Title: Poisson-Gamma Dynamical Systems with Non-Stationary Transition Dynamics
Venue: NeurIPS 2024
Accepted: no
Keywords: Bayesian non-parametrics

Negative-binomial processes

Dirichlet Markov chains
Abstract: Bayesian methodologies for handling count-valued time series have gained prominence due to their ability to infer interpretable latent structures and to estimate uncertainties, and thus are especially suitable for dealing with noisy and incomplete count data. Among these Bayesian models, Poisson-Gamma Dynamical Systems (PGDSs) are proven to be effective in capturing the evolving dynamics underlying observed count sequences. However, the state-of-the-art PGDS still falls short in capturing the time-varying transition dynamics that are commonly observed in real-world count time series. To mitigate this limitation, a non-stationary PGDS is proposed to allow the underlying transition matrices to evolve over time, and the evolving transition matrices are modeled by the specifically-designed Dirichlet Markov chains. Leveraging Dirichlet-Multinomial-Beta data augmentation techniques, a fully-conjugate and efficient Gibbs sampler is developed to perform posterior simulation. Experiments show that, in comparison with related models, the proposed non-stationary PGDS achieves improved predictive performance due to its capacity to learn non-stationary dependency structure captured by the time-evolving transition matrices.

Abstract

Bayesian methodologies for handling count-valued time series have gained prominence due to their ability to infer interpretable latent structures and to estimate uncertainties, and thus are especially suitable for dealing with noisy and incomplete count data. Among these Bayesian models, Poisson-Gamma Dynamical Systems (PGDSs) are proven to be effective in capturing the evolving dynamics underlying observed count sequences. However, the state-of-the-art PGDS still falls short in capturing the time-varying transition dynamics that are commonly observed in real-world count time series. To mitigate this limitation, a non-stationary PGDS is proposed to allow the underlying transition matrices to evolve over time, and the evolving transition matrices are modeled by the specifically-designed Dirichlet Markov chains. Leveraging Dirichlet-Multinomial-Beta data augmentation techniques, a fully-conjugate and efficient Gibbs sampler is developed to perform posterior simulation. Experiments show that, in comparison with related models, the proposed non-stationary PGDS achieves improved predictive performance due to its capacity to learn non-stationary dependency structure captured by the time-evolving transition matrices.

1 Introduction

In recent years, there has been an increasing interest in modeling count time series. For instance, some previous works [1, 2, 3] are concerned with how to learn the evolving topics behind text corpus (frequencies of words) over time. Some works [4, 5, 6, 7] try to predict global immigrant trends underlying international population movements. Count time series are often overdispersed , sparse , high-dimensional , and thus can not be well modeled by widely used dynamic models such as linear dynamical systems [8, 9]. Recently, many works [10, 11, 12, 13, 14, 15, 16] prefer to choose distributions of the gamma-Poisson family to build their hierarchical Bayesian models. In particular, these models enjoy strong explainability and can estimate uncertainty especially when the observations are noisy and incomplete . Among these works, Poisson-Gamma Dynamical Systems (PGDSs) [13] received a lot of attention because PGDS can learn how the latent dimensions excite each other to capture complicated dynamics in observed count series. For instance, a very inspiring research paper may motivate other researchers to publish papers on related topics [17]. The outbreak of COVID-19 in one state, may lead to the rapid rising of COVID-19 cases in the nearby states and vice versa [18]. In particular, PGDS can be efficiently learned with a tractable Gibbs sampling scheme via Poisson-Logarithmic data augmentation and marginalization technique [11]. Due to its strong flexibility, PGDS achieves better performance in predicting missing entities and future observations, compared with related models [9, 15].

Despite these advantages, PGDS still can not capture the time-varying transition dynamics underlying 35

observed count sequences, which are commonly observed in real-world scenarios [19]. For instance, 36

during the initial stage of the COVID-19 pandemic, the worldwide counts of infectious patients were 37 significantly affected by various local policies, government interventions, and emergent events [20, 38 21, 22]. The cross transition dynamics among the different monitoring areas were also evolving as 39 the corresponding policies and interventions changed over time. Hence, PGDS unavoidably makes a 40 certain amount of approximation error in capturing the aforementioned non-stationary count time 41 series, using a time-invariant transition kernel. 42 To mitigate this limitation, Non-Stationary Poisson-Gamma Dynamical Systems (NS-PGDSs), a novel 43 kind of Poisson-gamma dynamical systems with non-stationary transition dynamics are developed. 44 More specifically, NS-PGDS captures the evolving transition dynamics by the specifically-designed 45 Dirichlet Markov chains. Via the Dirichlet-Multinomial-Beta data augmentation strategy, the Non46 Stationary Poisson-Gamma Dynamical Systems can be inferred with a conjugate-yet-efficient Gibbs 47 sampler. Our contributions are summarized as follows: 48

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We propose a Non-Stationary Poisson-Gamma Dynamical System (NS-PGDS), a novel Poisson-gamma dynamical system with time-evolving transition matrices that can well capture non-stationary transition dynamics underlying observed count series.

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Three Dirichlet Markov chains are dedicated to improving the flexibility and expressiveness of NS-PGDSs, for capturing the complex transition dynamics behind sequential count data.

Fully-conjugate-yet-efficient Gibbs samplers are developed via Dirichlet-Multinomial-Beta augmentation techniques to perform posterior simulation for the proposed Dirichlet Markov chains.

Extensive experiments are conducted on four real-world datasets, to evaluate the performance of the proposed NS-PGDS in predicting missing and future unseen observations. We also provide exploratory analysis to demonstrate the explainable latent structure inferred by the proposed NS-PGDS.

2 Preliminaries

Let y ( t ) = [ y ( t ) 1 , · · · , y ( t ) V ] T ∈ N V be a vector of nonnegative count valued observations at time t . To capture the latent dynamics underlying count sequences, some previous works [23, 24] model the observations as

where p ( · ) is the observation likelihood function, and f ( · ) is an invertible link function that maps the parameters of observation component to continuous-valued latent variables x ( t ) ∈ R K . The latent factor x ( t ) evolves over time according to a linear dynamical system (LDS) given by x ( t ) ∼ N ( Ax ( t -1) , Λ -1 ) , where A is the state transition matrix of size K × K , and Λ = diag ( λ 1 , · · · , λ K ) is the inverse covariance matrix with λ -1 k determining the variance of k -th latent dimension. Han et al. [23] adopted the Extended Rank likelihood function to model count observations using LDS with time complexity O (( K + V ) 3 ) , which prevents it from practical applications for analyzing large-scale count data.

Recently, Acharya et al. [15] and Schein et al. [13, 16] developed Poisson-gamma family models for sequential count observations. Gamma Process Dynamic Poisson Factor Analysis (GP-DPFA) [15] models count data as y ( t ) v ∼ Pois( ∑ K k =1 λ k ϕ vk θ ( t ) k ) , where θ ( t ) k represents the strength of k -th latent factor at time t , and ϕ vk captures the involvement degree of k -th factor to v -th observed dimension. To ensure the model identifiability, we can impose a restriction as ∑ v ϕ vk = 1 , and thus place a Dirichlet prior over ϕ k = [ ϕ 1 k , · · · , ϕ V k ] T as ϕ k ∼ Dir ( ϵ 0 , · · · , ϵ 0 ) .

To capture the underlying dynamics, the latent factor θ ( t ) k evolves over time according to a gamma Markov chain as θ ( t ) k ∼ Gam( θ ( t -1) k , c t ) , where c t is the rate parameter of the gamma distribution to control the variance of the gamma Markov chains. Although GP-DPFA can well fit one-dimensional count sequences, it fails to learn how the latent dimensions interact with each other.

To address this concern, Schein et al. [13] developed Poisson-gamma dynamical systems to 83 capture the underlying transition dynamics. In particular, θ ( t ) k evolves over time as θ ( t ) k ∼ 84

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Gam( τ 0 ∑ K k 2 =1 π kk 2 θ ( t -1) k 2 , τ 0 ) , where π kk 2 represents how k 2 -th latent factor excites the k -th latent 85 factor at next time step, and ∑ K k =1 π kk 2 = 1 . 86

3 Non-Stationary Poisson-Gamma Dynamical Systems 87

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Real-world count time sequences are often nonstationary because the external interventional environments are always changing over time. The stationary PGDS with a time-invariant transition kernel fails to capture such time-varying transition dynamics. For instance, the transition dynamics behind COVID-19 infectious processes are time-varying, and highly affected by various interventional policies. Hence, to mitigate this limitation, we model the count sequences as

in which, the latent factors are specified by

where the multiplicative term δ ( t ) ∼ Gam( ϵ 0 , ϵ 0 ) and the transition matrices are time-varying as Π ( t ) ≡ [ π ( t ) kk 2 ] K k,k 2 =1 . As shown in Figure 2, to model the time-varying transition dynamics, we assume the whole time interval can be divided into I equally-spaced sub-intervals. The transition kernel behind complicated dynamic counts is assumed to be static within each sub-interval, while evolving over sub-intervals, to capture non-stationary behaviours. In another word, the proposed model allows the latent factors to evolve over time steps while the transition matrices change over sub-intervals but assumed to be stationary within each sub-interval, as shown in Figure 1. In particular, we let each sub-interval contains M time steps, and the i -th interval contains time steps { t | t = ( i -1) M +1 , · · · , iM } . We define i ( t ) as the function that maps time step t to its corresponding sub-interval.

Dirichlet-Dirichlet Markov processes. To capture how the underlying transition kernel smoothly evolves over sub-intervals, we first propose the Dirichlet-Dirichlet (Dir-Dir) Markov chain as

where π ( i ) k represents the k -th column of Π ( i ) , and the prior of the scaling parameter η is given by 114 η ∼ Gam( e 0 , f 0 ) . 115

The initial states are defined as θ (1) k ∼ Gam( τ 0 ν k , τ 0 ) . The prior for the transition ker116 nel of the first sub-interval is given by π (1) k ∼ Dir ( ν 1 ν k , · · · , ξν k , · · · , ν K ν k ) , where ν k ∼ 117 Gam( γ 0 K , β ) and ξ, β ∼ Gam( ϵ 0 , ϵ 0 ) . Note that the expectation and variance of the transition 118 kernel at i -th sub-interval can be calculated as 119

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respectively. The transition dynamics of i -th sub-interval inherits the information of the previous sub-interval, and also adapts to the data observed in the current sub-interval. The scaling parameter η controls the variance of the transition matrices.

The prior specification defined in Eq.(2) by rescaling the transition matrix at the previous sub-interval allows the transition dynamics to change smoothly, and thus might be insufficient to capture the rapid changes observed in complicated dynamics. To further improve the flexibility of the transition structure, two modified Dirichlet Markov chains are studied to capture the correlation structure between the dimensions of the transition matrices over time.

Dirichlet-Gamma-Dirichlet Markov processes. Wefirst introduce the Dirichlet-Gamma-Dirichlet (Dir-Gam-Dir) Markov chain to model the evolving transition matrices as

where we use ψ ( i -1) kk 1 k 2 to capture the mutation between two consecutive sub-intervals, and its prior is given by

and γ ( i ) k , c ( i ) k ∼ Gam( ϵ 0 , ϵ 0 ) . Compared with the construction defined by Eq.(2), the expectation of DirichletGamma-Dirichlet Markov chain is

This construction takes interactions among components of columns into account. Hence it will dramatically improve the flexibility of our model and thus better fit more complicated dynamics, compared with Dir-Dir Markov chains that only yield smoothing transition dynamics.

Poisson-randomized-gamma-Dirichlet Markov processes. By leveraging the Poisson-randomized gamma distribution [25], we introduce another type of time-varying transition kernels, which also model the interactions among components like Dir-Gam-Dir construction but may induce different properties such as sparsity. The Poisson-randomized-gamma-Dirichlet (PR-Gam-Dir) Markov chain can be formulated as

where RG1( · ) denotes the randomized gamma distribution of the first type. Similarly, for ψ ( i -1) kk 1 k 2 , γ ( i ) k , and c ( i ) k , the priors are given by

The diagrams of three Dirichlet Markov constructions are shown in Figure 3.

4 Markov Chain Monte Carlo Inference

In this section, we present the Gibbs sampler for the proposed NS-PGDS. We only illustrate the key points of the derivation and the details can be found in the appendix.

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Lemma 1 If y ∼ NB( a, g ( ζ )) and l ∼ CRT( y, a ) , where NB( · ) refers to negative-binomial distribution, CRT( · ) represents Chinese restaurant table distribution [26], and g ( z ) = 1 -exp( -z ) . Then the joint distribution of y and l can be equivalently distributed as y ∼ SumLog( l, g ( ζ )) and l ∼ Pois ( aζ ) [11], i.e.

where SumLog( l, g ( ζ )) = ∑ l i =1 x i and x i ∼ Log ( g ( ζ )) are independently and identically logarithmic distributed random variables [27].

where DirMult( · ) refers to Dirichlet-multimonial distribution. We sample the augmented variable q | n ∼ Beta( n, r · ) , where r · = ∑ K k =1 r k . According to [28], conditioning on q , we have n k ∼ NB( r k , q ) .

Sampling y ( t ) vk : Use the relationship between Poisson and multinomial distributions, we sample

Sampling ϕ k : Via Dirichlet-multinomial conjugacy, the posterior of ϕ k is

Sampling θ ( t ) k : To sample from the posterior of θ ( t ) k , we first sample the auxiliary variables. Setting l ( T +1) · k = 0 and ζ ( T +1) = 0 , we sample the augmented variables backwards from t = T, · · · , 2 ,

Let us define l ( t ) · k = ∑ K k 1 =1 l ( t ) k 1 k and ζ ( t ) = ln(1 + δ ( t ) τ 0 + ζ ( t +1) ) . After sampling the auxiliary 164 variables, then for t = 1 , · · · , T , by Poisson-gamma conjugacy, we obtain 165

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Sampling Π ( i ) : Weonly illustrate Gibbs sampling algorithm for PR-Gam-Dir construction, sampling algorithms for other constructions can be found in the appendix. We define M as the length of each sub-interval, and I as the number of intervals. For i = I, · · · , 2 , because ( l ( i ) 1 k , · · · , l ( i ) Kk ) and ( g ( i +1) · 1 k , · · · , g ( i +1) · Kk ) are multinomially distributed, where l ( i ) k 1 k = ∑ iM ( i -1) M +1 l ( t ) k 1 k refers to the summation of l ( t ) k 1 k over i -th sub-interval and same notation for other variables. By the definition of Dirichlet-multinomial distribution and Lemma 2, defining g ( I +1) k 1 k = 0 , we sample the auxiliary variables as ( q ( i ) k | -) ∼ Beta( l ( i ) · k + g ( i +1) · k , α ( i ) · k ) , then we have ( l ( i ) k 1 k + g ( i +1) · k 1 k ) ∼ NB( α ( i ) k 1 k , q ( i ) k ) . Then we further sample ( h ( i ) k 1 k | -) ∼ CRT( l ( i ) k 1 k + g ( i +1) · k 1 k , α ( i ) k 1 k ) . Via Lemma 1, we obtain h ( i ) k 1 k ∼ Pois( -α ( i ) k 1 k ln(1 -q ( i ) k )) . For Dirichlet-Randomized-Gamma-Dirichlet Markov construction defined by Eq.(4), we can equivalently represent it as

We define λ ( i -1) k 1 k ≜ γ ( i -1) k ∑ K k 2=1 ψ ( i -1) kk 1 k 2 π ( i -1) k 2 k for notation conciseness. By Poisson-gamma 176 conjugacy, we have ( α ( i ) k 1 k | -) ∼ Gam( g ( i ) k 1 k + ϵ α + h ( i ) k 1 k , c ( i ) k -ln(1 -q ( i ) k )) . If ϵ α > 0 , we can 177

sample the posterior of g ( i ) k 1 k via ( g ( i ) k 1 k | -) ∼ Bessel( ϵ α -1 , 2 √ α ( i ) k 1 k c ( i ) k λ ( i -1) k 1 k ) , where Bessel ( · ) 178 denotes Bessel distribution. If ϵ α = 0 , we sample g ( i ) via 179

where SCH( · ) denotes the shifted confluent hypergeometric distribution [16]. Defining g ( i ) k 1 k = 180 g ( i ) k 1 · k = ∑ K k 2=1 g ( i ) k 1 k 2 k , we first augment 181

then we obtain g ( i ) k 1 k 2 k ∼ Pois( γ ( i -1) ψ ( i -1) kk 1 k 2 π ( i -1) k 2 k ) . By Dirichlet-multinomial conjugacy, we have

̸

Specifically, we have α (1) k 1 k = ν k 1 ν k , if k 1 = k , and α (1) k 1 k = ξν k , if k 1 = k .

5 Related Work

Modeling count time sequences has been receiving increasing attentions in statistical and machine learning communities. Han et al. [23] adopted linear dynamical systems to capture the underlying dynamics of the data and leveraged Extended Rank likelihood function to model count observations. Some Poisson-gamma models assume that the count vector at each time step is modeled by Poisson factor analysis (PFA) [11] and leverage special stochastic processes to model the temporal dependencies of latent factors. For example, gamma process dynamic Poisson factor analysis (GP-DPFA) [15] adopts gamma Markov chains which assumes the latent factor of the next time step is drawn from a gamma distribution with the shape parameter be the latent factor of the current time step. Schein et al. [13] proposed Poisson-gamma dynamical systems (PGDSs), which take the interactions among latent dimensions into account and use a transition matrix to capture the interactions. Deep dynamic Poisson factor analysis (DDPFA) [29] adopts recurrent neural networks (RNNs) to capture the complex long-term dependencies of latent factors. Yang and Koeppl [30] applied Poisson-gamma count model to analyze relational data arising from longitudinal networks, which can capture the evolution of individual node-group memberships over time. Many modifications of PGDS have been proposed in recent years. Guo et al. [31] proposed deep Poisson-gamma dynamical systems which aim to capture the long-range temporal dependencies. Schein et al. [16] employed Poisson-randomized gamma distribution to build a new transition process of latent factors. Chen et al. [32] proposed Switching Poisson-gamma dynamical systems (SPGDS), allowing PGDS to select from several transition matrices, and thus can better adapt to nonlinear dynamics. In contrast to SPGDS, the number of transition matrices of the proposed NS-PGDS is not limited and thus can be adopted to analyze various complicated non-stationary count sequences. Filstroff et al. [33] extensively analyzed many gamma Markov chains for non-negative matrix factorization and introduced new gamma Markov chains with well-defined stationary distribution (BGAR).

6 Experiments

We conducted experiments for both predictive and exploratory analysis to demonstrate the ability of the proposed model in capturing non-stationary count time sequences. The baseline models included in the experiments are: 1) Gamma process dynamic Poisson factor analysis (GP-DPFA) [15]. GP-DPFA models the evolution of latent components as θ ( t ) k ∼ Gam( θ ( t -1) k , c t ) , in which each component evolves independently of the other components. 2) Gamma Markov chains on the rate parameter of gamma distribution (GMC-RATE) [33]. GMC-RATE adopts gamma Markov chains defined via the rate parameter of the gamma distribution to model the evolution of θ ( t ) k

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as θ ( t ) k ∼ Gam( α, β/θ ( t -1) k ) . 3) Gamma Markov chains on the rate parameter with hierarchical auxiliary variable (GMC-HIER) [33]. GMC-HIER models the evolution of latent components with an auxiliary variables as z ( t ) k ∼ Gam( α z , β z θ ( t -1) k ) and θ ( t ) k ∼ Gam( a θ , β θ z ( t ) k ) . 4) Autogressive beta-gamma procecss (BGAR) [34, 33]. BGAR is also a gamma Markov model. In contrast to the above models, there is a well-defined stationary distribution for BGAR. 5) Poisson-gamma dynamical system (PGDS) [13] takes interactions among latent dimensions into account, and models the evolution of θ ( t ) k as θ ( t ) k ∼ Gam( τ 0 ∑ K k 2 =1 π kk 2 θ ( t -1) k 2 , τ 0 ) .

The real-world datasets used in the experiments are: 1) Integrated Crisis Early Warning System (ICEWS) : ICEWS is an international relations event dataset, comprising interaction events between countries extracted from news corpora. For ICEWS dataset, we have T = 365 time steps and V = 6197 dimensions, and we set M = 30 . 2) NIPS : NIPS dataset contains the papers published in the NeurIPS conference from 1987 to 2015. We have T = 28 time steps and V = 2000 dimensions for NIPS dataset and we set M = 5 . 3) U.S. Earthquake Intensity (USEI) : USEI contains a collection of damage and felt reports for U.S. (and a few other countries) earthquakes. We use the monthly reports from 1957-1986 and have T = 348 , V = 64 and set M = 34 . 4) COVID-19 : This dataset contains daily death cases data for states in the United States, spanning from March 2020 to June 2020. For this dataset, we have V = 51 dimensions and T = 90 time steps and set M = 20 .

6.1 Predictive Analysis

To compare the predictive performance of the proposed model with the baselines, we considered two standard tasks: data smoothing and forecasting. For data smoothing task, our objective is to predict y ( t ) given the remaining data observation Y \ y ( t ) . To this end, we randomly masked 10 percents of the observed data over non-adjacent time steps, and predicted the masked values. For forecasting task, we held out data of the last S time steps, and predicted y ( T +1) , · · · , y ( T + S ) given y (1) , · · · , y ( T ) . In this experiment we set S = 2 . We ran the baseline models including GP-DPFA, PGDS, GMC-RATE, GMC-HIER, BGAR, using their default settings as provided in [15, 13, 33]. For the NS-PGDS, we set K = 100 for ICEWS, K = 10 for other datasets, and set τ 0 = 1 , γ 0 = 50 , ϵ 0 = 0 . 1 . We performed 4000 Gibbs sampling iterations. In the experiments, we found that the Gibbs sampler started to converge after 1000 iterations, and thus we set the burn-in time be 2000 iterations. We retained every hundredth sample, and averaged the predictions over the samples. Mean relative error (MRE) and mean absolute error (MAE) are adopted to evaluate the model's predictive capability, which are defined as MRE = 1 TV ∑ t ∑ v | y ( t ) v -ˆ y ( t ) v | 1+ y ( t ) v and MAE = 1 TV ∑ t ∑ v | y ( t ) v -ˆ y ( t ) v | respectively, where y ( t ) v indicates the true count and ˆ y ( t ) v is the prediction.

As the experiment results shown in Table 1, the NS-PGDS exhibits improved performance in both data smoothing and forecasting tasks. We attribute this enhanced capability to the time-varying transition kernels, which effectively adapt to the non-stationary environment, and thus achieve improved predictive performance. For some datasets (e.g. ICEWS) and tasks, the effectiveness of the Dir-Gam-Dir and Pr-Gam-Dir constructions does not be exhibited in the numerical results. However, these two constructions indeed induce more informative patterns compared with Dir-Dir construction, as shown in the exploratory analysis.

6.2 Exploratory Analysis 255

We used ICEWS and NIPS datasets for exploratory analysis, and chose the NS-PGDS with Dirichlet256 Dirichlet Markov chains for illustration. Figure 4(a) and Figure 4(b) demonstrate the top 2 latent 257 factors inferred by NS-PGDS from ICEWS dataset. From Figure 4(a) we can see that the main labels 258 are 'Iraq (IRQ)-United States (USA)", 'Iraq (IRQ)-United Kingdom (UK)", 'Russia (RUS)-United 259 States (USA)", and so on. This latent factor probably corresponds to the topic about Iraq war. Besides, 260 in Figure 4(a), there is a peak around March, 2003, and we know that the Iraq war broke out exactly on 261 20 March, 2003. In addition, the most dominant labels shown in Figure 4(b) are 'Japan (JPN)-United 262 States (USA)", 'China (CHN)-United States (USA)", 'North Korea (PRK)-United States (USA)", 263 'South Korea (KOR)-United States (USA)", and so on. We can infer that this latent factor corresponds 264 to 'Six-Party Talks" and other accidents about it. 265

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Figure 4(c) demonstrates the evolving trends of the top 5 latent factors inferred by the NS-PGDS from NIPS dataset, and the legend indicates the representative words of the corresponding latent factors. Clearly, the green and blue lines correspond to the latent factors of neural network research which started to decline from the 1990s. From the 1990s we see that the latent factors about statistical and probabilistic methods began to dominate the NeurIPS conference. In addition, the NS-PGDS also captured the revival of neural networks (blue line) from the 2010s. The above observations from the latent structure inferred by the NS-PGDS match our prior knowledge.

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Next, we explored the time-varying transition matrices inferred by the NS-PGDS. We chose NIPS dataset for illustratiuon, and set K = 10 and the interval length M to be 5. The time-varying transition matrices are shown from Figure 5(b) to Figure 5(f). At the beginning, matrices shown in Figure 5(b) and Figure 5(c) are close to identity matrices. Then the transition matrices tend to become block diagonal matrices with 2 blocks, as shown in Figure 5(d)-5(f). The representative words for latent factors in the first block are 'state-linear-classification", 'network-neural-networks", 'kernelimage-space", 'network-neuralnetworks", 'neural-networks-state".

The representative words for latent factors in the second block are 'image-sparse-matrix", 'kernelfirst block primarily captured the correlations among the research topics about neural networks.

supervised-random", 'matrix-sample-random", 'inference-prior-latent", 'state-policy-gamma". The The second block reflects that, from the 1990s, statistical learning and Bayesian methods began to dominate, and these topics are highly correlated. Figure 5(a) illustrates the transition matrix inferred

by the PGDS, which is averaged over all time steps. Compared with the NS-PGDS, the PGDS can 298 not capture the informative time-varying transition dynamics. We also analyzed the features of the 299 proposed Dirichlet Markov chains. The left column of Figure 6 demonstrates transition matrices 300 of the first four sub-intervals of ICEWS dataset inferred by the NS-PGDS (Dir-Dir). Because of 301 the Dir-Dir construction, the consecutive transition matrices smoothly change over time and thus 302 the NS-PGDS may lack sufficient flexibility to capture rapid dynamics. The middle column of 303 Figure 6 illustrates the transition matrices inferred by the NS-PGDS (Dir-Gam-Dir), which takes 304 mutations among latent components into account and captured more complicated patterns. Transition 305 matrices inferred by the PR-Gam-Dir construction are shown in the right column of Figure 6, these 306 matrices not only exhibited sufficient flexibility but also captured sparser patterns compared with the 307 Dir-Gam-Dir construction.

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7 Conclusion

The Poisson-gamma dynamical systems with time-varying transition matrices, have been proposed to 310 capture complicated dynamics observed in non-stationary count sequences. In particular, Dirichlet 311 Markov chains are constructed to allow the underlying transition matrices to evolve over time. 312 Although the Dirichlet Markov processes lack conjugacy, we have developed tractable-but-efficient 313 Gibbs sampling algorithms to perform posterior simulation. The experiment results demonstrate the 314 improved performance of the proposed NS-PGDS in data smoothing and forecasting tasks, compared 315 with the PGDS with a stationary transition kernel. Moreover, the experimental results on several 316 real-world data sets show the explainable structures inferred by the proposed NS-PGDS. For the 317 future work, we plan to design a method that can find the point of change and thus the length of each 318 sub-interval can be determined automatically instead of a constant. We also consider to generalize 319 Dirichlet belief networks by incorporating the proposed Dirichlet Markov chain constructions, which 320 allow the hierarchical topics to mutate across layers, and thus can generate more rich text information. 321 And we also consider to capture non-stationary interaction dynamics among individuals over online 322

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A MCMCInference

Notation. When expressing the full conditionals for Gibbs sampling, we use the shorthand '-' to denote all other variables. We use ' · ' as an index summation shorthand, e.g., x · j = ∑ i x ij .

In this section, we present a fully-conjugate and efficient Gibbs sampler for the proposed NS-PGDS. The sampling algorithms depend on several key technical results, which we will repeatedly exploit, thus we list them below.

Negative-binomial Distribution . Let y ∼ Pois ( cλ ) , and λ ∼ Gam( a, b ) . If we marginalize over λ , then y ∼ NB ( a, c b + c ) is a negative-binomial distributed random variable. We can further parameterize it as y ∼ NB( a, g ( ζ )) , where g ( z ) = 1 -exp( -z ) and ζ = ln ( 1 + c b ) .

Lemma 1. If y ∼ NB( a, g ( ζ )) and l ∼ CRT( y, a ) , where CRT( · ) represents Chinese restaurant table distribution [26], then the joint distribution of y and l can be equivalently distributed as y ∼ SumLog( l, g ( ζ )) and l ∼ Pois ( aζ ) [11], i.e.

where SumLog( l, g ( ζ )) = ∑ l i =1 x i and x i ∼ Log ( g ( ζ )) are independently and identically logarithmic distributed random variables [27].

Lemma 2. Suppose n = ( n 1 , · · · , n K ) and n | n ∼ DirMult( n, r 1 , · · · , r K ) , where DirMult( · ) refers to Dirichlet-multimonial distribution. We sample the augmented variable q | n ∼ Beta( n, r · ) , where r · = ∑ K k =1 r k . According to [28], conditioning on q , we have n k ∼ NB( r k , q ) .

Lemma 3. If y · = ∑ S s =1 y s , and y s i.i.d ∼ Pois( λ s ) , s = 1 , · · · , S . Then y · ∼ Pois( ∑ S s =1 λ s ) and ( y 1 , · · · , y S ) ∼ Mult( y · , ( λ 1 ∑ S s =1 λ s , · · · , λ S ∑ S s =1 λ s )) , where Mult( · ) represents multinomial distribution [35].

Sampling y ( t ) vk : Use the relationship between Poisson and multinomial distributions as described by Lemma 3, given observed counts and latent parameters, we sample

Then the distribution of y ( t ) vk is y ( t ) vk ∼ Pois( δ ( t ) ϕ vk θ ( t ) k ) . 433

Sampling ϕ k : Via Dirichlet-multinomial conjugacy, the posterior of ϕ k is 434

Marginalizing over θ ( t ) k : Note that y ( t ) v = y ( t ) v · = ∑ K k =1 y ( t ) vk and y ( t ) vk ∼ Pois( δ ( t ) ϕ vk θ ( t ) k ) . Then we define y ( t ) · k = ∑ V v =1 y ( t ) vk . Because ∑ V v =1 ϕ vk = 1 , we obtain y ( t ) · k ∼ Pois( δ ( t ) θ ( t ) k ) .

We start by marginalizing over θ ( T ) k , using the definition of negative-binomial distribution, we obtain

where ζ ( T ) = ln(1 + δ ( T ) τ 0 ) . Next, we further marginalize over θ ( T -1) k . To this end, we first sample 438 auxiliary variables 439

By Lemma 1, the joint distribution of y ( T ) · k and l ( T ) k can be expressed as 440

435

436

437

Via Lemma 3, we re-express the auxiliary variables as 441

Then we define l ( T ) · k = ∑ K k 1 =1 l ( T ) k 1 k . Leveraging Lemma 3 and ∑ K k 1 =1 π i ( T -1) k 1 k = 1 , we obtain 442

443 444

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446

Next, note that y ( T -1) · k ∼ Pois( δ ( T -1) θ ( T -1) k ) , if we introduce m ( T -1) k = y ( T -1) · k + l ( T ) · k , then we have m ( T -1) k ∼ Pois ( θ ( T -1) k ( δ ( T -1) + ζ ( T ) τ 0 )) .

Because the prior of θ ( T -1) k is gamma distributed, by the definition of negative-binomial distribution, we can again marginalize over θ ( T -1) k to obtain

where ζ ( T -1) = ln(1 + δ ( T -1) τ 0 + ζ ( T ) ) . Then we introduce auxiliary variables 447

And similar to the case for t = T , we can obtain 448

449

450

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Thus we have marginalized over θ ( T -2) k . Note that we can repeat this marginalization process recursively until t = 1 with ζ ( t ) = ln(1 + δ ( t ) τ 0 + ζ ( t +1) ) and m ( T ) k = y ( T ) · k to maginalize over all the θ ( t ) k .

Sampling θ ( t ) k : Via the above marginalization process, to sample from the posterior of θ ( t ) k , we first sample the auxiliary variables. Setting l ( T +1) · k = 0 and ζ ( T +1) = 0 , we sample the augmented variables backwards from t = T, · · · , 2 ,

And via Lemma 3, we obtain 455

We compute ζ ( t ) recursively via 456

After sampling the auxiliary variables, then for t = 1 , · · · , T , by Poisson-gamma conjugacy, we 457 obtain 458

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Sampling Π ( i ) : We define M as the length of each sub-interval, and I as the number of intervals. For i = I , by Eq.(10), ( l ( I ) 1 k , · · · , l ( I ) Kk ) is multinomial distributed. Thus by multinomial-Dirichlet conjugacy, we obtain

where l ( I ) k 1 k indicates the summation of l ( t ) k 1 k over I -th sub-interval, i.e. l ( I ) k 1 k = ∑ T t =( I -1) M +1 l ( t ) k 1 k .

Inference for Dirichlet-Dirichlet Markov chains. For Dirichlet-Dirichlet Markov chains, α ( i ) k 1 k = ηKπ ( i -1) k 1 k . By Eq.(10), ( l ( i ) 1 k , · · · , l ( i ) Kk ) is multinomial distributed. If we marginalize ( π ( i ) 1 k , · · · , π ( i ) Kk ) , ( l ( i ) 1 k , · · · , l ( i ) Kk ) will be Dirichlet-multinomial distributed. Thus by Lemma 2, for i = I , we first sample the auxiliary variables as

Similarly, by Eq.(18), ( h ( i ) 1 k , · · · , h ( i ) Kk ) is also Dirichlet-multinomial distributed. Thus for i = 468 I -1 , · · · , 2 , we sample the auxiliary variables as 469

where l ( i ) k 1 k = ∑ iM ( i -1) M +1 l ( t ) k 1 k refers to the summation of l ( t ) k 1 k over i -th interval. Via Lemma 2, 470 conditioning on q ( i ) k , we have 471

Then via Lemma 1, we obtain 472

Note that by Eq.(17), h ( i ) k 1 k is Poisson distributed and by Lemma 3, we obtain 473

474

In addition, note that

via Dirichlet-multinomial conjugacy, for i = I -1 , · · · , 2 , we obtain 475

476

Specifically, for i = 1 , we have

For sampling η , note that ( h ( i ) k 1 k | -) ∼ Pois( -ηKπ ( i -1) k 1 k ln ( 1 -q ( i ) k ) ) , i = I, · · · , 2 . Given the 477 prior η ∼ Gam( e 0 , f 0 ) , via Poisson-gamma conjugacy, we obtain 478

Inference for Dirichlet-Gamma-Dirichlet Markov chains. For Dirichlet-Gamma-Dirichlet Markov 479 chains 480

By Eq.(10), ( l ( i ) 1 k , · · · , l ( i ) Kk ) is multinomial distributed. If we marginalize ( π ( i ) 1 k , · · · , π ( i ) Kk ) , 481 ( l ( i ) 1 k , · · · , l ( i ) Kk ) will be Dirichlet-multinomial distributed. Thus by Lemma 2, for i = I , we first 482 sample the auxiliary variables as 483

Similarly, by Eq.(27), ( g ( i ) · 1 k , · · · , g ( i ) · Kk ) is also Dirichlet-multinomial distributed. Thus for i = 484 I -1 , · · · , 2 , we sample the auxiliary variables as 485

Via Lemma 2, conditioning on q ( i ) k , we have 486

Thus via Poisson-gamma conjugacy, we obtain 488

Marginalizing over α ( i ) k 1 k , and via the definition of negative-binomial distribution, we have 489

Then using Lemma 1, we sample 490

Then via Lemma 1, we obtain 487

491

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495

and obtain

If we define g ( i ) k 1 k = g ( i ) k 1 · k = ∑ K k 2=1 g ( i ) k 1 k 2 k , and augment

Using Lemma 3 and ∑ K k 1 ψ ( i -1) kk 1 k 2 = 1 , we have,

Thus by Dirichlet-multinomial conjugacy, for i = I, · · · , 2 , we can obtain 496

For sampling γ ( i -1) k , note that by Eq.(26) and ∑ K k 1 ψ ( i -1) kk 1 k 2 = 1 , we have 497

Thus via Poisson-gamma conjugacy, we obtain 498

By gamma-gamma conjugacy, we have 499

Inference for Dirichlet-Randomized-Gamma-Dirichlet Markov chains. For Dirichlet500 Randomized-Gamma-Dirichlet Markov chains, 501

which can be equivalently represented as 502

By Eq.(10), ( l ( i ) 1 k , · · · , l ( i ) Kk ) is multinomial distributed. If we marginalize ( π ( i ) 1 k , · · · , π ( i ) Kk ) , 503 ( l ( i ) 1 k , · · · , l ( i ) Kk ) will be Dirichlet-multinomial distributed. Thus by Lemma 2, for i = I , we first 504 sample the auxiliary variables as 505

Similarly, by Eq.(39), ( g ( i ) · 1 k , · · · , g ( i ) · Kk ) is also Dirichlet-multinomial distributed. Thus for i = 506 I -1 , · · · , 2 , we sample the auxiliary variables as 507

Via Lemma 2, conditioning on q ( i ) k , we have 508

If ϵ α > 0 , we can sample the posterior of g ( i ) k 1 k via 511

where Bessel ( · ) denotes Bessel distribution. If ϵ α = 0 , we sample g ( i ) k 1 k via 512

Then via Lemma 1, we obtain 509

where SCH( · ) denotes the shifted confluent hypergeometric distribution [16]. 513

Defining g ( i ) k 1 k = g ( i ) k 1 · k = ∑ K k 2=1 g ( i ) k 1 k 2 k , we first augment 514

516

and because ∑ K k 1 ψ ( i -1) kk 1 k 2 = 1 , we have

Thus by Dirichlet-multinomial conjugacy, for i = I, · · · , 2 , we have 517

518

By gamma-gamma conjugacy, we have 520

̸

Specifically, for i = 1 , we have α (1) k 1 k = ν k 1 ν k , if k 1 = k . And α (1) k 1 k = ξν k , if k 1 = k .

Sampling ν k and ξ : As we sample Π ( i ) , by the definition of Dirichlet-multinomial distribution, we obtain

where l (1) k 1 k = ∑ M t =1 l ( t ) k 1 k . In particular, with a little abuse of notation here, for Dir-Dir construction, we take g (2) · k 1 k = h (2) k 1 k . We first sample

̸

521

522

523

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525

Then we sample 526

We further introduce 527

̸

̸

̸

̸

̸

529

And by gamma-gamma conjugacy, we obtain 531

The full procedure of our Gibbs sampling algorithms are summarized in Algorithm 1, Algorithm 2 532 and Algorithm 3. 533

Algorithm 1 Gibbs sampling algorithm for NS-PGDS (Dir-Dir Markov construction)

Input: observed count sequence { y ( t ) } T , iterations J

Initialize the model's rank K , hyperparameters γ 0 , ϵ 0 , e 0 , f 0

Sample

(

t

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vk v,k

via Eq.(6).

y

{

}

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t

via Eq.(48). Update t =1 . .

ζ

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δ

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t

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end for

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end for

end for

Sample ξ , { ν k } k , β via Eq.(46), Eq.(47) and Eq.(49) respectively.

end for

Output posterior means:

(1:

k

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)

(

i

k

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do

}

k

{

l

and

{

l

{

θ

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via Eq.(8) and Eq.(9) respectively.

k

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k

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δ

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β

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Algorithm 2 Gibbs sampling algorithm for NS-PGDS (Dir-Gam-Dir Markov construction)

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Algorithm 3 Gibbs sampling algorithm for NS-PGDS (PR-Gam-Dir Markov construction)

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The main contributions of this paper are the constructions of Poisson-Gamma dynamical systems with non-stationary transition dynamics and the corresponding Gibbs sampler. The constructions can be found in sec.2 and sec.3 and the derivation of Gibbs sampler can be found in sec.4 and the appendix. The experiments have demonstrated the effectiveness and features of the proposed model.

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It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

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Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes]

Justification: As we discussed in the conclusion part, the length of each sub-interval is a constant and is treated as a hyper-parameter of the model. In the future work, we plan to design a method that can find the point of change and thus the length of each sub-interval can be determined automatically.

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Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

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The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work extends Poisson-Gamma Dynamical Systems (PGDSs) by considering non-stationary transition dynamics to effectively capture the evolving dynamics of observed count sequences.\n\nThe authors propose a model where the underlying transition matrices evolve over time, based on three (gradually more complex and flexible) Dirichlet Markov chains.\n\nFor inference of the model, the authors make use of the Dirichlet-Multinomial-Beta data augmentation to derive a fully-conjugate Gibbs sampler.\n\nExperiments showcase improved data-smoothing and forecasting performance of the proposed method across several real-world datasets.\n\nSTRENGTHS:\n- Extending PGDS models to accommodate time-varying transition dynamics is of interest and significant\n\n- The proposed variations of Dirichlet-Markov chains provide flexibility in capturing different modeling assumptions\n\n- Devising a closed-form Gibbs sampler for posterior inference of this model is significant.\n    - The attained expressions seem correct to the best of my knowledge, although I did not carefully double-check the mathematical details of the derivation.\n\nWEAKNESSES:\n- The main limitation of this work is the assumption that the transition kernel is static within each sub-interval: i.e., the authors consider that the kernel can only change at discrete instants, while is constant within each sub-interval.\n\nQUESTIONS:\n- Can the authors justify and explain their choice for only allowing discrete-time transition kernel changes?\n    - How can one determine the length of sub-intervals in practice? How did the authors determine these sub-intervals in their experiments?\n    - Would it be possible to accommodate sub-intervals of varying length, $M$, and what would be the implications?\n    - Would it be possible to consider a continuously changing transition kernel? What would be the implications for the model and/or the estimation procedure?\n\n- The different transition kernel evolution models proposed do not only differ in their flexibility to capture different phenomena, but also on their complexity:\n    - Can the authors elaborate on the number of learnable parameters of each model?\n    - What is the computational and statistical complexity associated with each?\n    - Results do not seem to provide data-smoothing and forecasting performance improvements: is the added flexibility worth the complexity?\n\n- Sections 5 seems to have quite an overlap with some of the preliminaries introduced in Section 2:\n   - Would it be possible to merge both, or is there a reason why these two should be self-contained in different sections?\n\nLIMITATIONS:\nThe authors address the main limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-10"}

comments: SUMMARY:
Existing PGDS models struggle with capturing the time-varying transition dynamics seen in real-world data. To address this, the submission proposed a non-stationary PGDS, allowing the transition matrices to evolve over time, modeled by Dirichlet Markov chains. Using Dirichlet-Multinomial-Beta data augmentation techniques, a fully-conjugate and efficient Gibbs sampler is developed for posterior simulation. Experiments demonstrate that the proposed non-stationary PGDS achieves improved predictive performance compared to related models.

STRENGTHS:
The proposed non-stationary Poisson-Gamma Dynamical System offers several notable advantages. 

Firstly, its ability to allow transition matrices to evolve over time addresses the limitation of state-of-the-art PGDS models in capturing time-varying transition dynamics, making it more suitable for real-world count time series. 

Secondly, the use of specifically-designed Dirichlet Markov chains to model the evolving transition matrices enhances the model’s capacity to learn non-stationary dependency structures. 

Thirdly, the application of Dirichlet-Multinomial-Beta data augmentation techniques facilitates the development of a fully-conjugate and efficient Gibbs sampler for posterior simulation.

WEAKNESSES:
I did not find any obvious weaknesses.

QUESTIONS:
None.

LIMITATIONS:
The authors discussed some future work directions in the conclusion.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-10

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work extends  Poisson-Gamma Dynamical systems (PGDS) to model non-stationary dynamics by replacing the constant transition matrix $\\Pi$ with a time dependent one $\\Pi^{(t)}$ and the original Dirichlet prior on the columns with three different Dirichlet Markov chain constructions. The manuscript describes am efficient Gibbs-sampler for inference.\n\nSTRENGTHS:\nThe work addresses a relevant problem of modeling non-stationary dynamics in count time series. The provided extension relative to the original PGDS is sufficiently novel. I lack deep enough understanding of some parts related to the sampler, therefore I cannot assess if the construction of the sampler required new ideas or was a mechanical extension of the sampler for PGDS  (this being the main reason for my lower confidence score.). I tend to assume new ideas were necessary.\n\nWEAKNESSES:\nMy major problem is the experiment evaluation. In Table 1 in the NIPS dataset we can see results like $14.014 \\pm 4.387$ bolded, over values like $14.706 \\pm 4.414$, or $17.105 \\pm 6.449$. \nIn ICEWS values like $0.214 \\pm 0.008$ over $0.215 \\pm 0.007$ , in USEI $4.596 \\pm 0.562$ over $4.703  \\pm 0.538$, in COVID $6.969 \\pm 1.107$ over $7.566 \\pm 1.095$. These are mainly smoothing results. In the light of this I am not confident in the statement “As the experiment results shown in Table 1, the NS-PGDS exhibits improved performance in both data smoothing and forecasting tasks.”.  We do not know how the confidence interval was computed, or how many repeats were made. The lack of statistical rigor in the evaluation stands in striking contrast with the sophisticated Bayesian model presented. \n\nBesides this, other possible problem with the evaluation is that the manuscript states that default paramerters were used for the benchmark methods “GP-DPFA, PGDS, GMC-RATE, GMC-HIER, BGAR”  while the present method used specific K based on the dataset. It is very hard to tell if this is a fair comparison or not.\n\nQUESTIONS:\n1)  The time series are divided  to equally-spaced sub-intervals, and a $\\Pi$ transition matrix is infered for all interval. Why not use a diferent matrix for all time points with a very slowly varying Markow chain. Is this decision was made due to computational reasons or due to modeling assumptions?\n2)  Please provide the parameters for the benchmark methods in the appendix to facilitate comparison, it is very time consuming for the reader to search all default parameters from the references.\n3) Please bold all indistinguishable results in Table 2 using a statistical test, and reassess your conclusions.\n4) What is the reason of duplicated  factors in the exploratory analysis like “neural-network-networks” and “network-neural-networks”. What the order of the words means? Strength of association?\n\nLIMITATIONS:\nNo specific limitation section was provided. The part on future work in the Conclusion can be interpreted as pointing out some limitations of the current model, but a specific limitation statement would be preferable.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}

comments: SUMMARY:
This paper introduces non-stationary Poisson-Gamma dynamical systems, an extension of Poisson Gamma dynamical systems with a dynamic transition matrix. Decomposing the time steps into equally spaced subintervals, the transition matrices evolve between sub-intervals, remaining static within sub-intervals. The authors introduce three options for transitions to occur. The authors derive a Gibbs sampling scheme for exact posterior inference using data augmentation techniques and showcase the effectiveness of their method through a series of predictive and qualitative results.

STRENGTHS:
This is a well-written, organized paper that is easy to read. The proposed method allows for exact posterior inference through Gibbs sampling. The authors exhibit extensive predictive results across 4 datasets, although their method only exhibits marginal improvement as compared to Poisson Gamma dynamical systems.

WEAKNESSES:
I'm not convinced that the magnitude of the author's contribution, nor the significance of the paper is strong enough to warrant acceptance, and the methods produce only marginally better results than that of Poisson Gamma dynamical systems. The qualitative results are not groundbreaking.

QUESTIONS:
How does inference scale with the number of sub-intervals in each of the three methods?
How does the user choose the number of sub-intervals? Can they be fit adaptively?
How much extra training time do the proposed methods add, as compared to Poisson Gamma dynamical systems?

LIMITATIONS:
The authors address the limitations of their work, stating intention to address these limitations (e.g. constant sub-interval lengths) in future work.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-12