I have a 3-dimensional embedded time series. How can I find the points/coordinates (x, y, z) where the 3d matrix of the time series intersects an arbitrary hyperplane. The problem is I don't have an equation for my embedded time series. Do I either find the closest points to the hyperplane and project them onto my hyperplane or do I find where one point crosses onto the other side to another point and then find the equation of that line and plug in my z-value to find the (x, y) coords? My plot looks like this:
Here's my current code for replicability:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from teaspoon.SP.tsa_tools import takens
import seaborn as sns
from datetime import datetime
import pandas_datareader.data as pdr
sns.set_style("darkgrid")
def fetch_data(symbol, from_date, to_date, cols_to_drop):
""" Fetch OHLC data."""
df = pdr.DataReader(symbol, "yahoo", from_date, to_date)
df.drop(columns=cols_to_drop, inplace=True)
return df
if __name__ == "__main__":
# Fetch OHLC Data #
symbol = "EURUSD=X" # ticker
from_date = datetime(2000, 1, 1)
to_date = datetime.now()
drop_columns = ["Adj Close", "Volume"]
df = fetch_data(symbol=symbol, from_date=from_date,
to_date=to_date, cols_to_drop=drop_columns)
# TAKEN'S EMBEDDING THEOREM #
taken_matrix = takens(np.array(df.Close), n=3, tau=62) # emb_dim =3, time delay=62
x_min, x_max = np.min(taken_matrix[:, 0]), np.max(taken_matrix[:, 0]) # x_min and x_max
y_min, y_max = np.min(taken_matrix[:, 1]), np.max(taken_matrix[:, 1]) # y_min and y_max
z_min, z_max = np.min(taken_matrix[:, 2]), np.max(taken_matrix[:, 2]) # z_min and z_max
# Method 1
x = np.array([x_min, x_max])
y = np.array([y_min, y_max])
xx, yy = np.meshgrid(x, y)
z = np.array([[1.4, 1.4], [1.4, 1.4]])
hyperplane = np.array([xx, yy, z])
hyperplane = np.reshape(hyperplane.T, (4, 3)) # 4 co-ords
fig = plt.figure(dpi=50)
ax = plt.axes(projection="3d")
ax.set_xlabel("x"); ax.set_ylabel("y"); ax.set_zlabel("z")
ax.plot(taken_matrix[:, 0], taken_matrix[:, 1], taken_matrix[:, 2], c="black", lw=0.6, alpha=0.8) # phase space
ax.plot_surface(xx, yy, z, alpha=0.2, color="seagreen") # Hyperplane
plt.show()
# Method 2
hyperplane1 = hyperplane[:-1, :] # 3 coordinates
p0, p1, p2 = hyperplane1
x0, y0, z0 = p0
x1, y1, z1 = p1
x2, y2, z2 = p2
ux, uy, uz = u = [x1 - x0, y1 - y0, z1 - z0] # first vector
vx, vy, vz = v = [x2 - x0, y2 - y0, z2 - z0] # second vector
u_cross_v = [uy*vz - uz*vy, uz*vx - ux*vz, ux*vy - uy*vx] # cross product
point1 = np.array(p1)
normal1 = np.array(u_cross_v) # hyerplane normal vector
d1 = -point1.dot(normal1) # computed for equation of plane
print('plane equation:\n{:1.4f}x + {:1.4f}y + {:1.4f}z + {:1.4f} = 0'.format(normal1[0], normal1[1], normal1[2], d1))
xx, yy = np.meshgrid(x, y)
z1 = (-normal1[0] * xx - normal1[1] * yy - d1) * 1. / normal1[2]
fig = plt.figure()
ax = plt.axes(projection="3d")
ax.plot_surface(xx, yy, z1, color="orange")
ax.plot(taken_matrix[:, 0], taken_matrix[:, 1], taken_matrix[:, 2], c="black", lw=0.6, alpha=0.8)
plt.show()
