"""
Approximating a Noisy Signal
----------------------------

In this example, we add noise to a sinusoidal signal and then approximate it using an exponential spline.
"""

import matplotlib.pyplot as plt
import numpy as np
import splinebox

# %%
# We generate example data with 100 points by evaluating a sine function
# at equidistant points over the interval from 0 to 15.
# To simulate real-world data, we add Gaussian noise with a standard deviation of 0.3.

N = 100
x = np.linspace(0, 15, N)
values = np.sin(x) + np.random.normal(0, 0.3, N)

# %%
# To approximate this data, we'll use an exponential spline, which can effectively model sinusoidal functions.
# We aim to use the smallest number of knots that still allows the spline to accurately approximate the signal.
# Empirical testing indicates that 8 knots provide a good balance between simplicity and accuracy in this case.

M = 8
basis_function = splinebox.Exponential(M)
spline = splinebox.Spline(M, basis_function, closed=False)

spline.fit(values[:, np.newaxis])

# %%
# To plot the spline we evaluate it at finely spaced parameter values.

ts = np.linspace(0, M - 1, 1000)
spline_y = spline(ts)
spline_x = np.linspace(x.min(), x.max(), len(ts))

plt.scatter(x, values, label="data")
plt.plot(spline_x, spline_y, label="spline")
plt.scatter(np.linspace(x.min(), x.max(), M), spline.knots, label="knots", marker="x", sizes=np.ones(M) * 150)
plt.legend()
plt.show()