Add effective_dim.py with participation_ratio function (#152)

Author: Faezehabibi

ngclearn/utils/analysis/effective_dim.py

@@ -0,0 +1,11 @@
+from jax import numpy as jnp
+
+def participation_ratio(Z):
+    Zc = Z - Z.mean(axis=0, keepdims=True)
+    cov = (Zc.T @ Zc) / (Zc.shape[0] - 1)
+
+    tr = jnp.trace(cov)
+    tr2_cov = tr * tr
+    cov2_tr = jnp.trace(cov @ cov)
+
+    return tr2_cov / cov2_tr if cov2_tr > 0 else float("nan")