Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [state-space-models]

It describes the probabilistic dependence between the latent state variable and the observed measurement.

0
votes
1answer
23 views

Custom Space State model using DLM in R

DLM package in R can model linear space state models of the form: I have a different category of equation which is also a linear polynomial equation of order 1 with constant coefficients. I would ...
0
votes
0answers
22 views

What's the proper name for these chain structured PGMs?

I'm trying to find previous work that has dealt with this type of PGMs, but don't know what to call them: a) "recurrent HMM"? $y_i$ are scalars and $x_i$ are discrete b) "triangle HMM"? again, $y_i$ ...
1
vote
0answers
20 views

Proving Matrix-Normal-Inverse-Wishart distribution is a conjugate prior for a Linear Model

How does one prove that the Matrix-Normal-Inverse-Wishart distribution is a conjugate prior for a Linear Model? This prior is a generalization of the Normal-Inverse-Wishart Distribution. By Matrix-...
1
vote
0answers
27 views

Hierarchical time series using DLM

I am developing a forecasting solution using R's dlm package and it is proving to be very useful for most of our requirements. However, I am also keen on sharing information among different time ...
2
votes
0answers
21 views

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 ...
0
votes
0answers
10 views

In a LGSSM how do we know that the prediction distribution is Gaussian?

I am trying to follow lecture notes regarding the Kalman Filter from a course taught at Stanford. The lecture notes can be found here. The linear Gaussian state space model (LGSSM) is introduced as ...
2
votes
0answers
36 views

Time varying representation of Okun's law

I've estimated a dynamic linear model to capture time varying parameters in an Okun's law type of model: I set the starting values for the state vector all equal to zero and estimate the system ...
1
vote
1answer
34 views

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
0
votes
0answers
17 views

What is the intuition between using shared covariance parameters or separate in state space models?

I apologise if my terms aren't very exact as I'm in a learning process here, but would appreciate if you could provide me some intuition of what are the pros and cons of two alternatives. I've got an ...
0
votes
0answers
9 views

Model for hormone levels over tissue cells

I have a certain type of biological data and I am unsure about how to model it. The data represent the amount of 3 hormones detected along 20 consecutive cells of a certain plant tissue. I think there ...
2
votes
0answers
37 views

Conditional mean and co-variance in $VAR(p)$ conditional on one lag only

Suppose I have a $p$'th order vector auto regression $$\vec Z_t = F_1\vec Z_{t-1}+F_2\vec Z_{t-2} + \cdots +F_p \vec Z_{t - p} + \vec \epsilon_t,\qquad \vec\epsilon_t\sim N_q(\vec0,Q)$$ where $...
0
votes
1answer
47 views

(Online) intuitive explanation of state space models

I have a similar question to the one in the link below: Intuitive explanation of state space models In the link they recommend the book by Commandeur and Koopman. I have this book already. I was ...
0
votes
0answers
55 views

Examples of state space models where the filtering problem can be solved analytically

Background A discrete-time, Markovian state space model takes the form \begin{align} \mathbf{y}_t&\sim p(\mathbf{y}_t\,|\,\mathbf{s}_t,\,\boldsymbol{\theta})\\ \mathbf{s}_t&\sim p(\mathbf{s}...
2
votes
0answers
75 views

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{\...
1
vote
0answers
15 views

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 ...
1
vote
0answers
74 views

Unable to recover time varying AR1 parameter from State Space model

I am trying to do a Time varying parameters regression. The equation is as follows: $y_t = a + b_t * x_{1t} + \epsilon_t$ Here a is fixed while $b_t$ is AR1. My state space equations are : There ...
0
votes
0answers
34 views

Linear regression of features inside a hidden Markov model?

I have an interesting little problem which I am trying to attack using HMMs. First, as usual, I am trying to do time-series segmentation/classification using a HMM. But the input to my HMM has an ...
1
vote
1answer
69 views

State-space model with contemporaneous effects

I have the following system of equations: $$ \begin{align} y_t^{(1)}&=y_t^{(2)}-x_t+\epsilon_t\\ y_t^{(2)}&=x_t+\nu_t\\ x_t&=\alpha x_{t-1}+u_t \end{align} $$ where $y_t^{(1)}, y_t^{(2)}$ ...
1
vote
2answers
101 views

Kalman Filter Derivation - Shumway / Stoffer

I'm going through the proof of the Kalman filter equations in Shumway, Stoffer - Time Series Analysis and its applications. Could someone please tell me how equation (6.26) is justified? How can we ...
1
vote
1answer
166 views

Estimating State Space Model Parameters

I'm having a bit of difficulty estimating parameters in DLM in R and I was wondering if I could get a bit of help with it. I have a system of equations given as: $p_{t} = m_{t} + s_{t}$ $m_{t} = m_{...
3
votes
1answer
89 views

State Space Model Form for Equations

I have a set of equations which I have to write in state space model form but unfortunately I'm having a bit of difficulty doing so. They are given as: $y_{t} = x_{t} + z_{t}$ $x_{t} = x_{t-1} + w_{...
2
votes
1answer
118 views

Deriving a filter like a Kalman filter from a non-Gaussian state space model

Assume we specify a state space model as $$Y_t = a X_t + W_t$$ and $$X_{t+1} = b X_t + V_t$$ where $b,a \in R$, $E[W_t] = E[V_t] = 0 \quad \forall{t }$ and $W_t $ and $V_t$ are indipendent for ...
1
vote
3answers
273 views

Simple explanation of dynamic linear models

I'm looking for a really simple explanation of what a dynamic linear model is as I need to explain this to a non-technical audience. I have looked around for examples but they are very maths heavy. I ...
0
votes
0answers
39 views

State space models with non stationary/unit root factors

state space model , I am trying to implement is as follows $$ Y_t= CY + FF* X_t + Ve_t$$ $$(X_t-m0)= GG (X_{t-1}-m0) +W\eta_t$$ I am enforcing GG to be to be diagonal for the base case. I am getting ...
1
vote
1answer
36 views

Why is Qk not included in the cost function that is optimised by the Kalman filter?

Assume the following linear discrete system: $x_k = Fx_{k-1} + w_{k-1}$ where $w_{k} \sim N(0, Q)$ $y_k = Hx_k + v_{k}$ where $v_{k} \sim N(0, R)$ One way to prove that the Kalman filter is optimal ...
-1
votes
2answers
38 views

Variance of a mixture of Normals with same $\sigma^2_i$

Let $Y\sim \sum^N_{i=1}\omega_iN(m_i,h^2 V)$. The text I'm reading states that $Var(Y)=(1+h^2)V$, when $m_i=\theta_i$, where $\theta_i$ are draws taken from $P(\theta|D)$, and $V=Var(\theta|D)$ I ...
0
votes
1answer
99 views

Convert a state-space model with exogenous input to one without

I have a state space model of the form \begin{align} x_{t+1} &= Ax_t + Bu_t + w_t\\ y_t &= Cx_t + Du_t + v_t \end{align} where $u$ is the exogenous input. Also, $ w_t \sim N(0, Q)$ and $v_t \...
0
votes
0answers
47 views

state space implementation using DLM FKF

state space model , I am trying to implement is as follows $$ y_t= CY + FF* X_t + Ve_t$$ $$(X_t-m0)= GG (X_{t-1}-m0) +W\eta_t$$ In DLM I am using following modification(because DLM does not allow ...
0
votes
0answers
74 views

How to approach SSM models for time series forecasting in general?

I have worked on SSM model using KFAS package (https://cran.r-project.org/web/packages/KFAS/KFAS.pdf) in R. Package suggests me to use one of the Box_Jenkins method to implement SSM. So we convert ...
4
votes
1answer
112 views

Coding resources: Accessible introductions to Bayesian Structural Time series?

Hello, all. I am asking this question in not necessarily a "subjectively recommend something for me" approach, but with a clear focus on just an accessible beginner's reference. My situation is I have ...
1
vote
0answers
51 views

R dynr: Having trouble setting up state-space model [closed]

I was looking around to flexibly implement state space models in R. I found dynr, but I am being frustrated to no end by its tad vague documentation and lack of ...
3
votes
2answers
140 views

A doubt on the notation of Frigola et al (2013) of Gaussian Processes(GP) for a State-Space model?

The picture above is from Frigola et al (2013) - Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC. In this paper, the authors later define $\mathbf{f}_t=f(\...
3
votes
1answer
86 views

Would a simple Gibbs, or a Metropolis-Hastings algorithm work for a State-Space model?

I'm wondering if a MCMC algorithm, in a Gibbs or a Metropolis-Hastings style, work for a State-Space model. Would I also be able to learn about the state variable and not just the parameters? I've ...
0
votes
0answers
192 views

ARIMA model for GDP

I am working through example 3.2.6 in 'Dynamic linear models with R' by Petris. I have download the quarterly deseasonalised USA GDP data located here: http://definetti.uark.edu/dlm/ (it's the data ...
1
vote
0answers
34 views

Assessing the fit of a state-space model in JAGS

I've been fitting a relatively complicated state-space model in JAGS and I want to do some basic model comparisons, including dropping parameters one at a time to assess their influence on the fit. ...
2
votes
0answers
198 views

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-...
1
vote
1answer
25 views

Introducing behavioural states into hidden markov model?

I have a hidden markov model which models movement. A map is split into even sized grids and the hidden states are the grids. I want to improve this model by adding behavioural states (so that ...
0
votes
0answers
106 views

Implementation of kalman filter with inner ARIMA non seasonal model

I am trying to write an application which impute some missing values on one time series signal. I have done it similarly in R using ImputeTS package but now need to do it similarly in Java. I just ...
0
votes
0answers
25 views

Splitting residuals in maximum likelihood estimation

I would like to estimate the parameters of a state space model with maximum likelihood. The model has non-smooth transition—I would like to split the estimated residuals into two groups dependent on ...
0
votes
0answers
49 views

How to predict changes in a signal based on other leading signals?

Goal: classify changes in green series using black series as predictor more specifically, create classification for when black series increased/decreased and n-seconds after black series moves then ...
1
vote
0answers
43 views

unscented kalman filter for non-linear state-space

I intend to use unscented kalman filter to estimate a non-linear state -space problem. latent factor $X_t$ in the formulation has usual VAR(1) specification $$X_t = \phi X_{t-1} +\epsilon_t$$ ...
2
votes
1answer
239 views

DLM representation of ARIMA models

I am working through example 3.2.6 in 'Dynamic linear models with R' by Petris. I have download the USA GDP data located here: http://definetti.uark.edu/dlm/ The example starts by estimating the ...
0
votes
1answer
31 views

State space model with third or more order trend

In state space model, a system model with first order trend is represented as $$ x_{t} = x_{t-1} + e_{t}, $$ where $x_{t}$ is system model, $e_{t}$ is system noise. Also, a system model with ...
0
votes
0answers
57 views

How does one approximate $\mu$ and $\sigma$ in an arithmetic Brownian motion using a Kalman filter?

My concern arises from the fact that in the following system: $x_k = (\mu, \sigma)^T = x_{k-1}$ $Y_k = Y_{k-1} + \mu + \sigma Z_k \quad Z_k \sim N(0,1)$ that I cannot separate the states I want to ...
1
vote
1answer
69 views

How do I write a state space model and how do you find the unknown parameters of phi, mu, and matrix A$_t,$ along with covariance matrices Q and R?

Consider a system process given by $x_t=-0.9x_{t-2}+z_t$,$t=1,2,…,n$ with observation $y_t=x_t+v_t$ where ${z_t}$ and ${v_t}$ are independent white noise with variances $σ^2$ and $σ_v^2$. Assume ...
1
vote
1answer
282 views

EM Algorithm seems to work, but Q is not monotonic. Possible reasons?

I have implemented Expectation maximization to fit some of the parameters of a linear Gaussian state space model using Kalman filtering / smoothing. The model is: $x(t) = Ax(t - 1) + w(t); w(t) \sim ...
0
votes
0answers
22 views

Correlated error terms in VAR and external observable AR as state equation

I ran into an estimation problem of a system that combines of a two-variable VAR(1) (entity level VAR(1)), and a AR(1) which represents the state. Can you help? Any suggestion would be appreciated. ...
1
vote
0answers
272 views

How to put an ARMA(2,2) model in state-space form

I am interested in an ARMA$(2,2)$ model with an additional input variable, which I want to put in state-space form. If $w_t$ is white noise, and $x_t$ is a known input, the model is given by: $$y_t ...
1
vote
1answer
60 views

Univariate Kalman filtering with factor in state-equation

I have a simple Kalman problem: how does one estimate the following local level univariate state-space model, but with some driving factor: ...
1
vote
1answer
251 views

State space models: Advantage of Stationary State Vector?

Consider a State Space Model, where the observed process is $Y_t$ $$ Y_t = B F_t + \epsilon_t \\ F_t = \Phi F_{t-1} + \nu_t $$ where the error terms are white noise. Later on, I want to compute the ...