Fourier Basis Mapping for Forecasting

• Fourier Basis Mapping (FBM) is a time-frequency learning framework that reinterprets Fourier coefficients into cosine and sine basis functions to retain both global cycles and local temporal details.
• It addresses limitations of traditional Fourier forecasting by resolving issues like inconsistent starting cycles, series length problems, and loss of temporal context.
• FBM variants—from linear to Transformer-based approaches—demonstrate competitive forecasting accuracy across diverse datasets by integrating explicit time-frequency mappings.

Searching arXiv for the FBM paper and closely related time-series forecasting work. Fourier Basis Mapping (FBM) is a time-frequency learning framework for time series forecasting that treats the Fourier transform explicitly as a basis-function expansion rather than as a purely spectral preprocessing step. In this formulation, the real and imaginary parts of Fourier coefficients are reinterpreted as coefficients of cosine and sine basis functions at tiered frequency levels, and are then mapped in a joint time-frequency space to produce forecasts. The framework is intended to address limitations attributed to existing Fourier-based forecasting methods—specifically inconsistent starting cycles, inconsistent series length issues, imprecise interpretation of frequency components, and loss of temporal information—while remaining plug-and-play with linear, MLP-based, and Transformer-based predictors by modifying only the first initial projection layer (Yang et al., 13 Jul 2025).

1. Conceptual basis and problem setting

FBM begins from a basis-functions view of the discrete Fourier transform. For a real-valued time series, the Fourier spectrum is not treated merely as amplitude or phase; instead, its real and imaginary parts are tied directly to cosine and sine bases. This yields explicit frequency features while preserving temporal characteristics.

The framework is motivated by two criticisms of prior Fourier-based forecasting pipelines. First, they are described as suffering from inconsistent starting cycles and inconsistent series length issues. Second, they are described as failing to interpret frequency components precisely and as overlooking temporal information. FBM addresses these issues by combining Fourier basis expansion with a mapping stage defined in time-frequency space, so that downstream prediction operates on representations indexed by both time and frequency rather than on a collapsed spectral summary (Yang et al., 13 Jul 2025).

A central claim of the framework is that the retained time index permits simultaneous access to global cycles and local spikes. This is operationally important for forecasting settings in which seasonal structure, trend, and short-horizon cross-channel interaction must be modeled with different inductive biases. FBM therefore functions both as a feature representation and as an architectural interface between spectral structure and temporal prediction.

2. Fourier-basis expansion and projection formulation

For a univariate real-valued series $X[n]$, $n=0,\ldots,T-1$, FBM starts from the standard DFT and IDFT:

$$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$

and

$$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$

Because $X$ is real, $H$ satisfies Hermitian symmetry, and the reconstruction can be rewritten in cosine-sine form. Splitting $H[k]$ into real and imaginary parts $H_R[k]$ and $H_I[k]$, the expansion is

$$X[n] = \frac{1}{T}H_R[0] + \frac{1}{T}H_R[T/2]\cos(\pi n) + \frac{2}{T}\sum_{k=1}^{T/2-1}\left[ H_R[k]\cos(2\pi kn/T) - H_I[k]\sin(2\pi kn/T)\right].$$

The paper then defines cosine and sine basis functions

$n=0,\ldots,T-1$0

and

$n=0,\ldots,T-1$1

together with coefficients

$n=0,\ldots,T-1$2

so that

$n=0,\ldots,T-1$3

This basis expansion is the formal entry point for FBM. Rather than discarding phase or amplitude, the method re-injects $n=0,\ldots,T-1$4 and $n=0,\ldots,T-1$5 into the time domain through the basis matrices $n=0,\ldots,T-1$6 and $n=0,\ldots,T-1$7. For $n=0,\ldots,T-1$8 channels, the paper describes the projection pipeline as follows: compute $n=0,\ldots,T-1$9 and $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$0, form shared basis matrices $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$1 and $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$2, and then construct the time-frequency tensor

$$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$3

The downstream projection network maps $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$4 to the forecast horizon. All variants are trained end-to-end with squared error

$$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$5

with MAE used as a secondary monitoring metric (Yang et al., 13 Jul 2025).

3. Time-frequency mapping and preservation of temporal locality

A defining property of FBM is that the representation $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$6 retains an explicit time index $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$7 in addition to the frequency index $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$8. The $$H[k] = \mathrm{DFT}(X)[k] = \sum_{n=0}^{T-1} X[n]\cdot e^{-i2\pi kn/T}, \qquad k=0,\ldots,T-1,$$9 entry is defined as the contribution of channel $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$0, time $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$1, and frequency $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$2. This means that downstream layers do not operate on a frequency-only tensor; they operate on a joint time-frequency representation.

For a linear projection $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$3, the forecast can be written as

$$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$4

In this form, the mapping explicitly respects both time and frequency indexing. The paper argues that because each $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$5 arises from multiplying $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$6 by $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$7 and $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$8 by $$X[n] = \frac{1}{T} \sum_{k=0}^{T-1} H[k]\cdot e^{+i2\pi kn/T}.$$9, any sliding window, patching operator, or attention mechanism over $X$0 still observes the relative timing of spectral components (Yang et al., 13 Jul 2025).

Within the proposed interpretation, this is the core distinction between FBM and pure Fourier-domain methods. The framework is not merely spectral feature extraction followed by prediction; it is a structured remapping of spectral coefficients back into a temporally indexed feature field. This suggests that FBM is designed to preserve temporal locality without abandoning global frequency structure.

4. Model family and architectural variants

All FBM variants share the same initial steps—DFT followed by basis expansion—and differ only in how the resulting tensor $X$1 is mapped to the forecast $X$2.

Variant	Mapping head	Defining characteristic
FBM-L	Single dense layer	Linear mapping on flattened time-frequency features
FBM-NL	Three fully connected layers with ReLU	Nonlinear refinement of explicit time-frequency features
FBM-NP	Patch + Transformer encoder	Patch-wise processing of $X$3 with position-in-time embedding
FBM-S	Seasonal, Trend, Interaction blocks	Synergetic three-block decomposition

FBM-L flattens $X$4 into a vector of length $X$5 and applies a single dense layer with no activation:

$X$6

The paper characterizes this as analogous to NLinear, but operating on time-frequency features.

FBM-NL uses the same flattening, followed by three fully connected layers with ReLU activations:

$X$7

This introduces depth while preserving the explicit FBM feature construction.

FBM-NP partitions $X$8 along the time axis into $X$9 patches of length $H$0, flattens each patch, projects it linearly to $H$1, adds position-in-time embedding, and processes the resulting patch embeddings with a small Transformer encoder stack before a final linear head. The paper states that this is identical to PatchTST’s decoder except that the input is $H$2 rather than raw $H$3.

FBM-S is the most specialized variant. It decomposes forecasting into three parallel paths—Seasonal, Trend, and Interaction—whose outputs are summed:

$H$4

This decomposition is the main architectural generalization within the FBM family (Yang et al., 13 Jul 2025).

5. Seasonal, trend, and interaction mechanisms in FBM-S

The Seasonal block is designed to exploit the sinusoidal structure already present in the basis expansion. The paper states that all basis functions $H$5 and $H$6 are sinusoids in $H$7. To extract seasonality, $H$8 is padded in time by $H$9 using the same $H[k]$0 basis, and a 1D rolling-window filter $H[k]$1 is applied:

$H[k]$2

This is also described as a learned weighting over time-frequency positions for repeated cycles.

The Trend block targets coarser scales. It may use patching as in FBM-NP, or flatten $H[k]$3 and feed it through either a single MLP layer with activation or a small Transformer block with patch tokens. Before projection, each patch is centralized; after projection, it is decentralized. The block may also incorporate multi-scale down-sampling of $H[k]$4 by averaging along time or summing along frequency, producing additional coarse trend views that are processed separately and summed.

The Interaction block is intended for short-term cross-channel coupling concentrated in the recent history. An input mask keeps only the last $H[k]$5 time steps, $H[k]$6, and an output mask may restrict the block to the first $H[k]$7 forecast steps. The resulting mini-tensor is centralized and then passed through a small MLP or one attention layer to produce $H[k]$8. The paper states that short-term TSF often benefits from cross-channel coupling, but only in the very recent window (Yang et al., 13 Jul 2025).

Several specialized techniques support these three blocks. Interaction masking restricts the temporal scope of the Interaction block. Centralization and decentralization normalize each patch with learned scale and shift parameters $H[k]$9:

$H_R[k]$0

$H_R[k]$1

Patching divides the time axis into $H_R[k]$2 equal windows and reduces initial projection size by factor $H_R[k]$3. Rolling-window projection implements the seasonal operation as a linear 1D convolution with kernel $H_R[k]$4. Multi-scale down-sampling generates lower-resolution versions of $H_R[k]$5 by averaging adjacent time steps with kernel $H_R[k]$6 or summing frequency bands with kernel $H_R[k]$7.

6. Empirical evaluation, scope, and related nomenclature

The reported experiments cover long-term TSF, short-term TSF, and M4 univariate forecasting. For long-term TSF, the datasets are ETT (ETTh1, ETTh2, ETTm1, ETTm2), Electricity (ECL), Traffic (TRA), Weather (WTH), Exchange, and PEMS (03,04,07,08), with input length $H_R[k]$8 and horizons $H_R[k]$9. For short-term TSF, the datasets are PEMS only, again with $H_I[k]$0 and $H_I[k]$1. For M4 univariate forecasting, the setup uses $H_I[k]$2 and $H_I[k]$3. The metrics are MSE and MAE for multivariate tasks, and SMAPE, MASE, and OWA for M4.

The baselines named in the paper span several model classes: NLinear for linear models; PatchTST, iTransformer, Autoformer, and Informer for Transformer-based approaches; N-BEATS and TimeMixer for MLP-based approaches; TimesNet and FiLM for CNN/RNN approaches; and FITS, FreTS, and CrossGNN for Fourier-based approaches.

The reported average results over all horizons and datasets are:

• For long-term TSF, FBM-S achieves MSE/MAE of $H_I[k]$4, while the best baseline, often FBM-L or FBM-NP, is approximately $H_I[k]$5.
• For short-term TSF on PEMS, FBM-S achieves $H_I[k]$6, with TimeMixer next best at $H_I[k]$7.
• For M4, FBM-S achieves OWA $H_I[k]$8.

The ablation studies described in the paper turn on and off the Seasonal block, the Interaction block, masking and centralization, the input mask $H_I[k]$9, the output mask $$X[n] = \frac{1}{T}H_R[0] + \frac{1}{T}H_R[T/2]\cos(\pi n) + \frac{2}{T}\sum_{k=1}^{T/2-1}\left[ H_R[k]\cos(2\pi kn/T) - H_I[k]\sin(2\pi kn/T)\right].$$0, multi-scale down-sampling, patching versus no patching in the Trend block, and linear versus nonlinear initial projection. The stated findings are that the Seasonal block improves over trend-only by approximately $$X[n] = \frac{1}{T}H_R[0] + \frac{1}{T}H_R[T/2]\cos(\pi n) + \frac{2}{T}\sum_{k=1}^{T/2-1}\left[ H_R[k]\cos(2\pi kn/T) - H_I[k]\sin(2\pi kn/T)\right].$$1–$$X[n] = \frac{1}{T}H_R[0] + \frac{1}{T}H_R[T/2]\cos(\pi n) + \frac{2}{T}\sum_{k=1}^{T/2-1}\left[ H_R[k]\cos(2\pi kn/T) - H_I[k]\sin(2\pi kn/T)\right].$$2, the Interaction block is critical for STSF and minor for LTSF, the best input mask is a very short window with $$X[n] = \frac{1}{T}H_R[0] + \frac{1}{T}H_R[T/2]\cos(\pi n) + \frac{2}{T}\sum_{k=1}^{T/2-1}\left[ H_R[k]\cos(2\pi kn/T) - H_I[k]\sin(2\pi kn/T)\right].$$3, multi-scale down-sampling benefits high-granularity data such as ETTm1, and patching with centralization outperforms the no-patching alternative. Under every configuration, the paper states that FBM-S matched or outperformed all ablation variants (Yang et al., 13 Jul 2025).

A separate point of terminology is that the acronym “FBM” is also used in another 2025 arXiv paper for the “Fourier Basis Density Model,” a probability model for band-limited distributions with closed-form density and cumulative distribution functions and a sampling method based on Discretized Approximate Ancestral Sampling (Fuente et al., 9 May 2025). That usage concerns density modeling and sampling rather than time series forecasting. The shared acronym therefore denotes two distinct research objects: Fourier Basis Mapping in forecasting, and Fourier Basis Density Model in probabilistic modeling.