Gaussian process (kernel-defined structure) - Nixtla

A Gaussian process specifies a series entirely through its covariance kernel: choose the kernel and you choose the smoothness, length scale, and periodicity of the samples. It is the most direct way to dial in a precise correlation structure, from rough and jagged to infinitely smooth.

Kernels $f \sim \mathcal{GP}(m, k), \qquad k_{\mathrm{rbf}}(r) = a^2 \exp\!\Big(\!-\frac{r^2}{2\ell^2}\Big), \qquad r = |t - t'|$ kernel selects the covariance function — rbf (infinitely smooth), matern_0.5 / matern_1.5 / matern_2.5 (increasingly smooth), or periodic (exact periodicity). length_scale sets how quickly correlation decays with distance, and amplitude sets the output scale.

import polars as pl
import matplotlib.pyplot as plt

from synforecast.generators import GaussianProcessGenerator

1. Kernel choice

The kernel is the whole model: it fixes smoothness, length scale, and periodicity. All four draws share a seed and a length scale, so the difference is entirely the covariance function.

rbf_df = GaussianProcessGenerator(
    engine="polars", min_length=200, max_length=200, freq="D",
    kernel="rbf", length_scale=15.0, seed=42,
).generate(n_series=1)

rough_df = GaussianProcessGenerator(
    engine="polars", min_length=200, max_length=200, freq="D",
    kernel="matern_0.5", length_scale=15.0, seed=42,
).generate(n_series=1)

smooth_df = GaussianProcessGenerator(
    engine="polars", min_length=200, max_length=200, freq="D",
    kernel="matern_2.5", length_scale=15.0, seed=42,
).generate(n_series=1)

periodic_df = GaussianProcessGenerator(
    engine="polars", min_length=200, max_length=200, freq="D",
    kernel="periodic", length_scale=15.0, period=30.0, seed=42,
).generate(n_series=1)

panels = [
    ("rbf (infinitely smooth)", rbf_df),
    ("matern_0.5 (rough, OU-like)", rough_df),
    ("matern_2.5 (twice differentiable)", smooth_df),
    ("periodic (period=30)", periodic_df),
]
fig, axes = plt.subplots(4, 1, figsize=(12, 9), sharex=True)
for ax, (label, df) in zip(axes, panels):
    ax.plot(df["ds"].to_list(), df["y"].to_list(), alpha=0.85, linewidth=1)
    ax.set(ylabel="Value", title=label)
axes[-1].set_xlabel("Timestamp")
plt.tight_layout()
plt.show()

2. Multiple series

Generate multiple independent GP realizations.

multi_gen = GaussianProcessGenerator(engine="polars", 
    min_length=150, max_length=150, freq="D",
    kernel="matern_2.5", length_scale=15.0, seed=42,
)
multi_df = multi_gen.generate(n_series=3)

fig, ax = plt.subplots(figsize=(12, 4))
for uid in multi_df["unique_id"].unique().to_list():
    series = multi_df.filter(pl.col("unique_id") == uid)
    ax.plot(series["ds"].to_list(), series["y"].to_list(), label=uid, alpha=0.8)
ax.set_xlabel("Timestamp")
ax.set_ylabel("Value")
ax.set_title("Multiple GP realizations (Matern 2.5)")
ax.legend()
plt.tight_layout()
plt.show()

Related generators

KernelSynth — random compositions of these kernels for pretraining corpora.

Cyclic — irregular oscillation without a fixed kernel.

Full parameters are in the generator reference.

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