Stationarity¶

Time series forecasting models require some form of temporal regularity in the data they learn from. The exact requirement depends on the modeling approach, but the practical consequence is the same: raw series with trends, seasonal swings, or changing variance violate these assumptions and produce unreliable forecasts. Stationarity transforms remove this time-dependent structure so the model sees well-behaved data.

Two distinct assumptions appear repeatedly in forecasting, and understanding how they differ clarifies when and why transforms are necessary.

The IID assumption (reduction forecasters). Scikit-Learn regressors assume training samples are independent and identically distributed (i.i.d.): each row is drawn from the same distribution, and knowing one row tells you nothing about another. When a reduction forecaster tabularizes a time series into feature/target rows, the resulting table should approximate this assumption. If the original series has a rising trend, the early rows have systematically lower targets than the later rows, so the rows are not identically distributed. A seasonal pattern creates the same problem at a different frequency. The regressor cannot distinguish "the data-generating process changed" from "the data is noisy," because it treats each row as exchangeable.

The stationarity assumption (native time series models). Classical econometric models (ARIMA, exponential smoothing, and their variants available through yohou-nixtla) operate directly on the ordered sequence of observations rather than on tabularized rows. These models assume stationarity: the joint distribution of any collection of time points depends only on the relative spacing between them, not on their absolute position. In practice, weak stationarity suffices: constant mean, constant variance, and autocovariance that depends only on lag. ARIMA enforces this by differencing the series until a unit-root test (ADF or KPSS) no longer rejects; exponential smoothing handles trend and seasonality through explicit state components.

Overlap. Both assumptions break down in the same situations (trends, seasonality, heteroscedasticity), and many of the same transforms fix both. Differencing a trended series removes the trend whether the downstream consumer is a Ridge regressor or an ARIMA model. A log transform stabilizes variance regardless of the modeling approach.

Distinction. IID is stronger than stationarity because it additionally requires independence: no autocorrelation between samples. A stationary AR(1) process is stationary but not IID, because each value depends on the previous one. In the reduction setting, the tabularized rows are not truly independent either (overlapping windows share observations), but the regressor still works well in practice because the features explicitly encode the temporal dependence that the IID assumption otherwise forbids. The key concern for reduction forecasters is identical distribution across rows (no trend or seasonal shift in the target), not literal independence. For native time series models, mild violations of stationarity are often tolerable because the model's own structure (autoregressive terms, seasonal components) captures the temporal dependence that stationarity permits.

Practical consequence. Whether you use a reduction forecaster or a native time series model, the same stationarity transforms help: they remove deterministic structure (trend, seasonality, heteroscedasticity) that violates the model's assumptions, leaving a residual that the model can learn from effectively.

Approaches in Yohou¶

Yohou provides two complementary approaches to stationarity: decomposition pipelines that model each component with a dedicated forecaster, and standalone transformers that apply invertible mathematical operations to the raw series.

Decomposition¶

The classical approach to time series decomposition splits a series into additive components:

y(t) = trend(t) + seasonality(t) + residual(t)

DecompositionPipeline automates this pattern. It accepts a list of (name, forecaster) tuples and fits them sequentially: the first forecaster models the full series, the second forecaster models the residuals left after subtracting the first forecaster's in-sample predictions, and so on. At prediction time, the component forecasts are summed to produce the final output.

A typical setup pairs a trend forecaster with a seasonality forecaster and a residual forecaster:

DecompositionPipeline([
    ("trend", PolynomialTrendForecaster(degree=1)),
    ("seasonality", PatternSeasonalityForecaster(seasonality=12)),
    ("residual", PointReductionForecaster(estimator=Ridge())),
])

The pipeline handles all the bookkeeping: computing residuals between stages, aligning time indices, and reconstructing the final prediction. Setting store_residuals=True exposes the intermediate residuals for inspection, which is useful for diagnosing whether the trend forecaster captured enough of the slow-moving level change before the seasonality forecaster tries to model what remains.

For multiplicative decomposition (where seasonal amplitude grows proportionally with the level), pass target_transformer=LogTransformer(). This converts the problem to additive in log-space, since log(trend * season * residual) = log(trend) + log(season) + log(residual).

Trend Estimation¶

Trend forecasters in the yohou.stationarity module estimate and remove slowly varying level changes. They convert datetime indices to numeric features internally, then fit a regression model to capture the trend shape.

PolynomialTrendForecaster fits a polynomial of configurable degree using ElasticNet regularization. With degree=1 it produces a linear trend; degree=2 gives a quadratic curve. Higher degrees are technically possible but risk overfitting: a cubic trend that wiggles through the training data will extrapolate wildly. For exponential trends, a more robust strategy is to combine degree=1 with target_transformer=LogTransformer(), which fits a linear model in log-space and produces exponential growth in the original scale.

Seasonality Estimation¶

Seasonality forecasters model repeating periodic patterns. The module provides two approaches with different trade-offs.

PatternSeasonalityForecaster extracts a discrete seasonal profile by averaging (or taking the median of) values at each position within the seasonal cycle. With method="average" and seasonality=12, it computes the mean January value, the mean February value, and so on, then tiles this fixed pattern into the future. The "median" method is more robust to outliers in individual years. The "naive" method simply repeats the last observed cycle and requires only one complete seasonal cycle of training data, while "average" and "median" require at least two. This approach works well when the seasonal shape is stable and the period aligns exactly with the data frequency.

FourierSeasonalityForecaster represents seasonality as a sum of sine and cosine waves at specified harmonics, fitted via ElasticNet regression. Fourier representation has two notable advantages: it handles non-integer seasonality (such as 365.25 days per year, accounting for leap years) and it produces smooth, differentiable seasonal curves rather than a piecewise-constant pattern. The harmonics parameter controls the complexity: more harmonics capture sharper seasonal features, while fewer harmonics produce a gentler curve.

Standalone Transforms¶

Not every situation calls for a full decomposition pipeline. Sometimes a single invertible transform is enough to make the residual well-behaved, especially when passed as target_transformer to a PointReductionForecaster. The transforms in yohou.stationarity fall into two categories: differencing-based (which remove trend and seasonality) and variance-stabilizing (which address heteroscedasticity).

Differencing¶

SeasonalDifferencing computes y(t) - y(t - s) where s is the seasonal period. With seasonality=1 this is ordinary first differencing, which removes a linear trend. With seasonality=12 on monthly data, it subtracts last January from this January, removing both the annual seasonal pattern and any trend that is roughly constant over one cycle. The first s values are consumed as history for the lag, so the output is shorter than the input. The transform is invertible: given the lagged values, the original series can be reconstructed exactly.

SeasonalLogDifferencing applies a log transform before differencing. Mathematically this computes log(y(t)) - log(y(t-s)), which equals log(y(t) / y(t-s)), the log-ratio between current and lagged values. This is the natural choice for series with multiplicative seasonality, where the amplitude of seasonal swings grows proportionally with the level.

SeasonalReturn and AbsoluteSeasonalReturn provide alternative formulations. SeasonalReturn computes (y(t) / y(t-s)) - 1, the percentage change relative to the seasonal lag. AbsoluteSeasonalReturn computes the raw difference y(t) - y(t-s), which is functionally similar to SeasonalDifferencing but offers a consistent API with SeasonalReturn, including an offset parameter for handling near-zero denominators.

All differencing-based transforms are stateful: they set observation_horizon equal to the seasonality parameter. This means the first s rows of the input are consumed as lag context rather than appearing in the output. When used as a target_transformer inside a reduction forecaster, the pipeline reserves these lag observations automatically before tabularization, so no manual adjustment is needed.

Variance Stabilization¶

Even after removing trend and seasonality, the residual may have non-constant variance. Financial returns, for instance, are roughly zero-mean but their volatility changes over time. Variance-stabilizing transforms compress the range of the data so that the residual variance is approximately uniform.

LogTransformer applies log(y + offset). It is the simplest variance stabilizer and works well for strictly positive series where larger values exhibit proportionally larger fluctuations. The offset parameter shifts the data to avoid taking the log of zero.

BoxCoxTransformer generalizes the log transform with a tunable power parameter lmbda. It computes \(((y + \text{offset})^{\lambda} - 1) / \lambda\) when \(\lambda \neq 0\), and \(\log(y + \text{offset})\) when \(\lambda = 0\). Setting lmbda=0.5 gives a square root transform; lmbda=1 is the identity. The Box-Cox family covers a broad range of variance-stabilizing behaviors, making it a good data-driven choice when the right transform is not obvious in advance. Like the log transform, it requires strictly positive input (after applying the offset).

ASinhTransformer centers each column by its median, scales by the Median Absolute Deviation (MAD), then applies the inverse hyperbolic sine: \(\operatorname{asinh}((y - \tilde{y}) / \text{MAD})\). During fit, it stores the per-column median and MAD so that inverse_transform can reverse the operation exactly. Unlike log or Box-Cox, asinh is defined for all real numbers: it handles zeros, negatives, and extreme outliers without issue. For large positive values it behaves approximately like \(\log(2x)\), compressing the upper tail. For values near zero it behaves approximately linearly, avoiding the singularity that plagues log transforms. This makes it a practical default when the data contains zeros or can go negative.

Trade-offs Between Transform Approaches¶

The two broad approaches (standalone invertible transforms and decomposition pipelines) represent different trade-offs between explicitness and flexibility.

Standalone transforms like SeasonalDifferencing and SeasonalLogDifferencing are the simplest path to stationarity: a single invertible operation removes a predictable pattern without introducing any additional model parameters. Their strength is transparency and inversion. Because the transform is fully reversible with a known formula, predictions in the differenced space map back to the original scale exactly. Their limitation is rigidity: differencing assumes the pattern to remove (trend or seasonal period) is stable and known in advance. A series whose seasonal amplitude changes over time, or whose seasonal period drifts, will still produce structured residuals after a fixed differencing step.

Decomposition pipelines via DecompositionPipeline take the opposite stance. Each component (trend, seasonality, residual) gets its own dedicated forecaster, which can adapt to whatever shape the component takes. A PolynomialTrendForecaster can model non-linear growth; a FourierSeasonalityForecaster can represent a seasonal pattern that changes smoothly across years. The cost is model complexity: each component forecaster introduces hyperparameters, increases fitting time, and requires the user to decide how many components to include. Inspecting intermediate residuals (via store_residuals=True) is essential for verifying that each forecaster removes the pattern it is supposed to remove without over-fitting.

Variance-stabilizing transforms occupy a different dimension from trend and seasonality removal. They address heteroscedasticity, not location shifts, and can be composed freely with either standalone differencing or decomposition pipelines.

The final arbiter is always out-of-sample accuracy. Comparing two or three candidate transforms using cross-validation reveals which actually produces a more predictable residual for your specific series and horizon. No heuristic about additive versus multiplicative seasonality substitutes for empirical evaluation.

References¶

Hyndman, R.J. & Athanasopoulos, G. (2021). Forecasting: principles and practice, 3^rd edition, OTexts. Chapters 3 (decomposition), 3.1 (transformations), and 9.1 (differencing).
Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press. Chapter 3 (stationarity and ergodicity).
Box, G.E.P. & Cox, D.R. (1964). "An analysis of transformations." Journal of the Royal Statistical Society, Series B, 26(2), 211-252.
Johnson, N.L. (1949). "Systems of frequency curves generated by methods of translation." Biometrika, 36, 149-176.

Connections¶

Stationarity transforms feed into the Reduction Forecasting pipeline as target_transformer parameters, and the decomposition approach is a complementary alternative to standalone transforms. Native time series models available through yohou-nixtla handle stationarity internally (ARIMA differences automatically, ETS models trend and seasonality as state components), so explicit transforms are optional when using those forecasters. The Preprocessing page covers non-stationarity transforms (scaling, windowing, imputation) that operate on features rather than targets. For how residuals reveal whether a stationarity transform has done its job, see Residual Diagnostics. The Interval Forecasting page discusses how traditional statistical models assume Gaussian residuals (a stronger condition than stationarity) while conformal prediction requires only exchangeability of calibration errors.

For practical recipes, see How to Apply Stationarity Transforms.

Interactive examples: Decomposition, Decomposition Variations, Fourier Tuning, and Stationarity Transforms.