# RANDU example

This example is adapted from Tyler et al. (2009) and illustrates the behavior of 
ICS on data generated by the well-known RANDU pseudo-random number generator.

The way in which RANDU generates its numbers is somewhat deterministic. Upon closer 
inspection, a structure can be seen in the data, which becomes clearer when the ICS 
method is applied. 
The example highlights how ICS is able to identify informative directions associated with 
these dependencies, in contrast to variance-based approaches such as PCA.

```python
from icspylab import ICS, plot_ics
from icspylab.distributions import generate_randu
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import matplotlib.pyplot as plt
```

The dataset is generated using the {func}`icspylab.distributions.generate_randu` function.

```python
X = generate_randu()
print("X shape:", X.shape)
```

```text
X shape: (400, 3)
```

The plot below illustrates the generated RANDU dataset in three dimensions.

```python
fig = plt.figure()
ax = fig.add_subplot(projection="3d")
ax.scatter(X[:, 0], X[:, 1], X[:, 2], s=2)
plt.show()
```

```{image} ../_static/randu_3d.png
:alt: 3D scatter plot of the RANDU dataset
:width: 400px
:align: center
```

In three-dimensional space, each point represents 3 consecutive pseudorandom values, and 
it is known that these points fall in one of 15 two-dimensional planes.
Applying ICS yields the following invariant components, which clearly exhibit those planes
on the last component. 

```python
ics = ICS(S1="cov", S2="tcovAxis", algorithm="standard")
X_ics = ics.fit_transform(X)
plot_ics(X_ics)
```

```{image} ../_static/randu_ics.png
:alt: ICS projection on the RANDU dataset
:width: 500px
:align: center
```

On the contrary, the underlying structure of RANDU is not visible on the first principal components. 

```python
# Compute the principal components
scaler = StandardScaler().set_output(transform="pandas")
X_scaled = scaler.fit_transform(X)

pca = PCA(n_components=2)
X_pca = pca.fit_transform(X_scaled)

# Plot the principal components
plt.figure(figsize=(5, 5))
plt.scatter(X_pca[:, 0], X_pca[:, 1], alpha=0.7)
plt.title("PCA projection of the simulated dataset \n(first two components)")
plt.xlabel("Principal component 1")
plt.ylabel("Principal component 2")
plt.axis("equal")
plt.show()
```

```{image} ../_static/randu_pca.png
:alt: PCA projection on the RANDU dataset
:width: 500px
:align: center
```