Theta Lines transformer

Computes Theta lines to be used in ThetaForecaster.

The Theta lines are obtained by modifying the distances between the points of the original time series. We apply coefficient θ (theta) to the second differences of the original observations, thus changing the local curvatures . Theta coefficient is a transformation parameter which creates series of the same mean and slope with that of the original data but different variances.

import matplotlib.pyplot as plt

from aeon.datasets import load_airline
from aeon.transformations.theta import ThetaLinesTransformer

y = load_airline()
transformer = ThetaLinesTransformer()
transformer.fit(y)
y_thetas = transformer.transform(y)

fig, ax = plt.subplots()
y_thetas.plot(ax=ax, figsize=(12, 7))
plt.legend(["theta=0", "theta=2"])

<matplotlib.legend.Legend at 0x1b808c49370>

Special cases: theta = 0 and theta = 1

Theta values 1 > theta > 0 reduce original time series curvatures. The smaller the value of Theta coefficient results in the larger deflation of time series.

Theta-line with theta=0 gives a linear regression line.

t = ThetaLinesTransformer([0, 0.25, 0.75, 1])
t.fit(y)
y_t = t.transform(y)

fig, ax = plt.subplots()
y_t.plot(ax=ax, figsize=(12, 7))
plt.legend(
    ["theta=0, linear regression", "theta=0.25", "theta=0.75", "theta=1, original ts"]
)
plt.ylim(0, 900)

(0.0, 900.0)

Theta-line with theta=1 returns original time series.

Theta values > 1 will increase local curvatures, and the larger theta coefficients will result in larger dilation.

t_1 = ThetaLinesTransformer([0, 1, 2, 2.5])
t_1.fit(y)
y_t1 = t_1.transform(y)

fig, ax = plt.subplots()
y_t1.plot(ax=ax, figsize=(12, 7))
plt.legend(
    ["theta=0, linear regression", "theta=1, original ts", "theta=2", "theta=2.5"]
)

<matplotlib.legend.Legend at 0x1b80996baf0>

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