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Questions tagged [state-space-models]

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

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27 views

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 ...
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9 views

Updating State-Space Holt-Winters Model Latent Variables

I'm trying to update the level at time t of a state-space, additive damped Holt-Winters model with a given ...
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0answers
13 views

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 ...
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60 views

DLM regression with parameter restriction

Good afternoon, I am attempting to fit a state space regression model of the form: $Y_{t} = \beta_{1}Y_{t-1} + (1-\beta_{1})[i^* + \beta_{2}X_{t}] + \epsilon_{1,t}$ $i^* = i^*_{t-1} + \epsilon_{2,t}...
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9 views

Learning a Hopfield network parametrizing a Hamiltonian vs RNN

I think of an RNN as parametrizing a vector field. Say we forget about the inputs, and instead just want to learn a non-linear state space model. To make it more concrete, perhaps we want to model a ...
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35 views

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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12 views

How would I model underlying probabilities of failure with this data?

I have a dataset with measurements of a machine taken at different timepoints. I'm assuming at every timestep, the underlying state of the machine is a number between 0 and 1, where 1 indicates the ...
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1answer
33 views

Parameter estimation in Dynamic Linear Models

I am currently developing a DLM of the following form $$\underset{k \times 1} {y_t} = \underset{k \times n}A \underset{n \times 1}{\theta_t} + \epsilon_t$$ $$\theta_t = \mu + \underset{n \times n}B\...
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1answer
30 views

likelihood of latent state space model

Im trying to calculate the likelihood function of my latent state space model. My model has Poisson observations $p(y_t|\beta_t;x_t) \sim \mathcal{Poiss}(z)$. where $z$ is the rate of the poisson ...
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1answer
19 views

How do Tile Coding offsets still cover full state space / affect edge cases?

Reading Sutton & Barto I’m having a hard time visualizing the implementation of the tile coding discretization of states. Specifically, if tilings are offset, how does this effect edge cases? For ...
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20 views

Using dlmMLE to estimate state space parameters

I have been trying to use the dlmMLE function from the R package dlm to estimate parameters ...
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29 views

AIC for latent variable models

I'm trying to use BIC/AIC for model comparison and want to know what the number of parameters is. The models I'm unsure about are linear Gaussian state space models with nonlinear observations. ...
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1answer
66 views

computing the distribution over the latent function values with the form of a GP predictive

If we have a latent state space $\mathbf{X}$ and the observations $\mathbf{Y}$ and the transition function between two states $\mathbf{x}_{t-1}$ and $\mathbf{x}_{t}$ is given by $\mathbf{f}$ which is ...
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35 views

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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1answer
27 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 ...
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1answer
29 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, $y_i$ ...
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43 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-...
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36 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 ...
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25 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 ...
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0answers
11 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 ...
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38 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 ...
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1answer
50 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
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18 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 ...
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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 ...
4
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1answer
54 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 $...
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1answer
57 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 ...
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1answer
250 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}...
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78 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{\...
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20 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 ...
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87 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 ...
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0answers
71 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
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1answer
73 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 $y_t^{(1)}, y_t^{(2)}$ ...
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2answers
108 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 ...
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1answer
208 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
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1answer
96 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_{...
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1answer
123 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 ...
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3answers
627 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 ...
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1answer
37 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 ...
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2answers
41 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 ...
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1answer
110 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 \...
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0answers
54 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 ...
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0answers
80 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
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1answer
145 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 ...
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0answers
62 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 ...
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2answers
312 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
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1answer
92 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 ...
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0answers
260 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 ...
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0answers
44 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. ...
4
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0answers
265 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-...
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1answer
26 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 ...