N-BEATS-I: Interpretable Time Series Forecasting

• N-BEATS-I architecture constrains each block to fixed polynomial and Fourier bases, enabling clear additive decompositions of trend and seasonality.
• The model’s doubly-residual stacking allows sequential backcasting and forecasting, ensuring interpretable signal separation and coefficient transparency.
• Experimental evaluations on clinical time series demonstrate competitive accuracy while providing actionable insights through precise trend analysis.

N-BEATS-I (Neural Basis Expansion Analysis for Interpretable Time Series Forecasting) is a specific configuration of the N-BEATS architecture, designed to yield interpretable, additive decompositions of time series forecasts into trend and seasonality components using fixed basis expansions. N-BEATS-I achieves interpretability by constraining block-level projections to known polynomial and Fourier bases such that each block’s output directly corresponds to a semantically meaningful subspace. This design enables rigorous inspection of the learned signal components at both block and aggregate levels and provides clean additive separation of trend and seasonality in the forecasted signal (Bhatti et al., 2023, Oreshkin et al., 2019).

1. N-BEATS-I Architecture and Functional Overview

N-BEATS-I is a fully-connected, residual deep network designed for point forecasting of univariate time series. The model is constructed from sequential “blocks” grouped into “stacks,” with each block responsible for both explaining (backcasting) a portion of the input history and contributing (forecasting) to the final time-ahead prediction. The doubly-residual stacking mechanism ensures that only unexplained history is passed forward to the next block and the process occurs independently within each stack.

The interpretable configuration consists of exactly two stacks:

• The Trend stack, which uses a fixed low-degree polynomial basis.
• The Seasonality stack, which uses a truncated Fourier basis of sines and cosines.

Each stack comprises several identical MLP blocks sharing weights and basis matrices. All blocks within the stack differ only in their inputs (successive residuals) and the coefficients they predict for the fixed basis functions.

The forward propagation in block $k$ of stack $s$ involves:

1. MLP forward pass: $\theta = \phi(x_{s,k-1})$
2. Basis projection: $f_{s,k} = V^f_s \theta$, $b_{s,k} = V^b_s \theta$
3. Residual update: $x_{s,k} = x_{s,k-1} - b_{s,k}$

The global forecast is then the sum of all partial forecasts:

$\hat{y} = \sum_{s=1}^2 \sum_{k=1}^{K_s} f_{s,k}$

where $K_s$ denotes the number of blocks in stack $s$.

2. Mathematical Basis Expansion and Stack Constraints

Interpretable blocks in N-BEATS-I restrict their outputs to fixed basis function classes. Each block’s forecast and backcast are linear combinations of either polynomial or Fourier basis vectors, with the expansion coefficients predicted by the block’s MLP head:

• Trend stack: Polynomial basis of degree $p$ over the forecast ($H$) and history ($T$) windows.

$t = [0, 1, \dots, H-1]^T/H$

$$f_{\text{trend}} = \sum_{i=0}^p \theta_{\text{tr}, i} t^i$$

(Equation (1) in (Bhatti et al., 2023))

• Seasonality stack: Fourier basis, using $m = \lfloor H/2 - 1 \rfloor$ harmonics,

$$f_{\text{season}} = \sum_{i=0}^m \theta_{\text{sea},i} \cos(2\pi i t) + \theta_{\text{sea},i+m+1} \sin(2\pi i t)$$

(Equation (2) in (Bhatti et al., 2023))

Backcasting uses an analogous basis over the look-back window.

All learnable parameters in a block consist solely of the coefficients $\theta$ for these fixed bases, guaranteeing that each block’s output lies strictly within its designated functional class.

3. Interpretability Mechanisms and Signal Decomposition

Interpretability in N-BEATS-I is achieved by architectural constraints:

• Stack-level separation: Each stack can only represent its designated function class (trend or seasonality) due to fixed bases.
• Coefficient transparency: All partial forecasts are direct sums of basis vectors weighted by $\theta$; each coefficient is semantically interpretable as the contribution of a specific polynomial term or harmonic component.
• Additive aggregation: The total forecast is decomposed as:

$$\hat{y} = \hat{y}_{\text{trend}} + \hat{y}_{\text{season}}$$

where $$\hat{y}_{\text{trend}} = \sum_{k} f_{\text{trend}, k}$$, $$\hat{y}_{\text{season}} = \sum_{k} f_{\text{season}, k}$$.

Inspection of individual $f_{\text{trend}, k}$ and $f_{\text{season}, k}$ reveals the incremental removal of polynomial and periodic components, respectively, from the signal as the network peels away and forecasts different aspects of the time series.

4. Experimental Evaluation Protocols

N-BEATS-I was applied to supervised forecasting of mean blood pressure (MBP) in sepsis and septic shock patients using the public eICU Collaborative Research Database (Bhatti et al., 2023).

Preprocessing steps included:

1. Missing value imputation (forward fill).
2. Segment extraction: 9h before diagnosis, split as 6h look-back (input) and 3h forecast horizon (target).
3. Moving average smoothing (window=3).
4. Filtering samples with near-constant MBP ($\text{std} \leq 0.025$).
5. Min-max scaling to [0, 190] mmHg.

The aggregate evaluation set comprised 4,020 samples from 1,442 patients.

Training employed:

• Generic and interpretable N-BEATS variants trained identically (typically Adam optimizer, learning rate $\sim$1e–3).
• Typical loss functions: MSE, MAPE, DTW.

Performance metrics:

Metric	N-BEATS-I Score	Unit
MSE	21.49 × 10⁻⁴	Scaled MBP
MAPE	8.6	%
DTW	18.2 × 10⁻³	Scaled MBP

Baseline for comparison: last-value persistence.

5. Clinical Interpretability Assessment and Outcome Analysis

Interpretability assessment targeted the extracted trend component $\hat{y}_{\text{trend}}$:

• Dynamic Time Warping (DTW) computed between $\hat{y}_{\text{trend}}$ and observed trend over the forecast window.
• Cases with largest DTW (worst trend mismatch) were analyzed with respect to clinical interventions, notably vasoactive drug infusions present after look-back but during the forecast window.
• Mortality rates were stratified: a higher rate (92%) was observed where actual and forecasted trend closely matched, compared to 84% where the trends diverged.

Figure-based analyses show the model’s forecasted trend continuation versus sharp real-world deviations induced by clinical intervention (e.g., rapid MBP elevation post-drug infusion), illustrating both the interpretability of trend prediction and how mismatches arise due to unmodeled events.

6. Comparison with Generic N-BEATS and Broader Implications

N-BEATS-I maintains competitive forecast accuracy compared to generic N-BEATS, despite its restrictions:

• Both models operate with deep fully-connected stacks, recurrent block structure, and residual connections.
• N-BEATS-I, however, is confined to trend/seasonal bases rather than arbitrary learned projections, enabling direct mapping between coefficient values and signal semantics.
• Model outputs provide classical “Trend + Seasonality” decomposition, with block-level transparency superior to unconstrained architectures.

A plausible implication is that such interpretable neural designs offer substantial benefits in clinical time series and any domain where segmentation of different signal mechanisms (slow drift vs. cyclic variation) is required for actionable downstream use and verification.

7. Historical Context and Future Directions

The interpretable configuration of N-BEATS was introduced in "N-BEATS: Neural basis expansion analysis for interpretable time series forecasting" (Oreshkin et al., 2019). Subsequent applications, including clinical forecasting in sepsis (Bhatti et al., 2023), demonstrate its operational utility where transparent decomposition is a primary requirement. Ongoing research focuses on extending basis selection mechanisms and further integrating external event modeling (e.g., abrupt interventions) to reduce forecast/actual mismatches and enhance the applicability of N-BEATS-I in domains characterized by exogenous shocks.