Questions tagged [state-space-models]

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

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Error: in ‘simulate’: argument “params” is missing, with no default [migrated]

Hey I am learning package 'pomp' recently. And I don't know why there is such an error in my code, the source is from paper https://kingaa.github.io/pomp/vignettes/pompjss.pdf ...
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Log innovation vs squared

I see some state space models specify their innovation process as log innovations and some squaring the term. For example, the examples in the R package DLM favours the use of log innovations when ...
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82 views

Are Kalman Filter recursions valid when the state noise has a singular covariance matrix?

Consider a Linear Gaussian State-Space Model where the states are denoted by $X_t$ and observations are denoted by $Y_t$: \begin{align} X_t &= A X_{t-1} + \epsilon_t, &&\epsilon_t \sim \...
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33 views

How to include seasonal effects into the system matrices of a state space model

I am working on learning state space models and am leaning heavily on this very helpful documentation. However, I'm really confused about the best way to include both a seasonal effect and dynamic ...
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58 views

Does the Markov property always hold for a state-space structure?

Markov Property: $p({\bf x}_t | {\bf x}_1, \ldots, {\bf x}_{t-1}) = p({\bf x}_t | {\bf x}_{t-1})$ Consider the following model for which the hidden states are ${\bf x}_t$ and the observations are ${\...
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1answer
21 views

How to infer state space parameters from an LSTM model?

I'm attempting to create a state-space model by training my time series data with an LSTM. I'm hoping the LSTM will capture non-linear phenomenon as opposed to a linear state-space model. The only ...
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21 views

How is the confidence metric of a state space model at a given state related to consumer demand for clickstream data?

I've been tasked to take over a project at my company where consumer demand for a given product is taken from the confidence metric of a given state of a state space model. The way it works is that ...
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21 views

Equivalency between ARMAX and state-space model

I am trying to understand the equivalency between ARMAX and the state-space model. I have read articles/websites with different conclusions on this topic. Some people claim ARMAX and state-space model ...
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17 views

Confidence intervals for state space models

i'm looking for on how to calculate the following ICs: smoothing, on-line filter and prediction on state space models. I'm not able to find any formula about them or any matlab command/class. Thanks....
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85 views

Kalman filter parameter estimation

From what I've known about Kalman filter, it requires all the parameters of the underlying state space model. Say the state space model is: $$\xi_{t+1} = F\xi_t + v_{t+1}$$ $$y_t = H\xi_t + w_{t}$$ ...
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Estimating a changing transit time between inputs and output

I work with a chemical process in which there is a time lag between the inputs (raw material quality and cooking parameters) and the output (final product quality). The problem is that the time lag ...
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58 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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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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21 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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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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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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50 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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45 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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32 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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21 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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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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32 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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74 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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64 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
30 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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30 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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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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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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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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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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41 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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59 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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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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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 ...
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56 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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69 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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470 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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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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101 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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101 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 ...
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1answer
77 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
113 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
227 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_{...
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104 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
134 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
1k 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
43 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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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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132 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 \...