Python: Hankel Matrices H0, H1, Generalised Eigenvalue of H0 and H1, and Singular Value Decomposition (SVD) used for the Matrix Pencil Method (MPM) and Singular Spectrum Analysis (SSA)

 

This process is the part of the Python code for Prony method introduced in the last post. The process of finding Hankel matrices, Generalised Eigenvalues, and Singular Value Decomposition (SVD) are the essential processes for both the modern Prony method (Matrix Pencil Method (MPM) and Singular Spectrum Analysis (SSA). 

 SSA is the non-parametric method which is said to be influenced by Prony method. For financial time series analyses, SSA is popularly used nowadays. SSA is better at capturing non-linearly time series with erratic fluctuations. SSA is popular due to the multi-resolution decomposition of time series into time trends and volatilities. 

The following Python codes operates the following actions:

- Downloads the foreign exchange rates of Euro based on USD

- Generating a time series variable EUR/USD

- Creating Hankel matrices H0 and H1 based on this time series

- Finding Generalised Eigenvalue and Eigenvectors of H0 and H1

- Singular Value Decomposition (SVD) of the Hankel matrix H0

 

# Execute this cell to have access to the API of Frankfurter App

import requests

import json

# importing necessary tools

import matplotlib.pyplot as plt

import networkx as nx

import numpy as np

import pandas as pd

import random

import statistics

import scipy.linalg

from numpy import linalg as LA

from scipy.stats import qmc

from scipy.optimize import newton

from scipy import stats

import statsmodels.formula.api as sm

import math

import cmath

from scipy.linalg import hankel

import scipy.odr

from scipy import odr

# Showing the plots bigger

plt.rcParams["figure.figsize"] = (12,6)

# defining complex number

i=complex(0,1)

i

# Connecting to the API https://www.frankfurter.app/docs/ to get the EUR/USD rates

url = "http://api.frankfurter.app/1999-01-01.."

# Write your code here

resp = requests.get(url,{'to':'USD'})

#print(resp.url)

#print(resp.status_code)

#print("Type:", type(resp.content))

#print(resp.content)

print("Type:", type(resp))

# Preparing for the right encoding

import ast

byte_str = resp.content

dict_str = byte_str.decode("UTF-8")

mydata = ast.literal_eval(dict_str)

print("Type:", type(mydata))

#mydata #show the accessed data

EURUSD=(mydata["rates"])

#EURUSD

# Sorting the data into the Python dictionary form

Dates_EURUSD=[]

Rates_EURUSD=[]

for key, value in EURUSD.items():

    Dates_EURUSD.append(key)

    for k,v in value.items():

        Rates_EURUSD.append(v)

EURUSD=dict(zip(Dates_EURUSD,Rates_EURUSD))

#EURUSD

#Rates_EURUSD

print("Length of data:",len(Rates_EURUSD))

# Converting the rate to natural logarithm

ln_Rates_EURUSD=np.log(Rates_EURUSD)

ln_Rates_EURUSD

#plt.plot(ln_Rates_EURUSD)   # Uncomment to show plot!

# Creating the log-difference (Daily return) of EUR/USD

d_ln_Rates_EURUSD=np.diff(np.log(Rates_EURUSD))

len(d_ln_Rates_EURUSD)

#plt.plot(d_ln_Rates_EURUSD)   # Uncomment to show plot!

# Obtaining the length which is half of the total length of this variable.

# This is needed for creating the matrix of this variable i.e. n/2 * n/2 matrix

X=d_ln_Rates_EURUSD.copy()

d=(int(len(X)/2-len(X)%2*0.5)) # Subtracting the remainder if it is an odd number

d

f=d_ln_Rates_EURUSD.copy()

n=len(f)

M=100  # sampling rate.

# M is supposed to be twice of the actual signal frequency.

# But, we have never known the true frequency of the financial data.

# So, it is assumed to be 100 because it seems to provide an appropriate answer

print(n) # Checking the total length of this variable.

# Creating Hankel matrix which is needed to

def mat_ge(samples):

    n_samples = len(samples)

    d = int(n_samples/2)

    H0=hankel(samples[0:-1])[0:d,0:d]

    H1=hankel(samples[1:])[0:d,0:d]

    return H0, H1

H0, H1 =mat_ge(f)

H=(H0,H1)

for h in range(len(H)):

    print(f'\n\n \033[1m Hankel Matrix H{h} \033[0m')

    panda_df = pd.DataFrame(data = H[h])

    display(panda_df)

# Finding the Generalised Eigenvalue of these Hankel matrices H0 and H1

print('\n\n Finding the Generalised Eigenvalue and Eigenvectors of these Hankel matrices H0 and H1 ')

Eig = scipy.linalg.eig(H0, H1)

Evalue= Eig[0]

Evectors = Eig[1:]

Evalue_real=Evalue.real

print('\n\n \033[1m Eigenvalue \033[0m')

panda_df = pd.DataFrame(data = Evalue)

display(panda_df)

print('\n\n \033[1m Eigenvectors \033[0m')

display(Eig[1:])

   

# Showing Verndermonde Matrix

Veig=np.transpose(np.vander(Evalue, N=None, increasing=True))

#print('\n\n Vandermonde Matrix of the eigenvalues')

#panda_df = pd.DataFrame(data = Veig)

#display(panda_df)

# Defnining the function plotting Singular Value Decomposition (SDV)

# This apprximate the number of the physical/true signals i.e. the number of true alpha s and phi s

def plot_SVD(A):

    SDV=LA.svd(A, full_matrices=True, compute_uv=True, hermitian=False)[1]

    Ln_SDV=[math.log(SDV[k]) for k in range(len(SDV)) ]

    plt.plot(Ln_SDV)

    plt.title('Singular Values')

    plt.xlabel('')

    plt.ylabel('Natural Log')

    plt.show

print('\n\n Singular Value decomposition (SVD) of the Hankel matrix H0')

plot_SVD(H0)

 

 

P.S. This following website shows the useful code to simulate Singular Spectrum Analysis (SSA) with Python

Introducing SSA for Time Series Decomposition
Python · MotionSense Dataset : Smartphone Sensor Data - HAR
https://www.kaggle.com/code/jdarcy/introducing-ssa-for-time-series-decomposition