section, i.e. the application part.

Keywords

  • symplectic geometry
  • symplectic principal component analysis (SPCA)
  • symplectic geometry spectrum analysis (SGSA)
  • symplectic geometry mode decomposition (SGMD)
  • symplectic entropy (SymEn)
  • chaotic time series
  • embedding dimension
  • feature extraction

1. Introduction

From the viewpoint of mathematical systems, the time series observed in physics are usually regarded as coming from the Lagrangian systems, also called the conventional systems. The systems can be analyzed by the conventional Euclidean geometry [1]. However, the systems in practice are usually nonlinear and complex. Thus, a lot of interesting time series in nature are complex due to nonlinear phenomena derived from nonlinear dynamical systems [2]. The nonlinear dynamical systems have been described by Hamiltonian systems and dealt with by using symplectic geometry [3]. Symplectic geometry is an even dimensional geometry living on even dimensional spaces. Different from the conventional Euclidean geometry that measures 1-dimensional lengths and angles, the symplectic geometry studies the metric properties (such as area) and can preserve the system structure in the phase space [4]. Apart from applications on the classical dynamical systems to solve the equation problems, symplectic geometry has been also used on the studies of nonlinear time series [5, 6, 7, 8].

According to Takens’ embedding theorem, a time series can be reconstructed into an attractor in phase space [9]. The reconstructed attractor is a geometrical object that can reflect the underlying dynamical system. In order to better understand the nature of the underlying system, the attractor and its properties are characterized in the phase space by various mathematical methods, such as dimension, fractal geometry, Lyapunov exponent, entropy and symplectic geometry [1, 5, 10, 11]. For dimension, fractal geometry, Lyapunov exponent, entropy, there are a more extensive discussion with mathematical details in some research literatures [12, 13, 14, 15]. Here, we only introduce how to apply symplectic geometry theory to extract the information from the reconstructed attractor and its application on physics, engineering and biomedical engineering.

Advertisement

2. Mathematical fundamental

2.1 Reconstruction of the system dynamics in phase space from a time series

The reconstruction from a time series of observation is the first and most crucial step in nonlinear time series analysis. It is also the basis of applications of symplectic geometry on time series analysis. Takens’ embedding theorem allows us to reconstruct an equivalent attractor of the underlying dynamical system by embedding one time series. The theorem proves that the reconstructed attractor has the same dynamical characteristics as the attractor of the original system if the embedding dimension mis sufficiently large. Let a time series of observation x1, x2, …, xn. nis the number of samples. The reconstructed attractor can be given in N-dimensional space RNby the time-delay embedding [5]:

X=X1X2Xm=x1x2xmx2x3xm+1xNxN+1xn,E1

where the number of dots in the attractor is m = n-N + 1, the embedding dimension is N. Xis also called as the trajectory matrix of the original system in phase space. The corresponding program is given by matlab software as follows:

----------------------------------

function matrixSignal = signalMatrix(x, N)

% ------Construct data matrix------

%

% Synopsis:

% matrixSignal = signalMatrix(x, N)

%

% Description:

% It constructs a data matrix from a time series as a column vector, i.e., a

% reconstruction attractor.

%

% Input:

% x a time series with the length n.

% N [1x1] Output dimension; N > 1 (default N = dim);

%

% Ouputs:

% matrixSignal [N x M] a data matrix (M = n-N + 1).

%

if nargin <2, N = 2; end

n = length(x);

M = n-N + 1;

matrixSignal = zeros(N,M);

for i = 1:N

matrixSignal(i,:) = x(i:M + i-1);

end

-------------------------------------

2.2 Hamilton matrix from the reconstructed attractor

In the symplectic spaces, Hamiltonian system is the analysis fundamental for the real physical processes [4, 5]. A real system should be first described by a suitable Hamiltonian system, i.e. an even dimensional matrix. For a time series, its Hamiltonian matrix Hcan be defined by using its reconstructed attractor X.

Definition 2.1Let Xbe a d-dimensional matrix in a real number field Rd. The matrix X¯can be given by removing the mean values of the columns of the X. We define the covariance matrix Aof the matrix X:

A=X¯X¯T.E2

Here, Ais a d × dreal number matrix.

Definition 2.2For a d × dmatrix A, the Hamiltonian matrix Hcan be defined:

H=A00AT.E3

Here, His a 2d × 2dmatrix.

2.3 Mathematical preliminaries in symplectic geometry

Symplectic geometry focuses on the study of area measure in symplectic space R2n. Its basic concepts and basic properties are related but different from those of a Euclidean geometry (see Table 1).

Geometry spaceSymplectic spaceEuclidean space
Space dimension2n-dimensionn-dimension
Unit matrixunit symplect matrix:
J2n=0+InIn0.
unit matrix:
In=100010001n.
Determinant of unit matrix|J2n| = 1|In| = 1
Product calculationsymplectic inner product <x, y>
xy=xJ2ny=xTJ2ny.
Inner product (x, y)
xy=xIny=xTIny=xTy.
Calculation measurearealength
Orthogonalityxy=xTJ2ny=0.xy=xTIny=xTy=0.
Space basisAdjoint symplectic orthonormal basis Q = {x1,x2, …, xm, y1, y2, …, ym}, m ≤ n; when determinant |Q| = 1, the basis Qis normal.Orthogonal basis W = {x1,x2, …, xm}, m ≤ n; when |W| = 1, the basis Wis normal.
Orthogonal matrixSymplectic matrix S
STJS=J.
Orthogonal matrix W
WTIW=WTW=I.
Analysis matrixHamiltonian matrix H
HT = JHJ.
Symmetry matrix A
AT = A = IAI.
Matrix transformationHamiltonian transformation
xHy=yHx.
Symmetry transformation
xAy=yAx.
Eigenvalues of the matrixThe eigenvalues of Hare ±μ.The eigenvalues μof Aare real.
Eigenvectors of the matrixThe eigenvectors of Hare symplectic orthogonal.The eigenvectors of Aare orthogonal.

Table 1.

The comparison between symplectic geometry and Euclidean geometry.

In Euclidean space, the inner product is denoted as the measure of the length. The unit matrix is I, i.e. the main diagonal elements are 1, and the other elements are 0. Corresponding to the unit matrix Iin Euclidean space, the unit matrix in symplectic space is defined as the unit symplectic matrix J, an even dimensional matrix:

J=J2n=0+InIn0,E4

The properties of the matrix Jhave:

J=1,E5
J2=I,E6
JT=J1=J,E7
JJ1=J1J=I.E8

Definition 2.3For any two n-dimensional vectors x2n × 1 and y2n × 1, the normal symplectic inner product is defined by using the inner product of Euclidean space:

xy=xJ2ny=i=1nxiyn+ixn+iyi=xTJ2ny.E9

The normal symplectic inner product is also denoted briefly as the symplectic inner product in a real vector space R2n. When n = 1, there is:

J2=0110,E10
xy=xJ2y=xTJ2y=x1y1x2y2=x1y2x2y1.E11

The symplectic inner product is a bilinear antisymmetric nonsingular cross product. In symplectic space, the length of any vectors is equal to 0. But there exists the concept of symplectic orthogonal cross-course.

Definition 2.4Let xand ybe a 2n-dimensional real vector. If their symplectic inner product is equal to zero, i.e.:

xy=xTJy=0,E12

then xand yare symplectic orthogonal. Otherwise, they are called as symplectic adjoint.

Definition 2.5If a vector set {x1,x2, …, xm, y1, y2, …, ym} in the real symplectic space R2n(m ≤ n)is an adjoint symplectic orthonormal vector set, then the vectors xiand yi(i = 1, …,m, xiR2n, yiR2n) satisfy

xiyj=xiTJ2nyj=aii0,i=j0,ij,E13
xi xj=0,E14
yi yj=0,E15

where i, j = 1, 2, …, m. It is called as an adjoint symplectic orthonormal basis in the 2n-dimensional symplectic space. If aii = 1, the vector set {x1,x2, …, xm, y1, y2, …, ym} is a normal adjoint symplectic orthonormal vector set (a normal adjoint symplectic orthonormal basis in the space R2n).

The orthogonal of the Euclidean space is different from the symplectic orthogonal. If vectors xand yin the space Rnare orthonormal, then they satisfy:

x y=xTy=0,E16

where x ≠ y.

If a vector set {x1,x2, …, xm}∈Rnis an orthonormal vector set, then any two vectors in the set satisfy:

xi xj=0,E17

where i, j = 1, 2, …, m, i ≠ j. Eq. (17) is similar to Eqs. (14) and (15). In the n-dimensional Euclidean space, the set {xi} is denoted as an orthonormal basis. If ||xi|| = 1, the orthonormal basis is a normal orthonormal basis.

Theorem 2.1Let {αi} be a normal adjoint symplectic orthonormal basis in a 2n-dimensional symplectic space Φ. Let the coordinates of any vectors βand γin Φbe {x1, x2, … xn, xn + 1, …, x2n}T and {y1, y2, … yn, yn + 1, …, y2n}T, respectively. Referring to the basis {αi}, the coordinates can be described as:

xi=βαn+i,xn+i=βαi,yi=γαn+i,yn+i=γαi,E18

where i = 1, 2, …, n. Then the symplectic inner product of βand γis as follows:

βγ=i=1nxiyn+ixn+iyi=xTJ2ny.E19

Thus, the symplectic inner product operation is transformed to the matrix operation of ordinary vectors or matrices by applying a normal adjoint symplectic orthonormal basis.

Definition 2.6Let Sis a 2n × 2nmatrix, if Ssatisfies:

JSJ1=ST,orSTJS=J,E20

then Sis a symplectic matrix and the determinant |S| = 1 or − 1. Meanwhile, the inverse matrix and the transpose matrix of a symplectic matrix are a symplectic matrix, respectively. The symplectic matrix Sis similar to an orthogonal matrix Win Euclidean space, like Eq. (20):

WTIW=WTW=I.E21

Theorem 2.2The product of sympletcic matrixes is also a symplectic matrix.

Proof:

Let Si(i = 1, 2, …, n) be a symplectic matrix. The product matrix M:

M=i=1nSi.E22

According to the above definition of symplectic matrix, there are:

JSiJ1=SiT,i=1,2,,nE23
J1J=I,E24
JMJ1=Ji=1nSiJ1=JS1S2SnJ1=JS1J1JS2J1JJ1JSnJ1=JS1J1JS2J1JJ1JSnJ1=S1TS2TSnT=S1S2SnT=MT,E25

Thus, the product of symplectic matrixes is also a symplectic matrix.

Definition 2.7If a 2n × 2nmatrix His a Hamiltonian matrix, then the matrix Hsatisfies the following properties:

JHJ1=HT,JHJ=HT,orJHT=JH,E26
xHy=yHx,E27

where xand yare 2n-dimensional vectors. In other words, if an even-dimensional matrix Hsatisfies these properties above, the matrix His a Hamiltonian matrix. In Euclidean space, a symmetric matrix Ais similar to a Hamilitonian matrix H, like Eqs. (26) and (27):

IAI=A=AT,E28
xAy=yAx.E29

Theorem 2.3Let a matrix Abe a n × nreal number matrix, if it can be built into a 2n × 2nmatrix Hin symplectic space in the following pattern:

A00AT.E30

Then the matrix His a Hamilton matrix.

Proof:

Let H=A00AT, then

JHJ1=JA00ATJ1=0InIn0A00AT0InIn01=AT00A=A00ATT=HT,E31

where Jis the 2n × 2nunit symplectic matrix. In terms of Definition 2.7, the matrix His a 2n × 2nHamiltonian matrix.

Theorem 2.4Let a 2n × 2nmatrix Hbe a Hamiltonian matrix. Then its properties keep unchanged at symplectic similar transform. That is, a Hamiltonian matrix Hthrough a series of symplectic similar transforms is still a Hamiltonian matrix.

Proof:

According to Definition 2.6, let the matrix Sbe a symplectic transform matrix. Then, the inverse matrix S−1 is also a symplectic matrix. For a Hamiltonian matrix H, let SHS−1 be the matrix Munder the symplectic similar transformation of the matrices Sand S−1. Thus,

JMJ1=JSHS1J1=JSJ1JHJ1JS1J1=STHTST=SHS1T=MT.E32

Therefore, Mis also a Hamiltonian matrix. Moreover, the matrix Mis similar to the matrix H. Therefore, the Hamiltonian matrix Hcan keep unchanged at symplectic similar transform in symplectic space.

The eigenvalues of a Hamiltonian matrix have the specific characteristics of the Hamiltonian matrix. However, the eigenvalues may be complex or repeated eigenvalues. In order to obtain the real eigenvalues of a Hamiltonian matrix H, symplectic QRdecomposition method is applied to deal with the Hamiltonian H:

  1. 1. Let a 2n × 2nmatrix Hbe (ATG; FA), then

    N=H2=ATGFA2,E33

  1. 2. Build a 2n × 2nsymplectic matrix Qand satisfy:

QTNQ=BR0BT,E34
B=b11b12b1nb21b22b2n0b3200bnn1bnn.E35

Here Bis an upper Hessenberg matrix. Besides, the matrix Qcan be a 2n × 2nHouseholder matrix.

  1. 3. Use the symplectic QRdecomposition method to obtain eigenvalues:

    μB=μ1μ2μn.E36

  1. 4. The eigenvalues of the Hamiltonian matrix Hwith multiplicity nare λi=μi, i = 1, 2, …, n; λn+i=λiis also an eigenvalue with multiplicity n.

In symplectic space, the symplectic QRdecomposition method allows the primary 2n-dimensional space transform into ndimensional space to resolve the eigenvalues of the Hamiltonian H, where the matrix Qis a symplectic unitary matrix. Thus, the consuming time of the calculation is only one fourth the number of floating-point operations. In general, one makes use of a Householder matrix instead of the matrix Q.

Theorem 2.5If a 2n × 2nmatrix Qis a Householder matrix, then the matrix Qis a symplectic unitary matrix.

Proof:

Let a Householder matrix Q

Q=Qkω=P00P,E37
P=In2ωωωω,E38
ω=00ωkωnT0,E39

where, ‘*’ means the conjugate transposition. Then, there is

P=P,E40
PP=P2=In2ωωωωIn2ωωωω=In,E41
QJQ=P00P0InIn0P00P=0PPPP0=0InIn0=J,E42
QQ=P00PP00P=PP00PP=In00In=I2n,E43

Therefore, the Householder matrix Qis a symplectic unitary matrix.

2.4 Mathematical fundamental on applications

2.4.1 Symplectic geometry spectrums of the reconstructed attractor from a time series

In symplectic space, the reconstructed attractor can keep its properties unchanged [5, 6]. Its symplectic geometry spectrums can be given by the symplectic geometry theory above. On the basis of Section 2.1 and 2.2, one can build a Hamiltonian matrix Mfrom a time series of the observation. Due to the structure characteristics of the matrix M, its eigenvalues can be calculated by the 2n-dimensional symplectic space reducing into n-dimensional space. In terms of Theorem 1.5, a 2n × 2nsymplectic Householder matrix Qcan be constructed. The matrix Pin the matrix Qcan be calculated by the matrix Ain the matrix M. The specific steps are as follows:

  1. Let Abe

E44

If the vector α2110, set S(1) be the first column vector of A:

S1=a111a211an11=a11a21an1,E45

then, there is:

α1=S12,E46
ρ1=S1α1E12,E47
ω1=S1α1E1ρ1,E48

where E(1) = (1, 0, …, 0)Tis a n × 1 unit column vector.

Then, the elementary reflective matrix P(1) can be calculated:

P1=I2ω1ω1T.E49

So, there is

A2=P1A=σ1a122a1n20a222a2n20an22ann2.E50

Continue to deal with A(2) by repeating the above steps, let S(2) be

S2=0a222an22.E51

Then,

α2=S22,E52
ρ2=S2α2E22,E53
ω2=S2α2E2ρ2,E54

where E(2) = (0, 1, 0, …, 0)Tis a n × 1 unit column vector.

Then, the elementary reflective matrix P(2) can be calculated:

P2=I2ω2ω2T.E55

Thus, we can get A(3) with all zeros elements except the first and second non-zero elements:

A3=P2A2=σ1a123a1330σ2a23300a33300an33a1n3a2n3ann3.E56

Repeat the same steps until A(n) becomes an upper triangle matrix, one can construct a Householder matrix Pas follows:

P=PnPn1P1.E57

Thus, a symplectic Householder matrix Qcan be built to make the Hamiltonian matrix Mtransform as an upper Hessenberg matrix, namely:

QMQT=P00PA00ATP00PT=PAPT00PATPT=B00BT.E58
μA=μB,E59

where μmeans the eigenvalue. The matlab program is as follow:

-------------------------------------

function [P, R] = householder (A)

% ------Solve Householder Transform Matrix------

%

% Synopsis:

% [P, R] = householder (A)

%

% Description:

% It solves a Householder matrix from a data matrix, i.e., a

% reconstruction attractor.

%

% Input:

% A [mRow x mCol] a data matrix.

%

% Ouputs:

% P [mRow x mRow] a Householder matrix

% R [mRow x mCol] an upper triangle matrix

[mRow, mCol] = size(A);

if mRow>mCol

A = A’;

[mRow, mCol] = size(A);

end

I_matrix = eye(mRow);

m = min([mRow, mCol]);

p = I_matrix;

for i = 1:m.

S = A(:,i);

if i > 1.

S(1:i-1) = 0;

end

alpha = sqrt(S’*S);

delta1 = S-alpha*I_matrix(:,i);

delta = sqrt(delta1'*delta1);

if delta==0

delta = eps;

end

omega = delta1/delta;

p = I_matrix-2*omega*omega';

A = p*A;

P = p*P;

end

R = A;

return

-------------------------------------

For the attractor matrix Xof a time series, its symplectic geometry spectrums SGSare calculated by the eigenvalues of the Ain descending order, that is:

SGSi=logσitrσi,E60
σ=μ2X=μA,σ1=μmax2,,σn=μmin2,E61

where i = 1, …, n. nis the dimension of the attractor X.

2.4.2 Embedding dimension estimate of the reconstructed attractor from a time series

To estimate the embedding dimension is usually the first step of nonlinear analysis [5]. For a time series, it is important to resolve a suitable embedding dimension of the observed system. Due to the measure-preserving charactistic of symplectic geometry, symplectic geometry spectrums can be used to estimate the embedding dimension of the system from a time series. With the increase of the dimension nin Eq. (61), the change of the symplectic geometry spectrums SGSin Eq. (60) tends to be flat at i = d(i∈(1,n))and enters the noise floor area, SGS1 > SGS2 > … > SGSd> > SGSd + 1 ≥ … ≥ SGSn,. That is, the eigenvalues exist σ1 > σ2 > … > σd> > σd + 1 ≥ … ≥ σn, then dis defined as the embedding dimension of the time series for the reconstruction system.

2.4.3 Symplectic entropy (SymEn) of a time series

Symplectic entropy(SymEn) is a kind of entropy measure for a dynamic system in symplectic space [16]. Based on the symplectic geometry spectrums, the SymEn measures the energy distribution in symplectic space of a dynamic system from a time series. The distribution of the energy of the system is described by the eigenvalues σin the relevant symplectic orthonormal bases of the symplectic space. In each base direction, the probability of the energy distribution can be given as follows:

pi=σii=1nσi,E62

where idenotes the ith base direction in the symplectic space, i=1npi=1, 0pi1.

Then,

SymEn=i=1npilogpi.E63

The matlab program is as follows:

----------------------------------

function SymEn = SymplecticEntropy(A)

[Q, R] = householder(A);

delta = diag(R);

sum_delta = sum(delta);

p = delta./sum_delta;

SymEn = −sum(p.*log(p));

Return

-------------------------------------

The SymEn value represents the uncertainty of the entropy about the underlying probability distribution of a dynamic system in symplectic space, called Symplectic Entropy.

2.4.4 Symplectic principal component analysis (SPCA) of a time series

Symplectic principal component analysis (SPCA) is a kind of principal component analysis (PCA) to map the dynamic system from a time series into the symplectic space [17]. Due to the preserving-measure nature of symplectic geometry, symplectic principal components elucidate the dominant features of a time series for an underlying system. The principal components corresponding to larger eigenvalues capture the key relationship between the variables in symplectic space. The components corresponding to smaller eigenvalues are regarded to relate primarily to the less important components or noise in the time series. The analysis of eigenvalues are also called as the symplectic geometry spectrums analysis (SGSA) [6, 18, 19]. The corresponding components are also regarded as symplectic geometry mode decomposition (SGMD) [7, 8, 20, 21]. According to the symplectic geometry spectrums above, if the number of the chosen symplectic principal components is k, the corresponding principal eigenvector matrix pcan be constructed by using the first keigenvectors of the matrix Pin the matrix Q. The corresponding principal eigenvalues are the first keigenvalues in the symplectic geometry spectrum. If k = n, p = P. Otherwise, pP. Then the reestimated attractor matrix X̂=ppTX, where pTXis defined the transformation coefficient matrix S. If piis the ith eigenvector in Pcorresponding to the ith eigenvalue σiin the symplectic geometry spectrum, Siwill be the ith principal component coefficients, or called the projection of the pith direction in the symplectic space:

Si=piTX=XTpi.E64

The corresponding pith principal component matrix X̂iis given as follows:

X̂i=piSi.E65

Then, the reestimated attractor matrix is equal to the sum of X̂i, i = 1,…, n.

X̂=i=1nX̂i.E66

The reestimated time series xris equal to the sum of each principal component, i.e. the sum of projections in different directions. If i = 1, the reestimated time series is a reduced noise data based on the first principal component.

Advertisement

3. Applications

Symplectic geometry theory has been applied to deal with a time series in fields of physics, engineering, biomedical engineering [6, 7, 8, 11, 16, 17, 18, 19, 20, 21, 22, 23, 24], since Lei et al.(2002) first proposed a symplectic geometry method to estimate the appropriate embedding dimension from a time series [5]. Here, we will introduce four research cases in terms of the above theorem and properties of symplectic geometry for the time series analysis.

Case 1:Embedding dimension estimation for Lorenz chaotic time series [5].

Lorenz chaotic system was accidentally discovered by Edward Norton Lorenz [25], an American meteorologist, in 1963 when he was studying weather forecast, and was known as the first chaotic attractor. Since then, people began to study chaos, a random-like phenomenon. Lorenz chaotic time series xcomes from Lorenz chaotic system, which is a three-dimensional dynamical system as follows [5]:

ẋ=σyxẏ=γxyxzż=bz+xy,E67

where σ = 10, b = 8/3, γ = 28. The state variable xis chosen as the analyzed data. The sampling interval is 0.005. The length nis 1000 points.

The attractor reconstructed from Lorenz chaotic time series xcan reflect the Lorenz system. Here, the dimension of the reconstructed attractor is estimated by the above symplectic geometry method. Let the embedding dimension dbe 3: 5: 23, where i = 1: d. The matlab program is as follows:

-------------------------------------

% Compute a Lorenz chaotic time series

% Example:

% state = [5 5 5];

% Ts = 0.005;

% N = 10000;

% y = calculate_lorenz(state, Ts, N);

% x = y(:,1);

function y = calculate_lorenz(state, Ts, N).

if nargin <1

state = [5 5 5];

Ts = 0.005;

N = 10000;

end

if nargin == 1

Ts = 0.005;

N = 10000;

end

if nargin == 2

N = 10000;

end

% set time span with specific times for the solution

T = 0:Ts:N*Ts;

% set a scalar relative error tolerance 'RelTol' (1e-3 by default).

% and a vector of absolute error tolerances 'AbsTol' (all components 1e-6% by default).

options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-5]);

% solve Lorenz chaotic system

[t,y] = ode45('lorenzeq1',T,state,options);

return

function ydot = lorenzeq(t,y)

% Lorenz equation

b = 8/3;

r = 28;

delta = 10;

A = [−delta delta 0;r − 1 -y(1);y(2) 0 -b];

ydot = A*y;

return

--------------------------------------

% Calculate the embedding dimension.

state = [5 5 5];

Ts = 0.005;

N = 10000;

y = calculate_lorenz(state, Ts, N);

x = y(:,1);

figure.

for N = 3:5:23

X = signalMatrix(x,N);

A = X*X’;

[Q, R] = householder(A);

delta = diag(R);

sum_delta = sum(delta);

d = log10(delta./sum_delta);

n = length(d);

plot(1:n, d, 'b*-')

hold on

end

ylabel('log10(\delta_{\iti}/tr(\delta_{\iti}))')

xlabel('{\itd} = 3:5:23')

axis([0 25–15 0])

-----------------------------------

Figure 1a shows the symplectic geometry spectrums SGSof xwithout noise according to the above equations based on symplectic geometry theory. We can see that the symplectic geometry spectrums turn abruptly into a flat area from i = 6, i.e. σ1 > σ2 > … > σ5> > σ5 + 1 ≥ … ≥ σd. So, the embedding dimension of the time series xcan be estimated at 6. But from the Figure 1b, we can see that it is difficult for the SVD method to determinate the embedding dimension from the time series x. The results indicate that the symplectic geometry method could better determinate the embedding dimension from a time series due to its preserving-measure properties.

Figure 1.

The embedding dimension estimation of Lorenz chaos series with no noise based on: (a) the symplectic geometry method; (b) the SVD method.

Case 2:Embedding dimension estimation for the surface EMG signal [5].

In the practical engineering research, a lot of time series data due to their complexity are considered to be nonlinear, such as the surface EMG signal in biomedical engineering. As a kind of non-invasive measure for the contracting skeletal muscles, the surface EMG signal reflects some information about the muscle, limb movements and loading of the bones and joints. It has been widely applied to assess biomechanical and motor control deficits and other functional disorders, as well as to diagnose neuromuscular problems. However, due to noise interference, the study of surface EMG signal is still a great challenge in biomedical engineering. Many researches indicate that the surface EMG signal is complex and nonlinear. The embedding dimension estimation of the surface EMG signal is usually critical to analyze its nonlinear features. As an example, we use the above symplectic geometry method to estimate the embedding dimension of the surface EMG signal during forearm supination. The length of the surface EMG signal is 1000 points. The data sampling frequency is 1 kHz. Figure 2a shows the raw surface EMG signal. Figure 2b gives the symplectic geometry spectrums SGSof the data in Figure 2a. From Figure 2b, the symplectic geometry spectrums SGSchange slowly at d = 6 and turn into noise floor with the increase of the index i. Then, the embedding dimension can be estimated at 6 for the surface EMG signal during forearm supination.

Figure 2.

The embedding dimension analysis of the surface EMG signal based on the symplectic geometry spectrums: (a) Typical surface EMG signal during forearm supination; (b) The symplectic geometry spectrums of the surface EMG data in (a), where abscissa is the analysis dimensiond = 3, 8, 13, 18, 23, ordinate isSGSi=logσi/trσi, where the indexi = 1:d.

Case 3:SymEn analysis of vibration signals on rolling bearings [11].

In the rotating machinery systems, it is extremely important for rolling bearings to detect faults from vibration signals. The Case Western Reserve University (CWRU) Bearing Data Center provides a website database for the vibration signals of bearings (http://csegroups.case.edu/bearingdatacenter /pages/welcome-case-western-reserve-university-bearing-data-center-website). From the website, the acceleration vibration data sets for 6205-2RS JEM of SKF deep-groove ball bearings are obtained to detect their fault categories. The corresponding sampling frequency is 12 kHz, the shaft speed 1730 r/min. The analyzed data sets include No.100 for normal condition(NC), No.212 for inner race fault (IRF), No.225 for rolling element fault (REF), and No.261 for outer race fault (ORF) at 12 o’clock position. The data of each set consist of the vibration signals at the housing of the drive end (DE) bearing and that of the fan end (FE) bearing, which the faults are at the drive end. The corresponding fault depth and diameter are 0.21 inches and 0.53 mm, respectively.

Symplectic geometry preserves the nature of a dynamic system under symplectic similar transformations. As an entropy measure in symplectic geometry, the SymEn value of a time series measures the lack of information in a dynamic system to reflect its properties. For the complexity of a rolling bearing, the SymEn estimate is applied to test its nonlinear characteristics from the vibration signals. Figure 2 shows the SymEn values of the vibration signals at the drive end and their surrogate data sets based on the null hypothesis of a Gaussian linear stochastic process. Here, the length of each data is 6000 points. The embedding dimension d= 7.

Meanwhile, the 39 sets of surrogate data are generated by the iterated amplitude adjusted Fourier transform (IAAFT) algorithm in the 95% confidence level [26]. From Figure 3, we can see that there are the significant differences between these SymEn values of the vibration signals of a rolling bearing and their surrogate data sets. The results indicate that the vibration data could contain nonlinear characteristics. The original vibration signals are not from a Gaussian linear stochastic process in the 95% confidence level but from a nonlinear dynamical system. It conforms that the rolling bearing system is a complex nonlinear dynamical system.

Figure 3.

The nonlinear analysis of vibration signals based on the SymEn method: (a) for the normal condition (NC); (b) for the outer race fault (ORF); (c) for the rolling element fault (REF); (d) for the inner race fault (IRF). The abscissa is the SymEn values of vibration signals and their surrogate data.

Due to the complexity of rolling bearings, it is often thought that the high dimensional features can better identify the faults of rolling bearings [27, 28, 29]. However, the SymEn method can availably extract the low-dimensional features to identify the faults of rolling bearings from vibration signals quite precisely. Figure 4 shows the four working states of rolling bearings, i.e., NC, ORF, REF, and IRF, based on 2-dimensional features. The abscissa is the SymEn estimates of vibration signals at the drive end. The ordinate is those estimates of vibration signals at the fan end. We can see that the four states are obviously different. There are 100% accuracies by RBF classifier for the four states of the rolling bearings. Figure 5 plots the histogram of error values between output classes and target classes for the SymEn estimates as features of vibration signals.

Figure 4.

The states analysis of rolling for bearings with the SymEn estimates.

Figure 5.

The analysis of error values identification accuracies of four states.

Case 4:Noise reduction analysis of vibration signals based on SPCA [17, 30].

In the practical engineering measurement, the vibration data of rolling bearings have often become contaminated with noise. The noise reduction is also beneficial to analyze the measured data. The SPCA method preserves the intrinsic nonlinear nature of the raw data. The symplectic principal components can better retrieve dominant patterns from the noisy data. For the vibration signals of rolling bearings, the first symplectic principal component is used two times continuously to reduce the noise in the data.

The specific analysis procedures are as follows:

  1. Build a Hamiltonian matrix from the measured data in terms of Eq. (1), Definition 2.1, 2.2 and Theorem 2.3;

  2. Use the Eq. (44)(59) to compute a symplectic Householder transform matrix Qfor the symplectic QRdecomposition in the SPCA method;

  3. Construct the first symplectic principal component eigenvector matrix p1;

  4. Calculate the first symplectic principal component coefficients S1, i.e.:

    S1=p1TX=XTp1;

  5. Get the first denoised data x1 from the reestimated matrix in the following:

    X̂1=p1S1;

  6. Let the first denoised data x1 into the first step, and repeat the above steps, then obtain the second denoised data x3.

Figure 6 shows the effect of denoising for the vibration signals of rolling element fault (REF), No.225 data in the CWRU database [11]. For the rolling element fault at the drive end, the fault state can be seen clearly by the second reducing noise (see Figure 6a). For the vibration signals at the fan end without faults, the periodical characteristics in the normal state can be shown after the two reducing noise (see Figure 6b).

Figure 6.

The two times denoising analysis for the vibration signals of rolling element fault (REF) in No.225 data from the CWRU database. (a) The abscissa is the number of data points; (b) the ordinate is the amplitude (v) of the data.

Moreover, the noise reduction method based on the symplectic geometry has been used to denoise several time series data of Lorenz chaotic system, duffing chaotic system, Chua’s chaotic system with noise, as well as the sunspot number [30]. The details can be found in literatures [17, 30].

Besides, the symplectic geometry method also further integrate other approaches to better investigate the fault extraction and identification for rotating systems, such as symplectic geometry mode decomposition [19] with power spectral entropy [7] as well as Lagrange multiplier [20], symplectic transformation based variational Bayesian learning [21].

Advertisement

4. Conclusions and future research

This chapter introduces the symplectic geometry theory in the research field of the time series analysis in view of the complexity of a time series. Corresponding to Euclidean geometry, the basic concepts and basic elements of mathematics of the symplectic geometry are given, such as the symplectic space, symplectic transformation, Hamiltonian matrix, symplectic entropy (SymEn), symplectic principal component analysis (SPCA), and so on. Based on the symplectic geometry theory, the symplectic geometry spectrum analysis (SGSA), the symplectic entropy (SymEn) method and the symplectic geometry mode decomposition (SGMD) method are demonstrated to investigate the principal characteristics of a time series in the symplectic space. Meanwhile, the corresponding matlab programs are given. At last, in order to facilitate readers to learn, use and develop the symplectic geometry method, some applications of symplectic geometry on time series analysis are presented, such as the embedding dimension estimation, nonlinear testing, fault diagnosis, as well as noise reduction.

The embedding dimension estimation is often the first step in nonlinear time series analysis. Case 1 and 2 show the embedding dimension estimation of Lorenz chaotic time series and the surface EMG signal based on symplectic geometry spectrum. Moreover, the symplectic entropy method is applied to detect the nonlinearity of vibration signals on rolling bearings and identify the faults of vibration signals on rolling bearings (see Case 3). Considering the noise pollution in the practical engineering measurement, to dispose of the noise problem is very necessary for the measured time series analysis. Case 4 uses the SPCA method based on symplectic geometry to investigate the denoise of the vibration signals for rolling element fault (REF) from the CWRU database.

Symplectic geometry provides a new research idea for data analysis in practice. Although the symplectic geometry theory has been developed and applied on the nonlinear time series analysis, the related research based on symplectic geometry still needs to be further developed. Many future challenges in the research of symplectic geometry theory and various applications on a number of diverse aspects need to be developed furtherly. This chapter is only to provide a snapshot of some current trends and future challenges in the research of symplectic geometry theory on the time series analysis.

Advertisement

Acknowledgments

This work was supported by Shanghai “Science and technology innovation action plan” bio-medicine science and technology support project (Grant No. 19441907400).

Advertisement

Conflict of interest

The authors declare no conflict of interest.

References

  1. 1. Tao T. An Introduction to Measure Theory. Graduate Studies in Mathematics, 2011, vol.126, published by American Mathematical Society. ISBNs: 978–0–8218-6919-2 (print); 978–1–4704-1187-9 (online). DOI:http://dx.doi. org/10.1090/gsm/126.
  2. 2. Kantz H, Schreiber T. Nonlinear time series analysis. Cambridge University Press, 1997.
  3. 3. Wimberger S. Nonlinear Hamiltonian systems. Chaptered in Nonlinear Dynamics and Quantum Chaos, 2014; 21–102.
  4. 4. Yao W, Zhong WX, Lim CW. Symplectic elasticity. World Scientific Press, 2009. DOI: 10.1142/6656
  5. 5. Lei M, Wang ZZ, Feng ZJ. A method of embedding dimension estimation based on symplectic geometry. Physics Letters A. 2002; 303(2–3): 179–189.
  6. 6. Xie HB, Dokos S, Sivakumar B, Mengersen K. Symplectic geometry spectrum regression for prediction of noisy time series. Physical Review E. 2016; 93: 052217.
  7. 7. Zheng Z, Xin G. Fault feature extraction of hydraulic pumps based on symplectic geometry mode decomposition and power spectral entropy. Entropy. 2019; 21: 476–1-24.
  8. 8. Jiang Y, Chen WZ, Li MY. Symplectic geometry decomposition-based features for automatic epileptic seizure detection. Computers in Biology and Medicine. 2020; 116: 103549.
  9. 9. Takens F. Detecting Strange Attractors in Turbulence. In: Dynamical Systems and Turbulence, eds. D. A. Rand and L. –S. Young, Lecture Notes in Mathematics. Warwick, 1980, vol.898. (Springer-Verlag, New York, 1980), 366–381.
  10. 10. Pincus SM. Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences USA. 1991; 88(6): 2297–2301.
  11. 11. Lei M, Meng G, Dong GM. Fault detection for vibration signals on rolling bearings based on the symplectic entropy method. Entropy. 2017; 19: 607–1-17.
  12. 12. Richman JS, Moorman JR. Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology Heart & Circulatory Physiology. 2000; 278: H2039-H2049.
  13. 13. Skokos Ch. The Lyapunov characteristic exponents and their computation. Chaptered in Dynamics of Small Solar System Bodies and Exoplanets. 2009; 63–135.
  14. 14. Vibe K, Vesin JM. On chaos detection methods. International Journal of Bifurcation and Chaos. 1996; 6(3): 529–543.
  15. 15. Fraser AM. Information and entropy in strange attractors. IEEE Trans. Inf. Theory IT. 1989; 35: 245.
  16. 16. Lei M, Meng G, Zhang WM, Wade J, Sarkar N. Symplectic entropy as a novel measure for complex systems. Entropy. 2016; 18: 412–1-18.
  17. 17. Lei M, Meng G. Symplectic principal component analysis: A new method for time series analysis. Mathematical Problems in Engineering. 2011; 2011: 793429.
  18. 18. Xie HB, Guo TR, Sivakumar B, Liew AW, Dokos S. Symplectic geometry spectrum analysis of nonlinear time series. Proceedings of The Royal Society A. 2014; 470: 20140409–1-17.
  19. 19. Cheng J, Yang Y, Li X, Pan HY, Cheng JS. An early fault diagnosis method of gear based on improved symplectic geometry mode decomposition. Measurement 2020; 151: 107140.
  20. 20. Pan HY, Yang Y, Zheng JD, Cheng JS. A noise reduction method of symplectic singular mode decomposition based on Lagrange multiplier. Mechanical Systems and Signal Processing. 2019; 133: 106283.
  21. 21. Pan HY, Yang Y, Zheng JD, Cheng JS. Symplectic transformation based variational Bayesian learning and its applications to gear fault diagnosis. Measurement 2019; 147: 106827.
  22. 22. Lei M, Wang ZH, Feng ZJ. The application of symplectic geometry on nonlinear dynamics analysis of the experimental data. In: 14th International Conference on Digital Signal Processing Proceeding; 1–3 July 2002; Santorini, Greece. 2002. v1–2, p.1137–1140
  23. 23. Niu X, Qu F, Wang N. Evaluating Sprinters' Surface EMG Signals Based on EMD and Symplectic Geometry. Journal of Ocean University of Qingdao. 2005; 35(1): pp125–129. (in Chinese)
  24. 24. Xie H, Wang Z, Huang H. Identification determinism in time series based on symplectic geometry spectra. Physics Letters A, 2005,342(1–2):156–161.
  25. 25. Lorenz EN. Deterministic nonperiodic flow. Journal of Atmospheric Sciences. 1963; 20: 130–141.
  26. 26. Schreiber T, Schmitz A. Improved surrogate data for nonlinearity tests. Physical Review Letters. 1996; 77: 635–638. DOI: 10.1103/PhysRevLett.77.635.
  27. 27. Yang J, Zhang Y, Zhu Y. Intelligent fault diagnosis of rolling element bearing based on SVMs and fractal dimension. Mechanical Systems and Signal Processing. 2007; 21:2012–2014. DOI:0.1016/j.ymssp.2006.10.005.
  28. 28. Prieto MD, Cirrincione G, Espionsa AG, Ortega JA, Henao H. Bearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neural networks. IEEE Transactions on Industrial Electronics 2013; 30: 3398–3407. DOI:10.1109/TIE.2012.2219838.
  29. 29. Wu S, Wu C, Wu T, Wang C. Multi-scale analysis based ball bearing defect diagnostics using mahalanobis distance and support vector machine. Entropy 2013; 15: 416–433. DOI:10.3390/e15020416.
  30. 30. Lei M, Meng G. A noise reduction method for continuous chaotic systems based on symplectic geometry. Journal of Vibration Engineering & Technologies. 2015; 3(1): 13–24.

Written By

Min Lei

Submitted: November 18th, 2019 Reviewed: November 9th, 2020 Published: December 8th, 2020