Deep Explicit Duration Switching Models for Time Series
Authors: Abdul Fatir Ansari
Konstantinos Benidis
Richard Kurle
Ali Caner Turkmen
Harold Soh
Alex Smola
Bernie Wang
Tim Januschowski
Venue: NeurIPS 2021 Poster (2021)
Keywords: state space models
switching dynamical systems
explicit duration models
time series
segmentation
forecasting

Abstract: Many complex time series can be effectively subdivided into distinct regimes that exhibit persistent dynamics. Discovering the switching behavior and the statistical patterns in these regimes is important for understanding the underlying dynamical system. We propose the Recurrent Explicit Duration Switching Dynamical System (RED-SDS), a flexible model that is capable of identifying both state- and time-dependent switching dynamics. State-dependent switching is enabled by a recurrent state-to-switch connection and an explicit duration count variable is used to improve the time-dependent switching behavior. We demonstrate how to perform efficient inference using a hybrid algorithm that approximates the posterior of the continuous states via an inference network and performs exact inference for the discrete switches and counts. The model is trained by maximizing a Monte Carlo lower bound of the marginal log-likelihood that can be computed efficiently as a byproduct of the inference routine. Empirical results on multiple datasets demonstrate that RED-SDS achieves  considerable improvement in time series segmentation and competitive forecasting performance against the state of the art. 

Deep Explicit Duration Switching Models for Time Series
Abdul Fatir Ansarf2*
Konstantinos Benidis
Richard Kurle'
Ali Caner Türkmen'
Harold Soh?
Alexander J. Smola'
Yuyang Wang'
Tim Januschowskil
Amazon Research
National University of Singapore
Correspondence to: abdulfatir@u nus edu, (kbenidis, (urler)@@mazon com
Abstract
Many complex time series can be effectively subdivided into distinct regimes that exhibit persistent dynamics. Discovering the switching behavior and the statistical patterns in these regimes is important for understanding the underlying dynam- ical system. We propose the Recurrent Explicit Duration Switching Dynamical System (RED-SDS), flexible model that is capable of identifying both state- and time-dependent switching dynamics. State-dependent switching is enabled by a recurrent state-to-switch connection and an explicit duration count variable is used to improve the time-dependent switching behavior. We demonstrate how to perform efficient inference using hybrid a algorithm that approximates the posterior the continuous states via an inference network and performs exact inference for the discrete switches and counts. The model is trained by maximizing Monte Carlo lower bound of the marginal log-likelihood that can be computed efficiently byproduct of the inference routine. Empirical results multiple datasets demonstrate that RED-SDS achieves considerable improvement in time series segmentation and competitive forecasting performance against the state of the art.
Introduction
Time series forecasting plays key role in informing industrial and business decisions 17 while segmentation is useful for understanding biological and physical systems [40 45 34 State
24 g
Space Models (SSMs) [66] are powerful tool for such tasks esseccilly when combined with since they provide a principled framework for time series modeling.
neural networks 442
12 13
One of the most popular SSMs is the Linear Dynamical System (LDS) 55 43), which models the dynamics of the data using a continuous latent variable, called state, that evolves with Markovian linear transitions. The assumptions of LDS allow for exact inference of the states (7); however, they are too restrictive for weal-world systems that often exhibit piecewise linear or non-linear hidden dynamics with finite number of operating modes regimes. For example, the power consumption of a city may follow different hidden dynamics during weekdays and weekends. Such data are better explained by Switching Dynamical System (SDS) [ (1), SSM with additional set of latent variables called switches that define the operating mode active at the current timestep.
Switching events can be classified into dime-dependent or state-dependent [33). Historically, emphasis was placed on the former, which occurs after a certain amount of time has elapsed in a given regime. While in vanilla SDS switch durations follow geometric distribution, more complex long-term
Equal contribution.
Work done during an internship at Amazon Research.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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temporal patterns can be captured using explicit duration models 40 9. As a recent alternative to -mee-eppnndency. recurrent state-to-switch connections [35] have been proposed that capture stateddepnddent switching, i.e., a change that occurs when the state variable enters a region that is governed by a different regime. For added flexibility, these models can be used conjunction with transition/emission distributions parameterized by neural networks [25] 19 13 30] Recent works, e.g., [13 30), proposed hybrid inference algorithms that exploit the graphical model structure to perform approximate inference for some latent variables and conditionally exact inference for others.
Despite these advances in repre- sentation and inference, model- ing complex real-world temporal phenomena remains challenging. For example, state-of-the-art state- dependent models (e.g., [K3) lack the capacity to adequately capture time-dependent switching. Empir- ically, we find this hampers their ability to learn parsimonious seg- mentations when faced with com- plex pattems and long-temn depen- dencies (see Fig. for an example). Conversely, time-dependent switching models are "open-loop" and unable model state-conditional behavioral transitions that are common in many systems, e.g., in autonomous or multi-agent sys- tems (35) Intuitively, the suitability of the switching model largely depends the underlying data-generating process; city power consumption may be better modeled via time-dependent switch- ing, whilst the motion of a ball bouncing between two walls is driven by its state. Indeed, complex real-world processes likely involve both types of switching behavior.
Conund Trih
Modsing
RED 408
100 120
Time
Figure I: Segments (colored bars the bottom) inferred by baseline with no Explicit Duration (ED) modeling vs. our RED-SDS for a time series from the dancing becs dataset (top). The baseline struggles to learn long-term temporal patterns, particularly during the "waggle" phase of the bee dance.
Motivated by this gap, we propose the Recurrent Explicit Duration Switching Dynamical System (RED-SDS) that captures both dependee and time- denendennd switching. RED-SDS combines the recurrent state-to-switch connection with explicit duration models for switches. Notably, RED- SDS allows the incorporation of inductive biases via the hyperparameters of the duration models to better identify long-term temporal patterns. However, this combination also complicates inference, especially when using neural networks to model the underlying probability distributions. To address this technical challenge, we propose hybrid algorithm that (i) uses inference network for the continuous latent variables (states) and (ii) performs efficient exact inference for the discrete latent variables (switches and counts) using newaard-backaand routine similar to Hidden Semi-Markov Models 48 9. The model is trained by maximizing a Monte Carlo lower bound of the marginal log-likelihood that can be efficiently computed by the inference routine.
We evaluated RED-SDS on two important tasks: segmentation and forecasting. Empirical results on segmentation show that RED-SDS is able to identify both state- and time demendependent switching patterns, considerably outperforiing benchmark models. For example, Fig. [ shows that RED-SDS addresses the oversegmentation that occurs with existing strong baseline (13]. For forecasting, illustrate the competitive performance of RED-SDS with an extensive evaluation against state-of- the-art models on multiple benchmark datasets. Further, we show how our model able to simplify the forecasting problem by breaking the time series into different meaningful regimes without any imposed structure. As such, we manage to learn appropriate duration models for each regime and extrapolate the learned patterns into the forecast horizon consistently.
In summary, the key contributions of this paper are:
RED-SDS, novel non-linear state space model which combines the recurrent state-to- switch connection with explicit duration models to flexibly model switch durations;
an efficient hybrid inference and learning algorithm that combines approximate inference for states with conditionally exact inference for switches and counts;
a thorough evaluation on a number of benchmark datasets for time series segmentation and forecasting, demonstrating that RED-SDS can learn meaningful duration models, identify both state- and time-dependent switching patterns and extrapolate the learned patterns consistently into the future.
2

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2 Background: switching dynamical systems
Notation.
Matrices, vectors and scalars are denoted by uppercase bold, lowercase bold and low- erease normal letters, respectively. We denote the sequence {yi,.... .T) by Y1.7, where Yr is the value of y at time In our notation, we do not further differentiate between random variables and their realizations.
Switching Dynamical Systems (SDS) are hybrid SSMs that use discrete "switching" states 2 index one of K base dynamical systems with continuous states X,. The joint distribution factorizes as
P(YITT,II,T XI.T, 21:)) II (x
where p(x] Xn, (1) p(x (==)) is the initial (continuous and discrete) state prior. The base dynamical systems have continuous state transition p(x,/x,-1+ 2x) and continuous or discrete emission p(yr(xe) that can both be linear or non-linear.
The discrete transition p<ze|ze-11 of vanilla SDS is parametrized by stochastic transition matrix A E IR K K where the entry a;j = (1. represents the probability of switching from state to state j. This results in "open loop" as the transition only depends on the previous switch which inhibits the model from learning state-dependent switching patterns 35 Further, the state duration (also known as the sojourn time) follows a geometric distribution 9 where the probability of staying in state i for d steps is p;(d) (1 - a;; )a This memoryless switching process results in frequent regime switching, limiting the ability to capture consistent long-term time-dependent switching patterns. In the following, we briefly discuss two approaches that have been proposed to improve the state-dependent and dime-dependent switching capabilities in SDSs.
30) address state-dependent switching by chang- ing the switch transition distribution to p(z X,-1; ) called the state-to-switch recurrence- implying that the switch transition distribution changes every step and the sojourn time no longer follows a geometric distribution. This extension complicates inference. Furthermore. the first-order Markovian recurrence does not adequately address long-term time-dependent switching.
Recurrent SDS. Recurrent SDSs (e.g.. 66
35
7
Explicit duration SDS. Explicit duration SDSs are family of models that introduce additional random variables to explicitly model the switch duration distribution. Explicit duration variables have been applied to both HMMs and SDSs with Gaussian linear continuous states; the resulting models are referred to Hidden Semi-Marko Models (HSMMs) (38 48). and Explicit Duration Switching Linear Gaussian SSMs (ED-SLGSSMs) p 40 I0, respectively. Several methods have been proposed in the literature for modeling the switch duration, e.g.. using decreasing or increasing count, and indiicaonn variables. In the following, we briefly describe modeling switch duration using increasing count variables and refer the reader to Chiappa [9] for details.
Increasing count random variables c; represent the run-length of the currently active regime and can either increment by or reset to 1. An increment indicates that the switch variable is copied over to the next timestep whereas a reset indicates regular Markov transition using the transition matrix A. Each of the K switches has distinct duration distribution Pk. a categorical distribution over (dming.... ddmax. where dmin and dmax delimit the number of steps before making a Markov transition. Following (40] 9 the probability of count increment is given by
(x(c) 1 px(c)
(2)
Edaes (b((d
The transition of count e, and switch z, variables is defined as
p(a|z=-== = k,(c-))== = Pr (C--)) 1 U/ (C-1)
if 4 = 4-1+1
(3)
if c =
p( = jl 21111 = i,0) = 852- if
4 >
(4)
(i,
if 4 1
where 5oond denotes the delta function which takes the value I only when cond is true.
Although SDSs with explicit switch duration distributions can identify long-term time-dependent switching patterns, the switch transitions are not informed by the inbibiiniiiiing their ability to
3

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model depeedentt switching events. Furthermore, to the best of our knowledge, SDSs with explicit duration models have only been studied for Gaussian linear states [0]
3 Recurrent explicit duration switching dynamical systems
In this section we describe the Recurrent Explicit Duration Switching Dynamical System (RED-SDS) that combines both state-to-switch recurrence and explicit duration modeling for switches in single non-linear model. We begin by formulating the generative model as a recurrent switching dynamical system that explicitly models the switch durations using increasing count variables. We then discuss how to perform efficient inference for different sets of latent variables. Finally, we discuss how to estimate the parameters of RED-SDS using maximum likelihood.
3.1 Model formulation
Consider the graphical model in Fig. (a); the joint distribution of the counts C, (1...., dmax h. the switches 2, E {1,..., K}, the states x, R., and the observations ya Rd, conditioned on the control inputs u, E RR, factorizes as
po(Y1.rs X1.7, R1.2, Gia (1:)) - ply x1)p(x1(az)) up)((//
I
T
(oxxlx))p(xxxxx-- u. M1)p(a Xt-1, (. w.)p(erlee-).ccc.c u.)
(5)
Similar to 40 99. we consider increasing count variables o to incorporate explicit switch durations every timestep and represent the
into the model, i.e., e can either increment by or reset run-length of the current regime. A self-transition is allowed after the exhaustion of d steps for flexibility. In the subsequent discussion we omit the control inputs u, for clarity of exposition.
We model the initial prior distributions in Eq. (5) for the respective discrete and continuous case as
p(=1) Cat(=1;
(6)
p(x, (2)) -x )..)
where Cat denotes a categorical and N multivariate Gaussian distribution. The transition distribu- tions for the discrete variables (count and switch) are given by
Uz (c--1) Una (Ct-1)
a =0-+11
p(c(22-11.--1)
(8)
if Ct =1=
: Cat(zz S, (f((x)1)))))) if G ==1²
if 4 >>>
where S, is the tempered softmax function (cf. Section 3.3 with temperature and fz can be linear function or a neural network. The probability of a count increment U): for a switch is defined
via the duration model Pk as in Eq. 2 The continuous state transition and the emission are given by
p(x, X
2x) = (x:;; (x.- 24), I F2 (x:-1, 2()),
(10)
Mx.) = fx))) (x:)),
(11)
where f2. fy' fy are again linear functions or neural networks.
The model is general and flexible enough to handle both state- and time-dependent switching. The switch transitions 2 + zt are conditioned on the previous state x;-1 which ensures that the switching events occur in "closed loop". The switch duration models Pk provide flexibility to stay long term in the same regime, allowing better capture time-der deeeddeenddnnt switching. We use increasing count variables to incorporate switch durations into our model as they are more amenable to the case when the count transitions depend on the control u, instance, decreasing count variables, another popular option [1]
36 9 deterministically count down from the sampled segment duration length 1. This makes difficult to condition the switch duration model on the control inputs. contrast, increasing count variables increment or reset probabilistically at every timestep.

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M++11
M---
Me--
M++1
h
h
he+s
X-++
he
hi+
X--1
X
Xe+1
Xe=1
X
Xte
(a) Generative Model
(b) Inference
Figure 2: (a) Forward generative model of RED-SDS. (b) Left: Approximate inference for the states X2 using inference network. h, is given by non-causal network and h² is given by causal RNN. Right: Exact inference for switch &5 and count & variables given pseudo-observations (highlighted in red) of x, provided by the inference network. (Shaded) circles represent (observed) random variables, diamonds represent deterministic nodes, and dashed lines represent optional connections.
3.2 Inference
Exact inference is intractable in SDSs and scales exponentially with time 32 Various approximate
inference procedures have been developed for traditional SDSs [14 21 6 while more recently inference networks have been used for amortized inference for all or subset of latent variables [25]
28 1 30). Particularly, Dong et al. 13 used inference network for the states and performed exact HMM- like inference for the switches, conditioned on the states, We take similar approach and use an inference network for the continuous latent variables (states) and perform conditionally exact inference for the discrete latent variables (switches and counts) similar to the forward backward procedure for HSMM 48 We define the variational approximation to the true posterior p(X1.T, æ1.7, G1.T ly::)) as (XITT, Zi:T, C1.T Y1.T) = (x(x).5 1.rl (y:r) (oo (=1:7, C1:T (y VI.7, X1:7) where & and 0 denote the parameters of the inference network and the generative model respectively.
Approximate inference for states. The posterior distribution of the states, 4o (x1.T yy.r), is approx- imated using an inference network. We first process the observation sequence yI.T using non-causal network such as a bi-RNN or a Transformer [46] to simulate smoothing by incorporating both past and future information. The non-causal network returns an embedding of the data h/rr which is then fed to causal RNN that outputs the posterior distribution g0 (x N1:T) = II, ( h.r) See Fig. (b) for an illustration of the inference procedure.
Exact inference for counts and switches. Inference for the switches ø1.7 and the counts cir be performed exactly conditioned on states X1:7 and observations y1.7- Samples from the approximate posterior X1:7 ~ ((1::T (Y:r) are used as pseudo-observations of x1:7 to infer the posterior distribution pe (11::T,,C11 (y 1:T, X. ). naive approach infer this distribution is by treating the pair (Cz, 2x) as a "meta switch" that takes K Kdmax possibles values and perform HMM-like -wwardd-ackwardd inference. However, this results in computationally expensive OTT-2 d procedure that scales poorly with dmax-- Fortunately, we can pre-compute some terms in the forward- backward equations by exploiting the fact that the count variable can only increment by 1 or reset to at every timestep. This results in an O(T + dmax)) algorithm that scales gracefully with d 9.. The forward a, and backward 8, variables, defined
ar((2, (x) =p(y X1s, 21, Gt), 21, ot),
(12)
Be(2,, = -
(13)
can be computed by modifying the forward-backwart recursions used for the HSMM 9 to handle the additional observed variables X1:t. We refer the reader to Appendix A.I for the exact derivation.
5

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3.3 Learning
The parameters fe, fe can be learned by maximizing the evidence lower bound (ELBO):
p(y1.T X1.7, 21.7, (1.T.
LELBO
log (/xirr/w.r)pp(een,s
(14)
(g(xrrly)))) log p(y1.r, X1:T)
((x1:T yI:T)
The likelihood term p(yIT, X1.T) can be computed using the forward variable ar (FT,, er) marginalizing out the switches and the counts,
by
P(Y)T,XIT) Σ OT (ET, er),
(15)
and the entropy term -Rggxilyy [log ((x:)IV::) can computed using the approximate posterior g(xi. (i)) output by the inference network. The ELBO can be maximized via stochastic gradient ascent given that the posterior g ((xiTTVVir) is reparameterizable.
We note that Dong al. 13 used lower bound for the likelihood term in Switching Non-Linea Linear Dynamical Systems (SNLDS); however, it can be computed succinctly by marginalizing out the discrete random variable (i.e., the switch in SNLDS) from the forward variable dr,, similar to Eq. (15). Using our objective function, we observed that the model was less prone to posterior collapse (where the model ends up using only switch) and we did not require the additional ad-hoc KL regularizer used in Dong et al. 13 Please refer to Appendix B.4 for a brief discussion on the likelihood term in SNLDS.
Temperature annealing. We use the tempered softmax function S, to map the logits to probabilities for the switch transition p(z, X1-1, Fe-1; 1) and the duration models Pk (d) which is defined as
exp
(o); Σ, exp
(16)
21
where is a vector of logits. The temperature T is deterministically annealed from a high value during training. The initial high temperature values soften the categorical distribution and encourage the model to explore all switches and durations. This prevents the model from getting stuck in poor local minima that ignore certain switches or longer durations which might explain the data better.
Related work
The most relevant components of RED-SDS are recurrent state-to-switch connections and the explicit duration model, enabling both for state- and time-dependent switching. Additionally, RED-SDS allows for efficient approximate inference (analytic for switches and counts), despite parameterizing the various conditional distributions through neural networks. Existing methods address only a subset of these features
discuss in the following.
The most prominent SDS is the Switching Linear Dynamical System (SLDS). where each regime is described by linear dynamics and additive Gaussian noise. A major focus of previous work has been on efficient approximate inference algorithms that exploit the Gaussian linear substructure (e.g., [2]] 44 14)). In contrast to RED-SDS, these models lack recurrent state-to-switch connections and duration variables and are limited to linear regimes.
Previous work has addressed the state-dependent switching by introducing a connection to the continuous state of the dynamical system [6] 35] 7 B0]. The additional recurrence complicates inference w.r.t. the continuous states; prior work uses expensive sampling methods in order to approximate the corresponding integrals [6] as part of message passing algorithm for joint
inference of states and parameters [35]. On the other hand, ARSGLS [30] avoids sampling the continuous states by using conditionally linear state-to-switch connections and softmax-transformet
Gaussian switch variables. However, both the ARSGLS and the related KVAE [19 can be interpreted as an SLDS with "soft" switches that interpolate linear regimes continuously rather than truly discrete
states. This makes them less suited for time series segmentation compared to RED-SDS Contrary to

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the aforementioned models, RED-SDS allows non-linear regimes described by neural networks and incorporates a discrete explicit duration model without complicating inference w.r.t. the continuous states, since closed-form expressions are used for the discrete variables instead. Using amortized variational inference for continuous variables and analytic expressions for discrete variables has been proposed previously for segmentation in SNLDS [3]. RED-SDS extends this via an additional explicit duration variable that represents the run-length for the currently active regime.
and segmentation [O 26]. For instance, BOCPD 2 is a Bayesian online changepoint detection model with explicit duration modeling. RED-SDS improves upon BOCPD by allowing for segment labeling rather than just detecting changepoints. The HDP-HSMM [26] a Bayesian non- con-parmettic extension to the traditional HSMM. Recent work [TT 36 has also combined HSMM with RNNs for amortized inference. These models Seiielss-eingg variants of HSMM-do not model the latent dynamies of the data like RED-SDS. Chiappa and Peters
Explicit duration variables have previously been proposed for changepoint detection
[10] proposed approximate inference techniques for variant of SLDS with explicit duration modeling. In contrast, RED-SDS more general non-linear model that allows for efficient amortized sirennce-llose--eee w.r.t. the discrete latent variables.
Experiments
In this section, we present empirical results on two prominent time series tasks: segmentation and forecasting. Our primary goals were to determine if RED-SDS (a) can discover meaningful switching patterns in the data in unsupervised manner, and (b) can probabilistically extrapolate a sequence of observations, serving as viable generative model for forecasting. the following, we discuss the main results and relegate details to the appendix.
5.1 Segmentation
We experimented with two instantia- tions of our model: RED-SDS (com- plete model) and ED-SDS, the ab- lated variant without statc-to-switch recurrence. We compared against the closely related SNLDS [13 trained with modified objective func- tion. The original objective proposed in [13) suffered from training difficul- ties: it resulted in frequent posterior collapse and was sensitive to the cross- entropy regularization term. Our ver- sion of SNLDS can be seen as a special case of RED-SDS with dmmax 1, i.e., without the explicit duration model- ing (cf. Appendix [B.4]. We also con- ducted preliminary experiments on soft-switching models: KVAE [19] and ARSGLS (30]. However, these models use continuous interpolation of the different operating modes which can- not always be correctly assigned to a single discrete mode, hence we do not report these unfavorable findings here (cf. Appendix (B.4). For all models, we performed segmentation by taking the most likely value of the switch at each timestep from the posterior distribution over the switches. As the segmentation labels are arbitrary and may not match the ground truth labels, we evaluated the models using multiple metrics: frame-wist wise segmentation accuracy (after matching the labelings using the Hungarian algorithm (9)).
going down
go MS
hpul
Reco 107uctio
Gound Trun
ED-808
Time
(a) Bouncing ball
moda
mode
mode
Reco solution
Ground True
BNLD9
ED-808
RED-005
125
142
Time
(b) 3 mode system
Gound Trush
SNLDS
ED-808
RED-808
28
100
120
Time
(c) Dancing bees
Figure 3: Qualitative segmentation results on the bouncing ball, 3 mode system, and dancing bees datasets. Background colors represent the different operating modes.
Normalized Mutual Information (NMI) [47]. and Adjusted Rand Index (ARD [3] (cf. Appendix B.2
7

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Table I: Quantitative results on segmentation tasks. Accuracy, NMI, and ARI denote the frame-wise segmenta- accuracy, the Normalized Mutual Information, and the Adjusted Rand Index metrics respectively (higher values are better). Mean and standard deviation are computed over 3 independent runs.
bouncing ball
3 mode system
dancing bees dancing bees (K=2)
SMLDS
0.97=0.00
0.82+0.08
0.44+0.01
0.63+0.02
Accuracy
ED-SDS (ours)
0,95+0.00
0.97=0.00
0.56=0.06
0.79=0.09
RED-SDS (ours)
0.97=0.00
0.98+0.00
0.73+0.10
0.91=0.00
SNLDS
0.83+0.01
0.63+0008
0.10#0.04
0.05+0.02
NMI
SDS (ours)
0.71=0.00
0.89=0.01
0.28+0.02
0.31=0.17
RED-SDS (ours)
0.81=0.00
0.91#0.01
0.48=0.07
0.60=0.09
SNLDS
0.90=0.01
0.67#0.11
0.10#0.03
0.07=0.02
ARI
ED-SDS (ours)
0.81#0.01
0.93+0.00
0.27+0.04
0.36+0.19
RED-SDS (ours)
0.88+0.00
0.95+0.01
0.53+0.11
0.68+0.111
We conducted experiments on three benchmark datasets: bouncing ball, 3 mode system, and dancing bees to investigate different segmentation capabilities of the models. We refer the reader to Appendix B.J for details on how these datasets were generated/preproosssed For all the datasets, we set the number of switches equal the number of ground truth operating modes.
Bouncing ball. We generated the bouncing ball dataset similar to 13 which comprises univariate time series that encode the location of ball bouncing between two fixed walls with constant velocity and clastic collisions. The underlying system switches between two operating modes (going up/down) and the switching events completely governed by the state of the ball, i.e., switch occurs only when the ball hits a wall. As such, the switching events are best explained by state-to- switch recurrence. All models able to segment this simple dataset well as shown qualitatively in (a) and quantitatively in Table We note that despite the seemingly qualitative equivalence, models with state-to-switch recurrence perform best quantitatively. RED-SDS learns to ignore the duration variable by assigning almost all probability mass to shorter durations (cf. Appendix B.5). which is intuitive since the recurrence best explains this dataset.
3 mode system. We generated this dataset from a switching linear dynamical system with 3 operating modes and explicit duration model for each mode (shown in Fig. 4 (a)). We study this dataset in the context of dime-depeneent switching-
operating mode switches after a specific amount of time elapses based its duration model. Both ED-SDS and RED-SDS learn to segment this dataset almost perfectly as shown in Fig. (b) and Table owing to their ability to explicitly model switch durations. In contrast, SNLDS fails to completely capture the long-term temporal patterns, resulting in spurious short-term segments as shown in Fig. 3 (b). Moreover, RED-SDS is able to recover the duration models associated with the different modes (Fig. 4 These results demonstrate that explicit duration models can better identify the time-dependent switching patterns in the data and can leverage prior knowledge about the switch durations imparted via the dman and dmax hyperparameters.
Dancing bees. We used the publicly- available dancing bees dataset [0 challenging dataset that exhibits long- term temporal patterns and has been
2
studied previously in the context of
time series segmentation [4] 39 [8]
The dataset comprises trajectories of six dancer honey bees performing the waggle dance. Each trajectory consists of the 2D coordinates and the heading angle of a bee at every timestep with three possible types of motion: waggle, turn right, and turn left. Fig. (c) shows that RED-SDS is able to segment the complex long- term motion patterns quite well. In contrast, ED-SDS identifies the long segment durations but often infers the mode inaccurately while SNLDS strug-
(a) True duration model
3
(b) Learned duration model
Figure 4: The ground truth duration model for the 3 mode system dataset (top) and the duration model learned by RED-SDS (bottom). The x-axis represents the durations from to 20 the y-axis represents the duration probabilities of the 3 modes pa(d), (dd).. and pa (d).

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Table 2: CRPS metries (lower is better). Mean and standard deviation computed over independent runs.
method achieving the result highlighted bold.
exchange
solar
electricity
traffic
wiki
DeepAR
0.019=0.002
0.440#0.004
0.062+0.004
0.138#0.001
0.855=0.552
DeepState
0.017=0.002
0.379#0.002
0.088=0.007
0.131=0.005
0.338=0.017
KVAE-MC
0.020=0.001
0.389#0.005
0.318=0.011
0.261=0.016
0.341#0.032
KVAE-RB
0.018+0.001
0.393+0.006
0.305#0.022
0.221+0.002
0.317=0.013
RSGLS-ISSM
0.01410.001
0.358+0.000
0.09140.004
0.206+0.00
0.345+0.010
ARSGLS
0.02240.001
0.371
0.154+0.005
0.175#0.000
0.283+0.006
RED-SDS (ours)
0.013+0.001
0.419#0.010
0.06620.002
0.129#0.002
0.318+0.006
gles learn the long-term motion patterns resulting in oversegmentation. This limitation of SNLDS is particularly apparent the "waggle" phase of the dance which involves rapid, shaky motion. We also observed that sometimes ED-SDS and RED-SDS combined the turn right and turn left motions into single switch, effectively segmenting the time series into regular (turn right and turn left) and waggle motion. This results in another reasonable segmentation, particularly in the absence of ground-truth supervision. We thus reevaluated the results after combining the turn right and turn left labels into a single label and present these results under dancing bees (K=2) in Table Empirically, RED-SDS significantly outperforms ED-SDS and SNLDS on both labelings of the dataset. This suggests that real-world phenomena are better modeled by a combination of state- and time-dependent modeling capacities via state-to-switch recurrence and explicit durations, respectively.
5.2 Forecasting
We evaluated RED-SDS in the context of time series forecasting on 5 popular public datasets available in GluonTS 4 following the experimental set up of [30]. The datasets have either hourly daily frequency with various seasonality patterns such as daily. weekly, or composite. In Appendix C.I we provide a detailed description of the datasets. We compared RED-SDS SDS closely related forecasting models: ARSGLS and its variant RSGLS-ISSM [30]; KVAE-MC and KVAE-RB, which refer to the original KVAE 19] and its Rao-Blackwellized variant (as described in 30) respectively; DeepState 422]; strong discriminative baseline that uses an autoregressive RNN (cf. Appendix C.4 for a discussion on these baselines).
and DeepAR
44
We used data prior to fixed forecast date for training and test the forecasts on the remaining unseen data; the probabilistic forecasts are conditioned on the training range and computed with 100 samples for each method. We used a forecast window of 150 days and 168 hours for datasets with daily and hourly frequency, respec- tively. We evaluated the forecasts us- ing the continuuu ranked probability score (CRPS) 337, proper scoring rule (22 (cf. Appendix C.2 The re-
(a) K =
3pp
sults are reported in Table 2 RED-SDS compares favorably competitively to the baselines on out datasets.
Figure illustrates how RED-SDS can infer meaningful switching patterns from the data and extrapolate the learned patterns into the future. perfectly reconstructs the past of the time series and segments an inter- pretable manner without an imposed
Figure 5: Segmentation and forecasting electricity time series for (a) K = 2 and K switches. The black vertical line indicates the start forecasting. The plots at second row of each figure indicate the duration model at the timestep marked by the corresponding vertical dashed lines.

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seasonality structure, e.g.. as used in DeepState and RSCLS-ISSM. The same switching pattern is consistently predicted into the future, simplifying the forecasting problem by breaking the time series into different regimes with corresponding properties such trend or noise variance. Further, the duration models several timesteps (the duration model is conditioned on the control u,) indicate that the model has learned how long each regime lasts and therefore avoids oversegmentation which would harm the efficient modeling of each segment. Notably, the model learns meaningful regime durations that sum up to the 24-hour day/night period for both K = 2 and K 3 switches. Thus, RED-SDS brings the added benefit of interpretabiity-bboll in terms of the discrete operating mode and the segment durations- while obtaining competitive quantitative performance relative to the baselines.
6 Conclusion and future work
Many real-world time series exhibit prolonged regimes of consistent dynamics as well as persistent statistical properties for the durations of these regimes. By explicitly modeling both state- and time-dependent switching dynamics, our proposed RED-SDS can more accurately model such data. Experiments on a variety of datasets show that RED-SDS algorithm that combines amortized variational inference with exact inference for continuous and discrete latent criialles-improvss upon existing models on segmentation tasks, while performing similarly to strong baselines for forecasting.
equipped with an efficient inference
One current challenge of the proposed model that learning interpretable segmentation sometimes requires careful hyperparameter tuning (c.g., dmin and dmax). This is not surprising given the flexible nature of the neural networks used as components in the base dynamical system. A promising future research direction is to incorporate simpler models that have predefined structure, thus exploiting domain knowledge. For instance, many forecasting models such as DeepState and RSGLS-ISSM parametrize classical level-trend and seasonality models in non-linear fashion. Similarly, simple forecasting models with such structure could be used as base dynamical systems along with more flexible neural networks. Another interesting application is semi-supervised time series segmentation. For timesteps where correct regime label is known, it is straightforward to condition on this additional information rather than performing inference; this may improve segmentation accuracy while providing an inductive bias that corresponds to an interpretable segmentation.
Funding disclosure
This work was funded by Amazon Research. This research is supported in part by the National Research Foundation Singapore under its Al Singapore Programme (Award Number: AISG-RP-2019. 011) to H. Soh.
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13


Reviews:
[{"review_id": "UWinAh9C_Nx", "reviewer": "Reviewer_Naxb", "summary": "The authors proposed a new state-space model which combines the previous two models: recurrent switching LDS and explicit duration together. The authors proposed an inference method to fit the model and applied the proposed method to several real datasets.", "questions": "", "limitations": "", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: The proposed method combines recurrent state-to-switch with explicit duration. The model is new, but the idea seems straightforward and not revolutionary.\n\nQuality: The paper did extensive experiments on multiple datasets qualitatively and quantitatively. The proposed model has a more complicated structure but doesn't seem to improve previous methods too much. The forecasting results are just comparable with the current methods. The segmentation results will be discussed below. \n1. Fig. 3, what are the inputs u for each dataset? Why SNLDS, ED-SDS, RED-SDS got very similar reconstruction, yet different segmentation results?\n2. For dancing bee dataset, check paper https://people.eecs.berkeley.edu/~jordan/papers/fox-etal-nips08.pdf. They also fit ARHMM & SLDS with enhanced self-persistence on the dancing bee datasets. The probability distribution of staying at the previous state still follows a geometric distribution. They don't have recurrent state-to-switch or explicit duration in their model, but the results are better than the one shown in the paper.\n3. Have you tried a more complicated segmentation task (e.g. number of switching states more than 3, like the one used in https://arxiv.org/pdf/1610.08466.pdf)? Current tasks in the paper are simple and seem not enough to show the superiority of having recurrent state-to-switch and explicit duration. Bouncing ball & 3 mode system are simulated data. Dancing bees data already has good (even better) SOTA results. \n\nClarity: The paper is clearly written.\n\nSignificance: As discussed above, the paper did comparisons with previous methods qualitatively and quantitatively, but doesn't seem to improve current methods too much. The tasks in real data seem not complicated enough. It would be good if the code can be open-source so that others can apply the proposed model to their own datasets.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "See the main review section above for suggestions for improvement. I don't foresee any negative societal impact of their work. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "Pn9doLfX2nX", "reviewer": "Reviewer_2NJA", "summary": "In this paper, the authors introduce RED-SDS an extension to switching dynamical systems (SDS) that combines state-dependent and duration-dependent transitions. ", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "# Strengths\nI think the combination of state-dependent and duration-dependent transitions is very cool, as it leads to a much more flexible model. For learning RED-SDS, the authors build upon previous work on variational inference for SDS [1]. Crucially, the authors demonstrate the gradient for$ \\underset{q(x_{1:T} \\vert y_{1:T})}{\\mathbb{E}}  [\\log p(y_{1:T}, x_{1:T})]$ can be computed more succinctly as opposed to [1]; moreover, this also leads to better training, removing the need of the regularizer that was used in [1]. The experiments section was also very well done and very thorough.\n\n# Weaknesses\nI noticed that details are missing for the experiments section. In neither the main text nor the appendix, are there details regarding the architectures being used i.e. number of hidden units and activation functions. Moreover, I don't see any details regarding the annealing schedule for the tempered softmax function.\n\n# Questions/Comments \n1) In line 556, shouldn't it be $ P \\in \\mathbb{R}^{K \\times (d_{max} - d_{min})}$, not $ P \\in \\mathbb{R}^{K \\times d_{max}}$?\n2) In line 602, since the latent dimension is 2, I don't think it qualifies as a univariate time series.\n3) In lines 619-629, the dancing bees dataset was described where it was stated that the data is comprised of 6 bees. Were the methods only trained on data from one bee or where each bee considered iid?\n4) In line 671, it was stated the dimensionality of the state variables was set to 4 for the bouncing ball and 3 mode datasets. I find this weird (and slightly concerning), as 1) the true latent dimension is known and 2) the data generating process is a subset of RED-SDS, thus setting the latent dimension to be the true dimension should suffice.\n5) In lines 769-773, the authors noted that they took the scaling of the data into effect by using the change of variables formula. This is interesting because the standard is just to use scale the data and treat it as if it wasn't scaled. I'm curious as to why the authors did this (Note, that the change of variables formula is only for one-to-one function; the sample average and standard deviation are not one-to-one).\n6) In line 776, the authors stated they considered a range of $K \\in [2, 5]$. I think it would be more appropriate to write it as $K \\in$ {$2, 3, 4, 5$}, as $K$ can only be discrete.\n\n# Conclusion\nI think this was a good paper overall! Only minor modifications are needed.\n\n# References\n[1] Collapsed amortized variational inference for switching nonlinear dynamical systems, https://arxiv.org/abs/1910.09588", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors adequately addressed the limitations and potential negative societal impact of their work.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "10 hours"}, "extra_text": {}}, {"review_id": "AXKDg2I5un", "reviewer": "Reviewer_ByG2", "summary": "The authors propose Recurrent Explicit Duration Switching Dynamical System model, that is used to detect state and time switching dynamics. The method falls to the category of generalized switching state space model (Ghahramani & Hinton 2000). In particular, they use recurrent state-to-state switching with explicit duration models and provide approximate inference algorithm that is using evidence lower bound. Authors verify they model on 3 segmentation task from time series with ground truth and forecasting task on 5 datasets. ", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Authors make a nice study, however I am not sure about the following:\n1. Related work on jump models\nThe authors are missing big part of related work related to Jump models. Jump models have been intensively used in financial machine learning and they also generalize hidden markov models by having abrubt and presistent change of state dynamics. \nSee Bemporad et al. (2018), that has proved that the HMM is a special case of a jump model. \n\nBemporad, A., V. Breschi, D. Piga, and S. Boyd. “Fitting jump models.” Automatica,\nvol. 96 (2018), pp. 11–21.\n\n2. How does their model tolerate more noisy datasets\nAuthors should try to see how they model behaves on very noisy datasets e.g. financial.\n\n3. Based on the results in table 2 for the forecasting, the method is performing the best only on 2 datasets. Do authors have intuition why?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "yes. ", "review_text": "", "extra_scores": {"ethical_concerns": "no ethical concerns. ", "needs_ethics_review": "No", "ethics_review_area": ["I don’t know"], "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper takes a nice idea (combining recurrent state-to-switch dependencies with semi-Markov duration distributions), pairs it with an amortized inference algorithm, describes the approach very well, and does a thorough set of experiments with reasonable baseline comparisons to recent methods. While it is a relatively simple idea, it is well executed and I think it will be a very nice contribution to the field. 

In private discussion, Reviewers Naxb and GyB2 agreed that they would increase their scores to a 6, which would yield two 6s and a 7, for an average of 6.33.  I strongly encourage the authors to make the promised changes in the final publication.
Decision: Accept (Poster)