Series-Symbol Data Generation

• Series-symbol data generation mechanism is a dual-modality approach generating paired time series and symbolic expressions that capture dynamic behavior and semantic structure.
• It employs a rigorous workflow with symbolic synthesis, structured time series sampling, and filtering to ensure valid, learnable, and diverse data representations.
• Leveraging synthetic pairs for pretraining, the mechanism enhances forecasting, classification, imputation, and anomaly detection, proving its broad applicability in time series research.

The series-symbol data generation mechanism is a synthetic dual-modality pipeline in which a numeric time series and a symbolic expression are generated as paired views of the same underlying system. In the formulation introduced for SymTime, time series are treated as numerical observations of complex dynamical systems, while symbolic expressions serve as semantic descriptors or mathematical models of those systems. The mechanism therefore constructs large-scale pairs $\big(X, f(\cdot)\big)$ together with outputs $Y=f(X)$, with the stated aim of addressing data scarcity and data imbalance in time series analysis while enabling pretraining that couples temporal patterns with symbolic semantics (Wang et al., 21 Feb 2025).

1. Conceptual Basis and Motivation

The mechanism is grounded in a specific semantic claim: a time series is a dynamic representation of an observed complex system, whereas a symbolic expression is an abstract semantic representation of that same system. On this view, the symbolic modality is not merely an auxiliary label. It is the semantic counterpart to the numeric series, designed to expose mathematical structure such as operators, variables, and compositional form alongside observable temporal behavior such as shape, trend, and seasonality (Wang et al., 21 Feb 2025).

This framing is introduced as a response to two difficulties in time-series foundation modeling. First, available time-series corpora are much smaller than those used in computer vision and natural language processing, partly because of privacy and collection constraints. Second, even relatively large pretraining corpora are described as imbalanced across domains, which can induce performance bias and weaken generalization. The series-symbol mechanism is therefore presented as a way to enable the unrestricted generation of high-quality time series data with corresponding symbolic descriptors, thereby expanding coverage of the time-series representation space and reducing reliance on scarce real data (Wang et al., 21 Feb 2025).

A recurrent misconception is to treat the symbolic side as simple annotation. The paper explicitly assigns it a stronger role: the symbolic expression is the semantic description or mathematical model of the same generating process that produces the numeric series. This suggests that the central innovation is not synthetic time-series generation alone, but the construction of semantically aligned series-symbol pairs.

2. Formal Definition and Generation Workflow

The formal structure of the mechanism is defined by three objects: a symbolic function $f(\cdot)$, an input series $X \in \mathbb{R}^{M \times L}$, and an output series $Y = f(X) \in \mathbb{R}^{N \times L}$. Here $M$ denotes the number of input channels, $N$ the number of output channels, and $L$ the series length. Each sample is a pair $\big(X, f(\cdot)\big)$, while the generated output is written as

$$Y = [y_1, y_2, \dots, y_N], \qquad y_i = f_i(x_1,\dots,x_M).$$

The dataset is constructed from many such pairs (Wang et al., 21 Feb 2025).

The workflow is specified as a deterministic sequence of stages applied to randomized components:

1. Choose input/output dimensions with

$Y=f(X)$0

where $Y=f(X)$1 and $Y=f(X)$2.

1. Sample a symbolic tree expression $Y=f(X)$3.
2. Sample an input series $Y=f(X)$4 using either mixed distributions or ARMA.
3. Normalize each channel.
4. Evaluate

$Y=f(X)$5

1. Reject invalid samples if a value falls outside the domain of $Y=f(X)$6, or if generated values exceed $Y=f(X)$7.
2. Traverse all input-output channels per seed; each expression is sampled only once per seed, and all channel combinations are covered.

This workflow makes the symbolic expression causally prior to the generated output: the symbolic tree is constructed first, the input series is then sampled, and the output series is obtained by forward evaluation of the expression on the sampled input. In that sense, the series-symbol pair is not a post hoc alignment but a jointly generated object.

3. Symbolic Expression Synthesis

Symbolic expressions are generated by a tree-based expression synthesis process. The expression is modeled as a tree with leaf nodes, binary nodes, and unary nodes. Leaf nodes contain variables and constants; binary nodes contain binary operators; unary nodes contain unary operators. The synthesis begins by selecting the input and output dimensions $Y=f(X)$8 and $Y=f(X)$9, then sampling the number of binary operators $f(\cdot)$0 from a uniform distribution, and then constructing a binary tree skeleton whose internal nodes are binary operators and whose leaves are symbolic variables or constants (Wang et al., 21 Feb 2025).

The operator families are fixed and explicitly enumerated. Binary operators are sampled from

$f(\cdot)$1

while unary operators are sampled from

$f(\cdot)$2

The generated tree is then read by in-order traversal. To increase diversity, unary operators are inserted randomly, and affine transformations are applied both to variables and to unary outputs:

$f(\cdot)$3

where $f(\cdot)$4 and $f(\cdot)$5 are random constants. This produces richer symbolic functions without changing the basic tree-construction principle (Wang et al., 21 Feb 2025).

The resulting symbolic space is therefore diverse along several axes at once: tree topology, operator composition, variable choice, constant injection, and affine perturbation. A plausible implication is that the mechanism attempts to make symbolic variability structurally compositional rather than merely lexical.

4. Numerical Series Sampling, Filtering, and Representation Coverage

The input series are not sampled as arbitrary noise. Two generation routes are used, both intended to better resemble time series. The first route is a mixed-distribution procedure. It first samples the number of component distributions

$f(\cdot)$6

then samples weights $f(\cdot)$7 and normalizes them so that

$f(\cdot)$8

For Gaussian components, the parameters are sampled as

$f(\cdot)$9

and each component is chosen either as

$X \in \mathbb{R}^{M \times L}$0

The second route is a randomly parameterized ARMA$X \in \mathbb{R}^{M \times L}$1 model:

$X \in \mathbb{R}^{M \times L}$2

with $X \in \mathbb{R}^{M \times L}$3 and $X \in \mathbb{R}^{M \times L}$4. To ensure stationarity, the AR coefficients satisfy

$X \in \mathbb{R}^{M \times L}$5

These two routes jointly provide heterogeneous marginal statistics and temporally correlated dynamics (Wang et al., 21 Feb 2025).

Quality control is enforced by filtering. Samples are discarded when inputs fall outside the valid domain of the symbolic expression, or when generated values become too large:

$X \in \mathbb{R}^{M \times L}$6

The paper further analyzes the generated data using the ADF test for stationarity and forecastability based on Fourier decomposition and entropy. It reports that ADF $X \in \mathbb{R}^{M \times L}$7-values are generally $X \in \mathbb{R}^{M \times L}$8, meaning the generated series are often non-stationary, and that forecastability is $X \in \mathbb{R}^{M \times L}$9 for tested samples, indicating that the series are not chaotic and are learnable. Representation-space coverage is also assessed using stationarity, forecastability, FFT mean, permutation entropy, seasonality, and Mann–Kendall trend, with Radviz visualizations reported to show large overlap with real-world datasets and growing coverage as more S2 data are added (Wang et al., 21 Feb 2025).

These evaluations are important because they directly address a common objection to synthetic time-series corpora: that randomly generated sequences may fail to exhibit learnable statistical structure. The reported stationarity and forecastability analyses are offered as evidence that the generated series are diverse without degenerating into unusable noise.

5. Dataset Scale and Role in SymTime Pretraining

The mechanism is instantiated at substantial scale. The paper states a maximum of 6 input channels and 12 output channels, a total of 25M series-symbol pairs, and a total cumulative series length of 50B. The dataset spans single-channel to multi-channel cases, and the appendix notes that each symbolic expression is sampled only once and that higher channel counts increase diversity of generated expressions and series (Wang et al., 21 Feb 2025).

SymTime uses this corpus as a dual-encoder foundation model. The time-series branch is a 6-layer Transformer trained with Masked Time Series Modeling, while the symbolic branch is a 6-layer DistilBERT trained with Masked Language Modeling. To align the modalities, SymTime computes projected similarities between [CLS] embeddings from the time-series and symbolic encoders and optimizes a bidirectional contrastive objective. Because masking can perturb the [CLS] representation, a momentum teacher is added through a momentum-distillation term. The final pretraining objective combines the masked reconstruction losses and the alignment losses as

$Y = f(X) \in \mathbb{R}^{N \times L}$0

with $Y = f(X) \in \mathbb{R}^{N \times L}$1 (Wang et al., 21 Feb 2025).

The data formatting is also specified. Time series are concatenated and split into non-overlapping patches with kernel size $Y = f(X) \in \mathbb{R}^{N \times L}$2 and step size $Y = f(X) \in \mathbb{R}^{N \times L}$3, giving up to

$Y = f(X) \in \mathbb{R}^{N \times L}$4

patches for the maximum channel setting; shorter sequences are zero-padded to 288 patches. Symbolic expressions are tokenized with maximum length 512. This operationalizes the claim that the series-symbol mechanism is not only a data generator but also a pretraining interface between temporal patches and symbolic tokens (Wang et al., 21 Feb 2025).

6. Empirical Effects, Limitations, and Relation to Broader Symbolic Generation

The reported downstream effect of the mechanism is that SymTime is competitive or state-of-the-art across five major time-series analysis tasks: long-term forecasting, short-term forecasting, classification, imputation, and anomaly detection. Several ablations are used to tie these gains specifically to the series-symbol design. Removing the symbolic branch degrades performance; removing either direction of contrastive loss, denoted as w/o T2S or w/o S2T, also hurts; and removing MLM or MTM likewise hurts. Performance improves as pretraining data scale increases across $Y = f(X) \in \mathbb{R}^{N \times L}$5, $Y = f(X) \in \mathbb{R}^{N \times L}$6, $Y = f(X) \in \mathbb{R}^{N \times L}$7, $Y = f(X) \in \mathbb{R}^{N \times L}$8, $Y = f(X) \in \mathbb{R}^{N \times L}$9, and $M$0, with some saturation around 40B on certain datasets. The model also exhibits zero-shot imputation on out-of-domain S2 data and on real datasets such as ETT, Electricity, and Weather, while t-SNE visualizations show clustering by symbolic operator class, including $M$1 near $M$2 and polynomial operations near each other (Wang et al., 21 Feb 2025).

A later paper, "Synthetic Series-Symbol Data Generation for Time Series Foundation Models," preserves the same central principle—randomly constructing symbolic equations, sampling multivariate input time series, and evaluating the equations to obtain correlated outputs—but adds an explicit theoretical framing through Takens’ theorem and symbolic dynamics, and reports a larger corpus of 40 million series-symbol pairs and about 50 billion total time steps (Wang et al., 9 Oct 2025). That paper also makes explicit several limitations: random tree-based generation can produce overly large or overly complex symbolic expressions, differentiation and integration are excluded because they can cause instability, overflow, or very slow sampling, and the generator is described as not yet fully general for all possible dynamical systems (Wang et al., 9 Oct 2025).

Within the broader literature, the series-symbol mechanism belongs to a wider class of systems that couple sub-symbolic generation with discrete symbolic structure. AbdGen grounds task-defined logical symbols into latent codebooks and uses abductive learning to align generation with logic programs (Peng et al., 2023). DiffMath conditions latent diffusion for handwritten mathematical expressions on RelAST triplets $M$3 derived from MathML, thereby using symbol identity, spatial relation, and nesting depth as structural priors (Pan et al., 18 Jun 2026). Symbol-LLM, in turn, constructs a symbolic data collection spanning 34 tasks and approximately 20 symbolic families, then injects symbolic competence through a two-stage tuning framework (Xu et al., 2023). These systems are not identical to the SymTime mechanism, but they clarify its position: the series-symbol pipeline is one instance of a broader neurosymbolic strategy in which symbolic structure is generated, aligned, or imposed alongside learned representations. A plausible implication is that its distinctive contribution lies in making symbolic expressions the semantic descriptors of synthetic time series rather than treating symbols only as downstream labels or control codes.