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Factored Level And Interleaved Ridge: a single-equation time series forecasting method.

Zero hyperparameters. One SVD. CPU only.

• #1 on Chronos Benchmark II (25 zero-shot datasets). Agg. Rel. MASE 0.678, Rel. WQL 0.716 — beats AutoARIMA (0.742) by 3.5%
• Matches PatchTST on GIFT-Eval (97 configs, 23 datasets). relMASE 0.838 (beats PatchTST 0.849), relCRPS 0.587 (ties PatchTST)
• ~1000 lines of pure NumPy/SciPy. No deep learning, no foundation models, no GPU.

• Pipeline
• Quick Start
• Installation
• Supported Frequencies
• How It Works
• Benchmark Results
• API Reference
• Design Principles
• Limitations
• Citation
• License

FLAIR reshapes a time series by its primary period, then separates what happens (level) from how it happens (shape):

y(phase, period) = Level(period) × Shape(phase)

Shape is structural (not learned), so it does not overfit. Level is a smooth, compressed series (one value per period instead of P values) forecast by Ridge regression. Two compressions happen simultaneously: summing P phases into one Level value reduces noise by ~√P, and forecasting Level requires only ⌈H/P⌉ recursive steps instead of H.

import numpy as np
from flaircast import forecast, FLAIR

y = np.random.rand(500) * 100  # your time series

# ── Functional API ───────────────────────────
samples = forecast(y, horizon=24, freq='H')
point   = samples.mean(axis=0)           # (24,)
lo, hi  = np.percentile(samples, [10, 90], axis=0)

# ── Class API (handy in loops) ───────────────
model   = FLAIR(freq='H')
samples = model.predict(y, horizon=24)

# ── With exogenous variables (weather, prices, holidays, ...) ─
X_hist   = np.column_stack([temperature, humidity, is_holiday])  # (n, 3)
X_future = np.column_stack([temp_fcst,   hum_fcst,   hol_fcst])  # (24, 3)
samples  = forecast(y, horizon=24, freq='H',
                    X_hist=X_hist, X_future=X_future)

# ── From pandas ──────────────────────────────
import pandas as pd
ts = pd.read_csv('data.csv')['value']
samples = forecast(ts.values, horizon=12, freq='M')

pip install flaircast

Or install from source:

git clone https://github.com/TakatoHonda/FLAIR.git
cd FLAIR
pip install .

BIC on the SVD spectrum selects the period that best supports a rank-1 structure. A P=1 null model (mean + noise) competes with every periodic candidate under the same BIC, so FLAIR rejects periodicity when the rank-1 fit does not justify the extra Shape parameters.

1. MDL Period Selection: BIC on SVD spectrum selects the primary period P from calendar candidates. A P=1 null model (mean + noise) tests whether periodicity exists at all
2. Reshape the series into a (P × n_complete) matrix. Dynamic DoF guard (n_train >= 2p) ensures the Ridge fit is stable
3. Shape = frozen global average of within-period proportions from the last K=2 periods
4. Level = period totals, denoised by Gavish-Donoho 2014 optimal Frobenius shrinkage (reuses the BIC SVD, no extra matrix decomposition)
5. Shape₂ = secondary periodic pattern in Level, estimated as w × raw + (1−w) × prior, where w = nc₂/(nc₂+cp). The prior is selected by BIC: first harmonic (2 params) when justified, flat (0 params) otherwise. Level is deseasonalized by dividing by Shape₂
6. Ridge on deseasonalized Level: Box-Cox → prior-centered reparameterization (random-walk prior) → intercept + trend + lags → LOOCV soft-average
7. Stochastic Level paths: bootstrap of LOOCV residuals, scaled by LWCP leverages per horizon step
8. Phase noise: scenario-coherent column sampling from the rank-1 residual matrix, with James-Stein per-phase bias shrinkage and horizon-adaptive deflation. Combined with Level paths: sample = Level_path × Shape × (1 + phase_noise)

Evaluated on the Chronos Benchmark II protocol (Ansari et al., 2024). Agg. Relative Score = geometric mean of (method / Seasonal Naive) per dataset. Lower is better.

Baseline results from autogluon/fev and amazon-science/chronos-forecasting.

GIFT-Eval. 7 domains, short/medium/long horizons, 53 non-agentic methods (no test leakage):

Standard benchmark from PatchTST, iTransformer, DLinear, Autoformer. Channel-independent (univariate) evaluation. MSE on StandardScaler-normalized data. Horizons: {96, 192, 336, 720}.

Average MSE across all 4 horizons:

FLAIR outperforms GPU-trained Transformers on 4 of 8 datasets (ETTh2, ETTm2, Weather, Traffic). Accuracy is higher on datasets with clear periodicity and lower on non-periodic series (Exchange).

Three compressions act simultaneously:

1. Noise reduction: summing P phases into one Level value reduces noise by ~√P
2. Horizon compression: forecasting Level requires only ⌈H/P⌉ steps instead of H, reducing error accumulation
3. Shape is frozen: Shape is a structural average, not a learned parameter, so it does not overfit

Generate probabilistic forecasts for a univariate time series.

Returns: ndarray of shape (n_samples, horizon). Probabilistic forecast sample paths.

from flaircast import forecast
samples = forecast(y, horizon=24, freq='H')
point   = samples.mean(axis=0)
median  = np.median(samples, axis=0)
lo, hi  = np.percentile(samples, [10, 90], axis=0)

# With exogenous variables
samples = forecast(y, horizon=24, freq='H',
                   X_hist=X_hist, X_future=X_future)

When X_hist=None (the default) the result is bit-identical to a call without the exog arguments.

Class wrapper. Useful when forecasting multiple series with the same frequency.

from flaircast import FLAIR
model = FLAIR(freq='D', n_samples=500)
for series, X_h, X_f in dataset:
    samples = model.predict(series, horizon=7, X_hist=X_h, X_future=X_f)

FLAIR accepts an arbitrary number of per-step exogenous columns. The columns are z-scored using training-window statistics, aggregated to the per-period (Level) timescale via period mean, and appended directly to the Level Ridge feature matrix. No new hyperparameters, no model selection — the existing LOOCV soft-averaged Ridge inside _ridge_sa handles regularization, so noise covariates are naturally damped without any explicit gating step. "One Ridge" is preserved.

• Recommended setup: at least a few dozen complete periods of training data (e.g. 60–90 days for daily exog, 60+ days for hourly exog) for stable coefficient estimates.
• Validated improvements: see validation/ for rolling-origin benchmarks. UCI Bike Sharing daily: MASE −9.4% (9/12 origins win). Jena Climate hourly: MASE −15.5% (19/24 origins win).
• Graceful degradation: passing pure-noise exog inflates MASE by less than 1% on average, with bounded worst-case behavior.
• Limitation: exog is coupled to the Level (per-period) factor only. Intra-period variation in X (e.g. hourly temperature within a daily period) is collapsed by the period mean and is not captured.

End-to-end walkthrough on the UCI Bike Sharing dataset:

FLAIR applies the Minimum Description Length principle at every scale:

• Non-periodic series: the Level × Shape decomposition provides no compression benefit when there is no periodicity (e.g., exchange rates). Use a dedicated non-periodic model instead
• Intermittent demand: series with >30% zeros are poorly served by the multiplicative structure. Croston-type methods are better suited
• Coarse exogenous resolution: X_hist / X_future are aggregated to the per-period (Level) timescale via period mean. Intra-period variation in covariates (e.g. hourly weather within a daily period) is dropped by design
• Short series: fewer than 3 complete periods forces P=1 degeneration (plain Ridge on raw series)

@misc{flair2026,
  title={FLAIR: Factored Level And Interleaved Ridge for Time Series Forecasting},
  year={2026}
}

Apache License 2.0