AR/MA estimation (ARIMA building blocks)

Durbin-Levinson AR estimation, innovations MA fitting and ARIMA forecasting building blocks.

arima fits AR(p) and MA(q) models in pure Mojo with zero Python imports — replacing statsmodels.tsa (AutoReg / ARIMA Yule-Walker fitting, innovations MA estimation, conditional-sum-of-squares likelihood). It complements timeseries_stats, which provides ACF/PACF diagnostics only; this module actually estimates coefficients. Model convention matches statsmodels (positive θ on lagged innovations), with the sample mean μ handled internally.

The modules

Module	What it does
arima	fits AR(p) and MA(q) coefficients in pure Mojo — Yule-Walker/Durbin-Levinson AR estimation, innovations-algorithm MA estimation, CSS log-likelihood scoring, and order-d differencing/undifferencing
timeseries_stats	ACF/PACF and ADF diagnostics plus rolling mean/std, exponential smoothing and Holt linear trend — describes a series, does not fit AR/MA coefficients

Functions

Function	What it does
ar_fit_yule_walker(series, p)	Fit AR(p) by Yule-Walker: Durbin-Levinson on the sample ACF (series de-meaned internally). Returns ARFit (phi, sigma2).
durbin_levinson(autocov, p)	Durbin-Levinson recursion solving the order-p Yule-Walker equations on given autocovariances γ(0..p).
ar_forecast(series, phi, steps)	h-step-ahead recursive AR forecasts; decays toward the sample mean for stationary φ.
ma_fit_innovations(series, q, max_iter)	Fit MA(q) via the innovations algorithm on the sample ACF. Returns MAFit (theta, sigma2).
innovations_algorithm(autocov, q, max_iter)	Innovations recursion (Brockwell & Davis) on given autocovariances; θ estimates converge to the MA(q) coefficients as iterations grow.
arma_loglik_css(series, phi, theta)	Conditional-sum-of-squares Gaussian log-likelihood at the profile MLE variance. Higher is better; compare fits with equal p.
difference(series, order)	Order-d differencing — the ARIMA "I" step.
undifference(diffed, initial)	Exact inverse of difference given the first d values of the original series.

Usage guidance

• Deterministic: the same series always produces bit-identical fits (asserted directly in the tests).

• Keep max_iter modest for the innovations algorithm — it is O(M³); 20–50 is typical and estimates stabilize quickly. ma_fit_innovations is sub-millisecond for M ≤ ~60 at any n ≤ 10k; do not pass max_iter ~ n.

• Degenerate inputs never trap: empty series or p/q ≤ 0 yield empty coefficient lists; an all-equal series (γ(0) ≈ 0) yields zero coefficients with sigma2 = 0.

• Round-trip identity: undifference(difference(x, d), x[0:d]) == x — exact (verified bit-for-bit on integer-valued floats for d = 1..3).

Verification

Tests pin hand-derived references, not library snapshots:

• Hand-solved Yule-Walker at p = 2 on γ = [1, 0.5, 0.2]: φ = (0.5333333333333333, −0.06666666666666667) = (8/15, −1/15) and sigma2 = 0.7466666666666666 = 11.2/15, matched to 1e-12.

• Hand-run innovations recursion on the exact MA(1) ACF (θ = 0.5): θ₁ = 42/85 = 0.49411764705882355 and sigma2 = 341/340 at M = 3 to 1e-12; at M = 30 the estimate converges to θ = 0.5, sigma2 = 1 within 1e-3.

• Seeded synthetic recovery (n = 2000): AR(1) with φ = 0.7 is recovered within 0.05 (overfit p = 2 gives |φ₂| < 0.08, the near-zero lag-2 PACF); MA(1) with θ = 0.5 is recovered within 0.08; both innovation variances land within 0.15 of the true 1.0.

• CSS log-likelihood matches hand-computed values −3.825292490936347 (AR(1)) and −3.806951651082006 (ARMA(1,1)) to 1e-9, and the true φ = 0.7 beats φ = 0 on the synthetic AR(1) data.

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