#!/usr/bin/env python
# coding: utf-8

# # 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.

# In[15]:


import matplotlib.pyplot as plt

from aeon.datasets import load_airline
from aeon.transformations.series.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"])


# ### 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.

# In[16]:


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)


# 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.

# In[17]:


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"]
)


# In[ ]: