Attempt to use Singular Spectrum Analysis (SSA) for Frankfurter App with Python

Singular Spectrum Analysis (SSA) is nowadays a popular analysis method for financial time series. 

SSA is similar to Prony's method (the analogy would describe them as cousins) which can decompose a time series with non-periodic signalling cycles and a trend (Not a waving signal). 

Autoregressive method has a limitation of distinguishing the physical terms and the noise terms of the volatility, and it is not well suited for the future forecast for a non-linear time series.

In contrast,  SSA and Prony method can decompose such time series with a multi-dimensional decomposition process to offer more detailed estimates than the other conventional methods.

The part implementing SSA in the following Python codes refers to MATLAB codes displayed in https://uk.mathworks.com/matlabcentral/fileexchange/58967-singular-spectrum-analysis-beginners-guide.

This is an attempt to implement SSA for Frankfurter App with Python.

The following Python codes will be transferable to analysing the other financial data.

# 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 matplotlib.gridspec as gridspec

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 import stats

import statsmodels.formula.api as sm

import math

from scipy.linalg import hankel

# For plotting with multiple axis with different scales

# https://stackoverflow.com/questions/9103166/multiple-axis-in-matplotlib-with-different-scales

from mpl_toolkits.axes_grid1 import host_subplot

import mpl_toolkits.axisartist as AA

# Showing the plots bigger

plt.rcParams["figure.figsize"] = (18,9)

# 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

f=ln_Rates_EURUSD.copy()

#f=d_ln_Rates_EURUSD.copy()

# Creating Hankel matrix which is needed to

n = len(f)

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

d

H=hankel(f[0:-1])[0:d,0:d]

#print(f'\n\n \033[1m Hankel Matrix 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(H)

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:])

# 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')

# Refers to the plot to determine the number of terms (NTERMS) to recover

#plot_SVD(H) # Uncomment to show the SVD plot

# Determining NTERMS

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

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

NTERMS=0

SlopeSVD=1 # Adjust accoording to the data type and its log(SV) plot

for k in range(len(Ln_SDV)):

    if Ln_SDV[k]>SlopeSVD:

        NTERMS+=1

        #print(NTERMS)

        continue

    else:

        break

# Generating the reconstruction components for NTERMS

RCs = []

for k in range(NTERMS):

    RCtemp=Evectors[0][:,k]

    RCs.append(RCtemp)

# Showing each reconstruction components

print('\n\n \033[1m Reconstruction Components for NTERMS \033[0m')

panda_df = pd.DataFrame(data = RCs)

display(panda_df)

# For plotting with multiple axis with different scales

# https://stackoverflow.com/questions/9103166/multiple-axis-in-matplotlib-with-different-scales

plt.rcParams["figure.figsize"] = (7*NTERMS,30)

gs = gridspec.GridSpec(50,10)

for k in range(NTERMS):

    ax = plt.subplot(gs[k, 0]) # row k, col 0

    plt.plot(RCs[k])

# # Uncomment the entire following lines to show the graph of comparison

# # Summing all the reconstruction components

# Sum=0

# SRCs=[]

# for k in range(len(RCs[0])):

#     for l in range(len(RCs)):

#         Sum=Sum+RCs[l][k]

#     SRCs.append(Sum)

# len(SRCs)

# # Showing the sum of the reconstruction components

# plt.rcParams["figure.figsize"] = (20,10)

# plt.plot(SRCs)

# plt.xlabel("Time")

# plt.ylabel("Estimated price")

# plt.show

# # Comparing the original time series and the reconstructed time series

# # Cutting the length of f

# fcut=[(f[2*k-1]+f[2*k])/2 for k in range(math.floor(len(f)/2))]

# fcut=f

# # Streching the length of RCs

# StrSRCs=[0]*(len(SRCs)*2)

# for k in range(math.floor(len(f)/2)):

#     StrSRCs[2*k]=SRCs[k]

#     StrSRCs[2*k+1]=SRCs[k]

# len(StrSRCs)

# #StrSRCs=SRCs

# ## Plottting graph with multiple y-axis with different labels

# # https://stackoverflow.com/questions/9103166/multiple-axis-in-matplotlib-with-different-scales

# host = host_subplot(111, axes_class=AA.Axes)

# plt.subplots_adjust(right=0.75)

# par1 = host.twinx()

# par2 = host.twinx()

# YLabelL="f"

# YLabelR="rcs"

# VariableL=fcut

# VariableR=StrSRCs

# host.set_xlabel("Year")

# host.set_ylabel(YLabelL)

# par1.set_ylabel(YLabelR)

# p1, = host.plot( VariableL, label=YLabelL)

# p2, = par1.plot( VariableR, label=YLabelR)

# host.legend()

# host.axis["left"].label.set_color(p1.get_color())

# par1.axis["right"].label.set_color(p2.get_color())

# plt.draw()

# plt.xticks(rotation='vertical')

# plt.show()

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