I am trying to use SVD and an Eigendecomposition for some data analysis using Dynamic Mode Decomposition. I am running into a simple problem of getting different results from Matlab and Python. I'm confused and don't know why Python is giving me wrong results/matrix values but everything looks (I think IS) correct.
So instead of using real data this time and looking at large data sets, I generated data. I will try to look at an eigenvalue plot after the eigendecomposition. I also use a delay embedding for the data because I will work with a data vector which is only (2x100), so I will perform a type of Hankel matrix to enrich the data with 10 delays.
clear all; close all; clc;
data = linspace(1,100);
data2 = linspace(2,101);
data = [data;data2];
numDelays = 10;
relTol= 10^-6;
%% Create first and second snap shot matrices for DMD. Any columns with missing
% data are not used.
disp('Constructing Data Matricies:')
X = zeros((numDelays+1)*size(data,1),size(data,2)-(numDelays+1));
Y = zeros(size(X));
for i = 1:numDelays+1
X(1 + (i-1)*size(data,1):i*size(data,1),:) = ...
data(:,(i):size(data,2)-(numDelays+1) + (i-1));
Y(1 + (i-1)*size(data,1):i*size(data,1),:) = ...
data(:,(i+1):size(data,2)-(numDelays+1) + (i));
end
[U,S,V] = svd(X);
r = find(diag(S)>S(1,1)*relTol,1,'last');
disp(['DMD subspace dimension:',num2str(r)])
U = U(:,1:r);
S = S(1:r,1:r);
V = V(:,1:r);
Atil = (U'*Y)*V*(S^-1);
[what,lambda] = eig(Atil);
Phi = (Y*V)*(S^-1)*what;
Keigs = diag(lambda);
tt = linspace(0,2*pi,101);
figure;
plot(real(Keigs),imag(Keigs),'ro')
hold on
plot(cos(tt),sin(tt),'--')
import scipy.io as sc
import math as m
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sys
from numpy import dot, multiply, diag, power, pi, exp, sin, cos, cosh, tanh, real, imag
from scipy.linalg import expm, sinm, cosm, fractional_matrix_power, svd, eig, inv
def dmd(X, Y, relTol):
U2,Sig2,Vh2 = svd(X, False) # SVD of input matrix
S = np.zeros((Sig2.shape[0], Sig2.shape[0])) # Create S matrix with zeros based on Diag of S
np.fill_diagonal(S, Sig2) # Fill diagonal of S matrix with the nonzero values
r = np.count_nonzero(np.diag(S) > S[0,0] * relTol) # rank truncation
U = U2[:,:r]
Sig = diag(Sig2)[:r,:r] #GOOD =)
V = Vh2.conj().T[:,:r]
Atil = dot(dot(dot(U.conj().T, Y), V), inv(Sig)) # build A tilde
print(Atil)
mu,W = eig(Atil)
Phi = dot(dot(dot(Y, V), inv(Sig)), W) # build DMD modes
return mu, Phi
data = np.array([(np.linspace(1,100,100)),(np.linspace(2,101,100))])
Data = np.array(data)
######### Choose number of Delays ###########
# observable (coordinates of feature points). Setting to zero means only
# experimental observables will be used.
numDelays = 10
relTol = 10**-6
########## Create Data Matrices for DMD ###############
# Create first and second snap shot matrices for DMD. Any columns with missing
# data are not used.
X = np.zeros(((numDelays + 1) * data.shape[0], data.shape[1] - (numDelays + 1)))
Y = np.zeros(X.shape)
for i in range(1, numDelays + 2):
X[0 + (i - 1) * Data.shape[0]:i * Data.shape[0], :] = Data[:, (i):Data.shape[1] - (numDelays + 1) + (i - 0)]
Y[0 + (i - 1) * Data.shape[0]:i * Data.shape[0], :] = Data[:, (i + 0):Data.shape[1] - (numDelays + 1) + (i)]
Keigs, Phi = dmd(X, Y, relTol)
tt = np.linspace(0,2*np.pi,101)
plt.figure()
plt.plot(np.cos(tt),np.sin(tt),'--')
plt.plot(Keigs.real,Keigs.imag,'ro')
plt.title('DMD Eigenvalues')
plt.xlabel(r'Real $\ lambda$')
plt.ylabel(r'Imaginary $\ lambda$')
# plt.axes().set_aspect('equal')
plt.show()
So in matlab and python, I get my eigenvalues to all sit on the unit circle (as expect) and I get precisely one, sitting at 1.
So the problem comes when I look at the matrices from SVD, they appear to have different values. The only matrix that is the same is the 'S or Sig' matrix. The rest will differ a number or +/- sign. The biggest thing that peaked my interest is the Atil matrix. In matlab, it looks like, [1.0157, -0.3116; 7.91229e-4, 0.9843] And python it looks like, [1.0, -4.508e-15; -4.439e-18, 1.0]
Now this may look slightly off due to numerical error possibly but when I look at real data and these differ, it messes up my analysis.
SVD of a non-square matrix is not unique in U and V. Even if you have a square matrix with non-zero, non-degenerate singular values, singular vectors in U and V are only unique up to a sign factor. https://math.stackexchange.com/questions/644327/how-unique-on-non-unique-are-u-and-v-in-singular-value-decomposition-svd
Moreover, Matlab (LAPACK + BLAS) and scipy.linalg.svd may use different algorithms for SVD. This can lead to the differences you have experienced.