---
kind: concept
slug: symbolic-aggregate-approximation
title: Symbolic aggregate approximation
summary: >-
  Turns a numeric series into short words over a small alphabet by averaging segments and binning them at Gaussian quantiles. Covers classic SAX, windowed SAX with dilation and stride, and a bag-of-words classifier that is deterministic and explainable. Author-reported: accuracy tied with ROCKET, but not the most accurate method.
tags: [time-series, classification, symbolic-representation]
papers: ["2311.18029"]
related: [time-series-classification, random-convolutional-kernels, explainable-ai, interpretable-machine-learning, feature-attribution]
status: draft
updated_at: 2026-10-19
---

## Overview

Symbolic aggregate approximation (SAX) replaces a numeric series with a short
string of symbols. It averages the series over fixed segments (piecewise
aggregate approximation) and maps each average to one of $\alpha$ symbols,
using breakpoints that split a standard normal into $\alpha$ equally likely
bins [arxiv:2311.18029 §3] [arxiv:2311.18029 eq. 1] [arxiv:2311.18029 eq. 2].
Once a series is a string of symbols, counting substrings gives a fixed-length
feature vector, which is the route the bag-of-words classifiers take.

The problem the windowed variants target is cost and repeatability. Earlier
bag-of-patterns classifiers use SAX over sliding windows and are deterministic
and interpretable, but their window-wise step has quadratic worst-case cost in
the series length when the window grows with the series [arxiv:2311.18029 §4.1]
[arxiv:2311.18029 §4.3]. The paper formalises window-wise SAX, which earlier
work used without a formal definition [arxiv:2311.18029 §4.1].

This page currently rests on one paper card. All results below are
author-reported.

## Variants

### Classic SAX: segment means and breakpoints

Split the series into equal segments, average each one, and map each average
to a symbol with Gaussian quantile breakpoints [arxiv:2311.18029 §3]
[arxiv:2311.18029 eq. 2]. The output is one word per series. The paper uses
this as the base step of the variants below. Card:
[BORF paper, arXiv:2311.18029](/papers/2311.18029).

### Windowed SAX with dilation and stride (wSAX)

Slide a receptive field of length $w$ across the signal. Its points sit $d$
apart, and fields start $s$ points apart [arxiv:2311.18029 §4] [arxiv:2311.18029 eq. 3].
Each field is split into $l$ segments, and each segment mean is z-normalised
with the mean and standard deviation of the whole field, then binned into
symbols [arxiv:2311.18029 §4.1] [arxiv:2311.18029 eq. 12]. Moving averages
compute every segment mean once, so the per-window cost does not grow with
window size [arxiv:2311.18029 §4.1] [arxiv:2311.18029 §4.3]. Card:
[BORF paper, arXiv:2311.18029](/papers/2311.18029).

### Bag of receptive fields (BORF)

Encode each field's $l$ symbols as one integer, reading the word in base
$\alpha$ with the first symbol most significant [arxiv:2311.18029 §4.2]
[arxiv:2311.18029 eq. 13]. Count the words over many configurations of
$(w, d, l)$ and store the counts as a sparse tensor indexed by series, signal
and word [arxiv:2311.18029 §4.2] [arxiv:2311.18029 §4.3]. The paper's default
heuristic sets window sizes from powers of two up to the series length,
dilations up to $\log_2 \log_2 n$, word lengths 2, 4 and 8, alphabet size 3
and stride 1 [arxiv:2311.18029 §5] [arxiv:2311.18029 tab. 4]. The counts pass
through an element-wise inverse hyperbolic sine and a ridge classifier
[arxiv:2311.18029 §5]. Card:
[BORF paper, arXiv:2311.18029](/papers/2311.18029).

The paper's explanation is a SHAP value per count, with saliency maps over
time. A contained word's contribution is spread over the points it came from,
and a word that is absent is shown by its typical shape from a background set
[arxiv:2311.18029 §4.5] [arxiv:2311.18029 eq. 14] [arxiv:2311.18029 fig. 3].

## When to use which

| Situation | Use | Why |
|---|---|---|
| You need the same model on every rerun | BORF | The paper reports that RDST and RSTSF change between runs, while BORF does not [arxiv:2311.18029 §5] |
| Each decision must trace back to a window of the signal | BORF with its SHAP maps | Each count has a contribution and a time saliency map [arxiv:2311.18029 §4.5] |
| Accuracy is the only goal | Not BORF | MR-Hydra and RDST are statistically better on the paper's 158 datasets, and the paper says accuracy is not its claim [arxiv:2311.18029 tab. 2] [arxiv:2311.18029 §5] |
| Long, low-dimensional series under a runtime budget | BORF | Second-best average runtime rank, tied with HYDRA [arxiv:2311.18029 §5] [arxiv:2311.18029 tab. 3] |
| Highly multivariate, short series | Benchmark before choosing BORF | Its mean runtime is far above ROCKET on FaceDetection and InsectWingbeat [arxiv:2311.18029 §5] |
| Several channels, some to be pooled together | BORF with per-group sums | On one EEG dataset, grouped sums beat separate or fully summed counts (0.61 vs 0.56 and 0.58). Single dataset [arxiv:2311.18029 §5] [arxiv:2311.18029 fig. 11] |
| Only a symbolic discretiser is needed | Classic SAX | The card records no benchmark for classic SAX on its own [arxiv:2311.18029 §3] |
| Irregularly sampled series | Not covered by the paper | The paper assumes equally spaced observations [arxiv:2311.18029 §3] |

Against other SAX-based methods, the paper reports BORF as the most accurate
and the fastest [arxiv:2311.18029 §5] [arxiv:2311.18029 fig. 7]. In its
ablation on 42 development datasets, the largest accuracy loss comes from
restricting word length to 2 (about 0.19). Dropping the arcsinh step costs
about 0.02 [arxiv:2311.18029 tab. 4].

## Implementation sketch

Language-agnostic pseudocode for the windowed core. Classic SAX is the
`sax_symbol` step.

```pseudocode
# Symbol of one value from its breakpoints  [§3] [eq. 2]
# Breakpoints: Gaussian quantiles into alpha equal-probability bins
def sax_symbol(value, breakpoints):
    # Count breakpoints strictly below the value  [eq. 2]
    symbol = 0
    for b in breakpoints:
        if value > b:
            symbol += 1
    return symbol
```

```pseudocode
# Windowed SAX word for each receptive field of one signal  [§4.1]
# Input: signal, window w, dilation d, stride s, word length l
def window_words(signal, w, d, s, l, alphabet_size, breakpoints):
    # Segment size; the paper assumes w is divisible by l  [§4.1]
    q = w // l
    words = []
    # One field per start point, step s  [§4] [eq. 3]
    for i in range(num_fields(len(signal), w, d, s)):
        # Points of the dilated field  [§4] [eq. 3]
        points = [signal[i * s + d * k] for k in range(w)]
        # Mean and population std of the whole field  [§4.1] [eq. 10]
        mu = mean(points)
        sigma = pop_std(points)
        word = 0
        for j in range(l):
            # Segment mean of this field  [§4.1] [eq. 6]
            seg_mean = mean(points[j * q : (j + 1) * q])
            # Z-normalise with the field's own statistics  [§4.1] [eq. 12]
            z = (seg_mean - mu) / sigma
            # Symbol, then append as the next base-alpha digit  [eq. 13]
            word = word * alphabet_size + sax_symbol(z, breakpoints)
        words.append(word)
    return words
```

Notes on where variants differ:

- **Flat windows.** The paper sets a threshold $\theta = 0.15$ for near-flat
  windows and reports that removing it costs about 0.013 accuracy and adds
  about 8 s on the development set [arxiv:2311.18029 §5] [arxiv:2311.18029 tab. 4].
  The sketch above divides by zero when `sigma` is 0, so an implementation
  needs its own rule. The paper does not give one for exactly zero.
- **Arcsinh.** Apply the inverse hyperbolic sine to the counts before the ridge
  classifier [arxiv:2311.18029 §5]. The paper does not state the ridge
  regularisation strength [arxiv:2311.18029 §5].
- **Vocabulary.** The paper defines the bag over a fixed set of patterns
  [arxiv:2311.18029 §4.2]. The library's default vocabulary is built from the
  training data instead; see the card's
  [Paper vs code](/papers/2311.18029) section.

## Contradictions

No disagreement between papers, since only one paper is on this page. The
paper and its library disagree on the following; each side is cited, and the
paper's version is the one this page follows:

- **Ties at breakpoints.** The paper assigns a value equal to a breakpoint to
  the lower symbol [arxiv:2311.18029 eq. 2]. The library assigns it to the
  upper symbol, as described in the card's Paper vs code section.
- **Flat-window rule.** The paper uses $\theta = 0.15$ and treats values at or
  below it as constant [arxiv:2311.18029 §5]. The library's default threshold
  is 0.0, and its comparison is strict; see the card.

## Papers

- [BORF paper, arXiv:2311.18029](/papers/2311.18029): defines window-wise SAX with dilation and stride, a moving-average speed-up, and a deterministic bag-of-receptive-fields classifier. Author-reported on 158 datasets: tied with ROCKET and MiniRocket on accuracy, second-best runtime.