Generative Learning for Financial Time Series with Irregular and Scale-Invariant Patterns
Authors: Hongbin Huang
Minghua Chen
Xiao Qiao
Venue: ICLR 2024 spotlight (2024)
Keywords: generative model
time series pattern recognition
diffusion model
financial time series
Primary area: generative models
TL;DR: We develop a novel generative framework, FTS-Diffusion, specifically for financial time series generation, exploring the underlying irregular and scale-invariant patterns.

Abstract: Limited data availability poses a major obstacle in training deep learning models for financial applications. Synthesizing financial time series to augment real-world data is challenging due to the irregular and scale-invariant patterns uniquely associated with financial time series - temporal dynamics that repeat with varying duration and magnitude. Such dynamics cannot be captured by existing approaches, which often assume regularity and uniformity in the underlying data. We develop a novel generative framework called FTS-Diffusion to model irregular and scale-invariant patterns that consists of three modules. First, we develop a scale-invariant pattern recognition algorithm to extract recurring patterns that vary in duration and magnitude. Second, we construct a diffusion-based generative network to synthesize segments of patterns. Third, we model the temporal transition of patterns in order to aggregate the generated segments. Extensive experiments show that FTS-Diffusion generates synthetic financial time series highly resembling observed data, outperforming state-of-the-art alternatives. Two downstream experiments demonstrate that augmenting real-world data with synthetic data generated by FTS-Diffusion reduces the error of stock market prediction by up to 17.9%. To the best of our knowledge, this is the first work on generating intricate time series with irregular and scale-invariant patterns, addressing data limitation issues in finance.

Published as a conference paper at ICLR 2024
GENERATIVE LEARNING FOR FINANCIAL TIME SERIES WITH IRREGULAR AND WWNDSSAAL--IEE SCALE-INVARIANT PATTERNS
IN MEMORY OF PROF. LI
Hongbin Huang, Minghua Chen', and Xiao Qiao' School Data Science, City University of Hong Kong hongbin. huang@my httpe/eyyyyyy.u.o cäu.hk,
hk
ABSTRACT
Limited data availability poses major obstacle in training deep learning models for financial applications. Synthesizing financial time series to augment real-world data is challenging due to the irregular and scale-invariant patterns uniquely associated with financial time series temporal dynamics that repeat with varying duration and magnitude. Such dynamics cannot be captured by existing approaches which often assume regularity and uniformity in the underlying data. We develop a novel generative framework called FTS- DTffDiiffsion that consists of three modules to model irregular and iaaaeinnvaiinnt patterns. First, present scale-invariant pattern recognition algorithm to extract recurring patterns that vary in duration and magnitude. Second, we construct diffusion-based generative network synthesize segments of patterns. Third, we model the temporal evolution of patterns order aggregate the generated segments. Extensive experiments show that FTS-Diffusion generates synthetic financial time series highly resembling observed data, outperforming satte-of-theaar alternatives. Two downstream experiments demonstrate that augmenting real-world data with synthetic data generated by FTS-Diffusion reduces the eтror of stock market prediction by up to 17.9%. To the best of our knowledge, this is the first work on generating intricate time series with irregular and cale-invariiat patterns, addressing data limitation issues finance.
INTRODUCTION
Researchers in financial economics have demonstrated intriguing potential for deep learning to solve complex problems in financial settings (Qin et al., 2017 Xu & Cohen 2018 Wu et al.) 2020 Manzo
& Qiao 2020 Huang& & Li 2021). However, a dearth of data and the low signal-to-noise ratio nature of financial data pose major obstacles that hinder the further development of deep learning in finance. Unlike the sciences, finance researchers cannot run experiments to obtain more data, so financial time series are limited by their existing history. Additionally, price and return data are subject to high levels of noise, making even more challenging to extract useful information from a limited dataset. Deep learning models trained on insufficient data are prone to overfitting and cannot be expected to perform reliably on unseen data.
To alleviate data scarcity, data augmentation techniques can be employed. Generative models that capture the properties of the underlying data-generating process would produce synthetic data that resemble observed data. Recently, deep generative modeling, especially generative adversarial net- works (GAN) (Goodfellow et al. 2014) and diffusion models (Ho et al. 2020), has made remarkable progress in multiple domains including image synthesis, reinforcement learning, and anomaly detec- tior They have also been applied to time series settings such as medical records, audio synthesis, power systems, and networked systems Despite these advances, modeling financial time series poses unique challenges that complicate the task and render existing models ineffective.
Corresponding authors: Minghua Chen and Xiao Qiao.
Papers images include Zhu et al, (2017). Turkoglu al. 2019). Dhariwal & Nichol
202 and Rombach (2022). and Eysenhach al.
2022p. Reinforcement learning works include Yu al.
2017] Janner
2022). Anomaly detection studies include /enati et al. (2018) and Akcay et al. 2019)
"Medical: /Esteban (Fatecal
al. (2017). Che et al. (207)) and etc. Audio: Kong et al.
2021
Leng et al. 2022 and
Power systems: /hhan al. 2018) and Chen et al. 2018.. Networked systems:
[in et al. (2020)

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(a) S&P 500
(b) ECG
(c) Solar Generation
(d) Power Consumption
Figure 1: Time-series data of (a) S&P 500 price (finance), (b) ECG (medical), (c) solar generation (renewable energy), and (d) power consumption (smart grid). Unlike other time-series data with explicit patterns, financial time series expresses complex patterns that are irregular and scale-invariant.
The time series studied in the extant literature of deep generative learning tend to exhibit some regularity. Patterns identified in these data appear at fixed or predictable increments in calendar time (e.g., heartbeats in ECG). Time series data that contain such regular patterns are amenable to modeling, as they allow the extraction of highly correlated features from similar repeating patterns. Although conceptually straightforward, identifying recurring patterns in financial time series proves difficult due to a lack of regularity. Instead, financial time series appear to contain more subtle patterns that repeat themselves with varying duration and magnitude, quality we refer to as scale-invariance. Irregularity and acale-invariance are hallmarks of financial time series that complicate their modeling and the synthesis of additional data. We illustrate these two properties in Fig. by comparing the S&P 500 Index, broad basket of U.S. stocks, to several regular series. The three regular series exhibit clear and consistent patterns that align with calendar time. In contrast, we do not observe neat patterns that adhere to a fixed frequency for the S&P 500. Instead, we can observe similar patterns (red and green circles) that exhibit scale-ivaariance. These patterns keep their basic shape but are shifted or stretched compared to each other 5l The unique properties of financial time series make data synthesis significantly more challenging task compared to that of well-behaved data. Effective time series data generation considering irregularity and scale-invariance remains largely an open problem.
To address this problem, we deconstruct financial time series generation into a prree-prong process: (i) pattern recognition to identify irregular and scale-invariant patterns, (ii) generation to synthesize segments of patterns, and (iii) evolution to connect the generated segments into a complete time series. We propose new generative framework. FTS-Diffusion to accomplish the pattem recognition- generation-evolle evolution process. We find that FTS-Diffusion is capable of generating synthetic financial time series that closely resemble observed data. We make the following contributions:
>> We identify and define two properties of financial time series: irregularity and (see Sec. B). We present novel FTS- Fff-Diffusion framework to model time series data exhibiting these properties. To the best of our knowledge, this is the first framework capable of generating challenging time series data that contain irregularity and scale-invariance. FTS-Diffusion may also be applied to other domains with data exhibiting similar properties.
The unique architecture of FTS-Diffusion is designed to handle irregularity and ocale-invariance There are three modules. The pattern recognition module is based on a new scale-invariant subse- quence clustering (SISC) algorithm (Sec. 4.1. By incorporating dynamic time warping (DTW), SISC is able to accurately identify and separate irregular and -cale-invariance patterns. The generation module consists of a diffusion-basee based network to synthesize the scale-invariant segments conditional on the patterns learned by SISC (Sec. 4.2). 'hee evolution module is made up of a pattern transi-
tion network that produces the temporal evolution of consecutive patterns, capturing the dynamic relationship among them (Sec.
> We demonstrate the effectiveness of FTS-Diffusion in capturing real-world financial data and we illustrate the value of the generated data for downstream applications (Sec. 5 Patterns identified by FTS-Diffusion can be cross-verified with financial domain knowledge (Lo et al., 2000), and experimental results from three real-world datasets show that realistic financial time series among several alternative models. We explore the usage of the generated data for the downstream task of predicting stock prices. Augmenting limited real-world data with synthetic samples from FTS-Diffusion reduces the predictive error by up to 17.9% across the datasets.
generates the most
Forr more details, Appendix A provides an in-depth discussion on the properties of financial time series.
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Repeating at ineeular frequencies
Repeating at foxdd frequency
Repeating with varying scales Patlon
Repeeting with speciii scales much
Melulale W W. IMW
Irregularity v.s. Regularity
(b) Inver-nnv v.s. Scale Dependence
Figure 2: (a) Irregular patterns with indeterminate recurrence intervals vs. regular patterns repeated at fixed frequency. (b) Scale-invariant patterns showing similarity after proper duration/magnitude scaling vs. scale-dependent patterns showing similarity given specific duration or magnitude.
These results shed light on the capability of FTS-Diffusion to improve the accuracy and reliability of deep learning models in financial applications. Codes are available in supplementary materials.
2 RELATED WORK
Advances in deep generative modeling have shown promise to generate time series data in various problem domains, particularly using VAEs-, GANs-, and diffusion-based models. We discuss the most relevant works in this section.
Time VAE Desai et al. 2021 presents VAE- framework to model the trend and seasonality in time series. RCGAN (Esteban et al. 2017) and MV-GAN (Brophy (020) use GANs for learning medical records. Several GAN variants are employed to model time series power systems (Zhang et al. 2018 Chen et al.) 2018). TimeGAN (Yoon et al.) 2019) develops a general framework to embed time-series data into a latent space with an autoencoder network and subsequently learn the latent representation with GANs. QuantGAN (Wiese et al. 2020) provides GAN-base network to decomposition.
capture long-range dependencies financial time series under the volatility CSDI (Tashiro al., 2021) proposes score- based diffusion model primarily designed for imputation, with an unconditional variant that can be also used for time series generation. DiffWave (Kong et al.) 2021 and BinauralGrad (Leng al. 2022) generate waveform time series with diffusion models.
The above approaches can model time series with regular patterns but struggle with more complex series characterized by irregularity and scale-invariance, central features in financial time series. The identification of latent patterns in financial time series is challenging, and it is difficult for generative model without auxiliary information distinguish between these diverse distributions. In our study, we decompose the financial time series generation into pattern recognition generratio---- evolution process, enabling better modeling the irregular and scale-invariant properties. addition, diffusion probabilistic models have been shown to achieve better quality and training stability than the classical GAN and VAE models (Dhariwal Nichol, 2021 Wang al. [02T). Hence, we design our generative model leveraging the denoising diffusion probabilistic model (DDPM) (Ho et al. 2020).
3
PROBLEM STATEMENT
3.1 UNIQUE CHARACTERISTICS OF FINANCIAL TIME SERIES
The irregular and scale-invariant patterns financial time series are difficult for existing models that assume regularity and uniformity to capture. Fig. illlstraaess these repeating temporal dynamics with non-deterministic intervals and varying duration and magnitudes. The typical technique of dividing time series into fixed-interval segments in existing approaches is likely to result in snapshot of either a fraction of a pattern or a mixture of multiple patterns.
time series. A time series X = {x1..... EM consists of M segments, Im = {Im.1,.... ".m.......Im.... }. The length of the entire time series is T - > >m=1 fm- Fm is sampled from conditional distribution f(|pp, a., 88) dependent on the pattern E P, whose duration is scaled by a and magnitude scaled by 3. This way, xmm. will be statistically similar to its underlying pattern p while allowing for adjustments in duration and magnitude. To model the dynamics across patterns, we employ a Markov chain. Each tuple (p, a, ()) is state, and the state transition probabilities Q(p oj, B311 (pi 0g, 3,) describe the stochastic
We propose novel framework to model irregular and
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Pattern Evolution Module
Segmentation
Next Patter
Duration
Sampling
mmy Synthetic Time Series
Magnibude
m
SISC
Time Series X
Synthotic Subsequerce
Pattern Recognition Module
Pattem Generation Module
Figure 3: The architecture of our proposed FTS-Diffusion. The pattern recognition module identifies the scale-invariant patterns within the entire financial time series using our proposed SISC algorithm. Subsequently, the pattern generation module synthesizes the segments with diffusion-bart network conditioned the patterns. Finally, the pattern evolution module connects the generated segments to construct a synthetic financial time series following the transition between consecutive patterns.
transition from one pattern to the next. Our setup is reminiscent of applications of the Markov property in financial time series (Dueker 1997. Bai & Wang 2011; Somani et al. 2014). The novelty in our approach is that we use a Markov model to capture the transition of three specific aspects of the time series: pattern, duration, and magnitude, whereas existing work attempts to recover some unspecified latent properties of a time series.
3.2 PROBLEM STATEMENT
We seek to operationalize the structure laid out in Sec, 3.1 When faced with time series, we have no knowledge the segments {I Jm=1 the set scale-invariant patterns P, or the scaling factors and 3 that transform reference pattern into its more realistic counterpart. We also do not know the transition probabilities (pp, ay 3g Ips as B.). Our goal is to develop a data-driven framework accomplish the following:
(Pattern Recognition) identify the patterns and learn the recurrent structures P, and group segments into clusters according to their corresponding patterns E P;
(Pattern Generation) learn the distribution f(-lp, a, 3), v/p E P;
(Pattern Evolution) learn the pattern transition probabilities Q(pj, ajljp 0z, B:).
The three components above allow us to generate financial time series by (i) determining the allocation of patterns using the pattern transition probabilities, and (ii) generating each segment from the corresponding pattern with the appropriate duration and magnitude scaling factors. Our three-pronged framework dedicated to identifying and modeling the irregular and scale-invariant patterns observed financial time series the first of its kind in the literature.
4 OUR PROPOSED FTS-DIFFUSION FRAMEWORK
In this section, present FTS-Diffusion framework. Fig. 3 provides illustration. FTS- Diffusion consists of three components: pattern recognition module, pattern generation module, and pattern evolution module. Next, we introduce each module and how they work together.
4.1 PATTERN RECOGNITION: IDENTIFYING IRREGULAR AND SCALE-INVARIAN PATTERNS
We propose a novel Scale-Invariant Subsequence Clustering (SISC) algorithm to partition the entire financial time series into segments of variable lengths and group them into K distinct clusters. The segments within the same cluster exhibit similar shapes after proper scaling in duration and magnitude. The centroid of each cluster then represents scale-inve inve--ivarrant pattern in the financial time series.
The idea is similar to the traditional K-Means clustering (Hartigan & Wong 1979), which primarily clusters segments of identical length and thus falls short in our context due to inability to handle
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candidate lengths
compare
K centroids
segment Xyp
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(a) Pattern Recognition
(b) Pattern Generation
(c) Pattern Evolution
Figure 4: Key designs in each module of our FTS- ITS-Diffussoo: (a) pattern recognition module: SISC greedy segmentation; (b) pattern generation module: the patter-conditioned diffusion network paired with the scaling autoencoder; (c) pattern evolution module: Markov transition.
segments of varying lengths and magnitudes. Instead of separating the entire time series into equal- length segments as is commonly done, we adaptively determine the optimal segment lengths through a simple yet effective greedy segmentation strategy. Specifically, as illustrated in Fig. ((a) we compare the segments of candidate lengths I E [min max] with the cluster centroids within a normalized
space at each evaluated position = Σr= 4.. The length 1 that minimizes the distance to the nearest centroid is considered the optimal segmentation for the current segment 2m = -xa44= i.e.
1" arg min d(Xext)))) EpEP. 1E[lminndmaxe)
(1)
The first key component in our design is a distance metric .(: that is robust to varying lengths and magnitudes and hence properly measures the difference between subsequences. Classical metrics, such as Euclidean distance, fail to provide accurate mcasurements due to their limitations in comparing variable-length sequences. In contrast, we employ dynamic time warping (DTW) to calculate the minimum distance across all pointwise alignments between two segments:
(T, = min AEA
(4,((a,)))
where A denotes the alignment between two sequences in the set of all possible aligmments A, and A (x, y) [0 yj li is the pointwise distance matrix between two normalized sequences æ and y- well-suits our purpose to identify similarities segments with similar shapes but varying duration and magnitudes. With the DTW metric as d(.;)), we apply the greedy segmentation strategy from the start to the end of the time series. Upon completing the greedy segmentation across the entire time series, proceed with the standard K-means clustering process. Each segment is assigned to the nearest centroid, then the centroids are updated based on current cluster assignments. This process iterates until cluster assignments stabilize or a pre- mre-detemmnne number of iterations is reached.
The second key component in our design is the initialization of the cluster centroids. The random initialization typically used in standard clustering methods often yields suboptimal results. To alleviate this issue, we design a wise initialization for more informed initial centroids. Our initialization begins by randomly selecting one segment from all available segments of a pre-specified length, which could be either the minimum or maximum length in practice, to serve as the first centroid. Afterward, we choose the subsequent centroids from the rest, with the selection weight being proportional their distances to the closest centroid within the chosen set. This means that segments located farther from their nearest centroid have higher probability of being the next choice. We repeat this process until K centroids have been initialized. This method ensures diverse set of centroids spreading across the data space, promoting an efficient start of our SISC algorithm.
We highlight that our SISC algorithm is the first in the literature designed to identify scale-invariant patterns. The computational complexity of SISC is OTK.maax), which is linear to the length of the entire time series. The pseudo-code of SISC and the selection of parameters, such the range of segment lengths and the number of clusters K, are detailed in Appendix B.I The learned patterns are cross cross-validade validated with the technical patterns in the financial literature (Lo al. 2000 in Appendix (.2 To better verify the effectiveness of our proposed SISC algorithm, we conduct a thorough investigation using simulated time series data in Appendix B.3
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4.2 PATTERN GENERATION: LEARNING CATTERN-CONDITOON TEMPORAL DYNAMICS
We develop a pattern generation module, 0 in Fig. 3 to synthesize the segments of patterns. The goal is to generate new segments that mimic the temporal dynamies within the observed segments. Considering the financial time series as a collection of scale-invariant patterns, the data-generating process can be interpreted as capturing the distribution of the reference patterns and transforming them with proper scales in duration and magnitude. Accordingly, we instantiate this data data-generating process using two dedicated networks for the two tasks, as discussed below.
The first network is the scaling autoencoder (AE) for learning the transformation between variable- length segments z and fixed-length representations x after we capture the reference pattern repre- sentation using the patter- centenn-oondiioned diffusion network. The encoder of the scaling AE stretches the variable-length segments into fixed-length representations that align with the dimension of refer- ence patterns. The decoder, the other hand, is responsible for reconstructing the variable-length segments from the fixed-length representations.
The second network is the pattern-conditioned diffusion network for simulating diffusion-denoising process perturbing the pattern representations gradually by adding noise over N steps (diffusion) and removing the noise gradually recover the original representation (denoising). The diffusion process is achieved by a pre-specified procedure of incrementally adding Gaussian noise step by step, while the denoising process is approximated by a neural network that learns the removing noise each step, i.e, the denoising gradient. Approximating the stepwise denoising gradients is equivalent to learning the mapping from a latent Gaussian space to the pattern space. Consequently, given Gaussian noise, we can generate a pattern representation. The continuous nature of the Gaussian space implies that we can sample an infinite amount of Gaussian noise and produce corresponding new pattern representations. We build our diffusion network based on the denoising diffusion probabilistic model (DDPM) (Ho et al. [ In detail, we apply the following diffusion process at each step to corrupt the representation into noise:
q<>'|=²11> N /1- -
where 3 represents the magnitude of the segments. Thereafter, we design conditional denoising process that recovers the target segments from a prior Gaussian noise conditioned on the reference patterns over the reversed N steps:
Po mn( (a))= = N(aa" poo (T),
where Hg is proportional to ep representing the neural network that learns the denoising gradient at each step (see Appendix C. Note that the superscript i denotes the step in the diffusion and denoising process.
We jointly train the pattern-conditionss diffusion network and the scaling AE using the standard supervised learning with the segments identified in Sec. 4.1 as training data. As depicted in Fig. 4(b) the observed segments are encoded and perturbed to noise by the encoder in the scaling AE and the diffusion process in the nattern-conditionn dashed arrows, reverses this by denoising and decoding the segments from noise through the denoising process in our diffusion network and the decoder in scaling AE, During this process, that (i) the diffusion and denoising gradients are consistent at each step, and (ii) the reconstruction successfully reproduces the observed segments. Therefore, the objective contains the reconstruction loss between the observed and reconstructed segments for the scaling AE and the unweighted variant of the variational lower bound (ELBO) (Ho et al. 2020) for the pattern-conditioned diffusion network:
diffusion network. The generation process, marked with
must ensure
[())- Ex... mm
[|] Im
&m.l? 12 + E.g. [[le' eø (x i, p) [
(5)
where e² is the noise added in the corresponding diffusion process at step i.
During the generation phase, new segments can be created by exclusively applying the denoising process in the pattern-conditionel diffusion network and the decoder in the sealing AE.
4.3 PATTERN EVOLUTION: LEARNING THE TRANSITION BETWEEN CONSECUTIVE PATTERNS
As mentioned in Sec. 3.1 we model the transition states (encompassing patterns, lengths, and
magnitudes) between consecutive generated segments using a Markov chain. Once the transition
We present the preliminary of DDPM and our network structures in Appendix
and C.2 respectively.

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states determined, we obtain evolution series of patterns, somehow addressing the irregularity in the financial time series. This ensures that the consecutive generated segments maintain the essential temporal correlations observed in real- world financial data. To capture the Markov-chain modeled temporal dynamics across patterns, we introduce a pattern evolution network 6, with the network structure in Appendix(... to learn the temporal evolution of the states between consecutive segments More specifically, the network learns the probability of the next pattern along with its corresponding length and magnitude, given the current state (because of the Markov property):
(pm+11, 0m+1, 6(Pm, Om, Pm))
(6)
where (fm+1: 1 0m+11 Bm+1) denotes the next pattern and its scales in length and magnitude,
The pattern evolution network is trained to optimize the following objective:
L(Φ) Es,,, (frempppppm pm++1) + ||am+1 -m++ + 113m+1 Bm+1112
where (c represents the cross-entropy.
4.4 PUTTING EVERYTHING TOGETHER: SYNTHESIZING ENTIRE FINANCIAL TIME SERIES
We regard patterns as the basic building blocks of generation. Accordingly, PTS-Diffusion produces synthetic time series on a wattern-y--ptteen basis.
Given an initial segment sampled from the historical data, it generates the successive segments by employing the pattern generation module and the pattern evolution module iteratively, as outlined in Algorithm 2 in Appendix D.2 At each position m, the pattern evolution network predicts the next pattern Pm+11 its length-scaling factor 4m and magnitude-scaling factor 3m+1. With these states, the pattern generation module generates the next segment 2m-11 The synthetic time series then grows as more segments are generated and appended. This procedure repeated until the entire time series reaches the desired total length.
5 NUMERICAL EXPERIMENTS
We conduct numerical experiments to evaluate the performance of our FTS-Diffusion compared with alternatives, i.e., whether the generated data resemble real data and are useful for downstream tasks.
5.1 DATA AND EXPERIMENTAL SETTING
We run experiments on three different types of financial assets with varying characteristics: the Standard and Poor's s 500 index (S&P 500), the stock price of Google (GOOG), and the corn futures traded on the Chicago Board of Trade (ZC=F), Detailed data settings are given in Appendix E. In finance, it is known that the raw asset prices follow a non-stationary random walk and not well-behaved for statistical models. Instead, the returns, i.c., closing price changes in consecutive time in- tervals, remain relatively constant statistical properties (such mean and variance) over time. Thus, we compare the return series generated by FTS-Diffusion those by repre- sentative baselines: RCGAN (Esteban et al.) 2017). TimeGAN (Yoon et al.) 2019), and CSDI (Tashiro et al. 2021). whose details are in Appendix
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Figure 5: Stylized facts of real and generated S&P 500 over 10 years: the heavy-tailed distribution (fat tails compared to the Gaussian in density and QQ-plot, the first two columns) and decaying auto- correlations in absolute return (the last column).
E.2
'We do not estimate transitions using traditional Markov models. Our NN-based approach avoids unwieldy transition matrices, generalizes well to unseen scenarios, and handles non-linear dependencies adeptly.
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Table 1: Generated return distributions compared to observed data. The KS and AD statisties floored/capeed at 0/1 and 0.01/0.25, respectively. A higher value indicates better goodness of fit. Variation in the test statistic across multiple runs shown with a +/- range.
S& P500
GOOG
7C=F
Model
KS
AD
KS
AD
KS
AD
RCGAN
.1891.006
.073:1.00
.185..006
.068...004
.179:.006
.0652.005
TimeGAN
.293+.001
.115#.006
.288+.007
.108=.005
.287=.007
.103=.005
CSDI (Generative)
.168#.003
.069:.002
.156+.001
.067=.003
.157=.003
.065=.00s 003
FTS-Diffusion
.3271.003
.1281.003
1.244.004
.119...002
.3251.003
.121 1121:.003
5.2 PROPERTIES OF THE SYNTHETIC TIME SERIES
The synthetic financial time series should inherit the stylized facts (Cont 2001| Barberis & Shleifer 2003) of asset returns, and resemble the distribution of observed data to a high degree of fidelity.
Stylized facts of financial time series. The empirical properties of financial time series have been studied extensively in the literature, which is often referred to stylized facts (Cont 2001 Barberis & Shleifer 2003). The empirical studies reveal that asset returns have heavy tails, and the autocorrelation of absolute returns decays slowly
time. We assess whether the synthetic time series adhere to these stylized facts in Fig. 5 Indeed, the synthetic series exhibit significant heavy tails in their distribution and gradual decay the autocorrelation of absolute returns, conforming to the aforementioned stylized facts. These results suggest that our approach is capable of generating synthetic financial time series that preserve the essential properties of observed data.
Distribution comparison. We also evaluate the discrepancy between the distribution of the synthetic time series and that of observed data, using the Kolmogorov-Sminot son son-Darling (AD) test as evaluation metrics. These tests estimate the goodness of fit between the synthesized distribution and the distribution of actual returns. For both tests, a larger test statistic indicates higher degree of similarity between the distributions. The KS test is more sensitive to differences in the center of the distribution, whereas the AD test more aware of the tails of the distribution. Table demonstrates that our TTS-Diifusion learns quantitatively closer distribution to the observed data, compared to other baselines. This result further confirms the efficacy of our approach in generating financial time series that resemble the observed data. More quantitative results using other metrics are included in Appendix E.3
(KS) test and the Ander-
5.3 DOWNSTREAM PREDICTION ANALYSIS OF THE SYNTHETIC TIME SERIES
We expand "Training Synthetic, Test Real" (Esteban et al. 2017 Jordon et al. 2018) and design two new settings to evaluate the usefulness of the synthetic data for downstream tasks. Specifically, focus the task of prediction and implement This structure is a prevalent choice in the literature (Yoon et al., 2019) Jeon et al. 2022 Remlinger (et al.) 2022). The downstream model is employed to predict the next data point the series, using the 64 previous historical values as input (see Appendix [E.4] for the additional five-day ahead prediction). We compute the mean absolute percentage error (MAPE) averaged over multiple runs.
LSTM- downstream predictive model.
Training on Mixture, Test on Real (TMTR). In this setting, we train the downstream predictive model on a dataset that combines observed and synthetic data in different proportions. For instance, a dataset with mixing proportion of (30%, 70%) would be composed of 30% of data sampled from the observed data and 70% of data synthesized by the generative model. We test the predictive model on the test set sampled from the observed data which had not been seen by the generative model. If the synthetic data resemble the observed data, the predictive power of the downstream model trained on datasets with different mixing proportions should remain similar. Fig. (6(a) shows the results of the TMTR experiment for the one-day forecast on the three assets. The predictive accuracy is remarkably consistent across all mixing proportions, when synthetic data generated using In comparison, the predictive accuracy deteriorates (large MAPEs) as the proportion of observed data decreases, when synthetic data are generated using RCGAN, TimeGAN, CSDI.
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75
10000 20000 Augmented Size
10000 20000 Augmented Size
10000 20000 Augm ented Size
10000. 20000
Syn. Proportion (%)
Augmented Size
(a) TMTR
(b) TATR
Figure 6: Prediction errors of the downstream model trained under the TMTR and TATR settings. Our FTS-diffusion maintains a comparable level of prediction accuracy across all mixing proportions of synthetic data and reduces the prediction errors by augmenting the observed dataset. Solid lines and shaded bands in each subfigure represent the average error and the 95% confidence interval over multiple runs, respectively. Dashed lines in each TATR test mark the initial prediction errors.
Thus,
is capable of generating synthetic time series sufficiently similar actual data to uphold the performance of downstream prediction task, whereas other models cannot.
Training Augmentation, Test on Real (TATR). We initialize the training set with limited observed data. We then iteratively append additional synthetic data and evaluate the resulting performance of the downstream predictive model for a one-day ahead forecast. The results in Fig. 6(b) show clear downward trend the prediction error as more synthetic data from FTS-Diffusion the training set. Appending 100 years of synthetic data reduces the MAPE by 17.9%. 15.3%, and 17.4% on the three assets, respectively. In contrast, the prediction error either increases or largely remains the same when synthetic data are generated by other baselines. These results indicate that ETS-Diffusion can effectively alleviate the problem of data shortage by augmenting the training set with sufficient synthetic samples. Supplementary experiments are provided in Appendix E
added to
6 CONCLUDING REMARK
We present FTS- "TS-Diffusion generative framework, for synthesizing financial time series with irregular and ialee-invariant patterns. We break down the challenging financial time series generation into a pattern
generation-evolutiin scheme. To facilitate this process, we design three dedicated modules: (i) a pattern recognition module leveraging our proposed SISC algorithm carefully designed to identify these patterns, (ii) pattern generation module using a diffusion-basse network to synthesize the segments of patterns, and (iii) pattern evolution network to assemble generated segments with proper temporal evolution between consecutive patterns. Experimental results confirm the effectiveness of Difffuifusion in synthesizing financial time series that resemble observed data in distribution and their usefulness for downstream tasks. To the best of our knowledge, this the first work in generating intricate yet crucial time series that encompass irregular and scale-invariant patterns, holding the potential for diverse applications across domains beyond finance.
This work offers new perspective on complex (financial) time series generation from its irregular and scale-invariant properties. promising direction for future research would be to extend our work to more challenging problem settings, e.g., the multivariate modeling that encompasses interactive dependencies across multiple time series. Our approach can handle potential distribution shifts arising from changes in the number of patterns, and an extension may be able to address shifts in transitions between patterns. One could also strengthen the theoretical and empirical guarantee of the generation quality. We leave these ideas for future research.

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ACKNOWLEDGEMENT
The authors wish to dedicate this work to the memory of Prof. Duan Li, whose insights and guidance have been instrumental to the early stage of this work.
This work is supported in part by General Research Funds from Research Grants Council, Hong Kong (Project No. 1 12002233, 21500422, and 11500823), InnoHK initiative, The Government of the HKSAR, Laboratory for Al-Powered Financial Technologies, and Shenzhen-Hong Kong-Macau Science & Technology Project (Category C, Project No. SGIX2222053011 1030266). The authors would also like to thank Mark Huang, Blair Hull, Ray Iwanowski, Alexander James, Carric Wang, Qi Wu, Hao Pan, and the anonymous reviewers for their helpful comments.
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A FURTHER COMPARISON OF FINANCIAL TIME SERIES AND OTHER REGULAR SERIES
As discussed in Sec. financial time series exhibit more complex distribution than other time series. In particular, the inherent patterns in financial time series are irregular and scale-invariant. We have thoroughly discussed the ccale-invariance in Sec. D and To better illustrate the irregular property of financial time series, we display the distributions of financial time series and other time series sampled at different frequencies by ridge plots as Figure 7 Financial time series elearly exhibit distinct distributions for each month in 2022, well as for each year spanning from 2017 to 2021. In contrast, other time series show similar distributions regardless of different sampling frequencies. The results exemplify that the distributions of the subsequences in financial time series tend to be diverse over time, reflecting the irregular property.
Distribution fron of 5EP5000 2022
Disribution of BCG Reconds (205)) Deior routinn So Generation in Weeks
Duy burken Power Consumption ns Wodos
2220
2000
2500
(a) S&P500 Monthly
(b) ECG 20s
(c) Solar Weekly
(d) Power Weekly
Dlaribution sar500 frice 2017-2003
Distribution 5 Reconds (600)
Diambut on of Solar Gane rntiaaiion Montha Danauzior Fower Coraumpion in Hontra
2017
500e
32dea
poice
aodce
(e) S&P500 Yearly
(1) ECG 60s
(g) Solar Monthly
(h) Power Monthly
Distributions of time series sampled with different frequencies: (a.e) S& P500 (finance), (b.f) ECG (medical), (c.g) solar generation (renewable), and (d.h) power consumption (smart grid). Financial time series, such as S&P 500, exhibit different distributions sampled different frequencies. Whereas, the distributions of other time series are similar regardless of the sampling frequency, respectively.
Figure 7:
B MORE INFORMATION ABOUT OUR PATTERN RECOGNITION MODULE
B. SUPPLEMENTARY TECHNICAL DETAILS OUR SISC ALGORITHM
The pseudo-code of our SISC algorithm is presented as Algorithm I As introduced in Sec. 4.1 SISC is performed with two main stages, (i) initializing the cluster centroids (Cluster Initialization in Algorithm 1 and (ii) segmenting and clustering the subsequences into K clusters using greedy strategy (Greedy Segmentation and Clustering in Algorithm l
In numerical experiments in Sec. 5] leveraging domain knowledge in finance, set the minimum and maximum segment lengths as 10 and 21, respectively, focusing on the atom-like short-term patterns commonly observed (Lo et al. (2000). Applying the elbow method (Thorndike 1953), empirically determine the K's for three financial assets, which are 14, 11, and 11, respectively.
B.2 VERIFICATION OF LEARNED PATTERNS WITH PREDEFINED COUNTERPARTS FINANCE
Employing the pattern recognition module, we successfully learn patterns in financial time series in data-driven manner. The majority of the learned patterns are consistent with predefined ones with particular shapes and formulaic definitions from technical analysis (Lo et al. 2000). Table 2 presents the consistent patterns along with their terminologies from technical analysis. The learned pattern, shown in the first row and second column of Table 2 completely coincides with the classical Inverse Head-and Bond--nd-Shoon Shoulders (IHS) via visual observation. Despite some sight noise, the other learned patterns exhibit similar shapes to the predefined ones. Surprisingly, the learned patterns sensitive
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Algorithm 1 SISC Algorithm
Require: Time series X, pre-determined number of clusters K, minimum and maximum subse-
quence length /min/max. maximum iterations mar mar_iters
P t-6
2: Prepare candidate centroids {Xxe++lmxx T==
T--maxx
3: Randomly select the first centroid po from the candidates
4: P. P.oppend(po)
5: while P.size K do
> Cluster Initialization
6: Compute the distance to the nearest chosen centroid for each remaining candidate
7: Set the probability of each candidate proportional to the above distance
8: Randomly select the next centroid Pt: with the above probability
9:
' (Pappend(pu)
end while
11: iter +0
while iter mar iitess do
Greedy Segmentation and clustering
13:
S #
15:
while 1 < T do 1" 4- arg me (Mminn S.append(1")
DTW (XWW(xt+1)) p),
17:
18:
+-
19:
end while
20:
iter iter
21:
Update P
end while
23:
Return P,S
minor changes in temporal dynamics. For instance, the Descending Rectangle (DR), the last one in the second column, is slightly different in its trend. Our pattern recognition module is capable of distinguishing such slight differences. This cross-verification indicates that the unsupervised algorithm is able to identify genuine recurring patterns and significantly raises the level of confidence our data-driven approach.
Moreover, we have revealed some intriguing findings. Our pattern recognition module focuses on identifying the irregular and scale-invariant patterns in the financial time series, while technical analysis defines the patterns trading signals to forecast future trends. Therefore, if the future trend predicted by technical analysis is exactly highly correlated with the predefined patterns, is possible for them emerge together in our clustering. In other words, our learned patterns may incorporate the future trend into the corresponding predefined patterns from technical analysis. As an example, the Rectangle Top (RTOP), the last in the first column indicates decreasing future trend in technical analysis, and our result covers such a declining curve. This provides new insights into technical analysis, by revealing patterns that combine the trading signal with future trends.
However, not all learned patterns match predefined patterns. We are faced with the dilemma of whether include these unmatched patterns data generation. As we will show later, downstream experiments reveal the benefit of including unmatched patterns. While there is long tradition of technical analysis, the predefined patterns rely on human identification. The unmatched patterns are likely capturing important components of financial data, too nuanced for humans to see.
B.3 INVESTIGATION WITH SIMULATED TIME SERIES DATA
In this investigation, we explore the capabilities our pattern recognition module in detail, in particular for the proposed SISC algorithm. To effectively demonstrate its functionality, conduct series of experiments using various simulated time series data that we have manually created. These simulated data comprise multiple scale-invarian patterns that define, exhibiting varying lengths and magnitudes. The key advantage of these simulated data is that we have full control over the data-generating process. This control allows know the ground truth against which can
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Table 2: Patterns learned by FTS-Diffusion consistent with predefined ones from technical analysis (Lo et al. 2000).
Terminology
Standard
Learned
Terminology
Standard
Learned
Shouldes (IIS)
Inverse ShouddSs (IIIS)
M
W
Triangle Tops (TTOP)
Triangle Bottoms (TBOT)
W
Double Tops (DTOP)
Double Bottoms (DBOT)
W
Ascending Rectangle (AR)
Descending Roctanglc (DR)
Rectangle Top (RTOP)
compare and evaluate the model performance. Therefore, this setup enables us to rigorously evaluate the module under variety of controlled scenarios.
More specifically, we construct three distinct simulated time series, each being designed with different configurations of recurring scale-invariant patterns. These configurations are as follows: (i) one pattern only, and (ii) multiple patterns. For each configuration, we establish the corresponding standard pattern(s). Subsequently, each pattern transformed into diverse segments that maintain similar shapes but exhibit varying lengths and magnitudes. This transformation involves stretching the pattern to variable lengths and magnitudes as well as integrating a certain degree of noise. These segments are then combined together to form an entire time series with a length of 10000 data points, following a pre-specified transition between consecutive patterns. Consequently, we keep hold of full control over the simulated time series, including the ground- truth patterns, the number of patterns, the segmentations, and the transition states.
We assess the performances of our SISC algorithm these simulated data from two primary perspectives, whether it accurately learns the cluster centroids that represent the identified patterns and whether it learns the correct segmentation relative to the ground truth. These also constitute the key objectives of our pattern recognition module. The ground truth centroid of each cluster is the barycenter of segments transformed from the same corresponding pattern. To quantitatively measure the discrepancy between each centroid Caisc learned by SISC and the ground truth Creat, we design a per-unit DTW eиrоr as DTW (Creal, Csise)/ L over a normalized space, where denotes the length of ground truth centroids. In this context, the normalized space implies that the magnitude of each point the centroid rescaled fall within the range of [0.1]. This metric describes the average deviation per unit length between the learned centroid and the ground truth, as measured in the time- warped space. A lower value suggests a higher degree of similarity between the learned centroid and the ground truth. The upper bound of this metric is 1, which represents complete dissimilarity between the learned and true centroids. For evaluating the segmentation, we utilize the Jaccard similarity coefficient, also known the Intersection over Union (IoU), to compare the segmentation saiec learned by SISC and the ground truth Sreal- It is defined as, |sreal n Ssteel//|ssea U $ siieel, the size of the intersection divided by the size of the union of the sample sets, which is floored and capped O and 1, respectively. A higher statistic means a greater similarity between the learned segmentation and the ground truth, indicating more accurate segmentation by the SISC algorithm. Next, we will present the experimental results for each simulated time series.
One pattern only. This setup is designed to verify the ability of our SISC algorithm correctly identify and segment a single pattern from the simulated time series. In this scenario, the simulated data has only one simple pattern, depicted as the standard pattern in Fig 8(b) This standard pattern is manipulated into devise segments, each with distinct lengths and magnitudes following pre-specified transitions between consecutive segments. These segments are then merged to construct a complete time series, as illustrated in Fig ((a) After performing SISC on this simulated time series, the ground truth centroid these segments (Fig&(()) and the counterpart learned by SISC (Fig S(d)) show a significant degree of similarity, achieving a very low per-unit DTW error of 0.009. Furthermore, the Jaccard similarity coefficient of the resulting segmentation 0.938, suggesting that 93.8% of the points in the simulated time series are correctly segmented. These results verify that our SISC algorithm successfully identifies the target pattern and correctly segments the simulated time series containing one pattern only.
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Simulated time series
(a) Simulated time series
Standard
Ground-truth
SISC
(b) Standard pattern
(c) Ground-truth
(d) Learned SISC
Figure 8: Investigation of SISC: one-pattern scenario. (a) Simulated time series containing one scale-invariant pattern only; (b) predefined standard pattern; (c) ground-truth centroid; (d) centroid learned by SISC.
Multiple patterns. Our focus on this configuration is to further validate the effectiveness of our SISC algorithm in learning the simulated time series that involves four patterns in Fig. The date-gengg process is the same as that in the one- pattertteen scenario but produces a variety of segments corresponding to multiple patterns. The learned centroids by SISC comparable to the ground truth, with an average per-unit DTW error of 0.01 over four patterns. Meanwhile, the learned segmentation achieves Jaccard similarity coefficient of 0.784. This score indicates that a significant majority of data points are well placed into the correct intervals. The points that are incorrectly segmented are predominantly located on the boundaries of the segments. Importantly, these boundary misclassifications do not significantly impact the general shape of the corresponding segments. Therefore, despite these minor discrepancies the segment boundaries, our SISC algorithm demonstrates its robustness when applied to simulated data consisting of multiple patterns.
We believe this investigation offers valuable insights into the pattern recognition module, particularly our proposed SISC algorithm, and confirms its potential applicability and reliability in more complex real-world scenarios.
C MORE INFORMATION ABOUT OUR PATTERN GENERATION MODULE
C.1 PRELIMINARY OF DENOISING DIFFUSION PROBABILISTIC MODEL
Diffusion models incorporate a forward diffusion process <<<1<x-- that gradually corrupts the target data by adding noise over N steps and backward demoising process po (Ix") that learns
the reverse procedure recover the target data, where i ~ 21,122.... N) denotes the i-th diffusion step. The diffusion process is typically pre-specified:
(xx I:N
200) = II
| (x'1))
g(x
(8)
==1
qix² x 11= = Naa';; v1 o --
(9)
The denoising process is approximated with neural network as follows:
Po O:NN = p(xx) II po (ax 1|20",
(10)
Po -x(=-11(=")) (mo=== (i) i),a'')
(11)
where the neural network learns the noise gradient between steps and g is predefined parameter.
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mrm
Simulated time series
(a) Simulated time series
Standard p1
Ground-truth pl
SISC p1
Standard p2
Ground-truth p2
SISC p2
Standard p3
Ground-truth p3
SISC p3
Standard p4
Ground-truth p4
SISC p4
(b) Standard pattern
(c) Ground-truth
(d) Learned by SISC
Figure 9: Investigation of SISC: multiple-patterns scenario. (a) Simulated time series containing
multiple scale-invariant patterns; (b) predefined standard patterns; (c) ground-truth centroids; (d) centroids learned by SISC.
The denoising diffusion probabilistic model (DDPM) (Ho et al. 2020) introduces nice property that enables us sample x' an arbitrary step during the diffusion process, using the reparameterization trick. Let od of and o' - of, the resulting closed form
q (x)) Niw';; Vo " 8² I).
(12)
Using the same property, we can also obtain:
1 - v
po(2 1)
(xr
co (1)..
(13)
V 23
Therefore, (Ho et al. 2020) demonstrated that training diffusion models with the following simplified objective empirically yields better results:
L EiNLL,),, Ille' c0 1).
(14)
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C.2 IMPLEMENTATION DETAILS OUR CATTERN-CONDITIN DIFFUSION NETWORK AND SCALING AUTOENCODER
Our pattern-conditooedd diffusion network utilizes six residual temporal convolutional (TCN) blocks to capture the internal temporal dynamics within pattern segments. Each block mainly comprises two temporal convolution layers. Time embeddings for each diffusion step are constructed with a fully-connected layer positioned at the top of each block. We set the number of diffusion steps to N 100.
Our scaling AE be implemented with two layers of LSTMs GRUs.
We jointly train these two networks following the procedure in Sec. 4.2 using the Adam optimizer with learning rate of 5e 04. We set the batch size to 32. The ppprr-prrameterss are determined empirically following the common techniques in the literature.
D MORE INFORMATION ABOUT OUR PATTERN EVOLUTION MODULE
D.1 IMPLEMENTATION DETAILS OF THE PATTERN EVOLUTION NETWORK
In practice, we treat modeling the next pattern as a multi-category classification, while the learning of the length-scaling factor and the angnitude-scalingg factor 3 regression. Note that treating the estimation the length as a classification task is also feasible. Hence, our pattern evolution network models the Markov transition of state (p. a, () between consecutive segments by a fully-connected neural network with three corresponding outputs. We train this network using the Adam optimizer with learning rate of 4e 04 over 1000 epochs. The haper-parameters are determined empirically.
D.2 PSEUDO-CODE OF THE SAMPLING PROCESS
We provide the pseudo-code of the sampling process in our PTS-Diffusion as Algorithm We commence the creation of a new synthetic time series by initializing the first segment, which sampled from the observed data. After the initialization, subsequent segments are produced iteratively through the following procedure. the pattern evolution module o. With these states, the next segment is generated by the pattern generation module 0. This newly generated segment is then appended the synthetic time series. This iterative process is repeated until the synthetic time series reaches the desired length.
each iteration, the transition states of the next segment are first
predicted using
Algorithm Data synthesizing procedure incorporating the pattern generation module and pattern evolution module.
Require: Pattern generation module 6, pattern evolution module 6, latent patterns P, terminal series
length T
X 50
2: Initialize æe E X
3: X Kappend(aag
4: +0
5: while len( X) do
6: Pin: Om, Bm Transition.Saaes((aa
7:
(pm+1, 0m,,1 3m+1) t (m Om, 8m
8:
Im+1 + 0(pm+1, am+ Bm+1)
9:
X. xappend((mmt))
10: m <- m +1
end while
12: Return X
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E SUPPLEMENTARY DETAILS ON EXPERIMENTS
E. DATA SETTINGS
As introduced in the main paper, we conduct our experiments on three assets: the Standard and Poor's 500 index (S&P 500), the stock price of Google (GOOG), and the corn futures traded on the Chicago Board Trade (ZC=F). The S&P 500 data covers the period from 1980-01-01 to 2020-01-00 The data for GOOG spans from 2005-01 1005001-01 01-01 to 2020-01-00 -0.1.. And the ZC=F data ranges from 2001-01-01 to 2020-01-01.
We employ 80/20 train-test split strategy, using the first 80% for training and the remaining 20% for testing. Importantly, neither TTS-Diffusion nor downstream models in our subsequential experiments have seen the test sets during the training phase, and all of our evaluation is out-of-sample basis.
E.2 IMPLEMENTATION DETAILS OF BASELINES
RCGAN. In our experiment, we apply the conditional version of RCGAN (Esteban et al. 2017). with time information (e.g., date) the input conditions.
TimeGAN. TimeGAN Yoon et al. 2019) first learns an embedding of the target data with an autoencoder network and then models the latent distribution with a GANs network. Consequently, the synthetic time series can be constructed by using the decoder to expand the generator output into the original data space. In the original paper. TimeGAN takes multi-diieensonal prices (including the volume and high, low, opening, closing, and adjusted closing prices) as inputs. However, our experimental settings focus on univariate time series. For fair comparison, we implement the TimeGAN following the original paper but with one-dimensional input of the return rate of daily closing prices.
CSDL CSDI (Tashiro et al., (021) is proposed for time series imputation. As indicated in its original paper, the unconditional variant of CSDI can be utilized for time series generation. Thus, we implement the unconditional variant of CSDI for financial time series generation.
All of the baselines are designed to synthesize the -dentical-interval samples. Thus, we partition the entire financial time series into equal-length subsequences using the classical segmentation approach. The length of each subsequence is set as 21, aligning with the setting of maximum length in our PTS-Diffusion.
E.3 QUANTITATIVE ANALYSIS USING ADDITIONAL METRICS
In addition to the Kolmogorov-Smimov (KS) test and the Anderson-Darling (AD) test employed in our primary experiments, there exist numerous other effective tests to evaluate the 'goodness of fit' between the distributions of different time series. Here, include two more widely- metrics: the nonparametric Sops-Singletto (ES) test and the Wasserstein Distance (W). The ES test, frequently used in the field of econometrics, is sometimes preferable the KS test in instances that do not assume a continuous distribution. A larger ES statistic represents a higher similarity between the distributions of the two samples. The Wasserstein distance measures the minimum amount of probability mass that needs to be moved between two probability distributions. A smaller distance indicates a closer match between the distributions. Table3] comparison of generated returns using these metrics. According to these distribution tests, our FTS TTS-Diffusion outperforms other baselines as well.
Moreover, we incorporate several fundamental metrics prevailing in finance, including the Sharpe ratio (SR), Sortino ratio (SoR), and Calmar ratio (CR). The Sharpe ratio measures the average return eamndd in excess of the risk-free rate per unit of volatility, which helps investors understand the return an asset compared to its risk. The Sortino ratio, akin to the Sharpe ratio but with a focus on downside risk, evaluates the excess return against its downside deviation. The Calmar Ratio compares the average compounded rate of return to the maximum drawdown risk, offering insight into the potential losses an asset might experience. These summary statistics represent fundamental characteristics of financial asset returns. If the synthetic time series closely resemble the observed data, these ratios of the synthetic data would align closely with those of the real data. In other
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Published as a conference paper at ICLR 2024
Table 3:
Generated return distributions compared observed data using more metrics. Variation in the test statistic across multiple runs is shown with a +/- range.
S&P500
GOOG
ZC=F
Model
ES f
ES f
W4
ES f
W W+
RCGAN
.044 +.0007
.099 +.0005
.005 +.0006
.009 +.0005
.004 +.0006
.009 +.0005
TimeGAN
.066 +.00
.005 =000
.005 +.0006
.005 =00004
.005 #.005
.005 =.005
CSDI
.04 1.0003
.05 1.0003
.044 1.0004
.066 1.0003
.05 1.0003
.006 1.0003
FTS-Diffusion
.008 +.0003
.003 +000
.007 +:0003
.003 0000
.008 0003
.003 0.00
words, the discrepancies in these statistics between the actual returns of the financial asset and their comparable generated counterparts should be relatively minor. We denote these discrepancies between the actual assets and synthetic data Asn. ASaR. and ACR, respectively. For instance, ASr calculated as the absolute difference between the Sharpe ratio of actual data and that the synthetic data, i.c., ASR = the results of differences in these metries between the actual data and the generated synthetic data. The synthetic data generated by FTS-Diffusion yield the lowest deviation, thereby exhibiting the most comparable statistics of the fundamental metrics closely aligned with those the actual data. This provides further evidence to support that our FTS-Diffusion replicates the fundamental characteristics of the actual financial assets.
Sharpe Ratio (snnthett)t Table 4 presents
Table 4: Difference in Sharpe ratio, Sortino ratio, and Calmar ratio between the generated returns and the actual returns of the corresponding financial assets.
S&P500
GOOG
ZC=F
Model
AsR
Asot
ACR
ASR
AsoR
ACR
An
ASoR
AcR
RCGAN
0.141
0.214
15.659
0.149
0.252
22.784
0.178
0.316
30.372
TimeGAN
0.111
0.189
12.775
0.177
0.249
22.067
0.136
0.203
26.450
CSDI
0.133
0.186
17.424
0.178
0.273
24.611
0.152
0,241
31.188
FTS- DisfDiffusion
0.019
0.034
0.294
0.020
0.037
0,302
0.069
0.092
0.614
E.4 SUPPLEMENTARY DOWNSTREAM EXPERIMENTS
For comparative purposes, we utilized naive prediction approach benchmark. This method simply uses the values from the previous day to make its predictions. this context, the mean absolute percentage errors (MAPEs) for the three datasets are 0.046, 0.053, and 0.057, respectively, These results serve as baseline for evaluating the performance of more sophisticated forecasting models.
In Sec. 5.3 we have demonstrated the effectiveness of the synthetic time series generated by our STS-Diffusion in the downstream experiments focused one-day ahead prediction. To further substantiate our approach, we extend our evaluation to a more complex task of multiple-day ahead prediction. For this extended evaluation, employ the same LSTM- neural networks the downstream prediction model. Fig. shows the results of the TMTR and TATR experiments for five-day forecasts of S&P 500. Our FTS-Diffusion successfully maintains its performance, proving its robustness and versatility.
E.5 DIFFERENT SETTINGS OF TRAIN/TEST SPLIT AND ROLLING WINDOW
Throughout the main body of our paper, we have evaluated our FTS-Diffusion using an 80/20 train/test split strategy. To ensure the robustness of our model and further investigate its performance under different circumstances, we extend our evaluation to include 70/30 and 60/40 train/test split strategies. In addition to the train/test split strategies, we further probe the robustness of our model by utilizing rolling windows of different sizes for testing, particularly sizes 32 and 48. This allows us to verify that the downstream model maintains stable performance when trained on different temporal slices of synthetic data.
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Published as a conference paper at ICLR 2024
0.30
SGP 500
FTS-Diftusion
RCGAN
TimeGAN
CSDI
TiraGAM
5645500
0.36
0.05
0.05
0.05
0.08
0.32
20
40
60
00
20000 Augmented Size
10000
20000 Augmentedd Size
10000
10000
20000
20000 Augmented Size
10000
Spithoti Rrsprrtion
Augmented
(a) TMTR
(b) TATR
Figure 10: Prediction errors of the downstream model for five-day ahead prediction trained under the TMTR and TATR settings. Our FTS- TTS-Diffusion successfully maintains its performance as in the one-day head prediction.
The outcomes of these extended downstream experiments, which utilize additional train/test split strategies and diverse rolling window sizes, are presented in Fig. II As more synthetic data generated by our FTS-Diffusion method are incorporated into the dataset, prediction errors are observed to decrease. This result demonstrates that augmenting the training dataset with synthetic data produced by our FTS-Diffusion method indeed bolsters the training process and the subsequent performance of downstream models. Such outcomes underscore the versatility and robustness our FTS-Diffusion approach across various operational contexts.
70-30 Split
60-40 Split
Rolling Window 32
Rolling Window 48
0.05
0.05
0.05
0.05
20000 Augmented Size
10000
20000 Augmented Size
10000
20000 Augmented Size
10000
10000 20000 Augmented Size
Figure 11: Prediction errors the downstream model trained on the augmented dataset with synthetic data generated by our FTS-Diffusion under TATR using different train/test split strategies and rolling windows.
These results demonstrate that our approach maintains its capability generalize across different train/test splits.
E.6 CONPLEXITY AND RUNTIME ANALYSIS
We conduct an analysis comparing the computational complexity and runtimes our approach against those of the baselines, with the results provided in Table In this context, the term 'ommputational complexity refers to the number of parameters in the neural networks. The 'runtimes' contain the times taken for training the model (training runtime) and generating the synthetic time series (inference runtime). Due to its multi-module nature, our FTS-Diffusion has a higher complexity. This represents trade-off between high fidelity and complexity. Our model achieves superior generation quality at the expense of somewhat increased complexity.
Table 5: Comparison of computation complexity and runtime. Number of Parameters
Model RCGAN TimeGAN
Training Runtime Inference Runtime
31190
23 min.
1.72 sec.
65302
49 min.
1.86 sec.
CSDI FTS-Diffusion
143284
97 min.
2.61 sec.
171209
131 min.
4.27 sec.
21


Reviews:
[{"review_id": "0rFgZp1i3d", "reviewer": "Reviewer_dpJk", "summary": "The paper proposes a new generative framework called FTS-Diffusion for synthesizing financial time series data that exhibit irregular and scale-invariant patterns. The framework consists of three modules: Pattern recognition module, Pattern generation module and Pattern evolution module. The paper demonstrates that FTS-Diffusion can generate synthetic financial series that closely match the distribution and properties of real data. The synthetic data also improves performance when used for downstream prediction tasks. The proposed framework is the first capable of generating complex time series with irregular and scale-invariant patterns. It addresses the problem of data shortage and improves reliability of deep learning models in financial applications.", "questions": "1. It seems the current experiments only use price returns as input. It's standard in industry to use a lot more derived features from price data / order book data, so I am wondering whether it's easy to incorporate other features into the current framework.\n2. In Figure 6(a), it's very surprising that regardless of the \"synthetic proportion\", the performance is about the same, meaning that the synthetic data is almost as good as the original data, which is very surprising. Do we have any hypothesis on why the synthetic data can be as good as the original data?\n3. If it's possible, it would be good to include code in the \"Supplementary Material\" for better reproducibility.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "Originality: The paper proposes a novel 3-module framework specifically designed to model irregular and scale-invariant patterns. The scale-invariant pattern recognition algorithm SISC is also a novel technique.\n\nQuality: The paper is well-written and easy to follow. However, I have some concerns regarding the experiment part, which I will discuss in details in the \"Weaknesses\" section.  \n\nClarity: The problem is clearly motivated through examples and definitions. The method description provides sufficient details to understand the key designs. The writing clearly conveys the core ideas and contributions.\n\nSignificance: The work addresses an important problem in financial time series modeling and generation. It has high practical significance for improving reliability of deep learning in finance under data shortage. The framework could potentially generalize to other irregular and scale-invariant time series beyond finance.", "weaknesses": "1. The main weakness of this paper is its experiment part. Specifically, the data / training details are not clear, but it's very important in financial time series forecasting as it's very easy to have \"label leakage\". Is it possible to include more data / training details? In particular, what data are used to train each module / model, what's data splitting strategy, what data are used for testing?\n2. It would be better to include some standard metrics like Sharpe Ratio in the experiment section.\n\nI'd like to raise my score once my concerns regarding experiments are addressed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZQSEpowmgX", "reviewer": "Reviewer_XZvW", "summary": "This paper proposes the FTS-Diffusion framework for capturing irregular and scale-invariant patterns in financial data. The framework consists of three parts: pattern recognition, pattern dependent generation and pattern evolution prediction. Experiments show that the data generated by this framework outperforms existing generation methods in terms of distribution matching and downstream task performance.", "questions": "1. Whether the pattern combination found by FTS-Diffusion has good generalization to overcome the serious distribution shift problem in financial scenarios\n2. Can FTS-Diffusion be used for direct prediction?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The irregular and scale-invariant characteristics of stock data compared with other time series data are explained clearly, and it is a relatively novel perspective.\n2. The design description of the three modules of FTS-Diffusion is clear and reasonable.\n3. The experimental results exhibits promising performance of synthetic data generation with FTS-Diffusion in terms of distribution matching with real data and downstream prediction performance.", "weaknesses": "1. The experiments are only conducted on  several types of stock data and did not include more scenarios in the financial field.\n2. Lack of ablation experiments. Three modules, and the effectiveness of some detailed designs in each module have not been effectively explained\n3. No rolling window experiment was performed, the results on a single train/test split are not reliable enough, and some detailed experimental settings (hyper-parameters, train/test split details) are not explained", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UiEFmgJZjA", "reviewer": "Reviewer_SkxH", "summary": "Learning from financial time series data is considered challenging due to limited data availability. \nThe present work proposes FTS-Diffusion, a generative framework to select features and model intrinsic patterns of univariate financial time series data with the aim to overcome data shortage. The model consists of the following three building blocks (1) recognition of irregular and scale-invariant patterns in each time series sample (2) generation of synthesized segments of patterns and (3) evolving of the latter over time.", "questions": "1. (p.3) I suppose the length of the entire time series is $T = \\sum_{m=1}^{M} t_m - t_{m-1}$ ?\n2. (Eq. (3)) $\\beta \\leq 1$ ?\n3. (p.9 Training on Augmentation) In case of a *one-day* ahead forecast I would have expected comparison to a very *naive* approach, e.g., using the value of the previous day as prediction; especially in reference to the Markov property. Can you additionally include a multiple-day forecast?\n4. Can you assess on computational complexity and runtime of the approach compared to the baselines?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "This method introduces a pattern-centric approach including clustering techniques with well-known variational inference techniques and solving a classification+regression task.  It is original to the best of my knowledge. Further, the paper is well positioned to existing work, the content is well organized and, to the best of my knowledge, has no spelling or grammatical flaws. Empirical evidence is accessed through experiments on three real-world financial assets; the results are state-of-the-art. I appreciate Figure 3 since it provides a concise overview of the proposed methods.", "weaknesses": "1. (Pattern Conjecture) The reasons given to support the claim that univariate financial time series data (in general) embody recurring complex patterns over time appear vague and lack thorough evaluation. For example, the authors consider only a single financial data set (i.e., the S&P 500 index), and the analytical approach seems very fine-grained and not very general. It would be helpful to have insights into the clustering results, i.e., indicating the actual number of classes as well as the number of items per class, or showing multiple centers, etc. And even though Appendix A/B broadens some aspects there remains a lack of rigor in the arguments/results.\n\n2. (Pattern Evolution) I apologize, but I lack understanding of the learning aspects of the pattern evolution network $\\phi$. How do you extract the information used from the training data? I assume my question refers to the *Info()* function in $\\texttt{line 6} \\$ of *Algorithm 2*. Unfortunately, I can't find any more detailed information about it.\n\n3. (Evaluation and Reproducibility)\nOverall, I am not convinced that the proposed pattern-centric approach really adds complementary information to learning financial time series data. I suppose this is primarily due to deficiencies in the evaluation of the pattern recognition module. I also miss a detailed description of the included datasets and results reproducibility is very limited due no code submission.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (spotlight)