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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 0.696 vs. Chronos-Bolt-Base 0.791 (205M params, GPU)
• Best statistical method on GIFT-Eval (97 configs, 23 datasets). relMASE 0.857, relCRPS 0.610
• ~500 lines of Python. Dependencies: numpy + scipy

• 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)

# ── 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 .

When multiple candidates exist, FLAIR uses BIC on the SVD spectrum (MDL principle) to select the period that best supports a rank-1 structure.

1. MDL Period Selection: BIC on SVD spectrum selects the primary period P from calendar candidates
2. Reshape the series into a (P × n_complete) matrix
3. Shape₁ = Dirichlet posterior mean per context (context = period_index % C). Shrinks toward the global average when data is scarce
4. Level = period totals
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 → NLinear → intercept + trend + lags → soft-average GCV
7. Stochastic Level paths: LOO residuals are injected into the recursive forecast. Errors propagate through the Ridge lag dynamics naturally. Mean-reverting series saturate; random-walk series grow as √step. No scaling formula needed
8. Phase noise from SVD Residual Quantiles: E = M − fitted gives phase-specific relative noise. Sampled scenario-coherently: all phases within the same forecast step share one historical period's residual pattern, preserving cross-phase correlation. Combined with Level paths: sample = Level_path × Shape₁ × (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. FLAIR outperforms Chronos-T5-Small (46M params) on 14 of 25 datasets in point forecast accuracy.

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 3 of 8 datasets and 11 of 32 individual settings. 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 fixed: Shape is a Dirichlet posterior, 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)

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

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

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
• No exogenous variables: the core API does not accept external features (calendar events, prices, promotions)
• 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