S-Map

S-Map (Sequential Locally Weighted Global Linear Map) is a prediction method that fits a local linear model at each query point, with weights that decay exponentially with distance.

How It Works

For each query point q:

1. Compute distances from q to all library points
2. Assign weights: _w_i = exp(-theta _ d_i / d_mean)*
3. Fit a weighted linear regression (with Tikhonov regularization) to predict the target

The theta parameter controls locality:

theta	Behavior
0	Global linear model (all points weighted equally)
Small	Weakly local — smooth transition
Large	Strongly local — only nearby points matter

Usage

from edmkit.smap import smap

predictions = smap(X, Y, Q, theta=3.0)

Parameters

• theta: Locality parameter (>= 0). Start with values in [0, 1, 2, 3, 5, 8].
• alpha: Tikhonov regularization strength (default: 1e-10). Increase if the system is ill-conditioned.
• mask: Boolean mask to exclude library points.
• use_tensor: Use tinygrad backend for GPU acceleration.

Detecting Nonlinearity

Compare prediction skill across theta values. If performance improves substantially for theta > 0, the system exhibits nonlinear dynamics:

from edmkit.smap import smap
from edmkit.metrics import mean_rho

for theta in [0, 1, 2, 4, 8]:
    preds = smap(lib, target, query, theta=theta)
    rho = mean_rho(preds, observations)
    print(f"theta={theta}: rho={rho:.4f}")

Batched Operations

Like simplex projection, S-Map supports batched inputs with a leading batch dimension:

# X: (B, N, E), Y: (B, N, D), Q: (B, M, E)
predictions = smap(X, Y, Q, theta=3.0)  # -> (B, M, D)