Autoregressive logistic processes

1989 ◽  
Vol 26 (03) ◽  
pp. 524-531 ◽  
Author(s):  
Barry C. Arnold ◽  
C. A. Robertson
Keyword(s):  
Markov Process ◽  
Stochastic Model ◽  
Moving Average ◽  
Higher Order ◽  
Invariant Distribution ◽  
Absolutely Continuous ◽  
Joint Distributions ◽  
Moving Average Processes ◽  
Stationary Markov Process

A stochastic model is presented which yields a stationary Markov process whose invariant distribution is logistic. The model is autoregressive in character and is closely related to the autoregressive Pareto processes introduced earlier by Yeh et al. (1988). The model may be constructed to have absolutely continuous joint distributions. Analogous higher-order autoregressive and moving average processes may be constructed.

Autoregressive logistic processes

1989 ◽  
Vol 26 (3) ◽  
pp. 524-531 ◽  
Author(s):  
Barry C. Arnold ◽  
C. A. Robertson
Keyword(s):  
Markov Process ◽  
Stochastic Model ◽  
Moving Average ◽  
Higher Order ◽  
Invariant Distribution ◽  
Absolutely Continuous ◽  
Joint Distributions ◽  
Moving Average Processes ◽  
Stationary Markov Process

A stochastic model is presented which yields a stationary Markov process whose invariant distribution is logistic. The model is autoregressive in character and is closely related to the autoregressive Pareto processes introduced earlier by Yeh et al. (1988). The model may be constructed to have absolutely continuous joint distributions. Analogous higher-order autoregressive and moving average processes may be constructed.

The distributional structure of finite moving-average processes

1988 ◽  
Vol 25 (02) ◽  
pp. 313-321 ◽  
Author(s):  
ED McKenzie
Keyword(s):  
Moving Average ◽  
Markov Property ◽  
Covariance Structure ◽  
Time Reversibility ◽  
Linear Processes ◽  
Joint Distributions ◽  
Moving Average Processes ◽  
Standard Models ◽  
Non Gaussian ◽  
Analysis Of Time Series

Analysis of time-series models has, in the past, concentrated mainly on second-order properties, i.e. the covariance structure. Recent interest in non-Gaussian and non-linear processes has necessitated exploration of more general properties, even for standard models. We demonstrate that the powerful Markov property which greatly simplifies the distributional structure of finite autoregressions has an analogue in the (non-Markovian) finite moving-average processes. In fact, all the joint distributions of samples of a qth-order moving average may be constructed from only the (q + 1)th-order distribution. The usefulness of this result is illustrated by references to three areas of application: time-reversibility; asymptotic behaviour; and sums and associated point and count processes. Generalizations of the result are also considered.

The distributional structure of finite moving-average processes

1988 ◽  
Vol 25 (2) ◽  
pp. 313-321 ◽  
Author(s):  
ED McKenzie
Keyword(s):  
Moving Average ◽  
Markov Property ◽  
Covariance Structure ◽  
Time Reversibility ◽  
Linear Processes ◽  
Joint Distributions ◽  
Moving Average Processes ◽  
Standard Models ◽  
Non Gaussian ◽  
Analysis Of Time Series

Analysis of time-series models has, in the past, concentrated mainly on second-order properties, i.e. the covariance structure. Recent interest in non-Gaussian and non-linear processes has necessitated exploration of more general properties, even for standard models. We demonstrate that the powerful Markov property which greatly simplifies the distributional structure of finite autoregressions has an analogue in the (non-Markovian) finite moving-average processes. In fact, all the joint distributions of samples of a qth-order moving average may be constructed from only the (q + 1)th-order distribution. The usefulness of this result is illustrated by references to three areas of application: time-reversibility; asymptotic behaviour; and sums and associated point and count processes. Generalizations of the result are also considered.

Stable Competitive Dynamics Emerge from Multispike Interactions in a Stochastic Model of Spike-Timing-Dependent Plasticity

2006 ◽  
Vol 18 (10) ◽  
pp. 2414-2464 ◽  
Author(s):  
Peter A. Appleby ◽  
Terry Elliott
Keyword(s):  
Synaptic Plasticity ◽  
Stochastic Model ◽  
Higher Order ◽  
Spike Timing ◽  
Competitive Dynamics ◽  
Spike Timing Dependent Plasticity ◽  
Interaction Function ◽  
Dependent Plasticity ◽  
Order Interaction ◽  
Synaptic Dynamics

In earlier work we presented a stochastic model of spike-timing-dependent plasticity (STDP) in which STDP emerges only at the level of temporal or spatial synaptic ensembles. We derived the two-spike interaction function from this model and showed that it exhibits an STDP-like form. Here, we extend this work by examining the general n-spike interaction functions that may be derived from the model. A comparison between the two-spike interaction function and the higher-order interaction functions reveals profound differences. In particular, we show that the two-spike interaction function cannot support stable, competitive synaptic plasticity, such as that seen during neuronal development, without including modifications designed specifically to stabilize its behavior. In contrast, we show that all the higher-order interaction functions exhibit a fixed-point structure consistent with the presence of competitive synaptic dynamics. This difference originates in the unification of our proposed “switch” mechanism for synaptic plasticity, coupling synaptic depression and synaptic potentiation processes together. While three or more spikes are required to probe this coupling, two spikes can never do so. We conclude that this coupling is critical to the presence of competitive dynamics and that multispike interactions are therefore vital to understanding synaptic competition.

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