r"""
Weighted least-squares regression
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

This example begins with linear regression in the real domain and then builds up
to show how linear problems can be thought of in the complex domain.

Useful references:

- https://stats.stackexchange.com/questions/66088/analysis-with-complex-data-anything-different
- https://www.chrishenson.net/article/complex_regression
"""
import numpy as np
import matplotlib.pyplot as plt
from regressioninc.linear.models import add_intercept, OLS

# %%
# One of the most straightforward linear problems to understand is the equation
# of line. Let's look at a line with gradient 3 and intercept -2.
params = np.array([3])
intercept = -2
X = np.arange(-5, 5).reshape(10, 1)
y = X * params + intercept

fig = plt.figure()
plt.scatter(y, X)
plt.xlabel("Independent variable")
plt.ylabel("Dependent variable")
plt.tight_layout()
fig.show()

# %%
# When performing linear regressions, the aim is to:
#
# - calculate the coefficients (coef, also called parameters)
# - given the regressors (X, values of the independent variable)
# - and values of the observations (y, values of the dependent variable)
#
# This can be done with linear regression, and the most common method of linear
# regression is least squares, which aims to estimate the coefficients whilst
# minimising the squared misfit between the observations and estimated
# observations calculated using the estimated coefficients.
X = add_intercept(X)
model = OLS()
model.fit(X, y)
print(model.estimate.params)