Evaluation is key: a survey on evaluation measures for synthetic time series

The evaluation of generative models for the synthesis of time series and other data types, such as images or text, is considered an open problem by many researchers [4, 11‐13]. To gain a wider awareness of the current standing in related fields of data synthesis, we provide a selection of works on generative adversarial networks (GANs), sequential data in general, software libraries and frameworks, tabular data, audio, and text.

A few years ago, Xu et al. [14] already set out to address the problem of how to “evaluate the evaluation metrics” for GANs in particular, with a focus on images. Firstly, they introduced six generator-agnostic measures, namely inception score (IS),Fréchet inception distance (FID), Wasserstein-1 distance (WD), mode score, kernel 112 maximum mean discrepancy (MMD), and classifier two-sample test (C2ST). Furthermore, the authors briefly describe four conditions they consider necessary for measures to adhere to. These are distinctiveness, robustness to transformations, efficiency, and detecting overfitting. An experiment is conducted for each condition and presented measure to check if the latter suffices said conditions. In a concluding discussion, the strengths and weaknesses of the six measures are outlined. While insightful, the list of measures considered is minimal and outdated. Many novel measures have been presented in the meantime, especially data type-agnostic ones, which can be adapted to any type of data. Overall, our focus is pointed toward time series, not images.

In a recent article, Brophy et al. [4] reviewed state-of-the-art GAN models for time series generation, augmentation, and imputation. They provide background on the workings of this learning paradigm and a classification of methods. The work primarily covers the latest popular architectures but also features a short section on evaluation strategies for this type of model. The overview contains some elementary measures, mostly applicable only to individual time series but not datasets. As presented there, it is a listing of popular measures used, but neither a complete nor critical one. Furthermore, the categorization of knowledge is limited to “quantitative” and “qualitative”.

Lately, Eigenschink et al. [15] proposed an evaluation framework for (mainly deep) generative models for synthesizing all kinds of sequential data: Text, audio, video, and time series. They aimed to overcome the isolated evaluation of generators in these areas and present a universally applicable set of criteria to check. Namely, they argue for five categories: Representativeness, novelty, realism, diversity, and coherence. The paper then goes into the current usage of measures in and the relevance of each category for the different data types. However, their table of measures for synthetic time series is very sparse, with six works reviewed and 13 measures found and limited to healthcare and mobility.

Recently, Borji [11] surveyed recent developments in GAN evaluation measures research, updating a previous paper on the topic [16]. Again, the work is not restricted to the time series domain but to one class of generation methods. Moreover, the focus is on image generation as it dominates research on synthetic data in general. It furnishes an extensive array of measures accompanied by concise explanations and numerous generated images. The measures are organized into sections based on their similarity rather than grouping them by applicable data types. No additional structure is provided. Thus, those relevant to time series need to be tediously searched for, while most measures we found were not covered there. Lastly, our analysis is tailored toward time series.

For the field of tabular data synthesis, Dankar et al. [17] introduced a scheme of four abstract criteria generated data may be tested on to demonstrate utility to end users: attribute fidelity, bivariate fidelity, population fidelity, and application fidelity. For each criterion, a representative measure was selected by the authors based on popularity and consistency after reviewing relevant literature. Still, only tabular data is considered here. Most of those measures cannot trivially be applied to time series.

Another related field of synthetic data is audio generation. Deep learning (DL) techniques have been used excessively to generate artificial music inspired by human-made samples. In a survey by Ji et al. [18], past developments and future directions of DL-based generators are presented, including typical evaluation measures of the domain. The authors found that there is no unified evaluation criterion, neither present nor to be expected, as music is a form of art and thus made to appeal to humans. Instead, there are various suggestions for how to approximate human evaluation.

The division of evaluation strategies into human assessment and machine-computed measures can equally be found in text generation. In a survey by Fatima et al. [19], however, the evaluation by human language and domain experts is attributed little importance. This is due to the intensive labor and cost involved as well as subjectivity and susceptibility to human error, while quantitative measures are correlated well enough with human perception. Another study conducted by Iqbal and Qureshi [12] also sees a variety of evaluation measures present. However, it concludes that evaluation is still an open research problem, as many methods poorly correlate with human assessments.

Assefa et al. [20] presented a review of the current standing and developments of synthetic data generation in finance, which addresses the evaluation of the similarity of real and generated datasets, among other things. They mention four works relevant to evaluating time series, while our survey not only considers these works but is domain-independent, includes far more measures, and analyses them.

Recently, Figueira and Vaz [13] published a survey on the generation and evaluation of tabular data with a focus on GANs. In a short section on evaluation measures, they list only seven that are also applicable to time series data, which our work contains as well. They do not provide sources for some of the measures and refrain from significant further analysis.

Above, we identified several mismatches between this survey and prior works. Most address specific domains or data types other than time series [12, 14, 18, 19], while a transfer of findings is non-trivial. Another one addresses the evaluation of sequential data generation, but in too broad a scope to provide detailed insights for time series [15]. Others are tailored towards images or are data type-agnostic but limit their scope to GANs [4, 21]. Compared to the reviewed articles, our study goes into more detail in the time series domain, both in terms of works covered and subsequent organization of knowledge.

This section briefly defines the key terminology used in this work.

In some works, the notation “metric” is used instead of measure. However, we opted for the latter since the former encompasses mathematical characteristics that might only hold for some of the proposed measures found in the literature.

For convenience, we reference additional literature on statistical and deep learning-based data synthesis by [22] and [23], respectively. Examples of classic time series models include SARIMA and Holt-Winter [24], while deep learning models often utilize recurrence [25], adversarial learning [26], or autoencoding [27].

For instance, the additional input can be a class label or, specifically for time series, the previous time step \(\textbf{x}_{t-1}\).

In practice, a time series is simply a finite set \(X = \{ \textbf{x}_t \mid t \text { is a point in time} \}\) of observations \(\textbf{x}_t \in \mathbb {R}^d, d \in \mathbb {N}^+\), typically some measurement or event. For our purposes, we implicitly assume an ordering on X given by time t. Often, the observations are also equidistant, but not necessarily. Still, some generators and measures assume this. Also see Fig. 1 for illustrative examples.

We typically denote a general data space by \(\mathcal {X}\), an unspecified data object by \(x \in \mathcal {X}\), a time series by \(X \in \mathbb {R}^{l \times d}\), and one of its channels \(X^c \in \mathbb {R}^{l \times 1}\). Furthermore, \(D, D_r, D_g \subset \mathcal {X}\) are datasets and \(f: \mathcal {X} \rightarrow \Omega\) an embedding function from the data space into some other feature space \(\Omega\), usually \(\Omega = \mathbb {R}^d\). Lastly, many evaluation measures depend on distance functions \(\delta : \mathcal {X} \times \mathcal {X} \rightarrow \mathbb {R}_0^+\) between two time series. While the authors of the reviewed works exclusively used the Euclidean distance (ED) and dynamic time warping (DTW), respectively, it should be mentioned that some measures might improve in accuracy or execution time when combined with other time series distance functions or time series models. Examples include Bag-of-SFA-Symbols (BOSS) [28], complexity invariance distance (CID)time series shapelets [29], complexity invariance distance (CID) [30], and maximum shifting correlation distance (MSCD) [31].

We conducted thorough searches for previous publications related to the evaluation of time series generation and synthesis. To this end, we applied the technique of Webster and Watson [32] to conduct a comprehensive literature review. The process consisted of four steps as described below:

Keywords and Data Sources The first step consists of a broad search for statement candidates based on the following keyword sets and their permutations:leading to search queries such as “time series synthesis”, “data synthesis evaluation”, or “generative adversarial network time series generation”. We used Google Scholar,¹ the IEEE Xplore,² and ACM DL digital libraries.³

Synthesizing Knowledge Certainly, there are many ways to approach the analysis of the collection of measures found. Hence, we list the nine dimensions considered in this study in Table 1 and briefly justify each choice. They are ordered by appearance. For someone interested in finding a suitable measure quickly, this may serve as a guidepost. We suggest starting with the dimension of highest personal priority, going to the respective location in this article, and making an initial selection of measures. Afterwards, the reader may return to this table, select the next-most relevant dimension, and refine the selection repeatedly until satisfaction.

Below, we provide an introduction to each distribution-level measure found. This list is sorted alphabetically to foster a faster look-up for non-sequential reading. A summary can be found in Table 2.

Agreement rate. This measure tracks the usability of the synthetic data in a downstream machine learning (ML) task, namely an improvised classification [33]. Figure 2 depicts the computation of this measure with the help of a flow chart. Generally speaking, an ML algorithm A is trained on synthetic data \(D_g\) and evaluated on real \(D_r\) to judge \(D_g\)’s practical value. More specifically, each data sample is annotated by setting one of its own features (an attribute of tabular data, a channel for time series, etc.) as a class label. If the model is conditional, this step is redundant. Then, the real dataset is split into train and test components, \(D_r = (D_r^{train}, D_r^{test})\). Afterwards, a classifier is trained separately on \(D_r^{train}\) and the synthetic \(D_g\), yielding two models, \(M_r\) and \(M_g\), respectively. Finally, the agreement rate \(r_{\text {agree}}\) is defined as the fraction of \(D_r^{test}\) for which both models make the same class assignments. More formally, it iswhere \({\textbf {1}}\{\cdot \}\) is the indicator function. The rate falls in [0, 1], with higher being better. An embedding is implicitly given by adjusting the classifier’s input layers for differently shaped inputs.

Algorithm comparison. Given real and synthetic datasets \(D_r\) and \(D_g\), respectively, Lin et al. [34] first create train-test-partitions \(D_r = (D_r^{train}, D_r^{test}), D_g = (D_g^{train}, D_g^{test})\). For a downstream task T, of which one assumes it might be a good indicator for the utility of the synthetic data, they select a group \(\mathcal {A}\) of algorithms and an appropriate performance score s for T. Now, algorithm comparison is a measure that evaluates a synthesis on whether it preserves the ranking within \(\mathcal {A}\) with regard to s when applied to task T. To give an example, T might be the classification of time series, the group of algorithms be given by \(\mathcal {A} = \{\) multi-layer perceptron, linear support vector machine, naive Bayes, decision tree, logistic regression\(\}\), and s be the F1 score. Note that this setting is only applicable to labeled datasets, that is, conditional generators. A flow chart illustrating the procedure is given in Fig. 3. To apply this measure, they train each algorithm on \(D_r^{train}\) and \(D_g^{train}\), which returns two sets of trained models \(\mathcal {M}_r\) and \(\mathcal {M}_g\). Second, Lin et al. [34] evaluate all models \(M_r \in \mathcal {M}_r\) on \(D_r^{test}\) and \(M_g \in \mathcal {M}_g\) on \(D_g^{test}\). As a result, they obtain the same number of scores, grouped in two sets as well. To this pair of sets, one can apply the Spearman rank correlation coefficient, which is the Pearson correlation coefficient (PCC) between the ranks of each model’s score within each group. As a result, we get a scalar value \(r_s \in [-1,1]\), indicating to what degree each algorithm performs equally on the generated data relative to the other algorithms, compared to their performance on real data. The higher it is, the better.

Annealed importance sampling (AIS) is a quantitative measure for evaluating log-likelihoods for decoder-based generative models [35]. By that, we mean parameterized neural networks defining a generative distribution by transforming samples from some simple distribution to the data manifold. AIS is a Monte Carlo algorithm commonly used to estimate normalizing constants, and employed here to estimate the log-likelihood \(\log p(\textbf{x}_{test})\) the model assigns to a held-out test sample [36]. The measure requires an embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\) as well as a held-out test set (AIS \(\in \mathbb {R}\), the higher the better).

Approximate entropy (ApEn). This is a channel-wise entropy measure specifically for time series used to determine their regularity and complexity. For more details, we refer to [37]. In [38], the measure is applied to all available real and synthetic data, while Leznik et al. [9] sample subsets \(R \subseteq D_r\) with \(|R|=500\) and \(G \subseteq D_g\) with \(|G|=10\). The latter approach is used to guide GAN training. ApEn yields a score for each channel of every sample in both the real and synthetic dataset. To retrieve a final measure, the individual scores are aggregated first via calculating the mean for each channel over all samples in each dataset separately, and then applying the squared difference between scores on real and synthetic time series over the channels. A smaller value indicates better performance.

Augmentation test. This measure is designed to probe the usefulness of the synthesized data \(D_g\) for augmenting the real data \(D_r\) [39]. To this end, the authors split the real data into training, validation and test sets \(D_r = (D_r^{train}, D_r^{val}, D_r^{test})\). The generator only gets to learn from the training set to create \(D_g\). Next, the augmented dataset \(D_{aug}\) is created by merging \(D_r^{train}\) with \(D_g\), \(D_{aug} :=D_r^{train} \cup D_g\). Furthermore, an ML task is defined to measure the usefulness of \(D_{aug}\) and an appropriate algorithm A selected. Training is conducted on \(D_{aug}\) to learn a model \(M_1\) and separately on \(D_r^{train}\) for a baseline model \(M_2\), while validation is performed on \(D_r^{val}\) in both cases. Validation and testing is conducted with respect to some measure s appropriate for the selected task. In the context of their work, the authors chose a classification task as they had a labeled dataset. In principle, other tasks that do not require labels are also imaginable. For instance, Jeha et al. [40] employ prediction, but without reference to [39]. The performance of \(M_1\) and \(M_2\) is measured in terms of precision and area under ROC curve (AUROC) on the test set. For \(D_g\) to be useful for augmentation, we expect \(s(M_1) > s(M_2)\), the greater the difference the better. The applicability to labeled datasets depends on the choice of the task.

Average cosine similarity (ACS). This measure compares pairs of real and synthetic time series with respect to their cosine similarity and computes a score for each class by averaging [41]. In this setting, class labels for both datasets are required. Let \(D_r\) and \(D_g\) be real and synthetic datasets. Instead of calculating the cosine similarity directly on the time series, the authors define an embedding \(f: \mathbb {R}^{l \times d} \rightarrow \mathbb {R}^{7 \cdot d}\) for time series of length l and dimensionality d, which extracts a vector of seven features from each of the d channels and concatenates them. Namely, these are median, mean, standard deviation, variance, root mean square, maximum, and minimum. For each class c, ACS is defined bywhere \(D_r^c, D_g^c\) denote the samples of class c in dataset \(D_r\) and \(D_g\), respectively (ACS\(_c \in [-1,1]\), the higher the better).

Average euclidean distance (AED) focuses on the distribution of amplitudes in the frequency domain within the synthetic dataset [42]. The goal is to ensure preservation of inter-channel correlation within the synthetic time series. The measure targets those with two channels exclusively. To do so, they first transform all synthetic samples into the frequency domain and extract the most likely amplitude in both channels of each sample. Then, they interpret the two amplitudes of each sample as a coordinate in the 2D plane. This allows them to compute the distance to the line through the origin with slope 1. However, the calculation may fail on time series without any seasonality.

Average Jensen-Shannon distance (JS distance) computes the distance between the distribution of each feature over the real dataset and the distribution of the corresponding feature over the synthetic dataset [41]. More precisely, the measure assumes a partition of the datasets into (the same) classes, with the score being computed for each class separately. To this end, they first define the feature vector f(X) for each time series X via \(f: \mathbb {R}^{l \times d} \rightarrow \mathbb {R}^{7 \cdot d}\), which is the same embedding used for ACS above. The seven features extracted are identical as well. Now, one can determine the distance between the distributions of each of the 7d features and take the average to arrive at a score \(\text {AJSD}_c\) for class c asJSD\('\) is an approximation of the conventional Jensen-Shannon divergence (JS divergence) on distributions, taking just two sets of samples. Furthermore, \(D_r^c, D_g^c\) denote the samples of class c in dataset \(D_r\) and \(D_g\), respectively. It holds \(\text {AJSD}_c \in \mathbb {R}_{\ge 0}\), the lower the better.

Average Wasserstein distance (AWD) compares the distribution of amplitudes in the frequency domain within the real dataset to that of the synthetic dataset [42]. The goal is to ensure the diversity with respect to the syntheses’ periodicity. To this end, fast Fourier transform (FFT) is used to determine the most likely period in each channel of each sample. Afterwards, the WD between the amplitudes extracted from real dataset on one side, and synthetic dataset on the other, is calculated for each channel, and finally averaged across channels. For the resulting real-valued score holds lower is better. However, the calculation may fail on time series without any seasonality and an order over the samples in both datasets is required, but not provided.

Classification accuracy score (CAS) is an evaluation method for conditional generative models proposed in [43]. Sometimes, they are also denoted by \(g_\theta (x|y)\), where the typically random input z is left aside, and \(g_\theta\) is viewed as producing samples x conditioned on an additional input y, here, a tag or class label. The idea is to train another model on the produced dataset for a downstream task, in this case classification using the class labels assigned during generation. The accuracy of this model on a held-out, real test set is the CAS for the generator. Compared to the score computed with a classifier trained on the real dataset, this measure tells if the utility of the synthesis is on par with the original. The entire procedure is depicted in Fig. 4. The closer the two are, the better the synthesis. Furthermore, it provides insight into problems with generating specific classes.

Classifier two-sample test (C2ST) is no measure specialized on the evaluation of synthetic data, but rather assesses whether two sets of data points are sampled from the same distribution [44]. As the name suggests, this is realized through a binary classifier \(c: \mathcal {X} \rightarrow [0,1]\) combined with a hypothesis test. The basic procedure is also illustrated in Fig. 5. Specifically for the evaluation of generative models, these two sets are real dataset \(D_r\) sampled from P and synthetic set \(D_g\) sampled from Q. First, the authors split off a substantial part of \(D_r\) to train the generator on, denoted by \(D_r^{gen}\). Afterwards, they split the remaining part of \(D_r\) and \(D_g\) into two parts each, one for training the classifier and the other for calculating the p-value of the hypothesis test based on the classification accuracy of c. This leaves them with \(D_r = (D_r^{gen}, D_r^{train}, D_r^{test}), D_g = (D_g^{train}, D_g^{test})\) with \(|D_r^{train}| = |D_g^{train}|, |D_r^{test}| = |D_g^{test}|\). Additionally, each data point in \(D_r^{train}, D_r^{test}\) is assigned label 0 and, similarly, \(D_g^{train}, D_g^{test}\) label 1. After training c, one applies the model to \(D_{test} :=D_r^{test} \cup D_g^{test}\) and obtain classification accuracywhich acts as the two-sample test statistic. z is a data point, l its label, and \({\textbf {1}} \{ \cdot \}\) the indicator function. If indeed \(P = Q\), \(A_{test}\) should be around \(\frac{1}{2}\), indicating a random assignment of samples to classes. For a predefined significance level s, one can now compute p-value \(\hat{p} = \text {Pr}(A \ge A_{test} \mid P = Q)\), that is, the probability that the test returns an accuracy A as high as it was, given that \(P = Q\) holds. Finally, if \(\hat{p} < s\), one rejects this hypothesis, otherwise we can confidently assume the equality. Besides this final result, C2ST has the additional advantage of interpretable intermediary results. On the sample-level, one can see which samples are identified with high confidence and thus cause the generator trouble, and which are already close to real data. On the distribution-level, one can also relate directly to \(A_{test}\) as a score to compare to other generators’ performance. Besides, an embedding is implicitly given by the classifier.

Computational complexity. With this term, we refer to a group of measures tracking the efficiency of generators concerning resource requirements and costs incurred. Although applicable on a sample-level in many cases, the usual approach is to compute measures for many synthetic samples and break down the result by providing an expected time/resources/cost per sample. The scope can be limited to the generation process itself or may include training or preparation of the generator. Bindschaedler et al. [33] evaluate their model with respect to time taken for inference, producing over one million samples in total. They specify the experimental setup as well as the parameters tested. Kulkarni et al. [45] provided the CPU time of model training and sample inference each. Clearly, the goal is to provide generators with fast and reliable convergence during resource-aware training while allowing quick and easy generation of high-quality samples. Note that this is irrelevant to the end user of the synthetic data but can still be a limiting factor for many creators of synthetic data, which might indirectly affect quality.

Confusion matrix. This measure requires labeled real and synthetic datasets with a matching set of classes, that is, the same labels in both cases [46]. By aggregating the result of a downstream classification task in a confusion matrix, one can quickly determine classes of generated samples that cause the generator most trouble synthesizing. On the other hand, classes that are overwhelmingly correctly recalled by the classifier indicate that the generative model learned what sets such samples apart from other classes. Figure 6 contains two example matrices. First, a classifier is trained and validated on the real data. Afterwards, one samples a synthetic dataset and infers the class label for each sample therein. Finally, we can compute and visualize the confusion matrix. Generally, a concentration of probabilities on the main diagonal is desirable. The classifier model is expected to compute its own data embedding.

Context-FID is an evaluation measure that aims to quantify the similarity of real and synthetic time series distributions following the example of the existing FID score for image synthesis. Specifically, this approach replaces the image feature extractor of regular FID, InceptionV3 [47], with the encoder component of an unsupervised time series embedding model [48]. After the embedding model is trained for each dataset separately, it infers latent space encodings for a batch of real and synthetic samples, respectively. This is followed by computing the Fréchet distance [49] between these two subsets. Figure 7 depicts the essential idea. Unfortunately, the authors do not provide details on the training procedure for the embedding model. In any case, the lower the score, the closer the two distributions presumably are and, therefore, the better the synthesis, with 0 being the best [40].

Correlation structure. This measure compares the correlations found in multivariate time series of the real dataset to those of the synthetic dataset. Initially, Remlinger et al. [50] described it as the “term-by-term mean squared error (MSE) between empirical correlation from reference samples on one side and from generated samples on the other side”. However, this is very vague, as it remains unclear which correlation is used, how it is applied, and how the MSE are aggregated. While Boursin et al. [51] propose covariance matrix for correlation and average as aggregation, it still does not specify the arrangement of input vectors for the covariance from the three dimensions samples, time steps, and channels. Hence, as code is unavailable as well, a definition cannot be given here.

Coverage. Naeem et al. [52] proposed this measure to improve upon the recall measure by a change of perspective from regions around synthetic samples to such around real samples as follows. The measure is calculated via the indicator function \({\textbf {1}}\{ \cdot \}\) on sample level, which is then aggregated over all real samples \(x \in D_r\). The function returns 1 if there exists a synthetic sample \(\hat{x}\) within a k-dimensional ball B around the real sample, else 0. Its radius is given by the distance to the K th-nearest neighbor (dNN\(_K\)) in \(D_r\). An example is given in Fig. 8. Formally, coverage C is defined byLike recall, coverage falls into the range [0, 1] where 1 is best and requires an embedding \(f: \mathcal {X} \rightarrow \mathbb {R}^k\) into Euclidean space. The measure is not to be confused with the category of the same name.

Data-copying test (C\(_T\) ) checks a particular type of overfitting behavior of generator \(g_\theta\) to the real data [53]. Data copying refers to the tendency to reproduce minimal variations of a subset of the data instead of covering the entire true data distribution. In this regard, it differs from over-representation of a certain data region. In preparation, they split the real dataset into a train set, which may be presented to the generator, and a held-out test set for evaluation only, \(D_r = (D_r^{train}, D_r^{test})\). Additionally, they sample the generator to create a synthetic dataset \(D_g\). Assuming that overfitting manifests itself in synthetic samples that are generally too close to training data, the measure employs a hypothesis test on the average distance between the train and test datasets, respectively, the train and synthetic datasets. \(H_0\) suggests that these are approximately even. To improve regional awareness, the authors apply the test on each set of a data space partition. Without further details, this finally yields a measure \(C_T(D_r^{train}, D_r^{test}, D_g)\). The only requirement is a distance function on the data points. For time series, this may be DTW, for instance. In principle, one can use embeddings for data spaces that do not support an adequate distance calculation. Furthermore, it holds \(C_T \in \mathbb {R}\), where 0 is optimal, \(C_t \ll 0\) signals data copying, and \(C_t \gg 0\) implies model underfitting.

Density is intended to improve the precision measure by putting less weight on outliers in the real data. Naeem et al. [52] calculate the measure via indicator function \({\textbf {1}}\{ \cdot \}\) on sample level, which is then aggregated over all real samples x and finally synthetic samples \(\hat{x}\). The function returns 1 if the synthetic sample is within a k-dimensional ball B around the real sample with dNN\(_K\) as radius, else 0. An example is given in Fig. 8. In summary, one obtains density D byUnlike precision, density may take values beyond 1, \(D \in \mathbb {R}_{\ge 0}\). However, it still holds that higher is better and an embedding \(f: \mathcal {X} \rightarrow \mathbb {R}^k\) is required.

Dependence scores subsume two scores designed to compare the dependence properties of the real and synthetic time series [54, 55]. Note, however, that both of them are only applicable to an individual, univariate real time series \(X \in \mathbb {R}^l\) and a set \(D_g\) of its syntheses. First, denote the autocorrelation \(\text {Corr}: \mathbb {R}^l \times \mathbb {R}^{l} \rightarrow [-1,1]\) of the time lag \(\tau \le l\) between the current time step t and a previous step \(t - \tau\) of time series X for all possible t. Formally, they define the autocorrelation function (ACF) asBoth scores build on these intra-time series correlations. The first variant, \(s_\text {ACF}\), uses the ACF as defined above, the second one, \(s_\text {LE}\), modifies the function slightly by computing the correlations between the squared lagged times series and the time series itself, that is, \(\text {Corr}(X_{t-1}^2, X_t), \dots , \text {Corr}(X_{t-\tau }^2, X_t)\). Hence, one obtains for the ACF variant the definitionThe formulation of \(s_\text {LE}\) is analogous using the modified autocorrelation function (\(s_\text {ACF}, s_\text {LE} \in \mathbb {R}_{\ge 0}\), where smaller is better).

Discriminative score is based on the performance of a binary classifier on a combined synthetic-real dataset [26]. First, each sample from real and generated data is labeled either “real” or “synthetic”, depending on where it was taken from. A dataset with two classes is created from these labeled samples and split into train and test sets again. Then, the authors train a simple two-layer long short-term memory (LSTM) network in standard supervised fashion to classify the merged dataset. They report its accuracy on the held-out test set minus 0.5 as discriminative score. Hence, it ranges from an optimal 0.0 to a worst 0.5. We depict this procedure in Fig. 9. An embedding is implicitly given by adjusting the classifier’s input layers for differently shaped input. An application to labeled data is impractical.

Distribution of reconstruction errors (DRE). Based on the nearest neighbor approach, this measure is intended to detect a generator just memorizing and reproducing noisy versions of its training data. While others do this by calculating the dNN\(_K\) (for \(K=1\)) for all synthetic samples, Esteban et al. [56] suggest a supposedly less computationally expensive approach. The idea is to put the actual \(D_g\) aside and explicitly generate the nearest synthetic neighbor for all \(x \in D_r\), and then test if the neighbors of the train set samples of \(D_r\) are systematically closer than those of held-out test samples \(D_r^{test}\). This is implemented by minimizing the reconstruction error \(\mathcal {L}_{r(x)}\) between x and a generated neighbor, given bywhere \(K: \mathcal {X} \times \mathcal {X} \rightarrow \mathbb {R}\) is the kernel function used in MMD. Moreover, \(g_\theta\) is the (potentially) parameterized generator function and we have \(z \in Z\) as its input. However, finding z such that \(g_\theta (z)\) is the approximate nearest neighbor still requires performing this optimization to approximate convergence for each x. Only then, given these \(\hat{x} = g_\theta (z)\), the authors can compare the results for \(D_r^{train}\), which \(g_\theta\) was trained on, with \(D_r^{test}\), which serves as a baseline. Namely, they test if the distribution of the reconstruction errors over \(D_r^{train}\) is significantly different from that over \(D_r^{test}\). If that is the case and the average reconstruction error is lower for the train than the test set, one can be confident that the generator memorized the data. In order to test the hypothesis of divergence, they employ the Kolmogorov-Smirnov two-sample test with a predefined significance level. A produced p-value below this level supports the hypothesis, while a good, generalizing \(g_\theta\) should fail the test.

Distribution visualization. With this term, we refer to a group of seven evaluation measures that employ some data transformation, feature extraction, or representation learning technique to map the data into a low-dimensional space. The image of both datasets in this space can then be visualized and inspected by a human judge. Hence, this is not about the visual assessment of individual time series but the dataset collectively. Usually, the synthesis is considered successful if the arrangement of real sample images matches that of synthetic samples. If they diverge strongly in shape or sample density, the generator performed poorly. The most prominent examples of such mappings are principal component analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) [26]. For convenience, we depict such a visualization on the right in Fig. 10. Without going into further detail, we just mention other measures here: QQ-plots and scatter plots are used in [57] and [45], respectively, to visually compare the similarity of sets of geographic trajectories. Pastor-Serrano et al. [58] visualize the latent space distribution of their autoencoder model. Wang et al. [39] propose another approach called dimension-wise probability, which is similar to [59]. Both Pan et al. [60] and Xu et al. [61] evaluate their methods by computing and visualizing probability density functions (PDFs) of marginal distributions.

Distributional metric. The measure proposed in [55] compares real data distribution P and synthetic Q via their respective empirical probability density functions (PDFs) \(f_P\) and \(f_Q\). These are determined using a binning approach. Let d be the dimensionality of the time series and \(0 \le c < d\) be a channel. Further, let \(\mathcal {B}_c = \{ B_c^1, \dots , B_c^n \}\) be a binning of the real time series dataset \(D_r\) for channel c, such that \(\forall B \in \mathcal {B}_c, X \in D :\left| \{ {\textbf{x}_t} \in X \mid {x_t^c} \in B \}\right| \approx 20\) for suitable n. Naturally, this requires the time series to be sufficiently long to define a reasonably accurate density function over the binning. The construction of \(\mathcal {B}\) is not further specified. Instead, they define the empirical PDF \(f_c: \mathcal {B}_c \rightarrow R_{\ge 0}, B_c \mapsto |B_c|\) for each c separately. The authors follow this procedure for both real dataset \(D_r\) and its synthesis \(D_g\), which gives them functions \(f_c^r\) for P and \(f_c^g\) for Q, respectively. However, they reuse the bins defined for \(D_r\) on \(D_g\). This gives them a measure \(M_{\text {epdf}}\) for the absolute difference of all these pairwise empirical distributions in the form ofIt holds that \(M_{\text {epdf}} \in \mathbb {R}_{\ge 0}\), where lower is better.

Distributional scores. The term refers to two closely related measures to capture the propensity of the generator to synthesize extremal values [55]. However, it is defined in a way that allows direct application to real time series \(D_r\) with \(|D_r| = 1\), that is, the comparison of (an arbitrary number of) synthetic time series to a single original, for instance when the latter is extremely long, while each synthesis is relatively short. Let \(f \in \{ \text {skew}, \text {kurtosis} \}\) denote the higher moment used in the measure. The remainder of their calculation is identical. For real time series \(X \in D_r\) and synthetic dataset \(D_g\), define the respective measure by\(X_c, \hat{X}_c\) denotes the cth channel of the respective time series. \(\text {DS}_f \in \mathbb {R}_{\ge 0}\), lower is better.

Duality gap. Based on concepts from game theory, duality gap is a distribution similarity measure for guiding and evaluating GAN models. Although data type-agnostic, it relies on the presence of a generator and discriminator positioned as opponents in a zero-sum game. In this context, duality gap measures the sub-optimality of both entities’ performance compared to an equilibrium, a game state in which no entity can increase its reward unless the opponent behavior changes as well. The game objective, which measures their performance, was left generic by the authors. For instance, it might be the accuracy of a separate binary classifier trained on real and generated data. However, this also means that researchers need to agree on a universal classifier architecture to make results comparable between publications. Figure 11 contains the specifics. The measure is always a non-negative real number, 0 is optimal. Note that duality gap is tailored towards GANs and primarily indicates a model’s convergence or divergence, not the quality of generated samples [62]. Recently, Sidheekh et al. [63, 64] proposed two variants called perturbed and proximal duality gap, which are more accurate than the plain version, especially in cases where the two-player game need not converge to a Nash equilibrium for the generator to model P.

Feature-based correlation analysis. In order to analyze the correlation structure between the channels of individual time series, Seyfi et al. [42] utilize a feature extraction tool called pycatch22 [65]. Specifically, it computes 22 numerical features for each channel of all multivariate time series in both \(D_r\) and \(D_g\), which are expected to be of equal length and dimensionality \(d > 1\). Afterwards, for each pair of channels \((X^i,X^j)\) with \(1 \le i < j \le d\), a \(22 \times 22\)-matrix of Pearson correlations between the extracted features is computed for real and synthetic dataset each. This is across the sample dimension. Now, for fixed (i, j), the matrices for real and synthetic data are compared with mean absolute error (MAE), MSE, Kendall’s \(\tau\), Spearman’s rank correlation, as well as Frobenius norm. The final score is the quintuple of these statistics averaged across all channel pairs (i, j), where lower is better.

FID was proposed in [66] as evaluation measure for synthetic images. It utilizes an inception model for extracting features from the samples. The model learns a Gaussian distribution with mean m and covariance matrix C for the real data \((m_P, C_P)\) and synthetic \((m_Q, C_Q)\). To these “inceptions”, the Fréchet distance \(d_f\) is applied:The resulting score measures the similarity between the two Gaussians and therefore, by approximation, between the real and synthetic data distribution. Generally, it is FID \(\in \mathbb {R}_0^+\), where lower is better [67].

Hedging effectiveness. In the financial world, hedging refers to the effort to reduce potential losses at the expense of gains achieved in transactions or speculation. In order to evaluate different time series generators with respect to their ability to produce useful synthetic option prices, Boursin et al. [51] propose hedging effectiveness. The measure is based on the accuracy of a neural network that was taught hedging strategies using the synthesized option prices, compared to the potential losses without hedging and the help of synthetic data. The score falls in the range [0, 1], where greater is better.

Improved precision \((p_i)\) builds on the notion of precision measures for two distributions introduced in [68]. This alternative definition, however, is more straightforward and based on the binary question: “Does a given synthetic sample lie in the support of P?”. This idea is illustrated on the left in Fig. 12. In this regard, the support is estimated using the union of regions around all samples of the respective dataset. Practically, the authors utilize k-dimensional balls B around real samples. Furthermore, the question gives rise to the indicator function \({\textbf {1}}(\cdot )\) seen in the definition of coverage and density before. The same is true for the distance to the Kth nearest neighbor, which serves as B’s radius. Although Sajjadi et al. [68] have chosen another syntax, we stick to the ball notation for consistency. For real dataset \(D_r\) and synthetic equivalent \(D_g\), we define \((p_i)\) asThe measure is calculated in the k-dimensional metric space, requiring a prior embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\). A neighborhood size of \(K=3\) and dataset sizes \(\left| D_r \right| , |D_g| \ge 50 \, 000\) are recommended for reliable scores, as indicated by tests on image data performed in [69].

Improved recall \((r_i)\). As the name suggests, this is the compliment measure to improved precision. In order to approximate the fraction of the real data distribution captured by the generator, Kynkäänniemi et al. [69] reduce the problem to an aggregate of binary decisions on the sample level, as seen before. Hence, the question proposed in this case is “Does a given real sample lie in the support of Q?”. This idea is illustrated in Fig. 12 on the right. We get a definition of improved recall exactly mirroring Eq. 13:Analogously, the measure is calculated in the k-dimensional, real-valued feature space, requiring a prior embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\) [69]. The recommendations \(K = 3\), \(\left| D_r \right| , |D_g| \ge 50 \, 000\) hold.

Intra-class distance (ICD). Given an additional sample-level distance d, this measure represents the average similarity of the samples within a dataset, in this case applied to a subset G of the generated time series \(D_g\), \(|G|=10\).The higher the distance the better, that is, more diverse the synthesis. Leznik et al. [9] use the ED for d.

JS divergence on marginals. Generally speaking, JS divergence measures the dissimilarity between two (empirical) distributions. Naturally, one can employ it on data distribution P and synthetic distribution Q. However, for joint distributions P, Q over a high dimensional space, this is computationally infeasible. Therefore, most approaches use an embedding step beforehand to map the data points into a lower dimensional space. Another way is to work with the marginal distributions of P, Q instead, at least in cases where they can be properly identified. An example where this works well is given in a paper on synthesizing human mobility trajectories by Ouyang et al. [57]. Therein, marginals are inferred from a semantic point of view. They define four distributions less complex than P, among them the visiting probability over all locations in the geographic area of interest. Each of the four distributions can be estimated using \(D_r\) for the real data and \(D_g\) for the synthetic. Moreover, the authors split \(D_r\) into two and compute the JS divergence on a held-out test set, which the generator has not seen during training, and the synthetic dataset. As a result, we get a divergence value for each marginal defined and thereby compare different generators on a statistical, yet informative level.

Length histogram. With this term, we title a measure for time series datasets and generators respectively containing and producing sequences of variable length. The intention is that the distribution of the lengths in both real and synthetic datasets should match. As the underlying space of realizations is finite and rather dense in practice, the approach chosen here is simply to compute the histogram of time series lengths in both datasets. By superimposing them in one figure, one can visually compare the histograms and find differences [34].

Manifold topology divergence (MTop-Div). Similar to JS divergence or WD, this measure represents the discrepancy between two distributions, P and Q. Here, these are data and model distribution, respectively. MTop-Div is different in that it is topology-based and views the real data and synthesis as manifolds, on which their respective datasets are point clouds. These manifolds are estimated using a simplicial complex, a concept of topology. To estimate the similarity between the two manifolds, Barannikov et al. [70] propose a mathematical tool called Cross-Barcode, which quantifies the evolution of topological features over multiple scales. The details require quite extensive mathematical explanations. Hence, we omit them here and refer to [70] instead. For the same reason, we provide a high-level depiction of the measure in Fig. 13. We know that \(MTop-Div \in \mathbb {R}_{\ge 0}\), where smaller is better. Since the measure operates on the k-dimensional real space, a data type-dependent embedding \(f: \mathcal {X} \rightarrow \mathbb {R}^k\) is required.

Marginal metrics. This is a combination of three classical statistics used to roughly compare the marginal distribution of each time step in the real dataset to its counterpart in the synthetic dataset [50]. Namely, these are the average, 95th percentile, and 5th percentile, which we refer to as \(s_1, s_2, s_3\), respectively, below. With that, define marginal metrics as the triplewhereAbove, \(X^c_t\) denotes the value in channel c at step t from dataset \(D_\delta\).

Maximum mean discrepancy (MMD). For a class \(\mathcal {F}\) of functions \(f: \mathcal {X} \rightarrow \mathbb {R}\), MMD is defined asThis statistic computes the difference between the mean function values of the real distribution on the one hand, and the model distribution on the other and returns it as a measure for dissimilarity [71]. Later, Esteban et al. [56] proposed a new approximation for P and Q denoted \(\text {MMD}_\approx\) using a kernel function \(K: \mathcal {X} \times \mathcal {X} \rightarrow \mathbb {R}\) applied to \(D_r, D_g\). This essentially boils down to learning an embedding and a dissimilarity function for time series in one. A lower score is better.

Maximum real to synthetic similarity (Max-RTS). This is meant to be an extreme-case similarity measure, determining the cosine similarity between the real and synthetic sample closest to each other [46]. A visual aid is provided in Fig. 14. The intuition is that individual synthetic samples must not get too close to real ones, in order to prevent data leakage into the synthesis and ensure generalization. Let \(D_r, D_g\) be the real and synthetic datasets, respectively. Then, Max-RTS is given bySince the cosine similarity is defined on vectors, an embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\) is required. We have Max-RTS \(\in [-1,1]\) and theoretically, the lower the better.

Membership inference attack (MIA). In general, this term refers to the process of determining whether a specific sample from the data distribution is present in the training set used for learning a given ML black-box model [72]. One way to approach this is by training an adversarial model to differentiate the behavior of the target model on inputs from the train set versus other samples. In other words, this can be used to measure the information leakage on sample membership by looking at the target model’s outputs. In this case, the goal is to infer (parts of) \(D_R\) given the synthesized data \(D_g\), as depicted in Fig. 15. [73] uses this attack as a means to detect presence disclosure. The work mentions a range of real samples, a threshold for the mean ED between “all” samples, and the number of generated samples being utilized. However, the work fails to provide more details on how the attack is conducted. If we were to guess, the author proposes to calculate the mean ED on \(D_r\) and filter those real samples that are closer to any \(\hat{x} \in D_g\) than the threshold allows. One may calculate a score in [0, 1] from this fraction of real samples. However, one needs access to \(D_r\) for this, which does not conform with the strict black-box assumption. Kulkarni et al. [45] are more specific and reference the location-privacy and mobility meter in [74]. This approach implements both a location-sequence attack and a membership inference attack. The former determines the accuracy with which an attacker can reconstruct trajectories, that is, time series capturing locations of an entity over time, in the real dataset. The latter tries to infer the identity of the entity contributing to a trajectory. However, these are domain-specific tests.

Memorization-informed Fréchet inception distance (MiFID). As the name suggests, this measure is an extension to the “classical” FID, addressing the issue of memorizing and regurgitating parts of the real dataset instead of producing novel samples. Hence, for real dataset \(D_r\) and synthesis \(D_g\), MiFID is given by the following combination of a memorization penalty \(m_\tau\) and the FID itself:To construct \(m_\tau\), Bai et al. [75] proposed a memorization distance s between the real and synthetic dataset. First, they fall back to the similarity between individual samples as per the cosine similarity, followed by a minimum over the real data and averaging across all synthetic samples. Formally, this is described byUsing this function, define \(m_\tau\) for an \(\epsilon > 0\) as\(\tau\) serves as a threshold above which similarity is considered overfitting and thus penalized inversely with respect to the memorization distance. Below this threshold, the FID is not modified. Choosing suitable values for parameters \(\tau\) and \(\epsilon\) is up to the user. For MiFID, a lower value is better. Analogous to FID, this extension operates on the real-valued vector space and therefore needs an embedding \(f: \mathcal {X} \rightarrow \mathbb {R}^k\) for all other data spaces.

Minimax loss is a derivative of the duality gap measure but with the motivation to assess synthesis quality and generalize to all kinds of generation methods. Based on the same idea of an equilibrium of two entities in a min-max game, the implementation rather comes across as merely evaluating the generated dataset through another, discriminative model very much like discriminative score does. More precisely, Grnarova et al. [62] suggest a split of real and synthetic data into three subsets each, one for training G, one for finding the “worst case discriminator” \(D_{worst}\), and one for determining the utility M for G. In this regard, \(D_{worst}\) is the best classifier network obtainable with a predefined architecture, that is, an optimized model. The abstract function M is given by the classification loss of \(D_{worst}\) on the third pair of real and synthetic subsets. The higher the loss, or inversely, the lower the classification accuracy, the better G performs. We assume that the classification model implicitly computes an embedding. The application to labeled datasets is impractical. CAS is more suited to this task, according to [62].

Multi-sequence aggregate similarity (MSAS) is a similarity measure for normalized time series built around an additional statistic \(f:\mathbb {R}^{l} \rightarrow \mathbb {R}\), which maps a column onto a scalar value [76]. Examples include length, mean, and standard deviation. Furthermore, it is restricted in its application by the assumption of a generator of the form \(g_\theta : \mathbb {R}^{l \times d} \rightarrow \mathbb {R}^{m \times d}\), where the input is a seed sequence of length l and d channels and the output one with length m and the same dimensionality. Hence, the algorithm can iterate the pairs \((X, \hat{X})\) of real time series and one of its syntheses, each channel separately. In each iteration, one computes \(f(X), f(\hat{X})\). In order to compare feature distributions, one applies the 2-sample Kolmogorov-Smirnov test on the set of features computed on the real samples and the one computed on the synthetic samples. This statistical test returns the probability that the two sets are taken from the same distribution. The scores for the columns are averaged to finally arrive at MSAS (\(MSAS \in [0,1]\), where higher is better).

Neural network divergence (NND) is a measure originating from the evaluation of image synthesis, but is at its core data type-agnostic [77]. Like WD or FID, it tries to estimate the discrepancy D(P, Q) of real and synthetic distributions using finite sets of samples \(D_r\) and \(D_g\), respectively. The idea is to use the loss of a neural network trained on discriminating \(D_r\) and \(D_g\) as empirical proxies for P and Q, respectively. The model architecture must be standardized across the generators for evaluation. In order to apply this measure, both real and synthetic data need to be split into train and test sets, \(D_r = (D_r^{train}, D_r^{test}), D_g = (D_g^{train}, D_g^{test})\). The generator may only use \(D_r^{train}\). Now, the neural network is trained on a supervised classification task, differentiating the two train sets. The classification loss during inference on the test sets is then used as an empirical discrepancy measure estimating D(P, Q). To foster generalization of the synthesis even for a rather small \(D_r^{test}\), Gulrajani et al. [77] propose to use proportionally bigger generated datasets, \(D_g^{train} \gg D_r^{train}\), \(D_g^{test} \gg D_r^{test}\) in order to detect an overfitting generator. This generalization aspect sets NND apart from prior statistical discrepancy measures like FID (\(NND \in \mathbb {R}_{\ge 0}\), and the smaller the better).

Number of statistically different bins (NDB) measures the degree to which a generator over-emphasizes particular modes of the data distribution within its synthesis, neglecting other, less prevalent regions [78]. This is commonly referred to as “mode collapse” and a major concern among GANs. The idea is to employ a two-sample hypothesis test assuming that in every region of the data space, real distributions P and Q should be equal with respect to a significance level of \(s = 0.05\). In preparation for this, the k-means clustering algorithm is applied to the real \(D_r\) in order to create a partition \(\Pi\), creating k subsets referred to as “bins”. The synthetic \(D_g\) is split into k bins in a similar fashion using the cluster centers of \(\Pi\). Alternatively, one can think of this binning as a discrete probability distribution. Given a sufficient number of samples falling into each bin, the distribution of real samples in bin \(\pi\), \(P_\pi\), can be compared to that of synthetic samples, \(Q_\pi\), using said hypothesis test. We get a score \(Z_\pi\) and can compute the number of bins, for which the score indicates a significant deviation of Q from P aswhere \({\textbf {1}}\{ \cdot \}\) is the indicator function. For variable k, it is \(NDB/k \in [0,1]\), where lower is better. Figure 16 depicts a high-level summary. Besides, one can also compare the relative sizes of corresponding bins \(P_\pi\) and \(Q_\pi\). The binning requires a distance function which may directly be applied in the data space (e.g., DTW for time series), or the feature space after an embedding.

NDB over-representation and under-representation (NDB-over/under). This measure is an adaption of the NDB measure in [78], intended as a complement to data-copying test [53]. The latter is unable to measure over- and under-representation of individual regions in the data space caused by an ill-fit generator, which can be countered with the help of NDB. To this end, the data space – or feature space after an embedding – is partitioned into regions. This time, the null hypothesis is the assumption that data distribution P and model distribution Q are equal in every region \(\pi\), \(H_0: P(\pi ) - Q(\pi ) = 0\). By applying a statistical test on \(H_0\), a region-specific score \(Z_\pi\) is calculated. By comparing each \(Z_\pi\) with a significance level s, one can determine the number NDB-under (resp. NDB-over) of regions under-represented (resp. over-represented) by the generator as followsFor short, we call this pair \(\mathrm{NDB-over/under} \in \mathbb {N}^2\) with (0, 0) being optimal.

Outgoing nearest neighbor distance (ONND). Let \(G \subseteq D_g\) be a random subset of the synthetic dataset and \(d \in \{ \textrm{ED,DTW} \}\) be a distance measure. For each real time series X, the ONND is given byBoth Arnout et al. [79] and Leznik et al. [9] use PCA as an embedding before employing the function, but this is optional. The smaller the distance, the better. To get a scalar score for \(D_r\), one can take the mean over all X, whereas in reviewed works, it is used during training.

Pairwise Pearson correlation (PPC). This measure aims to facilitate the visual comparison of real and synthetic time series datasets via heat maps of their internal pairwise correlation [5, 39]. Let D be a time series dataset with time series \(X \in \mathbb {R}^{l \times d}\) of length l and dimension d. Further, let \(X' \in \mathbb {R}^{l \cdot d}\) denote the concatenation of all channels of X. Now, they construct \(l \cdot d\) ordered sets of points across D, such that set \(S_i\) contains the value of \(X'\) at position i for the entire dataset, that is, \(S_i :=[ X'_i \mid X \in D ]\). At this point, one can compute the set C of pairwise correlations between these sets withIn this case, these are not the correlations between time series, but across sets of points, one from each times series in the dataset. They do the above for both real data \(D_r\) and synthetic \(D_g\) and get \(C_r\) and \(C_g\), respectively. Finally, they can arrange each one in an \(ld \times ld\) grid, visualize both using heat maps and compare the two. An examples is provided in Fig. 17.

Precision and recall for distributions (PRD). This measure does not compute a single score but rather a set PRD(P, Q) of precision-recall pairs between data distribution P and model distribution Q [68]. PRD is interpreted as a precision-recall curve using the points of the set farthest from the origin. Some examples are provided in Fig. 18. Although its computation is different from the formal definition and only an approximation of this curve is needed in practice, we try to convey the intuition here and thus stick to the definition. Intuitively, the precision component measures the probability that a random synthetic sample falls within the support of P and vice versa for recall. We say Q has precision p and recall \(r \in (0,1]\), with respect to P if there exist distributions \(\mu , \nu _P, \nu _Q\) such thatTherein, \(\mu\) is the shared probability mass, \(\nu _P\) denotes the remaining mass of P, and symmetrically, \(\nu _Q\) is the remaining mass of Q, all scaled accordingly. Now, PRD(P, Q) consists exactly of the pairs (p, r) satisfying Eq. 27. Both measure components fall into (0, 1] where higher is better. However, the approximation algorithm proposed by Sajjadi et al. [68] only works on discrete, finite feature spaces \(\Omega\). Hence, they use a histogram-based embedding \(f: \mathcal {X} \rightarrow \Omega\). Later, Simon et al. [80] simplified the definition and conceived a new algorithm exploiting type I and II error rates of a binary classifier trained on the merged real and synthetic datasets. As a result, the distributions can be compared in a real-valued, potentially infinite feature space. One can either use an embedding of the form \(f: \mathcal {X} \rightarrow \mathbb {R}^k\) or adapt the classifier to accommodate the input space instead.

Predictive score. Specifically proposed for time series data, predictive score measures the usefulness of the synthetic dataset for training one-step predictions. This task was chosen to stress the requirement of generative models to capture temporal correlations. Yoon et al. [26] suggest a simple LSTM-based model to perform the predictions. Training is conducted independently from the generator on the synthetic dataset, while the evaluation takes place on the real dataset by simply calculating the average prediction error across all time series. This averaged error – they use MAE – serves as a relative score for comparing the usefulness of different models. However, it is no absolute measure of quality. The predictive score lies in \(\mathbb {R}_{\ge 0}\), where lower is better.

Relative MMD test. Bounliphone et al. [81] proposed this general measure for data synthesis evaluation, meaning it is not specific to any data type. Relative MMD is a test of pairwise relative similarity between two synthetic datasets, \(D_g\) and \(D_g'\), on one side and a real dataset, \(D_r\), on the other. A hypothesis test based on MMD, which acts as the distance between embeddings of each distribution in a reproducing kernel Hilbert space, asserts whether \(D_g\) is significantly closer to \(D_r\). Hence, the resulting score is binary, stating which of the synthetic datasets is closer to the real one but without any absolute or more fine-grained information. Furthermore, it requires some embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\), which may simply be the concatenation of channels in the case of time series. Lastly, it assumes some real data to be held back from generator training.

Spatial Correlation. Let \(R \subseteq D_r\), \(G \subseteq D_g\) for a real and generated dataset [9]. Using these subsets, spatial correlation estimates the correlation between the channels of multivariate time series in the generated data compared to its original. For all X in both sets, they calculate all pairwise Pearson correlations PCC\((X^i, X^j)\) between channels i, j of X, \(i \not = j\). This gives us \(k = d(d-1)/2\) coefficients for dimensionality d, that is, the number of channels, which they average across each dataset. These k averages for real and synthetic datasets are now separately aggregated using the squared difference. Smaller is better. Xu et al. [61] also compare the correlation coefficient between channels. Specifically, they claim to take the “sum of the absolute difference of the correlation coefficients between channels averaged over time”. It remains unclear, however, how averaging over time can be conducted when the time dimension is needed for the correlation between channels. Additionally, the absolute difference can be applied between a coefficient from a real sample versus one from a synthetic sample, two samples from within one dataset, or the \(d \cdot d\) coefficients of one sample. In any case, their formulation is unclear as the explanation is limited to this one sentence. Besides, a smaller score is still better. Jarret et al. [10] propose this measure later as x-Corr score.

Synthetic to synthetic similarity (STS) measures the intra-class similarity between the samples synthesized by a conditional generator, that is, one that produces for each data point an accompanying label [46]. Let \(D_g = \{ (x_i,l_i) \}_i\) be a labeled synthetic dataset. The similarity score is independently calculated for each class and finally combined into a vector, one position for each class. Hence, for class c with size \(1< n_c < |D_g|\), they choose a complexity factor \(\gamma \in (0,1]\), and uniformly randomly sample \(\gamma \cdot n_c\) data points from c. Denote this set by \(S_c\). Now, they calculate the ACS between each \((x,c) \in S_c\) and five other random points \((x_1, c), \dots , (x_5, c) \in D_g\) in that class as followsSince one needs vectors for the cosine similarity, we can either apply the measure directly in case of univariate time series, for instance, or require an embedding \(f: \mathcal {X} \rightarrow \mathbb {R}^k\). The choice of \(\gamma\) depends on the computational resources and time available, and the desired accuracy of the score. During training, a low value seems appropriate, whereas final evaluation should warrant a larger proportion of samples. Since the goal is to ensure intra-class diversity and prevent mode collapse, values close to 1 should be avoided. However, Norgaard et al. [46] do not comment on which values of the codomain [0, 1] are desirable or acceptable. This is problematic since datasets are often scaled to [0, 1], heavily limiting the angle between data vectors.

Temporal correlation. Let \(R \subseteq D_r\), \(G \subseteq D_g\) for a real and generated dataset, respectively [9]. Using these subsets, temporal correlation estimates the correlation between time steps in the generated data compared to the original. Per channel, sample, and dataset, the discrete Fourier transform (DFT) between the time steps is calculated and scaled. Then, the authors extract the k largest values (peaks), where k is tuned manually on a per-dataset basis, and determine their distance. The aggregation method is the same as for ApEn. Smaller is better.

topN locations. Intended for use on mobility trajectories, that is, time series describing the location of some entity over time, this measure can be easily generalized to other use cases [45, 57]. The only condition is that we can identify a property or set of properties we are particularly interested in, and on that, each time series can be practically tested. Staying in the context of trajectories, this can be the time spent at or the number of visits to each location from a discrete, finite set of possible locations. We can view this as an embedding \(f: \mathbb {R}^{l \times d} \rightarrow \Omega\), where \(\Omega\) is the set of possible values for the property. Now, one can compute \(\omega _X = f(X), \omega _{\hat{X}} = f(\hat{X}) \; \forall X \in D_r, \hat{X} \in D_g\), where \(D_r\) and \(D_g\) are real and synthetic datasets, respectively. This opens up another distribution over the real and synthetic data, which the authors exploit by determining their discrepancy statistically or visually through plots. Assuming each \(\omega\) stands for a vector holding the number of visits of the time series to location l at position l, they take the sum over all vectors and select the N counts for the most frequently visited locations viaFinally, one can apply a discrepancy measure or plot the N values for both and judge their similarity.

Train on real, test on synthetic (TRTS) was proposed a few years ago by Esteban et al. [56]. Given a supervised task T applicable to the real data \(D_r\), the parameterized TRTS(T) determines the generator’s ability to mimic essential features of \(D_r\), such that a solution for the real data is also a solution for the synthetic one under T. Common choices for T include classification and prediction. In preparation for the measure, they split the real dataset into a train and test pair, \(D_r = (D_r^{train}, D_r^{test})\). The generator is trained on \(D_r^{train}\) following standard practice, as is the model M used to solve the task. Then, M is evaluated on both \(D_g\) and \(D_r^{test}\), which gives them two scores \(s_g, s_r\) for M, computed using an evaluation measure appropriate for T and M. Without loss of generality, we assume higher is better for the sake of simplicity. If \(s_g \approx s_r\) or – even better – \(s_g > s_r\), then one can safely assume that the samples in the generated test set encode features distinctive and similar enough to those in the real test set. If, however, \(s_g \ll s_r\), the generated samples deviate too much from those M has seen before in \(D_r^{train}\). A typical constellation is to use classification as T, a deep neural network as M, and a score like area under precision-recall curve (AUPRC) or AUROC. Strictly speaking, the need for an embedding depends on the choice of T, whereas typically, M provides its own integrated embedding. The applicability to classes in the datasets depends on T, for instance, classification requires labeled \(D_r\) and \(D_g\).

Train on synthetic, test on real (TSTR). With the above TRTS in mind, we can now consider the reverse case [56]. Similarly, given a supervised task T, the parameterized measure TSTR(T) determines the usefulness of a synthetic dataset \(D_g\) for T. Common choices for T include classification and prediction. In preparation for the measure, they split the real dataset into a train and test pair, \(D_r = (D_r^{train}, D_r^{test})\). The generator is trained on \(D_r^{train}\) following standard practice, whereas the ML algorithm M used to solve the task is independently trained on \(D_r^{train}\) and \(D_g\), yielding models \(M_1\) and \(M_2\), respectively. Then, both are evaluated on \(D_r^{test}\), which gives us two performance indicators \(s_1\) for \(M_1\) and \(s_2\) for \(M_2\) to compare. We assume higher is better for the underlying measure. If \(s_1 \approx s_2\) or \(s_1 < s_2\), then the synthesized data demonstrates high practical value with respect to T and can presumably replace \(D_r\) without a compromising model quality. Otherwise, a user knows that \(D_g\) lacks essential properties required for M to learn as well as possible with real data. To give one example tailored towards time series, T might be the prediction of the last steps in a time series, M an LSTM-based neural network, and s the average root mean squared error [38]. An embedding is usually not necessary, as most M provide their own integrated embedding. The applicability to classes depends on T, for instance, time series prediction requires no dedicated labels.

WD is defined aswhere \(\Pi (P,Q)\) is the set of all joint distributions \(\gamma (x,y)\) whose marginals are P and Q, respectively. P is the marginal on x, that is, \(\int _\mathcal {X} \gamma (x,y)dy = P(x)\), and similarly Q the marginal for y. We implicitly assume a metric space, for instance, \(\mathcal {X} = \mathbb {R}^n\). Figuratively speaking, \(\gamma\) is a function that says “how much” of one distribution needs to be moved from point x to point y in the metric space to transform it locally into the other. Then, the WD specifies the “cost” associated with the aggregated shortest distance to travel in order to transform p into q, point by point [82]. For more details and background, see the work of Villani [83]. In the context of time series synthesis, this measure was applied in [54, 84] to determine the discrepancy between real distribution p and synthetic q.

WD on cumulative distribution function (CDF). Lin et al. [34] applied this measure on time series data with values from a range of integers, while it can also be applied to those from the real-valued data domain that allow binning over the time steps. Let \(X \in \mathbb {N}^{l \times d}\) and \(\hat{X} \in \mathbb {N}^{l \times d}\) be real and synthetic uni- or multivariate time series from datasets \(D_r\) and \(D_g\), respectively. For each channel independently, one computes the Wasserstein-1 distance between the overall value distributions in both datasets. More specifically, the authors compute the empirical CDF for function \(f: \mathbb {N} \rightarrow \mathbb {N}\) counting the number of occurrences of some value \(x \in \mathbb {N}\) in all the time series of a set D. Doing this for both \(D_r\) and \(D_g\) gives us two CDFs, for which one can now determine the WD by taking the integrated absolute error between them. Doing this for each channel, this would amount to a vector \(\textbf{v} \in \mathbb {R}_{\ge 0}^d\) of distances. Of course, a visual comparison of their graphs is also possible.

Wavelet coherence score (WCS). Wavelet coherence is one of many ways to measure the correlation between two univariate time series [85]. It is especially useful to analyze non-stationary time series, that is, such with major changes in statistical characteristics over time like a moving mean caused by a trend. Like many other measures, it requires the transformation into a different feature space. Similar to DFT, this is also a frequency domain, with the continuous wavelet transform as mapping. The definition of wavelet coherence is quite complex and its explanation lengthy. We refer to the work of Grinsted et al. [86] for more details. Here, it is applied to a pair of channels at a time, one from a real, one from a synthetic time series. The immediate output of the wavelet coherence computation is a matrix wcoh with shape frequencies \(\times\) time steps. Hence, an intermediate aggregation step is required to obtain a scalar value \(wcoh_s \in \mathbb {R}\) for each pair of channels. To this end, one simply sums over the time steps and then takes the mean over the frequency axis. For entire datasets \(D_r\) (real) and \(D_g\) (synthetic), WCS is simply the average of all \(wcoh_s\) computed between real-synthetic pairs.with higher scores are better.

\(\beta\)-recall is a parameterized measure denoted by \(R_\beta\), where \(\beta \in [0,1]\) is the fraction of synthetic samples assumed to be “typical” for the generator [87]. It represents the fraction of real samples covered by those generator-typical data. More formally, \(R_\beta\) is the probability of a real sample falling within a region \(\mathcal {S}\). Therein, \(\mathcal {S}\) is the minimum volume subset of the support of the synthetic distribution that supports a probability mass of \(\beta\). We illustrate this concept on the left of Fig. 19. Single real samples are assigned a value of 1 if they fall within \(\mathcal {S}\), and otherwise 0. Implementation-wise, this is done by checking if there is at least one synthetic sample near the real sample. Although computed for individual real samples, \(R_\beta\) counts as distribution-level, as an entire synthetic dataset must be given to check this. Ordinary recall is a special case of this measure, which is instead computed for all \(\beta\). The individual scores are arranged as a recall curve and can be interpreted directly or aggregated to a single score called integrated \(R_\beta\) viaAlso, see Fig. 19 right for an example curve. It holds that \(I R_\beta \in [0,1]\), where higher is better. \(\beta\)-recall is data type-agnostic, operates on a fixed-size vector space, and therefore requires an embedding. By default, this is an LSTM-based autoencoder (AE) model.

Below, we provide an introduction to each sample-level measure found. This list is sorted in alphabetical order to foster faster look-up for non-sequential reading. A summary can be found in Table 5.

Authenticity measures the rate at which a generator produces samples that appear novel [87]. The underlying assumption is that the model creates samples randomly through one of two ways: With probability \(p_A\), a new sample is innovated, and with \(1 - p_A\), a memorized real sample with some added noise is presented. For a good model, \(p_A\) is close to one. The measure tries to determine which decision was made through a hypothesis test on the distances between two real samples, and the synthetic and a real sample. We denote authenticity by A, with \(A \in [0,1]\) on a distribution-level and \(A \in \{ 0,1 \}\) for samples, where \(A(\hat{x}) = 1\) classifies a sample as authentic. The generalization to a generated dataset \(D_g\) is given byand thus, the higher the score for the dataset, the more authentic it is. The measure is data type-agnostic, operates on a fixed-size vector space and therefore requires an embedding.

Cointegration tests. Cointegration is a statistical property that might be observed in a multivariate time series or a collection of univariate ones. It is present when all channels involved correlate in the long term, most likely following the same overall trend. These tests are quite popular in the financial world, for instance, for verifying a lasting correlation between the market values of two securities. Popular examples include the Engle-Granger test, which applies only to pairs of channels, and the Johansen test, which can handle arbitrarily many relationships and, therefore, channels. In the context of synthetic time series, Leznik et al. [38] employed it to check if a real time series and its synthesis follow the same trend. This implies that each synthesis has a unique original in the real dataset, which limits the applicability of these tests to generators of the form \(g_\theta : \mathcal {X} \rightarrow \mathcal {X}\). Although they do not mention their aggregation method, it can be assumed that a simple arithmetic mean is adequate.

Elementary statistics. We use the term to collectively refer to an open group of rather simple statistical properties that are defined on arbitrary real-valued data samples or time series in particular. Concretely, we identified the use of moments of different ordinal [59], covariance matrix [5, 61], and ACF [34, 38, 60, 93]. Comparison can be done via manual inspection of a plot or taking the difference between the score on the real and synthetic samples [59, 93]. In principle, these statistics can be easily adapted to the dataset level, that is, by averaging [34].

Feature-based distance. This is an aggregate measure of seven sub-scores, which are all calculated following the same procedure but use a different feature \(f: \mathbb {R}^l \rightarrow \mathbb {R}\) of the time series as embedding [95]. These are three deterministic components of time series (mean, trend, seasonality) and four stochastic features (standard deviation, skewness, kurtosis, lag-1 autocorrelation), all calculated on a per-sample basis. Furthermore, the measure assumes a generator of the form \(g_\theta : \mathbb {R}^l \rightarrow \mathbb {R}^l\), where each synthetic sample is deduced from an old one. Based on a scaled version \(f^s\) of some feature and with a pair \((X, \hat{X} = g_\theta (X))\) of real and synthetic time series, the authors define its sub-score aswhere the aggregate is the set of all seven \(d_f\). The feature-based distance is targeted at and tested on univariate time series.

Fréchet distance was originally introduced in [49] to calculate the similarity of two curves from the ordering and location of their points in \(\mathbb {R}^d\). Much later, Hazra and Byun [94] used it for the evaluation of synthetic time series by interpreting them as point sequences on such curves. This is done directly to a real time series on one, and its synthesis on the other side, implying that this version is applicable to generators of the form \(g_\theta : \mathcal {X} \rightarrow \mathcal {X}\) only. The score is always in \(\mathbb {R}^+\), where lower is better. In the following, we use this version when referring to Fréchet distance, although applications on the distribution level similar to FID and Context-FID are possible as well.

Incoming nearest neighbor distance (INND). Let \(d \in \{ \textrm{ED, DTW} \}\) be a distance measure, and \(R \subseteq D_r\) be a random subset of the real dataset [79]. For each generated time series \(\hat{X}\), the INND is given byThe authors use PCA as an embedding before employing the function, but this is optional. The smaller the distance, the better. To get a scalar score for \(D_g\), one can take the mean over all \(\hat{X}\).

Multivariate, dependent DTW (DTW\(_D\)) is a rather simple modification to the algorithm computing the standard DTW in order to accommodate multivariate time series. Dynamic time warping requires to calculate the distance d between the values at two time steps \(\textbf{x} \in X, \textbf{y} \in Y\), where X, Y are time series and \(\textbf{x},\textbf{y} \in \mathbb {R}^k\). d is commonly the squared Euclidean distance. In the univariate case, this comes down to \(d(\textbf{x},\textbf{y}) = (x_1-y_1)^2\). DTW\(_D\) allows to “warp” across the channels of X and Y by simply using the generalization to the k-dimensional space \(d(\textbf{x},\textbf{y}) = \left| \left| \textbf{x}-\textbf{y} \right| \right| _2^2 = \sum _i (x_i - y_i)^2\) [96]. Brophy [73] used the measure for evaluating a GAN producing multivariate time series. However, he failed to explain the extension of this sample-based measure to real and synthetic datasets \(D_r, D_g\). In cases where each synthesis \(\hat{X} \in D_g\) can be assigned an original \(X \in D_r\) due to the (probabilistic) relationship \(g_\theta (X, z) = \hat{X}\) for conditional input z, they find a straight-forward implementation asFor general \(g_\theta\), however, taking the average DTW\(_D\) for many randomly selected pairs \(X \in D_r, \hat{X} \in D_g\), if not all such pairs, is a reasonable and probable approach.

Nearest neighbor in training (NNT) is similar to INND in its formulation, but opposing in its purpose [34]. Where the INND tries to ensure that each generated sample stays true to the real data \(D_r\) via proximity to a real sample, this measure determines the degree to which the generator is memorizing the data \(D_r^{train} \subseteq D_r\) it was trained on. For a random synthetic sample \(\hat{x} \in D_g\), they select its K nearest neighbors in \(D_r^{train}\) with respect to the Euclidean distance viaLin et al. [34] chose \(K=3\) and repeated the procedure multiple times, if not for all \(\hat{x}\). One example is given in Fig. 20. A quantitative measure can be defined on the mean of the neighbor distances, while a qualitative one is given by visually comparing the plot of \(\hat{x}\) with those of the neighborhood. The Euclidean distance may require an additional embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\) for the data. Finally, the higher the score and visually different the plots, the better.

Pairwise measures. With this term, we refer to a group of measures m characterized by their common form \(m: \mathcal {X} \times \mathcal {X} \rightarrow \mathbb {R}\). They are only applicable to a pair of data objects, typically a real sample \(x \in D_r\) and its synthesis \(\hat{x} \in D_g\). In other words, we imply that each synthesis can be backtracked to a real sample from which it was deduced somehow. Such measures typically only consider direct divergences between the samples. A possible application scenario includes the generation of very few or even a single particularly long time series, where it is clear which original the synthesis is expected to mimic. Specifically, we found the five measures: PCC, root mean square error (RMSE), MAE, percent root mean square error (PRE) [94], and cross-correlation score [55]. An extension to a measure for entire datasets may be given by averaging over each pair to obtain a score or using visualizations such as plots and histograms.

Plausible deniability. Targeting data privacy, this measure is an independent extension to probabilistic generative models of the form \(g_\theta : \mathcal {X} \rightarrow \mathcal {X}\) that allow the efficient calculation or estimation of \(\text {Pr}\{ \hat{x} = g_\theta (x) \}\) for real sample x and its synthesis \(\hat{x}\) [33]. In Fig. 21, we provide a diagram depicting the steps of applying the measure. The idea is to take a high-utility generative model providing the synthetic samples and append a sample-level privacy test, which decides whether to release the sample or draw a new one from the generator. This way, high-quality data can be produced while making sure every synthesis adheres to a privacy policy. The test’s underlying concept is to allow a synthetic sample \(\hat{x}\) if and only if there exists an entire set of real data points \(D_x\) that could have caused \(\hat{x}\) with similar probability. As a consequence, an adversary can no longer infer with confidence which of these \(x \in D_x\) is actually part of the real dataset. Therefore, the creator of the synthetic set could always plausibly deny the membership of a particular data point.

Quadratic variation. This measure targets the time series generator’s ability to reproduce the temporal structure of each channel [50]. Quadratic variation originally comes from the analysis of stochastic processes and can be simplified for time series toIt is initially only applicable to individual univariate time series and needs further aggregation to be useful for the dataset-level. According to Boursin et al. [51], this is realized by computing the MSE between real samples on one side and synthetic samples on the other side. However, neither there nor in the original work is the order of the samples defined needed for the MSE or the aggregation method across channels.

Real to synthetic similarity (RTS). Proposed in [46], this measure compares the (approximate) similarity between real and synthetic samples to that among real samples. Similarity is determined by applying the cosine similarity to selected pairs of each dataset. To approximate the real-to-synthetic similarity for a synthetic sample \(\hat{x}\), they calculate its average cosine similarity to 10 random real samples \(x_1, \dots , x_{10} \in D_r\), resulting in the scoreThis can be used on individual samples to guide training or on a dataset-level for evaluation purposes at the end. As a baseline, they also calculate the average real to real similarity (RTR) for real dataset \(D_r\) usingThe RTR needs to be computed only once per real dataset. The goal are similar RTR and RTR scores, meaning \(\text {RTS} - \text {RTR} \approx 0\), where \(RTS, RTR \in [0,1]\). In case the time series happen to be univariate, one can apply the measure directly. Otherwise, an embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\) is required.

Realism score is intended as an adaption of the improved precision measure to the sample-level, in order to rank individual samples by their quality [69]. In practice, it approximates the distance of a synthetic sample to the real data manifold using a continuous extension of the binary indicator function \({\textbf {1}}\{ \cdot \}\) used in improved precision and improved recall. For a synthetic \(\hat{x} \in D_g\) and real dataset \(D_r\), realism score (R) is given bywhere\(D_r'\) is the half of \(D_r\) for which the distance to the K th-nearest neighbor is smaller than the median dNN\(_k\). This pruning of \(D_r\) strives to shrink the data manifold down to the densest region to increase robustness to outliers. Note that dNN\(_k\) also uses the Euclidean distance. The higher R the more realistic the sample, which effectively means that realism is determined by how close a synthetic is to a real sample after embedding \(f:\mathcal {X} \rightarrow \mathbb {R}^k\) .

Securities order marginals is a specialized measure tailored towards order streams observed in security transaction systems of stock exchanges [97]. It measures the quality of a generated sequence by computing five specialized statistics, targeting different channels like price and volume. Each statistic covers a marginal relevant to the financial domain, namely the distributions of security price, quantity bid/asked, inter-arrival time of orders, the evolution of the number of orders over time, and the ratio of bid-to-ask orders over time. For each statistic, real and synthetic samples are compared by calculating the Kolmogorov-Smirnov distance between the real marginal and synthetic marginal distribution. It is always in the interval [0, 1], where 0 is best. Consequently, the overall score for a pair \((X, \hat{X})\) is a quintuple of the similarities as expressed by the distances above.

Visual assessment. Using human judges as decision-makers on synthesis quality is a very simplistic and popular approach [4]. Indeed, entire works are built around the idea of making the assessment as clear and fast as possible using visualization tools. In the most basic form, evaluators may just look at a plot of each channel of the sequence. More complex approaches may account for their frequency domain using DFT, Wavelet transform [98], and z-transform [99] or specialized embeddings, extrapolating certain features of interest. Arnout et al. [79] propose a visualization framework that allows tracking of the appearance of individual samples during the training of the generation model and assessing the final result. In addition to ordinary plots, they also employ TimeHistograms [100] and Colorfields [101] to create more appealing embeddings. In two works, judgments are passed by domain experts and aggregated using mean opinion score [5, 39].

\(\alpha\)-precision. Mirroring \(\beta\)-recall, this is a parameterized measure denoted by \(P_\alpha\), where \(\alpha \in [0,1]\) is the fraction of real samples assumed to be “typical”, that is, no outliers [87]. It represents the fraction of synthetic samples resembling that fraction of real data. In other words, a high score ensures that the typical generator output is similar to typical real data, where typical can be interpreted very loosely on the one end or strictly on the other. Its formal definition and aggregation via mean absolute deviation is symmetrical to that of \(\beta\)-recall, that is, based on a minimum volume subset of the support of the synthetic distribution. Single real samples are assigned a value of 1 if they fall within the subset, and otherwise 0. Based on the mean absolute deviation between precision curve and fraction \(\alpha\), a summarizing score called integrated \(P_\alpha\) is given byAn example for the precision curve can be found in Fig. 19 center. \(I P_\alpha \in [0,1]\) where higher is better. Similar to \(\beta\)-recall, an embedding is required. By default, this is an LSTM-based autoencoder model.

In the following, we suggest grouping the 83 reviewed measures by evaluation criteria. Simply put, each criterion refers to a specific data property that each measure is designed to test. For instance, spatial correlation and MIA check different data characteristics to evaluate quality. This is why, commonly, a set of measures is used for evaluation, covering a variety of properties and, in turn, aspects of quality of synthetic time series. Currently, all these measures coexist in literature without structure or order, which prohibits researchers and practitioners from quickly and clearly identifying measures suitable to the properties they wish to test. Notably, Borji [11] previously addressed the problem for the evaluation of synthetic images through the structure of their article itself. Instead, we propose a taxonomy of evaluation criteria and measures, which provides an explicit and search-friendly solution to this hindrance. The result is depicted in Fig. 22 as a horizontal tree structure with the root on the left and the measures as leaves on the right. From the root outward, we arranged different criteria in boxed nodes, which themselves fan out to measures and sometimes other, subordinate criteria. Note that measures can also be subordinated under other measures if they test a more specific property than the general one, for instance, predictive score under TSTR. Additionally, we follow previous works [14, 39, 85] and incorporate another differentiation between measures into the figure based on color coding: Qualitative measures (orange) are characterized by their need for human interpretation of an intelligible but ambiguous result such as t-SNE. Quantitative measures (cyan), on the other hand, produce a numerical or categorical value, allowing a clear interpretation. In gray, we mark notable secondary purposes (i.e., criteria) that a measure may fulfill.

We constructed this taxonomy in three steps: First, we clustered the measures based on the properties they tested. In case a measure has secondary purposes, it may belong to multiple clusters. Then, we assigned each cluster a suitable criterion, which was already mentioned in the literature in most cases. Finally, we arranged the clusters in a hierarchy such that more general clusters (e.g., fidelity) subsume more specific ones (e.g., temporal dynamics). In the following, we explain the chosen criteria and the assignment of measures in top-down order.

We start with fidelity, which refers to an individual synthesized time series to preserve characteristics, patterns, and noise levels present in the underlying real time series, often associated with notions of resemblance or typicality. While we found most measures in this cluster to be moved into four sub-criteria, we assign AIS, which produces likelihoods for each time series, and TRTS, which is based on the performance of a real-trained model on synthetic data, to that criterion directly. Both represent broad indicators able to catch many facets of fidelity. Coherence is one sub-criterion by which we denote the consistency of the time series’ internal structure, especially along channel and time axes. Its only sub-criterion, temporal dynamics, contains AED, which is concerned with the preservation of periodicities within the synthetic time series. The second one is perceptual quality, meaning the aspects of quality perceptible by human judges during the inspection of synthetic data. We place visual assessment here.

With instance similarity, we opened a third cluster with eleven measures comparing a real time series to a synthetic one, determining the similarity on a sample-level. Directly assigned are ACS, correlation structure, feature-based distance, pairwise measures, RTS, and elementary statistics. ACS extracts and compares seven features from each time series individually. Similarly, feature-based distance extracts its own set of features, while correlation structure calculates the covariance matrix for each data instance, comparing real and synthetic time series pair-by-pair. Pairwise measures and RTS compare individual real and synthetic time series using different similarity functions. Elementary statistics typically compare the calculated statistical score on the dataset-level and sometimes between instances. As the remaining five measures all target temporal aspects of instance similarity, we introduced a separate sub-category called temporal dynamics once again. These measures are cointegration test, DTW\(_D\), Fréchet distance, quadratic variation, and WCS, addressing trend comparison, time warping, curve alignment, step-wise changes, and wavelet coherence, respectively.

Lastly, we place measures that interpret the fidelity of a synthetic sample as the proximity to its nearest real neighbors in an embedding space under criterion embedding proximity. We identified six measures following this description. To handle the two sides of PRD, we divided the measure into its two components and deemed precision suitable for evaluating fidelity. Its conceptual successors, improved precision, realism score, density, and \(\alpha\)-precision have the same purpose and thus belong here as well. INND is a special case in that it only takes the minimum distance to one real sample.

While fidelity focuses on sample quality, matching distributions requires real and synthetic data distributions P and Q to match in some aspects, if not overall. For instance, a generator may produce time series of exceptional quality but in the wrong proportions, such that P and Q still do not match. In this case, we differentiate three sub-criteria based on these aspects. First, there is a group dataset-level statistics of eight measures that compare P and Q with regard to some statistical characteristics, a weak demand given the complexity of typical time series datasets. For PCC, this characteristic is the Pearson correlation, provided qualitatively as a heat map for each dataset. MSAS does not extract one but multiple statistical characteristics like channel-wise mean or sequence length and subsequently compares real and synthetic datasets on these. As their name indicates, ApEn, temporal correlation, and spatial correlation all deliver a statistical value for the entire dataset. Along with dependence scores, we split off temporal correlation into a sub-criterion temporal dynamics. Dependence scores compare the ACF between real time series and their syntheses. Two measures remain, one of which being distributional scores, computing and comparing skewness and kurtosis for the single real and the set of synthetic time series. At last, we have the group of elementary statistics, such as statistical moments or ACF.

The next sub-criterion is called joint distribution, subsuming measures that view real P and synthetic Q as joint distributions of multiple marginals and compare the two with respect to one or more of these. In this cluster, we aggregated 11 measures. First, there is topN locations, which compares the distributions over bins of coordinates extracted from the real dataset on one, and the synthetic dataset on the other. The coordinates represent locations that can be visited in each step of the time series. The second one is length histogram, which is intended for the evaluation of synthetic time series of varying lengths. Here, the marginal distribution is length. AWD focuses on the various periodicities within the time series in each dataset, considers them as marginal distributions, and compares them using the WD. In the original work, they state to use it for diversity, but we find joint distribution to be the better-fitting cluster. WD on CDF first computes an empirical CDF over a discrete data space or binning of a real-valued data space. In the second step, the Wasserstein-1 distance is computed between the CDF of real and synthetic data, respectively. Since binning is typically required, the CDF only reflects a part of the actual distribution. Average JS distance extracts a set of features from each time series and interprets the values of each feature as a marginal distribution. After applying these steps to real and synthetic datasets, the JS divergences are determined and averaged. As one can imagine, JS divergences on marginals is a generalization of this concept, conceived by different authors and apparently without influence from one to the other. Distributional metric, similar to others before, is based on an empirical distribution over bins of data objects. As opposed to WD on CDF, for instance, it uses the PDF on the binnings for both real and synthetic data and computes the bin-wise absolute difference between the two. Nevertheless, it operates on marginal distributions and not the real ones. In the case of marginal metrics, the marginals of interest are the distributions of each time step. The last two, NDB and NDB-over/under, also use binnings to approximate the actual distributions, concentrate on the regional distributions and thereby define marginals. Securities order marginals compares the statistics of marginal distributions of financial time series such as price and volume.

By representativeness, we denote a criterion that requires the overall distributions to be similar, simply put, \(P \approx Q\). We reviewed a variety of measures that pursue this fundamental aspect of quality, among them the established, more general MMD, WD, or the qualitative distribution visualization approaches like t-SNE. We also found derivations like the relative MMD test. We also have C2ST, which uses a binary classifier model and a hypothesis test on its predictions, whereas the former methods use kernel functions and distances. Furthermore, five measures (NND, MTop-Div, duality gap, minimax loss, and discriminative score) use the loss or accuracy of a binary classifier model trained to distinguish real from synthetic samples. Their results depend solely on the capabilities of the classifier, and only what it considers relevant goes into the differentiation task. Finally, FID and its two variants are in this cluster. All three use feature extraction layers from neural models to map the data into a feature space before applying the Fréchet distance to compare the distributions. In the case of MiFID, FID is one of its components, and testing the distributions for a match its secondary purpose.

Coverage refers to the idea that the synthetic data as a whole should “cover” the entire region assumed by the real data distribution. It is occupied by four measures developed in sequence with the intention of improving on their respective predecessor, and a fifth one developed independently. The first in line is recall, which is one part of the composite measure PRD. Then, there are its successors, improved recall, coverage, and \(\beta\)-recall proposed in this order. They are all similar in that they use the proximity of real to synthetic objects in a feature space as an indicator of the generator’s ability to cover the entire support of the real data distribution. The fifth, ONND, is based on the distance of each real sample to its closest neighbors in the synthetic dataset. With our assessment of measures coverage and ONND, we disagree with their respective authors, who consider them to evaluate diversity primarily. Still, we acknowledge diversity as an aspect of quality but assign two what we believe to be more suitable measures: STS and ICD. STS measures and compares the average cosine similarity within real and synthetic datasets, respectively, and interprets the result as an indicator of diversity (or the lack thereof). ICD follows the same procedure but with a different similarity measure.

In the literature, the terms novelty and generalization are both used to describe a generator that produces samples beyond noisy versions of the training data. We differentiate the terms here and use generalization if reproducing some training samples is acceptable and novelty if each generated time series must be noticeably different from all training samples. We identified one measure suitable to test novelty directly, namely authenticity. It can be applied to each synthetic sample individually and even provide a binary decision on whether it is novel using a hypothesis test. The selection of measures for generalization is more extensive. MiFID employs a so-called memorization distance based on the cosine similarity between samples and penalizes the generator for producing samples too close to training samples. NNT computes the neighborhood of each synthetic sample to determine if its neighbors in the real dataset are too close. Intuitively, MMD just compares two distributions with respect to their overlap in density. However, Esteban et al. [56] consider an additional setting in which the discrepancy between the real train and a real hold-out set is compared to that between the synthetic set and the same hold-out set. We call this variant train-test MMD. What is more, they developed this idea further and conceived DRE for the same purpose. The data-copying test uses DTW to calculate the average distance between the train and synthetic set as well as between the train and the hold-out set. Afterwards, a hypothesis test determines if the average distance between the first pair of sets is significantly shorter than between the second. Max-RTS computes the most similar pair of real and synthetic samples and uses it as an indicator for generalization. Lastly, AIS tests generalization in a secondary role by comparing the average log-likelihood on training and test sets.

Privacy is an often-cited criterion for evaluation and a main reason for synthesizing data in the first place. Upholding privacy means reducing - if not eliminating - the risk of disclosure of sensitive information within the original dataset. MIA and plausible deniability set out to control and uncover the degree to which information on individual real data objects passes through into the generated set. As Max-RTS computes the closest pair of real and synthetic objects in terms of cosine similarity, it can be used as an indicator for the worst-case information leakage on a single data object.

Utility interprets the quality of synthetic data as its usefulness in a downstream (machine learning) task. The eight measures in its cluster are not directly assigned, but only TSTR, hedging effectiveness, and algorithm comparison. The remaining five are specializations of TSTR, a generic measure of utility. Downstream task T and the ML model for T are free of choice by default. By fixing T, we get a variant TSTR(T). In the case of time series, this is usually prediction and sometimes classification. Depending on T, another aspect of utility is evaluated each time. Except for minor implementation differences, predictive score is a variant of TSTR with prediction as T. Agreement rate, confusion matrix, and CAS use classification as task. Agreement rate includes a strategy for handling unlabeled time series data, which makes it slightly more general. The last variant is the augmentation test, where the task is augmentation of the real data using the synthesis in a predefined proportion. Algorithm comparison, however, not only utilizes one kind of ML model but several and bases its score on the similarity of the two performance rankings of these models, one on the real dataset and one on the synthetic set. Hedging effectiveness is different in that it does not train the same model on the real data as on the synthetic but uses a simple baseline for comparison.

Next, we analyze the usage statistics of these evaluation measures in the academic literature. We refrain from investigating the statistics of real-world adoption, as this would necessitate surveys conducted among practitioners. Instead, we discuss the impact a measure has on subsequent works depending on what kind of work it was proposed. To this end, we partition the collection of measures above into two sets based on whether they originate from work focused on the evaluation of synthetic data [69, 70] or on their generation [26, 92]. In the latter case, proposed measures are side contributions and directly employed in evaluating the presented generators. We call the first class of works Theory and the second Practice. Transitively, we apply the same distinction for the measures presented therein.

By reuse, we mean the number of works in Practice in which a measure was used minus the initial use. In other words, how often it was adopted. We find this statistic to be a more meaningful indicator for impact than citations, for instance, as these can instead reflect reuses of other measures in the work, references of the generator presented, or a citation for a survey article like ours. Following our approach, we list the number of reuses as well as the year of inception or first use, respectively, for each measure in Table 6. The latter allows us to take the age of the measure into account. The 28 measures assigned to Theory are listed first, and the 55 in Practice second and are additionally marked with *. Furthermore, note that we consider the measures of groups distribution visualization (QQ-plot, scatter plot, latent space visualization, PDF visualization, dimension-wise probability, PCA, t-SNE) and elementary statistics (ACF, covariance, moments) individually.

Based on these data, we immediately observe that most measures are never reused, some are used again once, and only a few have reached some level of popularity in the community. To support this further and differentiate Theory and Practice, we calculate four intuitive indicators of impact. To this end, let G be a set of measures, namely one of Total, Practice, and Theory. Further, we define r(m) as the number of reuses of measure m.

As the first indicator, we take the maximum number of reuses within each group,which serves as an indicator for the extremal case of impact. This value is only 2 for Theory, compared to 11 for Practice. The picture repeats for the next highest reuse values, showing that the most impactful measures belong to Practice.

Next, we compute the relative and normalized impacts \(I_r\) and \(I_n\), which consider the aggregate number of reuses in Theory and Practice, expressed as a fraction of total reuses in the first case, and a normalization by the group size in the second case.As a result, \(I_r\) attributes only 6.3% of the aggregated impact to Theory, the remaining 93.7% to Practice. Similarly, the average impact of a measure in Theory is only \(I_n(T)\approx 0.179\), but \(I_n(P)\approx 1.345\) for Practice. While this is considerably better, it still means that combined, measures are not even once reused on average.

As the final indicator, we consider the number of measures without any reuse and therefore impact on subsequent evaluations. Additionally, we adjust this number through division by the group size. The higher such a score is, the smaller its past relevance for the evaluation task. Formally, we define it asOverall, we find that over half of the reviewed measures made no impact in this regard, resulting in a normalized number of measures without impact of \(I_{no} \approx 0.643\). Moreover, this indicator is especially bad for the 28 measures from Theory, of which only three found reuse. Hence, its score is even worse with \(I_{no} \approx 0.893\). For Practice, the indicator is significantly better at just over half, \(I_{no} \approx 0.527\).

We summarize the results in Fig. 23. From these indicators and the high number and variety of measures itself, we draw two main insights: The reason for this situation is unclear to us and most likely requires a survey among the authors as to why they made the choices as we see them. Instead, we can only make fair guesses with the information at hand. First, a number of measures have only recently been presented, such that reuse in early works is impossible. Furthermore, the publications in Theory might not have reached the level of attention required to have a significant influence on interdisciplinary work such as medical informatics. Third, some of the proposed measures might be too complex for the audience of works from Practice, mainly reviewers who do not value such measures. For instance, MTop-Div and duality gap come to mind. Another reasonable explanation would be that generator developers do indeed consider them but regard these measures as unsuited and hence use their own or those used in works they compete with. An example of such an influential work from Practice is [26], which, proposed in 2019, introduced distribution visualization via t-SNE and PCA, discriminative score, and predictive score, all with high reuse rates.

Not all measures are equally suited to evaluate all kinds of time series data. Among other aspects, they differ in how length and dimensionality of the sequences and the sizes of the datasets involved are being handled. Here, we look at how the measures reviewed in this survey come off in terms of applicability, that is, if there are any limitations or recommendations for length, dimensionality, or size. Our findings are summarized in Table 7. For each measure, we provide information on time series length in the second column, on its dimensionality in the third, on the size of the real dataset in the fourth, and on the size of the synthetic set in the last column. The measures with identical entries are grouped and sorted alphabetically. If a measure can be applied arbitrarily with respect to one of these four aspects, we mark the respective cell as “no restrictions” (n.r.). In case the measure requires an embedding function upon which length and dimensionality of the possible input format depend, we indicate this by the abbreviation e.i.d. for “embedder input dimension”. Occasionally, the sequence length must remain constant or, to the contrary, vary across the input. If measures are more specific, we put the limitations in the respective columns. Furthermore, if the value is marked with an asterisk (*), it is our recommendation based on the above analysis; otherwise, it was taken from the original work. Note that these are just estimates, meaning a dataset size of 900 most likely causes no problem when the recommendation says \(\ge \! 1\)k*. For MIA and correlation structure, unfortunately, appropriate values are unknown. While lengths can vary, we implicitly assume \(\text {dim}(D_r) = \text {dim}(D_g) = \text {constant}\).

Our estimates (*) mostly follow the rationale that a certain amount of data must be available to reliably evaluate the synthetic data with the respective measure, especially since a portion of the dataset is to be held out from training the generative function. For measures that require a learned embedding or train a model for a downstream task with the data, we propose a minimum of 1000 samples per dataset. The same goes for measures that apply binning to estimate P and Q, such that some degree of statistical representativeness is maintained. For the same reason, we suggest a minimum length for distributional metric. For measures learning a model to distinguish real from synthetic data, the proposed minimum size is \(10\,000\). Securities order marginals, AED, and feature-based correlation analysis require multivariate time series with different restrictions on the number of channels. Visual assessment is an outlier in this regard, as we argue for upper bounds in length, dimensionality, and size of \(D_g\). This is based on the assumption that only time series of limited complexity and number can be visually judged reasonably.

To close out, we summarize our takeaways regarding the requirements for input data formats below. In terms of length, almost half the measures depend on the capabilities of the embedding model, and they themselves operate on fixed-size output vectors. 25 measures have no restrictions, while 10 require the time series to have constant length, and a few others have special requirements. The situation for dimensionality is very similar. Most measures either depend on the capabilities of the embedding model or impose no restrictions at all. A few may only be applied to either uni- or multivariate time series. The partition between sample-level and distribution-level measures continues over to the size of input datasets. More precisely, if restrictions apply, datasets may contain only one sample in the former and typically thousands of time elements in the latter case to have a dependable estimation of the underlying distribution. Furthermore, the size of \(D_r\) and \(D_g\) is usually expected to be similar in magnitude, with notable exceptions like MTop-Div and NND.