On the Constrained Time-Series Generation Problem
Authors: Andrea Coletta
Sriram Gopalakrishnan
Daniel Borrajo
Svitlana Vyetrenko
Venue: NeurIPS 2023 poster (2023)
Keywords: time-series
generative models
constrained optimization
machine learning
TL;DR: A set of generative models for synthetic time-series data that can satisfy soft and hard constraints.

Abstract: Synthetic time series are often used in practical applications to augment the historical time series dataset, 
amplify the occurrence of rare events and also create counterfactual scenarios.
Distributional-similarity (which we refer to as realism) as well as the satisfaction of certain numerical constraints are common requirements for counterfactual time series generation. For instance, the US Federal Reserve publishes synthetic market stress scenarios given by the constrained time series for financial institutions to assess their performance in hypothetical recessions.
Existing approaches for generating constrained time series usually penalize training loss to enforce constraints, and reject non-conforming samples. However, these approaches would require re-training if we change constraints, and rejection sampling can be computationally expensive, or impractical for complex constraints.
In this paper, we propose a novel set of methods to tackle the constrained time series generation problem and provide efficient sampling while ensuring the realism of generated time series.  
In particular, we frame the problem using a constrained optimization framework and then we propose a set of generative methods including 'GuidedDiffTime', a guided diffusion model. 
We empirically evaluate our work on several datasets for financial and energy data, where incorporating constraints is critical. We show that our approaches outperform existing work both qualitatively and quantitatively, and that 'GuidedDiffTime' does not require re-training for new constraints, resulting in a significant carbon footprint reduction, up to 92% w.r.t. existing deep learning methods.

Onn the Constrained Time-Series Generation Problem
Andrea Coletta J.P. Morgan Al Research London, UK
Sriram Gopalakrishan J.P. Morgan AI Research New York, USA
Daniel Borrajo
J.P. Morgan Al Research Madrid, ESP
Svitlana Vyetrenko
J.P. Morgan AI Research New York, USA
Abstract
Synthetic time series often used in practical applications augment the his- torical time series dataset, amplify the occurrence of rare events and also create counterfactual scenarios. Distributional-siminn similarrit (which we refer as realism) well as the satisfaction of certain numerical constraints are common requirements for counterfactual time series generation. For instance, the US Federal Reserve publishes synthetic market stress scenarios given by the constrained time series for financial institutions to assess their performance hypothetical recessions. Existing approaches for generating constrained time series usually penalize training loss to enforce constraints, and reject non-conforming samples. However, these approaches would require re-training if we change constraints, and rejection sam- pling can be computationally expensive, or impractical for complex constraints. this paper, we propose novel set of methods to tackle the constrained time series generation problem and provide efficient sampling while ensuring the realism generated time series. In particular, optimization framework and then we propose set of generative methods including "GuidedDiffTimee
frame the problem using constrained
guided diffusion model. We empirically evaluate work several datasets for financial and energy data, where incorporating constraints is critical. We show that our approaches outperform existing work both qualitatively and quantitatively, and that "GuidedDiffTime" does not require re-training for new constraints, resulting in significant carbon footprint reduction, up to 92% w.r.t. existing deep learning methods.
Introduction
In recent years, synthetic time series (TS) have gained popularity in various applications such as data augmentation, forecasting, and imputation of missing values 2 S. Additionally, synthetic TS are extremely useful to generate unseen and counterfactual scenarios, where we can test hypotheses and algorithms before employing them real settings (6) For example, financial markets, it can be very useful to test trading strategies unseen hypothetical markets scenarios, poorly tested algorithms can lead large losses for investors, as well as to overall market instability 7 S) In order to be useful, such hypothetical market stress scenarios need to be realistic i.e., the synthetic market TS need to have statistical properties similar to the historical ones. They also need to satisfy certain constraints supplied by experts how hypothetical market shock scenarios potentially unfold. For instance, in order to ensure financial market stability, the US Federal Reserve annually assesses the market conditions and publishes a set of constrained market stress scenarios that financial institutions must subject their portfolios to, in order to estimate and adjust for their losses in case of market downturns 9
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Time
Time Constrane
Constraitt Time
Constrant
High
(a) Unconstrained
(b) Trend
(c) Fixed value
(d) Global minimum
(c) Multivariate
Figure 1: An example of synthetic stock market time-series under different constraints: (a) uncon- strained generation; (b) time-series following trend constraint; (c) the final value of the TS has to hold a specific value: (d) the global minimum must be at given time point; (e) multi-variate TS where the High and Low dimensions have the maximum and minimum values, respectively.
Our work targets the problem of generating constrained TS that are both statistically similar to historical times series and match a given set of constraints. These constraints can be imposed by the underlying physical process that generates the data. For example, synthetic energy data should adhere to the principle of "energy conservation' [T]] Or, in the preceding example of the US Federal Reserve stress scenarios, constraints can be used to generate counterfactual synthetic TS with some
given conditions, c.g., stock market index decreases by 5% [
Existing work employs deep generative models (DGMs) to capture the statistical data 14 15], and additional constraints are usually introduced by penalizing the generative model proportionally to the mass it allocates to invalid data [16] 777 [8] or by adding regularization term the loss function 19 20 Other approaches condition the generative process by encoding constraints and feeding them into the model [10]: or they reject and re-sample TS that do not match the constraints (21). Finally, different line of work proposes special-purposs solutions, with ad-hoc architectures or sampling methods, which however tackle specific applications (not TS generation) [22 23 24 25]. In general, while most of these models are able to reproduce the real data statistics, complex constraints can still be challenging to be guaranteed. Most importantly, as DGMs incorporate constraints during training, change the constraints may require re-training since the learned distribution of DGM may no longer cover the target distribution, and thus even rejection sampling would not be effective 26
Related work:
properties and temporal dynamics of TS [ B
In this paper, we tackle the constrained TS generation problem for different constraint types, and present and compare set of methods that incorporate constraints into TS generation. First, approach in which we compile the problem into a constrained
we present an optimization-basee optimization problem (COP) 27 and generate synthetic data from explicit formulation of data properties and constraints. To overcome the need for an explicit definition of data properties, we employ a DGM that can implicitly capture data properties from training data. In particular, we introduce DiffTime, an approach based on conditional denoising diffusion models (28) where several constraints can condition the data generation. In addition, we show that any kind of constraint can be applied to diffusion models by penalizing the model proportionally to the constraint violation during training. This approach, called DiffTieee, shows good performance with efficient sampling, but requires re- re-training upon a new constraint. Finally, to increase computational efficiency and reduce the carbon footprint of the model 29,, we propose a guided diffusion model Muided-DiffTiig that does not require re-training upon changes in the constraints at inference, it can adjust the TS generation process based on the input constraints.
Our main contributions can be summarized as follows:
We formally define the constrained TS problem, and characterize different types of con- straints for synthetic TS
We propose and compare different approaches to generate synthetic TS. We evaluate their advantages and disadvantages with respect to different measures of performance and con- straint types. We show how COP can also be used for post-hoc fine-tuning of TS, such that synthetic TS generated by any DGM can be adjusted to guarantee constraint satisfaction.
We empirically demonstrate that our approaches outperform existing work in many TS domains, both qualitatively and quantitatively. We show that DiffTime outperforms existing state-of-art models in unconstrained TS generation, while Guided-DiffTine is the only solution where re-training for new constraints is not necessary, resulting in a significant carbon footprint reduction.

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2 Definitions and Problem Formulation
The constrained TS generation problem requires generating synthetic TS data, where each TS is defined in the sample space X = x where L is the length of the TS and K' is the number of features. Our goal to generate synthetic data such that the synthetic distribution approximates the input data distribution, and each TS also conforms to user-specified constraints. The problem input a tuple (ID = } consisting of a dataset ID of N time series x' X. E [1...V] and a list of constraints C that synthetic TS should conform to. The constraints include realism constraints (see Section 3.1 Henceforth, we will drop the sample index for x unless needed, and only keep the position and feature indices. We also shortly denote [K] & {O..... K} and [L] - {O....,.), L).
Constraints - like those in Figure I - are defined as tuples the form (t. f) E C. where t can be either soft or hard and differentiable function If the constraint type is hard, then f can be an inequality or an equality constraint. An inequality constraint is of the form f(x) I where x is the generated synthetic TS. An equality constraint is of the form [() O. Hard constraints are required to hold in a generated TS. Otherwise, the TS' is rejected. Soft constraints are of the form x - IR whose value we would like optimize (minimize) for. Therefore, by definition, soft constraints do not require sample rejection. The constraints can be defined with respect to individual synthetic TS samples &. or at the dataset level (distribution-relan define trend-lines (see Figure Tb)
constraints). As a type of soft constraint, we as a time series s E X. This constraint tells a generative method to minimize the L2 distance between the trend and the corresponding points of the synthetic TS. Formally, the synthetic TS & would optimized as to minimize ||s x||
Additionally, both soft and hard constraints can be categorized into local and global constraints. Global constraints are those that compare across all the points in the TS. For example, we can enforce that æ4,j S. 03,0; ((i,j) E [4] x [K] such that the maximum value is at 33.0- Local constraints are those that only refer only to subset of points. For instance, requiring (zi,j = 2.5) for given point (i,j) E [4] x [K]. We refer to this kind of constraint as fixed-point constraints (see Figure ( since they require that the value of the TS is fixed at that point to a specific value. The set of all -poit constraints is R, where each element Ti,y E IR and (i. E [L] x [K].
The aforementioned types of constraints are explicit. Additionally, the problem of synthetic data generation requires statistical similarity between the input and the synthetic datasets, which can either be built-in into the data generating method (e.g., by design GANs generate data that is distributionally similar to the input (2]) or specified explicitly as constraints in the model (e.g.. autocorrelation similarity can be an explicit constraint). The methods presented herein assume that the constraints are differentiable. This is needed for deep generative methods, such as diffusion models, where constraints need be incorporated into the training inference process. If the functions are differentiable, then straightforward approach (3 to incorporate them into the loss is:
loss(x, x) objectiiv_loss((xxX x) + (e,e Anh(()²
(1)
where g(x and h(x) are the inequality and an equality constraint respectively, which added as soft constraints into the loss function with penalty terms Xg and A However, incorporating constraints into the loss function may not guarantee constraint-conformin solutions, but good candidate or starting solutions that we can then fine-tune (i.e., adjust to guarantee constraints). If the constraints are not differentiable, one can approaches that compute the loss for such "rule" using perturbations [10]
3 Constrained Time-Series Generation Prooposd Approaches
In this section, we introduce several approaches tackle the constrained TS generation problem. In particular, we discuss their advantages and disadvantages, and how they handle different scenarios and constraints.
3.1 Constrained Optimization Problem (COP)
Our first model tackles the synthetic TS generation problem as a Constrained Optimization Problem (COP) in which we treat each point vi,j x as decision variable to be optimized. We will refer to this method simply as "ethome A COP problem is defined by an objective function that a solution is optimized for, and set of constraints that need to be satisfied by the solution.
3

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We can use COP in two ways, as a generative method, and as fine- tennit method. COP is used for generating synthetic TS, then we take as input real sample x from D as the starting TS (i.e., seed) for generation, and set the objective to maximize the difference between the seed and the synthetic TS. Formally, we maximize the 1.2 norm of the TS difference: obj(x x) = Il(x x)]l2- where x is the seed TS and & is the generated TS. COP can also be used for fine-tuning candidate solutions generated by other methods, such as by a diffusion model. When using COP as a fine-tuning method, the candidate solution generated by the other method becomes the seed TS, and the objective simply changes from maximizing the L2 difference to minimizing it; this is to preserve the information from the candidate TS and just search in the space of nearby solutions for one that satisfies all the constraints (if any failed). This can be helpful in fixing almost correct solutions, rather than using rejection sampling.
The constraints for the COP formulation come from C. Additionally, when using COP as a generative process we need to add constraints to satisfy the desired distributional properties that methods like GANs would implicitly handle, such as preserving the distribution of the autocorrelation of returns for stock data. This is done by constraining the COP solver to try and match the desired statistical properties of the seed TS; by matching the property at the sample level, we seek to match the distribution of that property at the dataset level. We do this by computing the target value from the seed TS and comparing it with the corresponding value from the synthetic TS. Specifically, constrain the COP to limit the magnitude of the error between the property value computed for & and x within an allowed amount (a budget for error tolerance). This is done by using an inequality constraint follows: e(2(X), =(x)) S O, where b is the budget set (b = 0.1 in experiments), z(-) is the function that computes the desired property, and e(-) is the error function that measures the error between the target and generated values. For example, for autocorrelation of returns in stock data, the TS property is a vector, so the e(-) is the L2-norm of the difference. If the COP solver cannot find a solution within the allowed error tolerance, we double the budget and 10 in our experiments). In the
repeat the process for up to fixed number of repeats (we set supplementary material, we show how distributional constraints can be learned directly from the input data, by training a Wasserstein-Gan -AN 31 32
and using the GAN critic in the objective function.
Once we define the objective function and constraints for COP, can employ one of the many solvers available to compute the synthetic TS. In our experiments, we use the Sequential Least Squares Programming (SLSQP) solver (7 in Scipy's optimization module 33
3.2 DiffTime- - Conditional Diffusion Model for Time Series generation
In the previous section, we introduced using COP to generate synthetic TS while guaranteeing the input constraints and data properties. However, such COP problems may be non-linear, and solving non-linear problem is in general difficult and computationally expensive, especially with multi-variate and long time-series (see Section 4. In this section, we introduce conditional diffusion model named DiffTime that leverages the latest advancements in score-based diffusion models [5] 34 28, 35) to generate synthetic TS. Our model can generate realistic TS and cope with Trend and Fixed Points constraints by conditioning the generative model.
Denoising diffusson models Denoising Diffusion models are latent variable models which are trained to generate samples by gradually removing noise- - denoising - from samples corrupted by Gaussian noise (28). These models approximate a real data distribution ((xo) by learning
model distribution po Po(xo) := J po(xx: (xp:)) dx where the latent variables X1:r are defined in the same space X' of the sample Xo. The training follows: forward process that progressively adds noise to the sample xo; and a reverse process where the generative process gradually denoises a noisy observation. The forward process is described with the following Markov chain with Gaussian transitions parameterized by B1.
q xo) = II ((4 x)-1),
q(x: xx-1) = N (VI- (11-Bxx-1188)
(2)
It admits the following close form q(x; Xo) =
Ni(x; Vārxo, (1 - where a := 1- 3r
and d; : I[ Og, which allows sampling x; at any arbitrary diffusion step t. The generation is

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performed by the reverse process defined as Markov Chain starting at p(xr) = (xx:; 0,1):
(Xu:)) == p(xx) II (x pe(xi-1(xx) == N (x Eg (x,,, f))
(3)
Following the formulation of Denoising Diffusion Probabilistic Models (DDPM) 28 we parameterize the reverse process as follows:
1
3r X, VI- &i
n4 ( (x,, t)
co(x),,)); Eo(x,, I²I, where 0² v
(4)
Vā;
where eo is a trainable denoising function that predicts e from x;, and the choice of 3 corresponds to the upper bound on the reverse process entropy [35]. This function is approximated through deep neural network trained according to the following objective:
E co(Vārxo + vI- ||
L(0) = E xo,
where i is uniformly sampled between and T, and the noise is Gaussian (O, I). The diffusion steps 7" and variances 3 control the expressiveness of the diffusion process and they are important hyperparameters to guarantee that the forward and reverse processes have the same functional form 35
Conditional diffusion models Our DiffTime model which is conditional diffusion model supports both trend and fixed point constraints that were defined in Section 2 To constrain a particular trend, we condition the diffusion process using trend TS X. Following recent work on conditional diffusion models (5), we define the following model distribution:
po (x)rr(s) == p(xr) II (x 8), (x xx,8) ==
4ss) Eo (Xt, 4|8)).
(6)
which we learn by extending the parametrization in Eq 4 with a conditional denoising function Eg:
B.
(x,, t |)) Vāe
x
v1 à (x,, |s)
where the Eg(xe, L. ss) = II In this formulation, the trend is provided during each diffusion step f, without any noise added to the conditioning trend. During the training, we extract the trend directly from the input TS Xo. which can be a simple linear or polynomial interpolation; during inference, the trend can be defined by the user at inference time. We recall that this a soft constraint, meaning that we do not expect the generated TS to exactly retrace the trend. In particular, during training, we provide trend that is a low-order polynomial approximation xo to avoid the model from copying the trend s. Figure T shows an example of the trend constraint.
Thus, the DiffTime training procedure minimizes the following revised loss function:
L( == Ex. //|l- calV + v1 â,e,t s)
(8)
Fixed Points. To satisfy the fixed point constraints, which are hard constraints, we modify the
reverse process of DiffTime to explicitly include them in the latent variables XI.r. We recall that R is the set of fixed point constraints, such that fixed point constraint ri. E R with (i.j) E [L] x [K]. Thus, at each diffusion step / we explicitly enforce the fixed-points values in the noisy time-series Xt, such that Y Ts.j E R, æz,j = Tı3 where Tı,j E Xt. This approach would guarantce that the generated TS have the desired fixed-point values. Most importantly, we experimentally validated that the forward process generates consistent neighboring points (around the constrained fixed- seedttooitt) which means that the synthetic samples conditioned by the fixed points, and preserve the realism of the original input data. During training, we randomly sample the fixed points from the input TS(x,) and require the diffusion process conform those fixed points. At inference, the fixed points can be provided by the user. Figure C shows an example of fixed point at the end of the TS, where the TS adapts to deal with the fixed point.
In the supplementary material, we provide additional details, network architecture, and the algorithm pseudo-codes.
5

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3.3 Loss-DiffTime Constrained generation with diffusion models
In DiffTime, we leverage conditional diffusion models to support trend and fixed values for generating TS. However, just by conditioning the model generation is not possible to encode all the constraints. common solution is to penalize the generative model proportionally to how much the generated TS violates the input constraint [36].
In this section, we propose Loss-Difflime where constraint penalty is applied to deal with any kind of constraint. The penalty function fe x + IR is added to the learning objective of the diffusion model, and evaluates whether the generated TS x meets the input constraint. We discuss the penalty function fc for constraints in Section and in Equation [] With fo in the loss, the greater the constraint violation is, the greater the model loss during training will be. However, the optimization problem in Eq. pieddicss the noise component for the sample Xu. into which we cannot directly feed to our penalty function. Moreover, we cannot apply fx(x a noisy sample x; as the constraints may be evaluated only on the final sample. Therefore, to apply our penalty function, we re- parametrize the optimization problem and force the diffusion model to explicitly model the final sample Xo at every step as follows:
L( = Ex,. [Ile co(x,, t s)]l + pfc(
(9)
where X, V0xXo + V1- GGE and Xo Vai X vI-a.
1-ac - 6/ (x+, 1). We consider that any
constraint in C' can be differentiable (as discussed in Section 2). So, we can train our diffusion model following Eq. B where is scale parameter used to adjust the importance of the constraint loss. The conditional information of the trend can be removed if we do not need to enforee any trend constraint. Figure d and Figure e show two examples of more complex constraints with Dofs-Diffimm
3.4 Guided-DiffTime Guided Diffusion models for constrained generation
The Loss-DiffTime model is now able to generate real TS while dealing with any constraint. However, notice two major drawbacks: 1) since translate constraints penalty terms in the loss, we need to re- re-train the model for new constraints; and 2) the diffusion models usually require several iterative steps T which can make slower and expensive for TS generation. Our final proposed approach, namely Giided-DiifTimm,, solves these two problems and can dramatically reduce the carbon footprint when using DGM for constrained TS generation. In particular, adopts a Denoising Diffusion Implicit Model (DDIM) 37 which requires fewer diffusion steps at inference. Moreover, by following the groundbreaking work of [3 39), which shows how to guide diffusion model using a noisy classifier, we demonstrate how a pre-trained diffusion model can be guided (conditioned) using gradients from differentiable constraints.
DDIM is a class of non-Markovian diffusion processes with the same training objective of classic DDPMs (28), but fewer diffusion steps to generate high-quality samples. In particular, DDIMs keep the same training procedure as DDPMs defined in Section 3.2 while the sampling can be accelerated by using the following re- paraaeetrrizatonn of the reverse process:
Xt-1 10==vor-1 01
X1
/1 âu cx)
V1 à of e0(xx; + ore
(10)
Vāa
where dg = 1 and different parametrizations of og lead to different generative processes. We set of = 0, E [o, T) to have deterministic forward process from latent variables to the sample Xy (since the noise term dfC is zeroed out). This deterministic forward process defines the DDIM which can use fewer diffusion steps to generate realistic samples. This diffusion steps are defined by sequence 7 of length V which is sub-sequence of [1, [1...., T) with the last value as T, i.e., TV T [37] For example, r 1, 4,9, T). Moreover, this parametrization a generalization of DDPM as setting o; = / 0; 14-1)/(1 - aa) V1 a describes the original DDPM [37] and the DDIM work showed that re- re-training of the DDPM model is unnecessary when we change the
value of a or the diffusion steps т.
Given the DDIM, we can then apply the recent results from guided diffusion models 38 39 to condition each sampling step with the information given by the gradients of the differentiable constraint fo (see Section 3.3) Algorithm shows the sampling procedure which computes the

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Algorithm/ 1 Guided-Diii Difife
Input: differentiable constraint fe : X' - I., scale parameter p
Output: new TS, Xo XT < sample from (0,I) for all from T to I do
? 4 (x(x,,t) t)
é & - pvl - dr Vx. fol Vāu (Xt- - EVI- -a))
X1-1 E v /
xx-v + v âs-
vāe
end for return XD
gradients w.r.t the input TS X2- We recall that the constraint is applied on the final sample Xo. computed according to the DDIM reverse process. Again, this approach does not require re-training of the original diffusion model deal with new constraints, which can be applied just at inference time. Hence, we reduce the carbon footprint of the model, and get a faster time-series generation.
4 Experiments
In this section we evaluate approaches, showing their advantages and disadvantages when applied to different domains and constraints, In particular, we follow the five scenarios shown in Figure while considering multiple real-world and synthetic datasets. For COP we use subset of the original TS as starting solution (seed), we leave in the supplementary material the analysis of different seeds.
Baselines We compare our approaches against existing TS generative models, including GT- GAN[12), TimeGAN []]. RCGAN [40), C-RNN-GAN [4I] a Recurrent Neural Networks (RNN) [I] trained with T-Foreing and P-Forcing 42 43
WaveNET [44), and WaveGAN 445) For the con-
strained scenarios, we extend the benchmark architectures to cope with constraints, by introducing a penalty loss 18, 16 or by conditioning the generation process. We also employ eijectionssaplling and fine-tuming with COP on their generated synthetic TS.
Datasets We consider three datasets with different characteristics such as periodicity, noise, correla- tion, and number of features: 1) daily stocks which uses daily historical Google stock data from 2004 to 2019 with open, high, low; close, adjusted close, and volume features [: 2) energy data from the UCI Appliances energy prediction dataset [46) containing 28 features with noisy periodicity and correlation; 3) sines a synthetic multivariate sinusoidal TS with different frequencies and phases [
Evaluation metrics For each experimental scenario, we evaluate the generative models and TS along different quantitative and qualitative dimensions. First, we evaluate the realism through discriminative score (1]. which measures how much the generated samples resemble (i.e., indistinguishable from) the real data using a post-hoc RNN trained to distinguish between real and generated samples. We evaluate the distributional-similaati
between the synthetic data and real data by applying t-SNE [47] on both real and synthetic samples; t-SNE shows (in a 2-dimensional space) how well the synthetic distribution covers the original input distribution. Then, we evaluate the usefulness of generated samples - how the synthetic data supports a downstream task such as prediction - by training an RNN on synthetic data and testing its prediction performance on real data (i.c.. predictive-score (1). To evaluate how the model satisfies different constraints, we introduce the following metrics: Perc. error distance which measures how much the synthetic data follows trend constraint by evaluating the L2 distance between the TS and the trend; satisfaction rate which measures the percentage of time synthetic TS meets the input constraints; the inference time measured as the average seconds required to generate new sample with given constraint; and finally the fine-tuning time which is the average time, seconds, needed to enforce constraints over a generated sample, using COP to fine-tune it.
We provide further experiments, including details on the baselines, datasets, metrics, and algorithm hyperparameters in the supplementary material.
7

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4.1 Unconstrained Generation
First, we compare the ability of DiffTime and COP to generate unconstrained TS against exist- ing benchmark datasets and algorithms. In Figure 2 we evaluate the realism with respect to the Iotributional-simmaattn where red dots represent the original TS and blue dots the generated TS. The figure shows that approaches have significantly better performance with better overlap between red and blue samples.
(a) COP
(b) DiffTime
(c) GT-GGAN
(d) TimeGAN
(c) RCGAN
(f) C-RNN-GAN
Figure t-SNE visualizations on multivariate stock data, where greater overlap of blue and red dots shows a better distributional-similaay between the generated data and original data. Our approaches show the best performance.
In Table we measure the usefulness and realism through the predictive and discriminative scores, respectively. Difflime consistently generates the most useful data with the best predictive score for both Sines and Stocks datasets, while keeping remarkable realism (i.e., discriminative score). COP generates excellent synthetic samples as well, indistinguishable from real data with the best discriminative score for both Stocks and Energy. However, we acknowledge COP is advantaged by the original time series as an input seed.
Table 1: Unconstrained Time-Series Generation (Bold indicates best performance).
Metric
Method
Sines
Stocks
Energy
DiffTime (Ours)
.133 + .006
.097 + .016
.455 + .004
COP (Ours)
.020 + .001
.050 + .017
.101 +: .019
Discriminative
GT-GAN
.012 + .044
.077 = .031
.221 + .068
TimeGAN
.011
.102 .021
.236 + .012
RCGAN
.022 .008
196+..02
.336 + -017
(Lower the Better)
C-RNN-GAN
.299 .040
.399 .028
.449 + 1011
T-Forcing
.455 1 .001
.226 + .035
.88 1 .004
P-Forcing
.227
.257 .026
.12
WaveNet
.158 1011
.32 1 .028
.397 1.00
WaveGAN
.013
.217 .022
.363 2012
DiffTime (Ours)
.093 + .000
.038 1 .001
.252 1000
COP (Ours)
.955 + .002
.041 + .001
.250 0003
GT-GAN
.097 0000
.040 + .000
.312
Predictive
TimeGAN
.093 .019
.038 + .001
.273 + .004
Score
RCGAN
.001
.040 + .001
.292 1 .004
C-RNN-GAN
.127 + .004
+ .000
.483 + 0005
(Lower the Better)
T-Forcing
.150 -.02
.038 + .001
.315 .005
P-Forcing
.116 + .004
.043 .001
.303 1 .005
WaveNet
.117 + .008
.042 .001
311 + .006
WaveGAN
.134 .03
.041 1 .001
.307 .00
Original
.094 .001
.036 1 .001
.250
4.2 Constrained Generation
We now evaluate the performance of our approaches against the constraints shown in Figure Dusing daily stock data. For univariate constraints (i.e., trend, fixed values, and global minimum) we consider only the Open value from the daily stock dataset. We consider as benchmarks the best three SoA approaches from Table D (i.e., GT-GAN, TimeGAN, and RCGAN), For trend- and fixed-values constraints we condition their generative process so that different trends and values can be used at inference time. For the other constraints, we add penalty term in the training loss [18 16
We report only the top 6 models, leaving the full evaluation the Supplementary material.

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Soft Constraints In table 2 we constrain the synthetic TS to follow a given trend, computed as 3-degree polynomial upproximation from the original samples. Our approaches generate synthetic data that are closer to the input trend, with the smallest relative distance (i.e., perc. error distance). Moreover, our approaches are among the hest in terms of realism and usefulness. the supplementary material we investigate the use of sinusoidal trends, including additional evaluation metrics.
Table 2: Soft Constraints (Trend) Time- Seriee Generation (Bold indicates best performance).
Algo
Discr-Score
Pred-Score
Inference- Time
Perc. Error Distance
COP (Ours)
0.01#0.01
0.20=0.00
0.73+0.05
0.01540
DiffTime (Ours)
0.01=0.01
0.20=0.00
0.02+0.00
0.018+0
GT-GAN
0.04+0.03
0.22+0.00
0.00=0000
1.378
TimeGAN
0.02+0.02
0.20::.00
0.00:00.00
0.073+0
RCGAN
0.02+0.01
0.20:a0.0
0.00=0000
0.071=0
Hard Constraints In Table we evaluate all the approaches against hard constraints (see Fixed Points, Global Min, and Multivariate in Figure [ For Global Min almost all approaches have great satisfaction rate. However, our approaches are above 0.90 and have the best discriminative and predictive score. Additionally, while 100% of the synthetic time-series generated by TimeGAN and RCGAN guarantee this type of constraint, they do not approximate the input distribution well(see Figure B). For most complex constraints like the multivariate one, the satisfaction rate drops for most of the benchmarks while for our GufdedDiffTime and COP the satisfaction rate is still very high, with great realism and usefulness. Finally, when employ the fixed point constraints, of the points at index and 18. All the benchmarks fail to satisfy these constraints, while we show instead that Diffimme is able always guarantee this constraint, by enforcing it during the diffusion steps. Most importantly, it achieves very good discriminative and predictive scores with low inference time. To summarize our results: COP achieves almost always the best realism and usefulness scores, but with higher inference time and using original input TS as seed (which makes the generated TS very similar the input data); diffusion models also very powerful with lower inference time and use random noise as input seed as opposed to a real TS (this gives us better variety in TS compared to COP); and GuidedDifTiime is able to enforce new constraints without any re-training yet achieving excellent performance.
fix the values
Table Hard Constraints Time-Series Generation (Bold indicates best performance).
Constraint
Algo
Discr-Score
Pred-Score
Inference- Time
Satisfaction Rate
Fine-Tuning Time
COP (Ours)
0.0250.011
0.20:0000
19.1+1.01
LO0000.00
0.00:40.00
GuidedDifTTiee (Ours)
0.0340002
0.21=0.00
0.03+0.00
3.01
LossDifTTmme (Ours)
0.22x0.03
0.3820.00
0.02a0.00
0.9920.00
6.0010.60
Global Min
GT-GAN
0.04+0.02
0.22+0.00
0.00::0.00
9.30=11.30
TimeGAN
0.0310.02
0.21.0.00
0.0020.00
1.00:0.000
0.0020.00
RCGAN
0.23+0.03
0.20:0.00
0.00:20.0
1.00=0.00
0.00:0.00
COP (Ours)
0.041:0.02
0.04:1000
2.17a0.10
1.0000.000
0,0010.000
Time (Ours)
0.08+60.0
0.04::010
0.15+0.00
31.0
Los:DifTTime (Ours)
0.3510.04
0.04.0.01
0. 0.1420.000
0.6920.01
57, 17,5555.0 .01
Multivariate (OHLC)
GT-GAN
0.2246.07
0.00::0.00
0.05=10.01
44.5+:1.01
TimeGAN
0.24x0.03
0.0520.00
0.00200.00
0.51=0.02
16. 1 1.30
RCGAN
0.3520.04
0.04:0.00
0.00:00.0
0.00=0.00
95. 12:403
COP (Ours)
0.0210.02
0.20:1000
0,5620.11
100000.00
0.00100.00
DiffTime (Ours)
0.04+00.3
0.21
0.01=0.00
100000000
0.00::0.00
GT-GAN
0.0410.03
0.21a0.00
0.00200000
0.00=0000
0.9920. 10
Twu Fixed Points
TimeGAN
0.03g0.01
0.20:0.00
0.00:0000
0.0020.00
0.8420.00
RCGAN
0.02+0.02
0.20:0.00
0.0010.00
0.00=0.00
0.8740.20
(a) (b) Guided DifTTime
(c) ListiiTTimme
(d) GT-GAN
TimeGAN
(f) RCGAN
Figure 3: t-SNE visualizations on Global-Min constrained data, where a greater overlap of blue and red dots implies better distributional-smiiaan between the generated data and original data. Our approaches show the best performance.

---

Conclusions
In summary, we defined the problem of generating synthetic TS data with both soft and hard constraints, and we presented set of novel methods. We evaluated our approaches on different datasets and we compared their performance against existing state-of-art methods. We showed that our approaches outperform existing work both qualitatively and quantitatively. Most importantly, we introduced GuidedDifTTime to handle new constraints without re-training, and we showed that the COP-method can be used to fine-tune candidate solutions. Please refer to the supplementary material for more details on experiments comparing the methods presented herein.
Disclaimer
This paper was prepared for informational purposes by the Artificial Intelligence Research group of JPMorgan Chase & Coand its affiliates ("JP Morgan"), and is not product of the Research Department of JP Morgan. JP Morgan makes representation and warranty whatsoever and disclaims all liability, for the completeness, accuracy or reliability of the information contained herein. This document is not intended as investment research investment advice, recommendation. offer or solicitation for the purchase or sale of any security, financial instrument, financial product or service, or be used in any way for evaluating the merits of participating in any transaction, and shall not constitute a solicitation under any jurisdiction or to any person, if such solicitation under such jurisdiction or such person would be unlawful.
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12


Reviews:
[{"review_id": "fVaDZrUc1q", "reviewer": "Reviewer_U4ao", "summary": "In this paper, the authors investigate the problem of time-series generation under different types of conditions. Since the existing methods usually require re-training when the conditions should be changed and the rejection sampling might require heavy computation, the authors use the conditional diffusion model to generate the time-series data with hard and soft constraints. The authors evaluate the proposed method and several datasets and obtain good performance.", "questions": "Please refer to the weaknesses.", "limitations": "Please refer to the weaknesses.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The problem of time-series generation that the authors investigate is interesting and important. And the authors devise a flexible framework for this problem. Moreover, the proposed method achieves ideal performance and is is environmentally friendly.", "weaknesses": "1.\tAccording to equations on Pages 4 and 5, the authors assume that the data points in each timestamp follow the Gaussian distribution and that the time-series data might be stationary. However, some time-series data, e.g., financial data, are influenced by unexpected policies, so the noise or the generation process of the time-series data might be changed, and the data become nonstationary. How can the proposed method address these challenges?\n2.\tSome implementation details of the proposed method are not clear. For example, what is the architecture of $\\epsilon_\\theta(x_t, t)$ in equation (4)? Can the authors provide some figures for the proposed method? \n3.\tAccording to Algorithm 1, the data are generated via a linear generation process, which might be a strong assumption. Can the proposed method be extended to other more challenging cases?\n4.\tThe compared methods might be too weak and the authors only consider one latest methods. It is suggested that the authors should consider more latest compared methods like [1][2][3].\n5.\tI think the authors should consider more metrics for the proposed method. For the trend constraint as shown in Figure 1(b), it is suggested that the authors should employ DTW to measure the distance between the original trend and the generated data. Moreover, since the authors consider the sinusoidal data, which is seasonal, it is suggested that the authors should employ spectral transformation methods like Fourier transform and compare the basis. \n\n[1] Deep Latent State Space Models for Time-Series Generation\n\n[2] Generating multivariate time series with COmmon Source CoordInated GAN (COSCI-GAN)\n\n[3] Towards Generating Real-World Time Series Data", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for pointing out aspects that were not clearly explained in the paper.\n\n**Q1/3** Our diffusion models use a Markov chain to convert a simple known distribution (i.e., a Gaussian) into the target (data) distribution using a diffusion process[1], without any strong assumptions on the target data distribution. Nevertheless, we agree with authors that time-series can have unexpected changes, and distributional shifts pose new specific modelling challenges[11]. We will investigate such scenarios in future work. Regarding to Algorithm 1 of Guided-DiffTime, we will clarify that we approximate $e_{\\theta}$ using a neural network. Moreover is it possible to incorporate recent work to operate the generative process into a latent space [6], to deal with more challenging scenarios. If instead, the reviewer is referring to Algorithm 1 of supplementary material, we will clarify that COP uses an optimizer that supports non-linear constraints (SLSQP optimizer).\n\n**Q2** We have now included an illustration showing the architecture of $e_{\\theta}(x_t, t)$ shared across our diffusion models (Figure 1 of PDF rebuttal). We will include such illustration in the main paper.\n\n**Q4** In our current effort, we have integrated *COSCI-GAN*[2] and *RTSGAN[3]* models for unconstrained time-series (TS) generation, as well as for constrained TS generation, specifically focusing on the OHLC constraint. For this constraint, we modified the training procedure by introducing a penalty loss, which penalizes the generative models proportional to how much the generated time-series violate the input constraint. While we have been able to replicate and adapt *RTSGAN*, we note that despite our efforts to replicate *COSCI-GAN* using the provided hyper-parameters and code from the paper, the obtained results are different from those of the original paper. We include preliminary results, and will continue our efforts to integrate such work. Lastly, we were not able to incorporate the work in [4] in our rebuttal due to limited time and difficulty accessing the source code, but plan to include it in the final version. \n\n- *Unconstrained generation (Stock Data)*. We report the quantitative metrics, Discr. and Pred. Score, in following table, while t-SNE analysis is shown in Figure 9 of PDF rebuttal. We also included additional metrics, as kindly suggested by the reviewers. Following the literature on financial TS [5], we analyse the distribution of price returns in Figure 3, and their autocorrelation distribution in Figure 4. *RTSGAN* achieves notable performance in terms of Discr. and Pred. scores, with good distributional similarity from the t-SNE chart in Figure 9. However, with respect to properties pertinent to financial data (Figure 3 and 4), the synthetic data has higher autocorrelation than real data, potentially stemming from multiple GRU layers; and more shallow return distribution (it is well known that asset returns have fat tailed distributions[5]). \n\n$\\begin{array}{l|lll}\n\\hline\n\\text{Algo} &  \\text{Discr-Score} & \\text{Pred-Score} & \\text{Inference-Time} \\\\\\\\ \\hline\n\\text{COP (Ours)} & .050 \\pm .017 & .041 \\pm .001 & 1.01 \\pm 0.00   \\\\\\\\\n\\text{DiffTime (Ours)} & .097 \\pm .016 & .038 \\pm .001 & 0.02 \\pm 0.00  \\\\\\\\\n\\text{COSCI-GAN} &  .412 \\pm .002 & .088 \\pm .000 & 0.00\\pm0.00 \\\\\\\\\n\\text{RTSGAN} & .024\\pm.007 & .036\\pm.000 & 0.00\\pm0.00 \\\\\\\\\n\\hline\n\\end{array}$\n\n- *OHLC Constrained generation (Stock Data)*.  For constrained TS generation, in Figure 10 of PDF rebuttal and in the table below, we can observe similar performance as in the unconstrained setting for *COSCI-GAN* and *RTSGAN*, both in terms of  distributional similarity, discr. and pred. scores. However, looking at the satisfaction rate (i.e., percentage of time-series respecting the input constraint), our methods outperform the two benchmarks. Most importantly, our Guided-DiffTime model stands out for its remarkable capacity to accommodate new constraints without any retraining, constituting a fundamental innovative contribution to the literature on generating TS data.\n\n$\\begin{array}{l|llll}\n\\hline\n&  \\text{Discr-Score} &   \\text{Pred-Score} & \\text{Inference-Time} & \\text{Satisfaction Rate} \\\\\\\\ \\hline\n  \\text{COP (Ours)} & 0.04 \\pm 0.02 & 0.04\\pm0.00 & 2.17 \\pm 0.10 & 1.00\\pm0.00  \\\\\\\\\n  \\text{GuidedDiffTime (Ours)} & 0.08 \\pm 0.00 & 0.04\\pm0.10 & 0.15 \\pm 0.00 &  0.72 \\pm 0.02  \\\\\\\\\n  \\text{LossDiffTime (Ours)} & 0.35 \\pm 0.04 & 0.04\\pm0.01 & 0.14 \\pm 0.00 &  0.69 \\pm 0.01  \\\\\\\\\n \\text{COSCI-GAN} &  0.45 \\pm 0.01 &  0.09 \\pm 0.00 & 0.00\\pm0.00 & 0.02 \\pm 0.00 \\\\\\\\\n \\text{RTSGAN} &  0.02\\pm0.01 &  0.04\\pm0.00 & 0.00\\pm0.00 & 0.54 \\pm 0.02 \\\\\\\\\n\\hline\n\\end{array}$\n\n**Q5.** Following the reviewer suggestions, we have introduced additional metrics to evaluate the synthetic TS for the sinusoidal trend-constrained generation. In the table below we included distances based on Dynamic Time Warping (DTW) and spectral transformation (L1 distance of Fourier basis). Thanks to the suggested metrics the results better highlight how some methods, like RCGAN, are able to capture the trend, even if shifted, which is also visible on Figure 6 of the supplementary material. Our DiffTime method has the best performance in these measures as well.\n\n$\\begin{array}{l|lllll}\n\\hline\n& \\text{Discr-Score} & \\text{Pred-Score} & \\text{L2 Distance} & \\text{DTW Distance}\n& \\text{Fourier-based Distance} \\\\\\\\ \\hline\n\\text{COP (Ours)} & 0.01\\pm0.01 & 0.20\\pm0.00 & 46.3\\pm32.9 & 35.8\\pm25.8 & 0.57\\pm0.57 \\\\\\\\\n\\text{DiffTime (Ours)} & 0.01\\pm0.01 & 0.20\\pm0.00 & 35.57\\pm16.99 & 27.57\\pm13.12 & 0.49\\pm0.57\n\\\\\\\\\n\\text{GT-GAN} & 0.04 \\pm 0.03 & 0.22 \\pm 0.00 & 1699.4\\pm1253.1 & 1692.5\\pm1253.9 & 1.74\\pm2.51\n\\\\\\\\\n\\text{TimeGAN} & 0.02 \\pm 0.02 & 0.20\\pm0.00 & 121.35\\pm61.30 & 87.29\\pm50.25 & 1.06\\pm1.11\n\\\\\\\\ \n\\text{RCGAN} & 0.02 \\pm 0.01 & 0.20\\pm0.00 & 124.82\\pm83.29 & 95.73\\pm72.62 & 0.70\\pm0.75\n\\\\\\\\ \\hline\n\\end{array}$\n\n**References in the global rebuttal comment**"}, {"review_id": "xf3NMsXTFD", "reviewer": "Reviewer_5wsv", "summary": "The authors propose multiple methods to solve the constrained time-series generation problem: Given a time-series and a set of (hard or soft, local or global) constraints, the goal is to generate a time-series which is realistic, adheres to the input constraints, yet sufficiently different from the original inputs. The authors propose multiple methods starting from simpler constrained optimization to more complicated diffusion model based solutions relying on recent advances in that space. The authors then compare their models to some state-of-the-art methods and demonstrate that their method's efficacy. ", "questions": "N/A", "limitations": "The authors have not mentioned any limitations of their proposed work. I would encourage them to add a discussion about the same. ", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "I really like this work. It is well-written, motivated and clear. \nI am not an expert in diffusion models, but the ideas, the problem formulation and the solution are interesting and make sense to me. \nThe authors do a good job at evaluating their methods and demonstrate superior performance in comparison to state-of-the-art methods. The authors use standard datasets to evaluate their methods.", "weaknesses": "I do not see any weaknesses in the proposed method. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the positive feedback, and for pointing out an aspect that was not sufficiently discussed. We agree that discussing the limitations of our work will enhance the overall quality of our paper and its impact on the academic community. We will add a more extensive discussion in the final paper, extending the discussion on tradeoffs that we currently have about our approaches.  Finally, we plan to discuss the training and convergence times, as well as any potential limitations can be found in replicating our work."}, {"review_id": "irOdH7DL0N", "reviewer": "Reviewer_DGpF", "summary": "This paper proposes multiple approaches, including a Constrained Optimization Problem (COP) method and a conditional diffusion model called DiffTime, to cope with the problem of generating constrained time series data that adheres to statistical properties and specific user-defined constraints. It also provides a comparative evaluation against existing time series generative models using real-world and synthetic datasets.", "questions": "I have a concern regarding the potential optimization issues that may arise when incorporating soft and hard constraints during the training of generative models. I wonder if additional constraints prolong the tuning process or affect the convergence of the models, requiring additional time to train effectively.", "limitations": "The presentation and writing require significant improvement.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This paper provides a strong motivation for studying the constrained time series generation problem by highlighting its practical applications and the need for constraint-compliant synthetic data. Additionally, it offers comprehensive experiments against existing models using real-world and synthetic datasets.", "weaknesses": "1. The clarity and definition of certain terms in the paper require improvement. For example, terms like \"seed\" in line 133 and \"desired property\" in line 145 need clear definitions to enhance understanding. Additionally, in line 154, what does “TS property is a vector mean?” I assume it refers to the autocorrelation of a time series with various lags. It is recommended that the authors focus on the presentation and ensure better clarity in their writing.\n\n2. I find the experiments regarding COP in unconstrained time series generation unclear. I am unsure how COP is applied in such settings and what the specific objective and generative model used are. \n\n3. An ablation study is lacking in the paper, particularly in relation to DiffTime. Although DiffTime is designed based on diffusion models, no diffusion models are included as baselines for comparison. Furthermore, would the performance of GAN baselines be improved with the soft and hard constraints?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the constructive suggestions provided by the reviewer, and we are happy to revise and improve our paper accordingly, incorporating the writing and presentation suggestions pointed out.\n\n## Weaknesses \n\n**Q1.** \nIn particular, we thank the reviewer for pointing out unclear terms and definitions. In the revised paper we will clarify and change any ambiguous terms:\n- By \"seed for generation\" (line 133) we mean a starting time-series (TS) sample which is modified to generate a new sample (data augmentation); \n - The \"desired property\" (line 145) is a function over the original TS data that defines the statistical properties we want to preserve in the generated sample. In line 151-152, we mention that this is a function z($\\cdot$) over the TS vector and, as the reviewer correctly supposed, auto-correlation is one example of such a function. We use the term \"property\" to keep the definition more general, as different TS properties can be specified and preserved. We will improve this section by giving also examples of different properties.\n\n**Q2.** \nWe will improve the  COP unconstrained generation section to make it more clear for the reader. The main idea behind the COP formulation for TS generation is the following: we can frame the task of generating a TS sample as optimizing the value of a set of ordered points that make up the TS sample, such that the sample satisfies domain properties (like auto-correlation).\nTo do this, we define the domain properties as constraints, and we define the objective as optimizing (maximizing) the distance between the generated sample (set of points) from a starting sample (taken from the dataset, or randomly generated) using the L2 distance between the two TS. With this generation problem setting, we can use existing COP solvers (e.g., SLSQP) to get new samples by solving those non-linear constraints and objectives. Thus the COP solver using the specified constraints and objective function becomes the generative process. There is no additional generative model needed. \n\nIn unconstrained TS generation with COP, the objective remains maximizing the difference between an input real TS and the generated TS sample (using L2 distance).  Even when unconstrained -- as in the user has not specified additional constraints-- the COP formulation for TS generation will have inequality constraints associated to preserving domain specific properties, such as auto-correlation, of the starting sample TS. The impact of different input TS (be it real or noise) is investigated in the supplementary material, Section G.3.\nFinally, we would like to highlight that COP has been specifically designed to easily support complex constraints with a straightforward manageable generation procedure, while our diffusion models can provide more sophisticated data generation.\n\n**Q3.** \nWe thank the reviewer for pointing out important missing experiments. We extended the ablation study for our diffusion model architecture, and we evaluated the impact of: \n- different number of diffusion steps $T$ (Supplementary material Section G.1); \n- different noise variance schedulers (Supplementary material Section G.2); \n- different model layers and hyper-parameters (discussed below);\n\nWe introduce the following variants of DiffTime:\n- *DiffTime-K-Heads* - we change the number of \\textit{attention heads}, from 1 to 8; \n- *DiffTime-LSTM* - we replace the convolutional layers using recurrent layers (i.e., LSTM) along the attention mechanism, which is particularly successful for imputation and interpolation of TS[7]; \n- *DiffTime-full-LSTM* - we replace all the convolutional and transformer layers by using LSTM layers, which is common for time-series generation[8]; \n- *DiffTime-full-CNN* - we replace the transformer layers using convolutional layers;\n\n$\n\\begin{array}{l|lll}\n\\hline\n\\text{Algo} &  \\text{Discr-Score} &   \\text{Pred-Score}  &   \\text{Inference-Time}   \\\\\\\\ \\hline\n\\text{DiffTime-1Heads} & 0.03\\pm0.02 &  0.21\\pm0.00 & 0.02\\pm0.01 \\\\\\\\\n\\text{DiffTime-4Heads} & 0.06 \\pm 0.02 &  0.21\\pm0.00 & 0.04\\pm0.0 \\\\\\\\\n\\text{DiffTime-8Heads} & 0.05 \\pm 0.03 &  0.21\\pm0.00 & 0.02\\pm0.01 \\\\\\\\\n\\text{DiffTime-LSTM}  & 0.06 \\pm 0.01 &  0.21\\pm0.00 &  0.02\\pm0.01 \\\\\\\\\n\\text{DiffTime-full-LSTM} &   0.50 \\pm 0.00 &  0.21\\pm0.00 & 0.02\\pm0.01 \\\\\\\\\n\\text{DiffTime-full-CNN}  &   0.14 \\pm 0.04 &  0.21\\pm0.00 & 0.03\\pm0.01 \\\\\\\\\n\\hline \n\\end{array}\n$\n\nThe results highlight the performance of the current architecture, which uses transformer layers, where the number of attention heads can be tuned according to the input dataset. \n\nFinally, we would like to highlight that the GAN-based benchmarks have been adapted to deal with soft and hard constraints: we conditioned the architectures on the soft-constraint, while we add a penalty term in the training loss for the hard constraints. The results in the main paper show improved satisfaction-rate (yet lower than those of our approaches), but slightly worst performance for hard-constraints in terms of discr. and pred. scores, and distribution similarity. We found that the GAN-based architectures are inherently unstable and harder to train for hard-constraints. \n\n## Questions\n\n**Q.1** We empirically noticed that:\n- soft-constraints (which are used to condition the model generation) often make the training faster, but they also increase the chance of over-fitting (for diffusion models) or having mode collapse (for GAN-based models).\n- hard-constraints enforced through a penalty term in the model loss, usually aggravate the training procedure. We experienced more training time, and more instability for GAN-based architectures. However, the training of our Guided-DiffTime is not affected by any constraints, as they can be added at inference time, making its training easier and faster.\n\n**References in the global rebuttal comment**"}, {"review_id": "SSix3h3ik0", "reviewer": "Reviewer_DrNe", "summary": "The paper proposes a model for generating synthetic time series data given user-specified constraints. The constraints can be either soft (e.g. following trend) or hard ((in)equalility). The first proposed method is framed as a Constrained Optimization Problem (COP), it inputs a seed time series which is either real or sythetic and maximizes or minimizes the L2 distance, respectively, while satisfying the constraints.\nThe second method, called DiffTime, is a conditional diffusion model -- where the soft constraint (trend) is conditioning the diffusion process. The hard constraint (fixed point) is incorporated in the reverse process.\nThirdly, general constraints are included in the diffusion loss s.t. the model reproduces the sample x_0 with all the constraints satisfied, at every diffusion step.\nFinally, sampling from a diffusion-based model can be guided with gradients that come from the constraints directly.", "questions": "- In COP-method, is L2 objective the best proxy for obtaining diversity/similarity? Could simple pointwise difference guarantee these properties?\n\n- Section 4.1, how is COP exactly \"trained\"? As pointed in this section, since it's using real data seed, the comparison is not really fair.\n\n- Can you guarantee that the final samples follow true (conditional) data distribution? I imagine some of the proposed methods can create samples that satisfy constraints and are unrelated to data.", "limitations": "Performance limitations are discussed.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper is well written. Approaches are straightforward and seem valid. Each extension in Section 3 comes naturally as an answer to a weakness of the previous approach. There is some novelty in the presented methods.", "weaknesses": "The originality of most of the approaches is somewhat limited. The first three approaches seem like simple baselines that are applying existing methods to constrained generation. Only the last approach with guidance cannot be directly applied to a GAN or some other generative model.\n\nMotivation to solve this problem. Although the paper mentions reasons to perform constrained generation, the experiments do not support this. A case study or showing benefits on a downstream task would significantly improve the argument.\n\nConserning the results, in Figure 2 and 3 t-SNE is not very convinving, and in general using t-SNE is not very rigorous. GT-GAN seems to be outperforming DiffTime in Table 1 (upper part).\nAn interesting baseline would be a naive approach that adds noise to data samples or mixes time series or does some other simple transformation. This would show how much is relative difference between methods significant.\nThe experiments would benefit from having more datasets and baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the positive and constructive feedback. We are happy to revise and improve the paper according to the reviewer's comments, and to better motivate our work and its novelty.\n\n## Weaknesses \n\n**Q.1** \nOur paper presents a set of methods for incorporating constraints during time-series (TS) generation, which is not directly addressed in the current literature.  The first approaches are indeed adaptations of existing models to provide them the ability of constrained TS generation. However, we believe these approaches provide the reader with a more natural narrative and motivation for our proposed new methods, showing how each method address weakness of the previous one. Moreover, some of these approaches achieve remarkable performance, and can be used for specific constraints or tasks.\n\n**Q.2** In the revised version we will add more examples of downstream tasks that can benefit from constrained synthetic TS. We recall that constrained TS generation allows to: \n- augment training datasets by amplifying the occurrence of rare events; \n- test algorithms/models on counterfactual scenarios. \n\n**Financial Market Example:** An example of a downstream task is the testing of existing trading or portfolio strategies from the literature [13,14] for specific stock market scenarios. Ideally, we would like to know how such strategies behave during flash-crashes [9] or market shocks [10, 12]. However, as such events occur rarely, there is shortage of data for such market regimes. Two recent examples are the flash crash of E-MINI SP500 futures [9] on 6th May 2010, or the COVID market crash on March 2020 [10]. While we do not know when and how such rare events will happen, we can use constrained TS generation to create thousand of possible market crashes, with different magnitude and duration, to evaluate the strategies. \nFigure 2 of the attached rebuttal PDF shows that we can generate synthetic market scenarios with similar crashes to E-MINI and COVID. It is worth to notice that, the real market crash data are not used during the diffusion model training, but we can still generate realistic crash scenarios with a specific *constraint* (i.e., from 10\\% to 30\\% price drops).\nThe synthetic market data can be used to test and evaluate such strategies, and study their behaviors when similar events actually happen.\n\n**Q.3** We thank the reviewer for suggesting new baselines for evaluating synthetic TS. We introduced the three new baselines that (1) add Gaussian noise to data samples; (2) mix (average) existing TS; (3) create a simple Brownian motion starting from a real TS. We report quantitative metrics in the table below, while t-SNE plots are shown in Figure 8 of the PDF rebuttal. It is interesting to note that only the mixed TS approach shows good distributional similarity, as averaging two real TS can preserve some distribution information. However, as we report in the Table, such baseline is not competitive with our methods.\n\n$\n\\begin{array}{l|lll}\n\\hline\n\\text{Algo} &  \\text{Discr-Score} & \\text{Pred-Score} \\\\\\\\ \\hline\n\\text{COP (Ours)} & .050 \\pm .017 & .041 \\pm .001   \\\\\\\\\n\\text{DiffTime (Ours)} & .097 \\pm .016 & .038 \\pm .001  \\\\\\\\\n\\text{Noised TS} & .491 \\pm .091 &  .038 \\pm .001 \\\\\\\\\n\\text{Mixed TS} & .186 \\pm .010 &  .041 \\pm .001  \\\\\\\\\n\\text{Brownian TS} &  .085 \\pm. 010 &  .038 \\pm .000  \\\\\\\\\n\\hline\n\\end{array}\n$\n\n\n## Questions \n \n**Q1** We agree with the reviewer that L2 distance may not necessarily be the best proxy for diversity, and that we can use any other distance-based metric. We will better clarify this in the paper, providing additional experiments and comparing different distance metrics. In particular, following the reviewer suggestion, we empirically evaluated L2 distance and point-wise distance. Figure 5 in PDF rebuttal shows that both the distance metrics preserve distributional similarity in the synthetic data, which we empirically evaluated using t-SNE. However, the L2 distance achieves better quantitative results, a shown below: \n\n$\n\\begin{array}{l|ll}\n\\hline\n\\text{Algo} &  \\text{Discr-Score} & \\text{Pred-Score} \\\\\\\\ \\hline\n \\text{COP L2-distance} & 0.017 \\pm 0.006 & 0.203 \\pm 0.001 \\\\\\\\ \\hline\n \\text{COP Pointwise-distance} & 0.021 \\pm 0.012 & 0.203 \\pm 0.002  \\\\\\\\ \\hline\n \\end{array}\n$\n\n**Q.2**\nCOP frames the task of generating a TS sample as optimizing the value of a set of ordered points that make up the TS sample, such that the sample satisfies domain properties (like auto-correlation). In particular, it starts from an initial sample TS (taken from the dataset, or randomly generated) and optimizes its values according to its objective (e.g., maximize the distance between the generated and initial sample), while respecting some constraints (be them statistical properties or additional structural constraints). This optimization approach avoids training, but usually involves longer inference times. While deep models use the real training set to learn the data distribution, COP uses the training set differently, as starting samples to optimize. Nevertheless, in the supplementary material (Section G.3) we show that COP achieves similar performance when the initial TS samples are generated by random Brownian motion, or by noising original data.\n\nFinally, we would like to highlight that our adaptation of COP for TS generation easily supports complex constraints, and the generation procedure is straightforward. On the other hand, our diffusion models can provide more sophisticated data generation by learning implicit patterns from data, as well as faster sampling. \n\n**Q.3**\nCurrently, we do not have any formal guarantee that the synthetic data follows the true conditional distribution, but we can only empirically evaluate that. We agree that this is an interesting discussion, and we will focus on this aspect in a future work.  \n\n**References in the global rebuttal comment**"}, {"review_id": "FkcEu7Ap5a", "reviewer": "Reviewer_5edF", "summary": "The authors propose a set of methods for generating synthetic time series, with or without constraints. They defined the different types of constraints possible, including soft and hard constraints.  They developed a COP approach for generating time series from a seed, then presented ways of adding constraints, and delivered an all-inclusive method based on diffusion models: GuidedDiffTime.", "questions": "1) What kind of constraints are not included in those you describe? How would you go about adding them and taking them into account?\n2) Have you thought about less \"traditional\" time series that look more like \"functions of time\"? More precisely, how would your method adapt to very different curve shapes? ", "limitations": "See questions.", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is very clear and pleasant to read. Everything is very well explained, including the context, the issues and the solutions implemented. The content is substantial and shows a lot of hard work on the part of the authors. The results have been tested on several databases and show the advantages and disadvantages of their solutions.", "weaknesses": "Nothing in particular.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "\nWe thank the reviewer for the positive and constructive feedback. We are happy to add more discussion on the type of constraints and time-series we currently support, and to extend our work in the future, following the reviewer suggestion.\n\n\n**Q.1**  Currently our work cannot cover constraints that involve logical functions over constraints and are not differentiable. For example, we cannot support exclusive-or constraints like ``either $f(x) \\leq  3$ or $g(x) \\leq 4.5$ but not both\". In fact, we cannot model such constraints to be provided as input to the generative models. We plan to address such limitation in future work. \n\n**Q.2** We thank the reviewer for the interesting comment. We currently have one experiment in which we consider a sine-wave dataset, as an example of time series that is a function of time. Some results associated to this can be found in Table 1 of main paper. Nevertheless, we acknowledge that in such examples we considered discrete time-series, i.e., we take observation from the sine function at specific regular times. Considering the different modelling challenges posed by continous functions, we plan to consider this scenarios for future work. Similarly, we will better investigate time-series with changing curve shapes (e.g., a time series that can be represented by a piece-wise continuous trajectory with a distributional shift occurring), in the future, including examples from existing work [11].  \n\n\n**References in the global rebuttal comment**"}]

Author rebuttal:
We would like to thank all the Reviewers for their constructive and valuable comments. We revised our work according to the reviewer's comments, and new results are included in this rebuttal, and attached PDF. In particular: 

$\bullet$ We included 2 new state-of-art benchmarks for time-series generation; 

$\bullet$ We introduced 3 baseline approaches to synthetic time-series generation;  

$\bullet$ We included more metrics to evaluate the generated synthetic time-series and their statistical properties; 

$\bullet$ We added more details on our diffusion model architecture, including a graphical representation; 

$\bullet$ We performed a more detailed ablation study on the diffusion model architecture; 

$\bullet$ We improved the study of COP approach, showing new experiments and its fine-tuning capabilities in more detail. By using COP for fine-tuning, after any generative model, we observe that we can double the constraint-satisfaction rate of generated time-series, while keeping the same distributional properties (see Figure 6 and Figure 7 of PDF rebuttal); 

$\bullet$ We provided more discussion and examples on how constrained time-series generation can be helpful in real-world problems; 

$\bullet$ We are incorporating the writing and presentation suggestions pointed out by reviewers in the final paper version. 
 
We are happy to discuss our work further, and we believe our manuscript has improved by incorporating reviewer suggestions, providing more valuable insights and results. We will integrate the new developments in the final paper version. A more detailed discussion is provided in each rebuttal. 

**Novelty:**
Finally, we would like to highlight that our approach provides realistic time-series generations for both constrained and unconstrained generation. In fact, while existing GAN-based architectures have comparable performance in some unconstrained generative scenarios, they are not directly suitable for constrained generation. Modifying the GAN loss function to incorporate constraints is not trivial, and can aggravate the inherent instability of GANs, and rejection-sampling is expensive for complex constraints. Instead, our work directly targets constrained generation. Our Guided-DiffTime can generate different constrained time-series, without any special (re)training, as constraints can just be introduced at inference time. This drastically reduces the computational cost and carbon footprint (as empirically shown in Table 6 of Supplementary material). Additionally, our COP approach can be used instead of rejection sampling to revise synthetic samples from any generative model. 



**All references of the rebuttal comments:**

[1] - J. Sohl-Dickstein, et al. "Deep Unsupervised Learning using Nonequilibrium Thermodynamics" ICML 2015.

[2] - A. Seyfi, et al.. "Generating multivariate time series with COmmon Source CoordInated GAN (COSCI-GAN)." NeurIPS 2022

[3] - H. Pei, et al. "Towards generating real-world time series data." ICDM 2021

[4] - L. Zhou, et al. "Deep latent state space models for time-series generation." ICML 2023.

[5] - JP. Bouchaud,  et al. Trades, quotes and prices: financial markets under the microscope. Cambridge University Press, 2018.

[6] - R. Rombach, et al. "High-resolution image synthesis with latent diffusion models." IEEE/CVF 2022.

[7] - S.N. Shukla et al. "Multi-Time Attention Networks for Irregularly Sampled Time Series." ICLR 2020.

[8] - Mogren, Olof. "C-RNN-GAN: Continuous recurrent neural networks with adversarial training." 

[9]  - Kirilenko, Andrei, et al. "The flash crash: High‐frequency trading in an electronic market." The Journal of Finance 72.3 (2017): 967-998.

[10] - Mazur, Mieszko, Man Dang, and Miguel Vega. "COVID-19 and the march 2020 stock market crash. Evidence from S\&P1500." Finance research letters 38 (2021): 101690.

[11] - J. Jia, A. Benson. Neural Jump Stochastic Differential Equations. Neurips 2019.

[12] - D. Cao, et al. "DSLOB: a synthetic limit order book dataset for benchmarking forecasting algorithms under distributional shift" 

[13] - S. Bartram,et al. "Machine learning for active portfolio management." The Journal of Financial Data Science (2021)

[14] - F. Ferreira "Artificial intelligence applied to stock market trading: a review." IEEE Access (2021)

Meta-review: This paper solves the problem of constrained time series generation problem, where several types of constrained e.g., hard and soft, local and global, are taken into account. The authors achieve the ideal performance and illustrate the effectiveness of the proposed method. After rebuttal, the authors address most of the problems of the reviewers. Some reviewers consider that this paper is well-written. The decision is accepted.
Decision: Accept (poster)