How to recognise whether an observed person walks or runs? We consider a dynamic environment where observations (e.g. the posture of a person) are caused by different dynamic processes (walking or running) which are active one at a time and which may transition from one to another at any time. For this setup, switching dynamic models have been suggested previously, mostly, for linear and nonlinear dynamics in discrete time. Motivated by basic principles of computations in the brain (dynamic, internal models) we suggest a model for switching nonlinear differential equations. The switching process in the model is implemented by a Hopfield network and we use parametric dynamic movement primitives to represent arbitrary rhythmic motions. The model generates observed dynamics by linearly interpolating the primitives weighted by the switching variables and it is constructed such that standard filtering algorithms can be applied. In two experiments with synthetic planar motion and a human motion capture data set we show that inference with the unscented Kalman filter can successfully discriminate several dynamic processes online.
Switching dynamic models are well-established in statistics @cite_15 @cite_9 , signal processing @cite_3 and machine learning @cite_4 @cite_2 @cite_21 . In contrast to these models, we define both, the dynamic models and the switching variables, using nonlinear dynamical systems with continuous states running in continuous time. This allows us to link our model more easily to computations implemented in analogue biological substrate such as the brain. Additionally, a formulation in continuous time allows us to easily perform time-rescaling of dynamical systems @cite_10 .
@cite_4 @cite_9 @cite_21 @cite_3 @cite_2 @cite_15 @cite_10
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Time varying problems usually have complex underlying structures represented as dynamic networks where entities and relationships appear and disappear over time. The problem of efficiently performing dynamic link inference is extremely challenging due to the dynamic nature in massive evolving networks especially when there exist sparse connectivities and nonlinear transitional patterns. In this paper, we propose a novel deep learning framework, i.e., Conditional Temporal Restricted Boltzmann Machine (ctRBM), which predicts links based on individual transition variance as well as influence introduced by local neighbors. The proposed model is robust to noise and have the exponential capability to capture nonlinear variance. We tackle the computational challenges by developing an efficient algorithm for learning and inference of the proposed model. To improve the efficiency of the approach, we give a faster approximated implementation based on a proposed Neighbor Influence Clustering algorithm. Extensive experiments on simulated as well as real-world dynamic networks show that the proposed method outperforms existing algorithms in link inference on dynamic networks. This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of state-space or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear time-series analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain. We present a switching Kalman filter model for the real-time inference of hand kinematics from a population of motor cortical neurons. Firing rates are modeled as a Gaussian mixture where the mean of each Gaussian component is a linear function of hand kinematics. A "hidden state" models the probability of each mixture component and evolves over time in a Markov chain. The model generalizes previous encoding and decoding methods, addresses the non-Gaussian nature of firing rates, and can cope with crudely sorted neural data common in on-line prosthetic applications. We introduce a new statistical model for time series that iteratively segments data into regimes with approximately linear dynamics and learns the parameters of each of these linear regimes. This model combines and generalizes two of the most widely used stochastic time-series models— hidden Markov models and linear dynamical systems—and is closely related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network (Jacobs, Jordan, Nowlan, & Hinton, 1991) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and therefore the exact expectation maximization (EM) algorithm cannot be applied. However, we present a variational approximation that maximizes a lower bound on the log-likelihood and makes use of both the forward and backward recursions for hidden Markov models and the Kalman filter recursions for linear dynamical systems. We tested the algorithm on artificial data sets and a natural data set of respiration force from a patient with sleep apnea. The results suggest that variational approximations are a viable method for inference and learning in switching state-space models.