Questions tagged [state-space-models]
It describes the probabilistic dependence between the latent state variable and the observed measurement.
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questions with no upvoted or accepted answers
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traditional state-space models and LSTMs
I am trying to understand the nature of LSTMs in relation to intuitions from traditional state-space models (e.g., Kalman filtering). The code below aims to simulate a simple univariate linear state-...
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Recurrent neural networks vs. State space models
I'm trying to understand the differences between RNNs and State Space Models (SSMs). I know that SSMs can take on different definitions depending on who you ask, but here I define it as in Learning ...
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AR(1) model with autoregressive intercept
Let us consider the following model:
$$
y_{t} = c_{t} + \alpha y_{t-1} + v_{t} \\
c_{t+1} = c_{0} + \beta c_{t} + w_{t}
$$
where $v_{t} \in \mathcal{N}(0, \sigma^{2}_{v})$ and $w_{t} \in \mathcal{N}(...
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ARIMA and SARIMA state space form
I need to write down a program that place ARIMA(p,d,q) and SARIMA models in state space form, however I cannot figure out the composition of the system matrices.
In the book of Koopman (pag. 54) the ...
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Possible limitations of Dynamic Bayesian Networks and valid alternatives
I just recently started to work again on Dynamic Bayesian Networks, and looking at the literature, it seems to me that they were popular in the '00, but less widely in use now, for what it concerns ...
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What state-space representation of VARMA is commonly used for fitting
What state-space representation of VARMA is commonly used for fitting?
Is Kalman filter + MLE approach used for fitting VARMA model as a common practice?
Does the choice of which state-space ...
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What is the difference between regression and state-space models?
I would like to know the differences between a regression model with autocorrelated errors and state space models (time series). When should each be used?
According to this lecture, regression (linear ...
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answer
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Marginal Covariance of State Vector in a Linear Gaussian State Space Model
Paper: A Unifying Review of Linear Gaussian Models by Roweis & Ghahramani
The generative model is the typical state space model written as
\begin{align}
\text{state transition equation: }{\bf x}_t ...
- 655
3
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0
answers
99
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Parameter dimensionality reduction in a Kalman filter framework
My problem is related to parameter identification with maximum likelihood in a Kalman filter. This framework consists of a multivariate set-up, wherein the unobserved components of the initial ...
- 31
3
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0
answers
456
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boostrap particle filter marginal likelihood
I want to calculate the marginal likelihood $p(y|\Theta)$ of the parameters of a Markov state space model with unknown parameters $\Theta$ that I am trying to estimate the marginal likelihood (...
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Bayesian estimation of Dynamic Regression with AR(1) parameters
I would like to draw (Bayesian) inference in a dynamic linear regression with regression parameters following independent AR(1) processes $\beta_{t,i} = \mu_i+\beta_{t-1,i}+w_{t,i}$. However, I ...
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time series model selection in exploratory research
I work in the field of behavioural interventions and I use dynamic models to gain better insight into (explain) the process of behaviour change and to help inform future behavioural interventions (...
3
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What graphical model formalism should I use to model state-space model?
I have a state-space model of a greenhouse control model that I'd like to transform into a probabilistic graphical model (to make it easier for non-technical managers to understand relationships among ...
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Gibbs sampler for local linear trend model
Question: Consider the local linear trend model given by:
\begin{align*}
y_t = \mu_t + \tau \varepsilon_t \ \cdots \ \text{Observation equation} \\
\mu_{t+1} = \phi \mu_t + \eta_t \ \cdots \ \text{...
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Rao-Blackwellising state space for a (marginalised) particle filter
I am starting to look at particle filtering for a problem that I have. In particular, I would like to reduce the dimensionality of the particles.
The model that I have is able to be partitioned. ...
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522
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Poisson State Space with AR(1) latent process
I have been trying to use sspir R package to estimate the following Poisson model:
$Y_{t}\sim Po(\exp(\lambda_{t}));$ such that $\lambda_{t}=X_{t}\beta
+\gamma_{t}$ and $\gamma_{t}=\theta\gamma_{t-1}+...
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votes
1
answer
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Slice sampling in Particle Gibbs with Ancestral Sampling
Bear with me as I am not from statistical background. My question is about the implementation of PGAS algorithm as given in Lindsten et. al 2014 concerning sampling in state-space models. The two ...
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answers
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State-Space Model with categorical exogenous variables
I have a state-space model:
$$
x_{t+1} = A x_{t} + \alpha_{t} \\
y_{t+1} = B y_{t} + \beta_{t}
$$
The observation model is parameterized by a categorical exogenous variable, $$ a = M_1\ or \ M_2 $$
...
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2
votes
0
answers
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State space model to invert moving average of AR1 process whose mean temporarily jumps up once
This is a follow up query of this question.
Here is the problem statement:
I have an AR1 process say x[t] whose mean jumps up in a given time period.
ie. $x[t]-\mu[t] = \phi (x[t-1] -\mu[t-1]) + \...
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1
answer
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Transforming/modelling a State-Space model as a Gaussian Process?
Is there a way to model, or represent/transform, a State-Space model as a Gaussian Process?
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1
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Fitting data to a reward based learning model?
I need some help thinking about modeling some data I have. Specifically, I am wondering if this dataset fits within the framework of reinforcement learning (I have only terse knowledge in this area), ...
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1
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Particle Filtering: Derivation that mean of weights is the marginal likelihood
I see everywhere the following (for the Bootstrap Filter)
$$
p(y_t \mid y_{1:t-1}) \approx \frac{1}{N} \sum_{i=1}^N W(x_{0:t}^i)
$$
where $W(x_{0:t}^i)$ are the normalized weights defined as
$$W(x_{0:...
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Algorithm for dynamic linear regression with stochastic volatility?
Is there any paper or textbook on how to estimate dynamic linear regression model with stochastic volatility?
The observation equation and state equation,
$$Y_t = \beta_t'X_t + \epsilon_{t}$$
$$\...
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votes
0
answers
335
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Proving/showing that the Markov property holds in discrete time Markov chain example
I am currently studying the textbook Introduction to Modeling and Analysis of Stochastic Systems, Second Edition, by V. G. Kulkarni. In a section on discrete-time Markov chains, the author presents ...
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75
views
How to compute dummy variable seasonality in Local Level Model?
I am attempting to understand the intuition behind the local level model that incorporates a seasonal component. I am currently reading an introductory book regarding state space modelling.
For the ...
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0
answers
317
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State Space Model for SARIMA
Objective:
I wanted to find the state space model for $SARIMA(3,2,1)(2,1,1)_5$. I simplified the expression and obtained the following:
B is backshift operator
$w_t$ is white noise
Equation:
$(1-\...
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2
votes
1
answer
90
views
Predictions after SMC
I have a statistical model given by
$$
y_t\sim p(y_t|x_t, \theta)\\
x_t\sim p(x_t|x_{t-1},\theta)\\
\theta\sim p(\theta)
$$
where $y$ is the only observed component. Using a sequential Monte Carlo ...
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2
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answers
513
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Eigenvalue decomposition of a covariance matrix using a fast Cholesky decomposition
Let $\mathbf{C}$ be a $n \times n$ covariance matrix and assume that
the LDL' Cholesky decomposition can be obtained efficiently. Can we
take advantage of this to obtain a fast eigenvalue ...
- 5,388
2
votes
0
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50
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State space estimation with state dependent state variance
I am estimating a state space of the following form
$$Y_t= A X_t + \epsilon_t$$
$$X_{t+1} = B X_{t} + \sigma \sqrt{( a-X_t)(X_t-b)} \eta_t$$
Considering the variance of the state error is state ...
2
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0
answers
249
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Multilevel dynamic linear models in R
I am interested in fitting a multilevel bayesian structural time series with a hierarchical structure of the dynamic regression coefficients. The reason I want to do this is is that I have a number of ...
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answers
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Decomposition of interest rate risk premia
I have a question on econometric modelling techniques for decomposition. I have three variables:
- V1 which is an indicator of an interest rate risk premia
- V2 which is an indicator of a credit risk ...
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2
votes
2
answers
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State space with lasso
Is it possible to incorporate lasso variable selection in the high dimensional state space model. If yes, is there any code or package available in R
2
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0
answers
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Infill likelihood for a continuously observed continuous-time process
Consider a continuous-time stochastic process $y(t)$ having the following linear
(Gaussian) state-space representation for $t \geq 0$
$$
\left\{
\begin{array}{c c l}
\text{d}{\boldsymbol{\...
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Are there any R code examples for estimating the state space vector in this case?
I couldn't make sure Whether the model I'm using is a local level model with multiplicative components (state vector $\times$ regressor vector) or a linear gaussian state-space model. And couldn't ...
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0
answers
315
views
Need help conceptualizing MLE for stochastic processes?
I recently learned how to perform some Maximum-Likelihood Estimation, and thought I had a fair grasp of it. For example, for the normal distribution where both $\mu$ and $\sigma^2$ are unknown for ...
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Bayesian estimation of an econometric State Space Model [Need suggestions or feedback]
I'm struggling to estimate the parameters $\theta_{t},\beta,\gamma,\sigma^{2}_{\epsilon},\sigma^{2}_{\eta}$ of the following model:
$y_{t}=\theta_{t}+\beta (y_{t-1}-\theta_{t-1})+\gamma' x_{t} + \...
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Best measure for multiple time series modelling prediction methods?
Newbie question, sorry. I have a highly seasonal monthly time series, predictable with no exogenous/independent variables and no obvious trend. I want to show that a suitable state space model (using <...
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votes
0
answers
317
views
Parameters estimation by MLE and Kalman filter
I am trying to estimate the parameters of a discrete nonlinear state space model using MLE and kalman filter:
\begin{equation}
\begin{aligned}
x_k & = f(x_{k-1},\theta)+q_{k-1}\\
y_k & = h(...
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answers
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Metropolis Hastings: What motivates the use of Metropolis-Hastings?
I am confused with metropolis hastings. This is a simple question. In the metropolis hastings, it is assumed that we know the un-normalised posterior, $\pi(x)$. We can obtain the density by ...
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Multiplicative gaussian state space model
I am wondering about the effectiveness or optimality of Kalman smoother algorithm for multiplicative state space model with gaussian errors. Can I still use the standard linear gaussian kalman ...
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State and input simultaneous estimation
I have a discrete nonlinear state space model :
\begin{equation}
\begin{aligned}
x_k & = f(x_{k-1},u_{k-1},\theta)+q_{k-1}\\
y_k & = f(x_k,\theta)+r_k
\end{aligned}
\end{equation}
$x_{k-1} \...
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answers
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views
Kalman Filter Forecasting converging?
I have implemented Kalman Filter to forecast 250 forward steps of some observations
The formula I used after recursively filtering on the in-sample set is
$y_{T+h|T} = HB^hz_{T|T}$
where H and B ...
- 149
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votes
0
answers
39
views
Disprove state space estimation
I am currently looking at a state space estimation procedure by Elliott & Timmermann (2016) which is apparently wrong. They suggest to (i) pick some H, F, Q and R, (ii) run the Kalman filter to ...
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votes
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State space model parameter estimate
I'm working on one project trying to reconstruct a sequence of multivariate signal data from another sequence of multivariate signal data. That is let $\{S_t\}_{t=1}^n$ be the first sequence of ...
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0
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391
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Unscented Kalman Filter transformations for a Poisson state-space model
I have count data which I'm trying to model using a state-space model where
$z_t \sim Poisson(exp\{F^\prime\ x_t\})$
$x_t \sim N(G\ x_{t-1}, R)$
Where $z_t$ are the observations and $x_t$ the ...
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votes
0
answers
111
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adaptive Kalman filtering
I am learning about Kalman filters/dynamic linear models/state-space models and I am interested in whether the following scheme is possible, in which I try to estimate distribution parameters ...
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0
answers
46
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Representation power of State Space Models
We can represent subclass of linear time invariant (LTI) systems with State Space Representation:
$$\dot X = AX + BU,$$
$$Y = CX + DU.$$
Also, nonlinear systems are formulated with generalized State ...
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votes
1
answer
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views
Difference between particle filter (PF) and recurrent neural network (RNN) for time series
Both method are used to estimate time series from data. The question is, when should I use one method or other? Is any advantage to use one instead of the other?
I know that in a PF there is a hidden ...
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votes
0
answers
399
views
Predicting a Chi-Square Process
Assume that $W(t)$ is a one-parameter stochastic process given by
$W(t) := X_1^2(t) + X_2^2(t)$ where $X_i(t)$ are independent copies of
a stationary gaussian process with known covariance function. ...
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votes
0
answers
104
views
Estimate latent states for a Bernoulli stace space model, when the latent states follow an AR(1) process
I am dealing with this model
$$y_t|\alpha_t \sim Bernoulli \left( \frac{\exp (\alpha_t)}{ 1+ \exp(\alpha_t)} \right) $$ with $\alpha_t = \phi \alpha_{t-1} + \epsilon_t,$ where $\epsilon_t \sim ...
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