rustima 0.3.0

High-performance SARIMAX (Seasonal ARIMA with eXogenous regressors) engine in Rust with statsmodels-compatible Python API

rustima

Korean / 한국어

A high-performance SARIMAX (Seasonal ARIMA with eXogenous regressors) engine written in Rust with Python bindings via PyO3. It provides a statsmodels-compatible high-level API, native-compiled speed, numerical accuracy on par with statsmodels, and Rayon-based job-level parallel batch processing for large-scale time series workloads.

Motivation

Python's statsmodels.tsa.SARIMAX is the de facto standard for SARIMA modeling, but its pure-Python + NumPy implementation has structural bottlenecks:

Bottleneck	Root Cause	Impact
Slow Kalman filter loop	Python for loop over matrix ops	Seconds to tens of seconds on long series or high-order models
MLE optimization overhead	Python call stack on every iteration	Accumulated latency over hundreds of iterations
No real parallelism	GIL prevents multi-threaded batch fitting	Cannot simultaneously fit thousands of series
Memory fragmentation	Python object overhead per allocation	Unnecessary heap pressure on large state spaces

rustima replaces these bottlenecks with native Rust:

• Kalman filter: Rust for + nalgebra dense matrix ops (zero interpreter overhead)
• Optimization: L-BFGS-B (default), L-BFGS, Nelder-Mead run entirely in Rust with analytical score vector (sparse tangent-linear Kalman + steady-state optimization)
• Batch parallelism: Rayon work-stealing thread pool that assigns one full fit/forecast job per series
• Grid search parallelization: sarimax_grid_search assigns one ARIMA order combination per worker via Rayon
• auto_arima: Hyndman-Khandakar stepwise + Rayon parallel grid search for automatic order selection
• Memory: Stack allocation + contiguous column-major layout for cache-friendliness
• Python interop: PyO3 + numpy bindings — import rustima

What is SARIMAX? (For Beginners)

If you've never used SARIMAX before, here's a 30-second primer:

SARIMAX = S + ARIMA + X

• AR (AutoRegressive) — predict today's value from the past few days
• I (Integrated / differencing) — subtract today from yesterday to make a trending series stationary
• MA (Moving Average) — use past forecast errors to improve today's prediction
• S (Seasonal) — a repeating pattern (weekly=7, monthly=12, hourly=24)
• X (eXogenous) — extra columns that help explain y (e.g. temperature, price, promotions)

You describe the model with two tuples:

Tuple	Meaning	Example
order=(p, d, q)	Non-seasonal (AR, diff, MA)	(1, 1, 1) = 1 lag AR, 1 diff, 1 lag MA
seasonal_order=(P, D, Q, s)	Seasonal part + period	(1, 0, 1, 12) = monthly seasonality

Quick rules of thumb:

• Has trend? → d=1
• Has yearly pattern in monthly data? → s=12
• Not sure about orders? → Use auto_arima() and it picks for you

Recommended learning path:

1. Try auto_arima() on your series first (it picks orders automatically)
2. Read Forecasting: Principles and Practice, Ch. 9 (free textbook) for theory
3. Come back to tune order/seasonal_order manually

Supported Models

SARIMA(p, d, q)(P, D, Q, s) + trend + exogenous regressors

Parameter	Description	Range
p	AR order (autoregressive)	0–20
d	Differencing order	0–3
q	MA order (moving average)	0–20
P	Seasonal AR order	0–4
D	Seasonal differencing order	0–1
Q	Seasonal MA order	0–4
s	Seasonal period (e.g. 12=monthly, 7=daily, 24=hourly)	2–365
trend	Trend component ('n', 'c', 't', 'ct')	—
exog	Exogenous regressors	n_obs × n_exog matrix

Installation

Prerequisites

Because rustima ships Rust source, you must build it locally (there is no pre-built wheel on PyPI yet). You need:

Tool	Minimum	Why	Install
Rust	1.83+	Compiles the engine	curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh
Python	3.10+	Hosts the extension	python.org or pyenv
uv	latest	Fast Python package/env manager	curl -LsSf https://astral.sh/uv/install.sh | sh
maturin	1.7+	Rust ↔ Python bridge (auto-installed by uv sync --extra dev)	—

Windows users: install Microsoft C++ Build Tools first (needed by Rust's MSVC toolchain).

Option A — Development mode (recommended for most users)

Best for: testing, Jupyter notebooks, running examples. Fast rebuilds on code changes.

git clone https://github.com/<your-org>/rustima.git
cd rustima/        # the repo has an inner `rustima/` package directory

# 1) Create virtualenv + install Python deps (numpy, polars, pytest, etc.)
uv sync --extra dev

# 2) Compile Rust engine in release mode and link it in-place
uv run maturin develop --release

After this you can run:

uv run python -c "import rustima; print(rustima.version())"

Option B — Build a redistributable wheel

Best for: deploying to another machine, CI, production.

cd rustima/
uv sync --extra dev
CARGO_TARGET_DIR=target_wheel uv run maturin build --release --out /tmp/wheels
uv pip install --force-reinstall /tmp/wheels/rustima-*.whl

Copy the .whl from /tmp/wheels/ to the target machine and pip install it there (must have the same OS + Python version).

Verifying the install

import rustima
import numpy as np

print(rustima.version())                              # "0.1.0"
y = np.random.randn(100).cumsum()
r = rustima.sarimax_fit(y, order=(1, 1, 1), seasonal=(0, 0, 0, 0))
print(f"converged={r['converged']}, AIC={r['aic']:.2f}")

If this prints without error you're good to go. See Troubleshooting at the bottom if it fails.

Using inside Jupyter Notebook

cd rustima/
uv sync --extra dev
uv run maturin develop --release

# Register the venv as a Jupyter kernel (run once)
uv run python -m ipykernel install --user --name rustima --display-name "rustima"

# Launch Jupyter
uv run jupyter notebook

In the notebook, select the kernel "rustima" from the top-right corner, then:

import rustima
from rustima import SARIMAXModel, auto_arima

Which API Should I Use?

rustima exposes two layers. Pick one based on what you're doing:

You want to...	Use	Why
Replace statsmodels.SARIMAX with a faster drop-in	SARIMAXModel (high-level)	Same class/method names, same output format, statsmodels-style .summary()
Let the library pick the order for you	auto_arima() (high-level)	Automatic (p,d,q)(P,D,Q,s) search, like pmdarima
Fit thousands of series in parallel	rustima.sarimax_batch_fit (low-level)	Skips Python object overhead, releases GIL
Try many orders on one series	rustima.sarimax_grid_search (low-level)	Rayon-parallel over order combinations
Inspect raw log-likelihood / residuals for research	rustima.sarimax_loglike / sarimax_residuals	Direct access to Kalman filter output

Rule of thumb: Start with SARIMAXModel or auto_arima. Drop to the low-level API only when you need to fit many series/orders at once.

Your First Forecast (5-minute walkthrough)

A complete end-to-end example. Copy-paste into a Python file or notebook.

import numpy as np
from rustima import SARIMAXModel, auto_arima

# ── 1. Simulate monthly sales data with trend + yearly seasonality ─────────
rng = np.random.default_rng(42)
n = 120  # 10 years of monthly data
trend = 0.5 * np.arange(n)                           # linear upward trend
season = 10 * np.sin(2 * np.pi * np.arange(n) / 12)  # yearly cycle (s=12)
noise = rng.normal(0, 1.0, n)
y = trend + season + noise

# ── 2. Let auto_arima pick the orders for you ─────────────────────────────
auto_result = auto_arima(y, s=12, trace=True)  # trace=True prints each try
print(auto_result.search_summary())
# >>> Best: SARIMA(0,1,1)(0,1,1)[12]  AIC=345.67  (evaluated 23 models)

# ── 3. Inspect the chosen model ───────────────────────────────────────────
model = auto_result.result              # a SARIMAXResult
print(model.summary())                  # statsmodels-style parameter table
print(f"AIC={model.aic:.2f}  BIC={model.bic:.2f}")

# ── 4. Forecast the next 12 months with 95% confidence intervals ──────────
forecast = model.forecast(steps=12, alpha=0.05)
df = forecast.to_dataframe()            # Polars DataFrame
print(df)
# shape (12, 5): step | mean | variance | ci_lower | ci_upper

# ── 5. Check residuals (should look like random noise) ────────────────────
diag = model.diagnostics()
print(f"Ljung-Box p-value: {diag['ljung_box_pvalue'][0]:.3f}  (>0.05 = good)")

What to look at in the output:

• converged=True → optimization succeeded
• Low AIC / BIC → better model fit (relative to other orders)
• Ljung-Box p > 0.05 → residuals look like white noise (good)
• ci_lower / ci_upper → uncertainty band; wider = less confident

Quick Start

Low-Level API (rustima)

import numpy as np
import rustima

y = np.random.randn(200).cumsum()

# 1. Fit model
result = rustima.sarimax_fit(y, order=(1, 1, 1), seasonal=(0, 0, 0, 0))
print(f"Converged: {result['converged']}, AIC: {result['aic']:.2f}")

# 2. 10-step ahead forecast
fc = rustima.sarimax_forecast(
    y, order=(1, 1, 1), seasonal=(0, 0, 0, 0),
    params=np.array(result["params"]), steps=10
)
print(f"Forecast: {fc['mean'][:5]}")

# 3. Residual diagnostics
res = rustima.sarimax_residuals(
    y, order=(1, 1, 1), seasonal=(0, 0, 0, 0),
    params=np.array(result["params"])
)

High-Level API (SARIMAXModel — statsmodels compatible)

from rustima import SARIMAXModel

model = SARIMAXModel(y, order=(1, 1, 1), seasonal_order=(0, 0, 0, 0), trend="c")
result = model.fit()

# Parameter summary (fast, no inference stats)
print(result.summary())

# With Hessian-based inference (std err, z, p-value, CI)
print(result.summary(inference="hessian"))

# Side-by-side Hessian vs statsmodels inference
print(result.summary(inference="both"))

# Parameter table as Polars DataFrame
pt = result.params_table(inference="hessian")
print(pt)  # shape: (k, 7) — name, coef, std_err, z, p_value, ci_lower, ci_upper

print(f"AIC: {result.aic:.2f}, BIC: {result.bic:.2f}, HQIC: {result.hqic:.2f}")

# Forecast with confidence intervals + Polars DataFrame
fcast = result.forecast(steps=10, alpha=0.05)
print(fcast.predicted_mean)
df = fcast.to_dataframe()  # Polars: step, mean, variance, ci_lower, ci_upper
ci = fcast.conf_int()          # (10, 2) array [lower, upper]
ci_90 = fcast.conf_int(0.10)   # recalculate with different alpha

# In-sample prediction
pred = result.get_prediction(start=0, end=210)
pred_df = pred.to_dataframe()  # Polars: index, predicted_mean

# Standardized residuals
residuals = result.resid

# Residual diagnostics (Ljung-Box, Jarque-Bera, heteroskedasticity)
diag = result.diagnostics()

auto_arima — Automatic Order Selection

from rustima import auto_arima

# Stepwise (Hyndman-Khandakar, default)
res = auto_arima(y, max_p=5, max_q=5, s=12, stepwise=True, trace=True)
print(res.summary())           # Full statsmodels-style summary with inference
print(res.search_summary())    # Short 3-line summary (order, IC, model count)
print(res.result.forecast(steps=12).to_dataframe())

# Grid Search (Rayon parallel — one fit job per order combination)
res = auto_arima(y, max_p=3, max_q=3, s=7, stepwise=False, criterion="bic")
print(res.summary())

# Search history (Polars DataFrame)
print(res.history_dataframe())

auto_arima key parameters:

Parameter	Default	Description
max_p, max_q	5	Max AR/MA order
max_P, max_Q	2	Max seasonal AR/MA order
s	0	Seasonal period (0=non-seasonal)
d, D	None	Differencing order (None=auto-detect)
trend	"n"	Trend: 'n', 'c', 't', 'ct'
criterion	"aic"	Information criterion: "aic", "bic", "hqic"
stepwise	True	False → exhaustive grid search (Rayon parallel)
trace	False	True → print each model evaluation result

Trend Support

# Constant (intercept)
model = SARIMAXModel(y, order=(1, 1, 1), seasonal_order=(0, 0, 0, 0), trend="c")
# Linear trend (drift)
model = SARIMAXModel(y, order=(1, 1, 1), seasonal_order=(0, 0, 0, 0), trend="t")
# Constant + linear (intercept + drift)
model = SARIMAXModel(y, order=(1, 1, 1), seasonal_order=(0, 0, 0, 0), trend="ct")

result = model.fit()
print(result.param_names)  # ['intercept', 'drift', 'ar.L1', 'ma.L1'] (trend='ct')

Exogenous Regressors

import numpy as np

X_train = np.column_stack([np.arange(200), np.random.randn(200)])  # (200, 2)
X_future = np.column_stack([np.arange(200, 210), np.random.randn(10)])  # (10, 2)

model = SARIMAXModel(y, order=(1, 0, 1), seasonal_order=(0, 0, 0, 0), exog=X_train)
result = model.fit()
fcast = result.forecast(steps=10, exog=X_future)

Profiled Kalman-GLS Trust-Region for weakly identified exog coefficients. When the exog block is weakly identified (flat likelihood in the β direction), the default optimizer can leave β in a poor basin. The profile-trust-region method (academic name: Profiled Kalman-GLS Trust-Region, equivalently innovations-form profile likelihood with trust-region BFGS) removes β from the nonlinear optimizer and solves it exactly via innovation-space GLS at each likelihood evaluation. The optimizer only sees [trend | AR | MA | sAR | sMA | sigma2]. On exogenous SARIMAX cases it tends to reproduce R stats::arima(method="CSS-ML") β estimates in sign and magnitude:

result = model.fit(method="profile-trust-region", maxiter=200)

If n_exog == 0 the method falls back to the regular trust-region path. When simple_differencing=True, the same differencing operator is now applied to both endog and every exog column so that the regression Δy_t = (Δx_t) β + ARMA noise is solved on a consistent pre-differenced dataset. See rustima/docs/profiled_kalman_gls_plan.md for the algorithm, parameter-level comparison against statsmodels and R, and init-scheme caveats.

Batched Kalman filter inside PTR. Each profiled-likelihood evaluation runs the Kalman filter on y and once per exogenous column x_j at the same ψ; the state-space matrices, predicted covariance, gain, and innovation variance are bit-identical across these passes. rustima's kalman_filter_batched shares the O(k²) covariance recursion once and runs only the cheap O(k) mean recursion per series. For r=2 exog the per-evaluation Kalman cost drops from (1+r) = 3 full passes to 1 full pass plus three cheap mean updates — about a 2× per-PTR-fit speedup. Implementation: rustima/src/kalman.rs::kalman_filter_batched, called from ProfiledSarimaxObjective::profile_beta_and_loglike.

Parallel stepwise neighborhood search. auto_arima(stepwise=True) evaluates the 8–10 candidates around the current best in a single Rayon-parallel rustima.sarimax_grid_search call rather than one-by-one. On a 12-core machine the inner trust-region BFGS routines run concurrently, so each Hyndman-Khandakar step finishes in roughly the time of the slowest neighbour rather than the sum of all neighbours.

Automatic CPU pinning on Apple Silicon (macOS). On import, rustima/python/rustima/__init__.py detects the P-core count via sysctl hw.perflevel0.physicalcpu and exports RAYON_NUM_THREADS=<P-core count> before the native extension loads. This avoids E-cores becoming the per-step critical path on heterogeneous chips (M-series), yielding a 20–35% wall-time reduction on compute-heavy SARIMAX fits. The behaviour is a no-op on Linux/Windows and on macOS systems where the user has already set RAYON_NUM_THREADS themselves.

Batch Parallel Processing

# Fit 100 series with job-level Rayon parallelism
series_list = [np.random.randn(200) for _ in range(100)]

results = rustima.sarimax_batch_fit(
    series_list, order=(1, 0, 0), seasonal=(0, 0, 0, 0)
)

for i, r in enumerate(results):
    print(f"Series {i}: converged={r['converged']}, AIC={r['aic']:.2f}")

# Batch forecast with per-series parameters
params_list = [np.array(r["params"]) for r in results]
forecasts = rustima.sarimax_batch_forecast(
    series_list, order=(1, 0, 0), seasonal=(0, 0, 0, 0),
    params_list=params_list, steps=10, alpha=0.05,
)

Grid Search Parallel Processing

# Fit multiple ARIMA orders via Rayon in one call
results = rustima.sarimax_grid_search(
    y,
    order_list=[(0,1,0), (1,1,0), (1,1,1), (2,1,1)],
    seasonal_list=[(0,0,0,0)] * 4,
    trend="c",
)
for r in results:
    if "error" not in r:
        print(f"{r['order']}: AIC={r['aic']:.3f}")

---

Architecture

System Overview

graph TB
    subgraph Python["Python Layer"]
        USER["User Code"]
        MODEL["SARIMAXModel<br/><i>python/rustima/model.py</i>"]
        AUTO["auto_arima<br/><i>python/rustima/auto.py</i>"]
        USER --> MODEL
        USER --> AUTO
    end

    subgraph PyO3["PyO3 Bindings — lib.rs"]
        FIT["sarimax_fit()"]
        LL["sarimax_loglike()"]
        FC["sarimax_forecast()"]
        RES["sarimax_residuals()"]
        BF["sarimax_batch_fit()"]
        BFC["sarimax_batch_forecast()"]
        BLL["sarimax_batch_loglike()"]
        GS["sarimax_grid_search()"]
        INF["sarimax_inference()"]
        DIAG["sarimax_diagnostics()"]
    end

    MODEL --> FIT
    MODEL --> FC
    MODEL --> RES
    MODEL --> INF
    MODEL --> DIAG
    AUTO --> GS
    AUTO --> FIT
    USER --> BF
    USER --> BFC
    USER --> BLL

    subgraph Rust["Rust Engine"]
        OPT["optimizer.rs<br/>L-BFGS-B + L-BFGS + Nelder-Mead"]
        KAL["kalman.rs<br/>Kalman filter / log-likelihood"]
        FCAST["forecast.rs<br/>h-step prediction + residuals"]
        BATCH["batch.rs<br/>Rayon par_iter()"]
        SS["state_space.rs<br/>Harvey representation T, Z, R, Q"]
        INIT["initialization.rs<br/>Approximate diffuse init"]
        SP["start_params.rs<br/>Hannan-Rissanen + CSS"]
        POLY["polynomial.rs<br/>AR/MA polynomial expansion"]
        PAR["params.rs<br/>Monahan transform"]
        SCR["score.rs<br/>Analytical gradient"]
        INFER["inference.rs<br/>Hessian / OPG"]
    end

    FIT --> OPT
    LL --> KAL
    FC --> FCAST
    RES --> FCAST
    BF --> BATCH
    BFC --> BATCH
    BLL --> BATCH
    GS --> BATCH
    INF --> INFER

    BATCH --> OPT
    BATCH --> FCAST
    OPT --> KAL
    OPT --> SP
    OPT --> SCR
    FCAST --> KAL
    KAL --> SS
    KAL --> INIT
    SS --> POLY
    SS --> PAR
    OPT --> PAR
    SCR --> KAL
    INFER --> KAL

    style Python fill:#3776ab,color:#fff
    style PyO3 fill:#f7a41d,color:#000
    style Rust fill:#dea584,color:#000

Model Fitting Flow

sequenceDiagram
    participant P as Python
    participant L as lib.rs (PyO3)
    participant O as optimizer.rs
    participant S as start_params.rs
    participant K as kalman.rs
    participant SS as state_space.rs
    participant PR as params.rs
    participant SC as score.rs

    P->>L: sarimax_fit(y, order, seasonal, trend)
    L->>L: Build SarimaxConfig (incl. trend)
    L->>O: fit(endog, config, method, maxiter)

    O->>S: compute_start_params(endog, config)
    S->>S: Differencing → Hannan-Rissanen / Burg+CSS
    S-->>O: initial θ₀

    O->>PR: untransform_params(θ₀)
    PR-->>O: unconstrained x₀

    loop L-BFGS-B iteration (default max 200)
        O->>PR: transform_params(x_k)
        PR-->>O: constrained θ_k
        O->>SS: StateSpace::new(config, θ_k, y)
        SS->>SS: Build T, Z, R, Q matrices + state_intercept(c_t)
        SS-->>O: state_space
        O->>K: kalman_loglike(y, ss, init)
        K->>K: Predict → Observe → Update loop
        K-->>O: loglike, scale
        O->>SC: score(y, ss, init, config, params)
        SC-->>O: analytical gradient
        O->>O: cost = -loglike
        Note over O: Convergence check (pgtol / factr)
    end

    alt L-BFGS-B fails
        O->>O: Nelder-Mead fallback (max 500 iter)
    end

    O->>PR: transform_params(x_final)
    PR-->>O: θ_final
    O->>O: AIC = -2·ll + 2·k<br/>BIC = -2·ll + k·ln(n)
    O-->>L: FitResult
    L-->>P: dict

Forecast Flow

sequenceDiagram
    participant P as Python
    participant L as lib.rs
    participant F as forecast.rs
    participant K as kalman.rs
    participant SS as state_space.rs

    P->>L: sarimax_forecast(y, order, seasonal, params, steps, alpha, trend)
    L->>F: forecast_pipeline(endog, config, params, steps, alpha)

    F->>SS: StateSpace::new(config, params, endog)
    SS-->>F: state_space (T, Z, R, Q)

    F->>K: kalman_filter(endog, ss, init)
    K->>K: Full Kalman filter over time series
    K-->>F: KalmanFilterOutput (filtered_state, filtered_cov, scale)

    F->>F: forecast(ss, filter_output, steps, alpha)

    loop h = 1 to steps
        F->>F: ŷ_h = Z' · â_h
        F->>F: Var_h = Z' · P̂_h · Z · σ²
        F->>F: CI_h = ŷ_h ± z(α/2) · √Var_h
        F->>F: â_{h+1} = T · â_h + c_{n+h}  (trend state_intercept)
        F->>F: P̂_{h+1} = T · P̂_h · T' + R·Q·R'
    end

    F-->>L: ForecastResult (mean, variance, ci_lower, ci_upper)
    L-->>P: dict

Batch & Grid Search Parallel Processing Flow

sequenceDiagram
    participant P as Python
    participant L as lib.rs
    participant B as batch.rs
    participant R as Rayon ThreadPool

    P->>L: sarimax_batch_fit(series_list, order, seasonal)
    L->>L: Convert Vec<Vec<f64>> (GIL held)
    L->>B: batch_fit(series, config, method, maxiter)

    B->>R: series.par_iter()

    par [Rayon parallel execution]
        R->>R: Thread 0: optimizer::fit(series[0])
        R->>R: Thread 1: optimizer::fit(series[1])
        R->>R: Thread N: optimizer::fit(series[N])
    end

    R-->>B: Vec<Result<FitResult>>
    B-->>L: results
    L-->>P: List[dict]

    Note over P,R: grid_search: same data + different order combos in parallel

    P->>L: sarimax_grid_search(y, order_list, seasonal_list)
    L->>B: grid_search_fit(endog, configs)
    B->>R: configs.par_iter()

    par [Rayon parallel — order combinations]
        R->>R: Thread 0: fit(endog, config[0])
        R->>R: Thread 1: fit(endog, config[1])
        R->>R: Thread N: fit(endog, config[N])
    end

    R-->>B: Vec<Result<FitResult>>
    B-->>L: results (with order keys)
    L-->>P: List[dict]

---

Python API Reference

Low-Level Functions (rustima)

rustima.sarimax_loglike

Computes log-likelihood at given parameters.

ll = rustima.sarimax_loglike(
    y,
    order=(1, 1, 1),           # (p, d, q)
    seasonal=(1, 1, 1, 12),    # (P, D, Q, s)
    params=np.array([0.5, 0.3, 0.2, -0.4]),  # [ar, ma, sar, sma]
    concentrate_scale=True,    # concentrate sigma2 in likelihood
    trend="c",                 # trend: "n", "c", "t", "ct"
)

rustima.sarimax_fit

Fits model via MLE.

result = rustima.sarimax_fit(
    y,
    order=(1, 0, 1),
    seasonal=(0, 0, 0, 0),
    enforce_stationarity=True,   # AR stationarity constraint
    enforce_invertibility=True,  # MA invertibility constraint
    method="lbfgsb",             # "lbfgsb" | "lbfgsb-multi" | "lbfgs" | "bfgs"
                                 # | "trust-region" | "profile-trust-region"
                                 # | "nelder-mead"
    maxiter=500,
    trend="c",
)

Return dict:

Key	Type	Description
params	list[float]	Estimated parameters [trend..., exog..., ar..., ma..., sar..., sma...]
loglike	float	Final log-likelihood
scale	float	Estimated variance (sigma2)
aic	float	AIC
bic	float	BIC
n_obs	int	Number of observations
n_params	int	Number of estimated parameters (incl. sigma2)
n_iter	int	Optimizer iterations (L-BFGS, NM) or function evaluations (L-BFGS-B)
converged	bool	True convergence, not just maxiter reached
method	str	Optimization method used

rustima.sarimax_forecast

Performs h-step ahead forecast with confidence intervals.

fc = rustima.sarimax_forecast(
    y,
    order=(1, 0, 0),
    seasonal=(0, 0, 0, 0),
    params=np.array([0.65]),
    steps=10,
    alpha=0.05,        # 95% confidence interval
    exog=X_train,      # past exog if model uses exog
    future_exog=X_future,
    trend="c",
)

print(fc["mean"])       # point forecast (list[float])
print(fc["ci_lower"])   # lower bound
print(fc["ci_upper"])   # upper bound
print(fc["variance"])   # forecast variance

rustima.sarimax_residuals

Computes residuals and standardized residuals.

res = rustima.sarimax_residuals(
    y,
    order=(1, 0, 1),
    seasonal=(0, 0, 0, 0),
    params=np.array([0.5, 0.3]),
    trend="c",
)

print(res["residuals"])                # innovations v_t
print(res["standardized_residuals"])   # v_t / sqrt(F_t * sigma2)

rustima.sarimax_batch_fit

Fits N series in parallel using a Rayon thread pool, with one complete fit job per series.

results = rustima.sarimax_batch_fit(
    series_list,
    order=(1, 0, 0),
    seasonal=(0, 0, 0, 0),
    enforce_stationarity=True,
    method="lbfgsb",
    maxiter=500,
    trend="c",
)
# Returns: list[dict] — same keys as sarimax_fit
# Failed series: {"error": "...", "converged": false}

rustima.sarimax_batch_forecast

Forecasts N series (each with different params) in parallel, with one complete forecast job per series.

params_list = [np.array(r["params"]) for r in results]

forecasts = rustima.sarimax_batch_forecast(
    series_list,
    order=(1, 0, 0),
    seasonal=(0, 0, 0, 0),
    params_list=params_list,
    steps=10,
    alpha=0.05,
)
# Returns: list[dict] with mean, variance, ci_lower, ci_upper

rustima.sarimax_grid_search

Fits a single series with multiple ARIMA order combinations in parallel via Rayon, with one complete fit job per configuration.

results = rustima.sarimax_grid_search(
    y,
    order_list=[(1,0,0), (1,0,1), (2,0,0)],
    seasonal_list=[(0,0,0,0)] * 3,
    enforce_stationarity=True,
    enforce_invertibility=True,
    trend="c",
    method="lbfgsb",
    maxiter=500,
)
# Returns: list[dict] — same keys as sarimax_fit + "order", "seasonal_order"

rustima.sarimax_inference

Computes Hessian or OPG-based inference statistics on fitted parameters.

inf = rustima.sarimax_inference(
    y, order=(1,0,1), seasonal=(0,0,0,0),
    params=np.array([0.5, 0.3]),
    method="hessian",   # "hessian" | "opg"
    alpha=0.05,
    trend="c",
)
print(inf["std_err"])   # standard errors
print(inf["z_stat"])    # z statistics
print(inf["p_value"])   # two-sided p-values
print(inf["ci_lower"])  # CI lower bound
print(inf["ci_upper"])  # CI upper bound

rustima.sarimax_diagnostics

Performs residual diagnostic tests.

diag = rustima.sarimax_diagnostics(
    y, order=(1,0,1), seasonal=(0,0,0,0),
    params=np.array([0.5, 0.3]),
    trend="c",
)
print(diag["ljung_box_stat"])     # Ljung-Box Q statistic
print(diag["ljung_box_pvalue"])   # Ljung-Box p-value
print(diag["jarque_bera_stat"])   # Jarque-Bera normality
print(diag["het_stat"])           # Heteroskedasticity test

High-Level Classes (rustima)

statsmodels-compatible Python wrapper using the Rust engine.

SARIMAXModel

from rustima import SARIMAXModel

model = SARIMAXModel(
    endog=y,                        # time series data
    order=(1, 1, 1),                # ARIMA(p, d, q)
    seasonal_order=(1, 0, 0, 12),   # (P, D, Q, s)
    exog=X,                         # optional: exogenous regressors
    trend="c",                      # trend: 'n', 'c', 't', 'ct'
    enforce_stationarity=True,
    enforce_invertibility=True,
)

SARIMAXResult

Returned by model.fit().

result = model.fit(method="lbfgsb", maxiter=500)

# Attributes
result.params          # np.ndarray — estimated parameters
result.param_names     # list[str] — e.g. ['intercept', 'ar.L1', 'ma.L1']
result.llf             # float — log-likelihood
result.aic             # float — AIC
result.bic             # float — BIC
result.hqic            # float — HQIC (Hannan-Quinn)
result.scale           # float — sigma2
result.nobs            # int — number of observations
result.converged       # bool — convergence status
result.method          # str — optimization method
result.resid           # np.ndarray — standardized residuals (lazy computed)

# Methods
result.forecast(steps=10, alpha=0.05)     # → ForecastResult
result.forecast(steps=10, exog=X_future)  # with future exog
result.get_forecast(steps=10, alpha=0.05) # alias (statsmodels compat)
result.get_prediction(start=0, end=210)   # → PredictionResult (in-sample + OOS)
result.summary()                          # → str (default: parameter table)
result.summary(inference="hessian")       # → str (std err / z / p / CI)
result.summary(inference="statsmodels")   # → str (statsmodels inference values)
result.summary(inference="both")          # → str (side-by-side comparison)
result.params_table(inference="hessian")  # → Polars DataFrame
result.diagnostics()                      # → dict (Ljung-Box, Jarque-Bera, het)

# Machine-friendly parameter summary dict
ps = result.parameter_summary(alpha=0.05, inference="hessian")

Parameter vector layout:

[trend(kt) | exog(k) | ar(p) | ma(q) | sar(P) | sma(Q)]

Parameter name convention (consistent with statsmodels):

Component	Name
Constant (trend='c')	intercept
Linear trend (trend='t')	drift
Constant + linear (trend='ct')	intercept, drift
Exogenous regressors	x1, x2, ..., xk
AR	ar.L1, ar.L2, ..., ar.Lp
MA	ma.L1, ma.L2, ..., ma.Lq
Seasonal AR	ar.S.L{s}, ar.S.L{2s}, ..., ar.S.L{Ps}
Seasonal MA	ma.S.L{s}, ma.S.L{2s}, ..., ma.S.L{Qs}
Variance	sigma2 (only when concentrate_scale=False)

ForecastResult

fcast = result.forecast(steps=10)

fcast.predicted_mean   # np.ndarray — point forecasts
fcast.variance         # np.ndarray — forecast variance
fcast.ci_lower         # np.ndarray — CI lower bound
fcast.ci_upper         # np.ndarray — CI upper bound

fcast.conf_int()       # np.ndarray (steps, 2) — [lower, upper]
fcast.conf_int(0.10)   # recalculate with different alpha
fcast.to_dataframe()   # Polars DataFrame (step, mean, variance, ci_lower, ci_upper)

PredictionResult

pred = result.get_prediction(start=0, end=210)

pred.predicted_mean    # np.ndarray — predictions (in-sample + OOS)
pred.to_dataframe()    # Polars DataFrame (index, predicted_mean)

AutoARIMAResult

from rustima import auto_arima

res = auto_arima(y, max_p=5, max_q=5, s=12)

res.result             # SARIMAXResult — best model fit result
res.order              # tuple (p, d, q) — best order
res.seasonal_order     # tuple (P, D, Q, s)
res.best_ic            # float — best information criterion value
res.criterion          # str — criterion used ("aic", "bic", "hqic")
res.history            # list[dict] — search history
res.summary()          # str — full statsmodels-style summary with inference
res.search_summary()   # str — short 3-line search summary
res.history_dataframe()  # Polars DataFrame — search history table

---

Core Algorithms

1. State-Space Representation (state_space.rs)

Converts SARIMA(p,d,q)(P,D,Q,s) to the Harvey (1989) representation.

State equation:

alpha_{t+1} = T * alpha_t + c_t + R * eta_t,    eta_t ~ N(0, Q)
y_t         = Z' * alpha_t + d_t + eps_t,        eps_t ~ N(0, H), H=0

State vector alpha dimension: k_states = k_states_diff + k_order, where

• k_states_diff = d + s*D (differencing states)
• k_order = max(p + s*P, q + s*Q + 1) (ARMA companion matrix dimension)

2. Kalman Filter (kalman.rs)

Standard Harvey-form Kalman filter for log-likelihood evaluation.

For each t = 0, ..., n-1:
  1. Innovation:           v_t = y_t - Z' * a_{t|t-1} - d_t
  2. Innovation variance:  F_t = Z' * P_{t|t-1} * Z
  3. Kalman gain:          K_t = P_{t|t-1} * Z / F_t
  4. State update:         a_{t|t} = a_{t|t-1} + K_t * v_t
  5. Covariance update:    P_{t|t} = (I - K*Z') * P * (I - K*Z')'  [Joseph form]
  6. Prediction:           a_{t+1|t} = T * a_{t|t} + c_t
  7. Covariance prediction: P_{t+1|t} = T * P_{t|t} * T' + R*Q*R'

Concentrated log-likelihood (concentrate_scale=true, default):

σ²_hat = (1/n_eff) * Σ(v_t² / F_t)
loglike = -n_eff/2 * ln(2π) - n_eff/2 * ln(σ²_hat) - n_eff/2 - 0.5 * Σ ln(F_t)

3. Analytical Score Vector (score.rs)

Computes ∂loglike/∂θ via tangent-linear Kalman filter (Koopman & Shephard, 1992; Harvey, 1989) in a single forward pass, avoiding the O(n_params + 1) cost of numerical differentiation. To our knowledge, this is the first Rust implementation of the analytical exact score for state-space models.

Performance optimizations:

• Sparse T: Exploits companion matrix T sparsity to reduce O(k³) → O(nnz×k) (~23x speedup at k=27)
• Steady-state detection: Once P converges, freezes dP/dz/dF and updates only da/dv, skipping O(k³×np) ops for all subsequent timesteps

4. Parameter Transform (params.rs)

Optimization in unconstrained space, evaluation in constrained space.

Stationarity constraint (AR) — Monahan (1984) / Jones (1980):

Unconstrained → PACF:  r_k = x_k / sqrt(1 + x_k²)
PACF → AR coefficients: Levinson-Durbin recursion

5. Optimizer (optimizer.rs)

Minimizes negative log-likelihood.

Strategy:

1. Starting values: Hannan-Rissanen (seasonal) or CSS-based estimation (start_params.rs)
2. L-BFGS-B (default): Bounded + analytical gradient, pgtol=1e-5, factr=1e7
3. L-BFGS-B multi-start: 3 perturbed restarts for robustness
4. L-BFGS: MoreThuente line search, grad_tol=1e-8, cost_tol=1e-12
5. Nelder-Mead fallback: Auto-switch on L-BFGS failure, 5% scale simplex
6. Information criteria: AIC = -2·ll + 2·k, BIC = -2·ll + k·ln(n), HQIC = -2·ll + 2·k·ln(ln(n))

6. Forecast (forecast.rs)

h-step ahead prediction from the Kalman filter's final state.

For each forecast step h = 1, ..., steps:
  ŷ_h = Z' · â_h                   (point forecast)
  F_h = Z' · P̂_h · Z · σ²          (forecast variance)
  CI_h = ŷ_h ± z_{α/2} · √F_h      (confidence interval)
  â_{h+1} = T · â_h + c_{n+h}       (state propagation + trend)
  P̂_{h+1} = T · P̂_h · T' + R·Q·R'  (covariance propagation)

7. Batch & Grid Search Parallelism (batch.rs)

Uses Rayon par_iter() for work-stealing parallelism at the independent-job level.

Batch processing (same order + different series):

• All series share the same SarimaxConfig (Clone, Send + Sync)
• Each worker independently runs one full StateSpace::new() → fit() / forecast_pipeline() pipeline for one series
• Failed series do not affect others (Vec<Result<T>>)

Grid search (same series + different orders):

• grid_search_fit(endog, configs) — parallel over order combinations via configs.par_iter(), one full fit per config

---

Benchmarks

All benchmarks run on macOS 15.1 (Apple Silicon arm64), Python 3.14, rustima 0.1.0 vs statsmodels 0.14.6.

Accuracy vs statsmodels

Both engines fitted with enforce_stationarity=True, enforce_invertibility=True, concentrate_scale=True, trend='n' on identical data.

Model	n	k	Max |Δparam|	|Δloglike|	|ΔAIC|
AR(1)	200	1	0.000124	0.0000	0.0000
AR(2)	300	2	0.001902	0.0007	0.0013
MA(1)	200	1	0.003862	0.0015	0.0031
ARMA(1,1)	300	2	0.004170	0.0048	0.0096
ARIMA(1,1,1)	300	2	0.026525	0.0011	0.0022
ARIMA(2,1,1)	400	3	0.833380	0.4884	0.9768
SARIMA(1,0,0)(1,0,0,4)	200	2	0.002865	0.0014	0.0028
SARIMA(0,1,1)(0,1,1,12)	300	2	0.054174	1.1886	2.3771
SARIMA(1,1,1)(1,1,1,12)	300	4	0.006335	0.0012	0.0025

Parameter accuracy within 0.006 and log-likelihood difference within 0.005 for most models.

High-Order Models — Accuracy vs statsmodels

16 models from non-seasonal ARIMA(4–5) to hourly (s=24) high-order SARIMA. rs_worse_by shows how much lower the rustima log-likelihood is versus statsmodels; negative (★) means rustima found a better optimum.

Model	k_states	rs_worse_by	Result
ARIMA(4,1,1)	5	+0.002	Pass
ARIMA(4,1,4)	6	-2.07	Pass ★
ARIMA(3,1,3)	5	+0.004	Pass
ARIMA(5,1,1)	6	~0	Pass
ARIMA(1,1,5)	7	~0	Pass
SARIMA(3,1,2)(2,1,1,4)	16	+0.86	Pass
SARIMA(2,1,1)(2,1,1,4)	15	+2.40	Pass
SARIMA(2,1,2)(1,1,1,7)	18	+0.003	Pass
SARIMA(3,1,1)(1,1,1,7)	18	+0.04	Pass
SARIMA(4,1,1)(2,1,1,12)	41	+0.07	Pass
SARIMA(4,1,4)(2,1,2,12)	42	+5.90	Pass
SARIMA(2,1,2)(2,1,1,12)	39	-2.46	Pass ★
SARIMA(3,1,1)(2,1,1,12)	40	+0.05	Pass
SARIMA(2,1,1)(2,1,1,24)	75	+1.35	Pass
SARIMA(2,1,2)(1,1,1,24)	52	+1.01	Pass
SARIMA(1,1,1)(2,1,1,24)	74	+0.32	Pass

16/16 pass. ★ models: statsmodels issued ConvergenceWarning and failed to converge, while rustima found a better solution.

---

Speed — Single Fit

Best-of-5 wall clock time (lower is better):

Model	rustima	statsmodels	Speedup
AR(1) n=200	0.0 ms	2.9 ms	66.8x
ARMA(1,1) n=300	0.1 ms	6.1 ms	41.6x
ARIMA(1,1,1) n=300	0.4 ms	7.6 ms	17.9x
ARIMA(2,1,2) n=400	1.5 ms	31.4 ms	20.4x
SARIMA(1,1,1)(1,1,1,12) n=300	122.6 ms	928.7 ms	7.6x
SARIMA(1,0,0)(1,0,0,7) n=365	0.5 ms	19.9 ms	38.0x
SARIMA(1,1,1)(1,1,1,24) n=2160	2547.8 ms	7364.5 ms	2.9x

Average 27.9x, max 66.8x. Non-seasonal models: 17–67x faster; high-order seasonal SARIMA: 2.9–7.6x faster.

Speed — Batch Fit (Rayon Parallel)

AR(1) n=200/series:

Batch Size	rustima	statsmodels	Speedup
10 series	0.2 ms	32.2 ms	165.7x
50 series	0.7 ms	157.3 ms	232.4x
100 series	1.2 ms	312.2 ms	269.8x

Speed — Grid Search Parallel vs Sequential

Scenario	Parallel (ms)	Sequential (ms)	Speedup
n=200, 9 combos	1.2	1.8	1.53x
n=500, 9 combos	2.0	4.3	2.16x
n=500, 25 combos	0.9	4.3	4.59x
n=2000, 9 combos	2.5	11.2	4.53x

Speed — auto_arima

Scenario	n	Mode	Time	Models	AIC
Daily s=7 (2yr)	730	stepwise	5.68s	23	1099.16
Daily s=7 (2yr)	730	grid (parallel)	4.52s	81	1099.16
Hourly s=24 (90d)	2160	stepwise	19.21s	17	3222.21
Hourly s=24 (90d)	2160	grid (parallel)	2.59s	16	3222.21
Hourly s=24 (30d)	720	stepwise	20.04s	17	1099.85
Hourly s=24 (30d)	720	grid (parallel)	1.66s	16	1246.49

Grid search with Rayon parallelization is up to 12x faster than stepwise on hourly (s=24) data.

---

Project Structure

rustima/
├── Cargo.toml                      # Rust dependencies and build config
├── pyproject.toml                   # Python package config (maturin)
├── build.rs                         # cc build script (compiles lbfgsb_c/)
│
├── src/                             # Rust engine (19 modules, ~11,960 LOC)
│   ├── lib.rs                       # PyO3 module entry point (11 Python functions)
│   ├── types.rs                     # SarimaxOrder, SarimaxConfig, Trend, FitResult
│   ├── error.rs                     # SarimaxError (thiserror-based)
│   ├── params.rs                    # Parameter struct + Monahan/Jones transform
│   ├── polynomial.rs                # AR/MA polynomial expansion (polymul, reduced_ar/ma)
│   ├── state_space.rs               # Harvey state-space T, Z, R, Q, c_t construction
│   ├── initialization.rs            # Approximate diffuse / DARE initialization
│   ├── kalman.rs                    # Kalman filter (loglike + full filter)
│   ├── score.rs                     # Analytical gradient (sparse tangent-linear + steady-state)
│   ├── css.rs                       # Conditional sum-of-squares log-likelihood
│   ├── inference.rs                 # Numerical Hessian / OPG inference
│   ├── start_params.rs              # Hannan-Rissanen + CSS initial parameter estimation
│   ├── optimizer.rs                 # L-BFGS-B + L-BFGS + Nelder-Mead MLE
│   ├── forecast.rs                  # h-step prediction + residuals + z_score
│   ├── batch.rs                     # Rayon-based batch/grid search parallel processing
│   ├── pipeline.rs                  # Shared helpers (kalman_eval, kalman_filter_full)
│   ├── lbfgsb_ffi.rs                # L-BFGS-B C FFI bindings (unsafe extern)
│   ├── lbfgsb_wrapper.rs            # Safe Rust wrapper around L-BFGS-B C solver
│   └── test_helpers.rs              # Shared test utilities (cfg(test) only)
│
├── python/
│   └── rustima/                     # Python package (native ext + high-level API)
│       ├── __init__.py              # Package exports (low-level + high-level API)
│       ├── model.py                 # SARIMAXModel, SARIMAXResult, ForecastResult, PredictionResult
│       ├── auto.py                  # auto_arima, AutoARIMAResult
│       └── rustima.*.so             # Compiled Rust extension (built by maturin)
│
├── python_tests/                    # Python integration tests (371 tests, 16 modules)
│   ├── conftest.py                  # pytest fixtures
│   ├── generate_fixtures.py         # statsmodels reference data generator
│   └── test_*.py                    # test_fit, test_forecast, test_batch, test_exog, etc.
│
├── lbfgsb_c/                        # L-BFGS-B C source (compiled via cc build-dep)
│   ├── lbfgsb.c                     # Main L-BFGS-B solver
│   ├── lbfgsb.h                     # Header
│   ├── linesearch.c                 # Line search subroutine
│   ├── linpack.c                    # LINPACK routines
│   └── miniCBLAS.c                  # Minimal CBLAS routines
│
├── tests/fixtures/                  # statsmodels reference data (JSON)
│
└── benches/                         # Criterion benchmarks
    ├── bench_kalman.rs              # Kalman loglike performance
    └── bench_fit.rs                 # Single/batch fit performance

Dependencies

Rust (Cargo.toml)

Crate	Version	Purpose
nalgebra	0.34	Dynamic matrix/vector operations (DMatrix, DVector)
argmin	0.11	L-BFGS, Nelder-Mead optimization framework
argmin-math	0.5	nalgebra integration for argmin
anyhow	1	Error context propagation
statrs	0.18	Statistical distributions
rayon	1.10	Data parallelism (work-stealing thread pool)
pyo3	0.28	Python C-API bindings
numpy	0.28	NumPy array zero-copy interop
thiserror	2	Error type macros
serde / serde_json	1	Test fixture serialization
cc	1	C compiler driver (build-dep, compiles lbfgsb_c/)

Note: L-BFGS-B is not a crate dependency — it is compiled from vendored C source (lbfgsb_c/) via cc and called through a safe Rust FFI wrapper (lbfgsb_ffi.rs + lbfgsb_wrapper.rs).

Python (pyproject.toml)

Package	Purpose
numpy >= 1.24	Array operations (runtime)
polars >= 1.0	DataFrame output (runtime)
ipykernel >= 7.2	Jupyter notebook integration (runtime)
pytest >= 7.0	Test framework (dev)
statsmodels >= 0.14	Reference result generation (dev)
scipy >= 1.10	Statistical utilities (dev)
pandas >= 2.0	Benchmark comparison utilities (dev)
matplotlib >= 3.7	Visualization for benchmarks/reports (dev)
maturin >= 1.7	Rust → Python wheel build (dev)

Development

# Rust unit tests (155 tests)
cargo test --all-targets

# Python integration tests (371 tests, wheel build required first)
uv run maturin develop --release
uv run python -m pytest python_tests/ -v

# Run a single Python test
uv run python -m pytest python_tests/test_fit.py::test_arima_111_fit -v

# Run a single Rust test
cargo test test_name

# Benchmarks
cargo bench

# Regenerate statsmodels reference fixtures
uv run python python_tests/generate_fixtures.py

Test Summary

Category	Tests	Coverage
Rust unit tests	155	types, params, polynomial, state_space, initialization, kalman, score, css, inference, start_params, optimizer, forecast, batch
Python smoke	14	import, version, basic API checks
Python fit	13	fitting, AIC/BIC, convergence, start_params, Nelder-Mead
Python forecast	9	forecast mean, CI, residuals vs statsmodels
Python input validation	64	param length, seasonal D/s, bounds, exog, NaN/Inf
Python batch	6	batch fit/forecast, parallel perf, error isolation
Python exog	14	exogenous regressors, future_exog, batch exog
Python multi-order accuracy	27	multi-order cross-validation vs statsmodels
Python high-order accuracy	20	ARIMA(4–5), SARIMA(4,1,4)(2,1,2,12), s=24
Python inference	70	Rust Hessian/OPG inference, statsmodels parity
Python trend	16	trend='n','c','t','ct' fit/forecast/residuals/summary
Python Polars	14	to_dataframe(), params_table(), PredictionResult, HQIC
Python auto_arima	19	stepwise, grid, history, criterion, summary
Python safety guards	44	edge case safety, bounds checking
Python simple differencing	22	simple_differencing=True path
Python matrix tier A	12	state-space matrix construction validation
Python prediction quality	7	in-sample/out-of-sample prediction accuracy
Total	526	155 Rust + 371 Python

Troubleshooting & FAQ

Install / Build problems

error: linker 'cc' not found / failed to run custom build command for 'rustima' You don't have a C compiler. On macOS run xcode-select --install; on Ubuntu sudo apt install build-essential; on Windows install MSVC Build Tools.

ModuleNotFoundError: No module named 'rustima' after maturin develop The Rust extension was built but Python can't find it. Check you are running Python inside the same venv:

uv run python -c "import sys; print(sys.executable)"
# Should point to .venv/bin/python inside the rustima/ folder

If you're in Jupyter, make sure you selected the "rustima" kernel (see Installation).

Jupyter notebook says rustima is missing even though uv run python works Your notebook kernel is pointing to a different Python. Re-register it:

uv run python -m ipykernel install --user --name rustima --display-name "rustima"

Then restart the kernel and pick "rustima" from the kernel menu.

error: can't find Rust compiler Install Rust: curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh, then source ~/.cargo/env.

Modeling questions

My model didn't converge (converged=False). What now?

1. If the model has exogenous regressors, try method="profile-trust-region" — it eliminates β from the nonlinear search by Kalman-GLS profiling and often matches R stats::arima β estimates more closely.
2. Try method="nelder-mead" (slower but more robust)
3. Increase maxiter=1000
4. Reduce the orders — high p, q, P, Q often fail to identify
5. Check your data has no NaN/Inf: np.isfinite(y).all() should be True

auto_arima takes forever on my hourly data (s=24) Use stepwise=False to switch to the Rayon-parallel grid search. On hourly data it's often 10x faster than stepwise (see benchmarks).

My forecast looks like a flat line Usually means d=0 on a trending series, or the model decided the best prediction is the mean. Try setting d=1 manually, or add trend='c' / trend='t'.

Results are slightly different from statsmodels Expected. rustima uses L-BFGS-B with analytical gradients (statsmodels uses L-BFGS with numerical gradients). Differences within |ΔAIC| < 2 are normal. See the accuracy benchmarks above.

How do I know which order / seasonal_order to use?

1. Easy path: just call auto_arima(y, s=<period>) and use what it picks
2. Theory path: ACF/PACF plots + unit-root tests — see FPP3 Ch. 9

When should I use simple_differencing=True? Only when you want to match R's default behavior or reproduce results with R-style AIC/BIC on a pre-differenced series. Default (False) matches statsmodels. When True, the same differencing operator is applied to both endog and every exog column (so the regression Δy_t = (Δx_t) β + noise is fitted on a consistently pre-differenced dataset). LL/AIC on this path uses n = n_obs − d − s·D effective observations.

Why does my exog SARIMAX LL differ from statsmodels on the simple_differencing=True path? Under matched enforce_stationarity=False, enforce_invertibility=False, rustima's Kalman log-likelihood at the same parameter vector is bit- identical to statsmodels. Under enforce=True, the two engines initialize the stationary state covariance with different schemes, which accumulates a likelihood-scale offset over n. Parameter estimates remain close. For like-for-like LL comparisons across engines on exogenous models, prefer the exact path (simple_differencing=False) where this divergence does not appear.

Performance

Why is the first fit slow but subsequent ones fast? Python imports, Rust JIT-friendly compilation of specialized monomorphizations, and Rayon thread-pool warmup. Warm up once before benchmarking.

Debug build is slower than statsmodels! Always build with --release. Without it, Rust runs with no optimizations and is ~10× slower than a release build.

Limitations

• Seasonal differencing D > 1 not supported (only D = 0 or 1)
• Maximum forecast steps: 10,000; alpha must be in (0, 1)
• State dimension capped at 1,024 (bounds memory and compute blow-up on extreme orders, but does not eliminate OOM risk in the absolute sense)
• auto_arima automatic differencing detection uses ADF test (scipy) or variance-reduction heuristic

License

GPL-3.0-or-later. See LICENSE for details.