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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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40 views

Small noise of state process and filtering

Assume we have a linear state-space model: $$ z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1}+ w_{k}. $$ We are interested in filtering, i.e. we aim to estimate $E[x_{n}|z_{0}, \dots, z_{n}]$. If the ...
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23 views

How to evaluate state space model?

I have a state space model which is basically a Kalman filter. The parameters of the Kalman filter are unknown and are estimated from data using EM algorithm. After I get these parameters, what are ...
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19 views

Fitting a Local Poisson model (Exponential Smoothing) [closed]

I am working through "Forecasting with Exponential Smoothing". I am stuck on exercise 16.4 on the part that states: The data set partx contains a history of ...
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1answer
27 views

General exponential smoothing to linear functions of past observations

I am just trying to derive an equation in "Forecasting with Exponential Smoothing" page 36 section 3.2. I am given the following $\hat{y}_{t|t-1} = \textbf{w}'x_{t-1}$ $\epsilon_{t} = y_t - \hat{y}...
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38 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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19 views

Can I model a trend and seasonal component for a stationary time series?

I modelled quarterly german inflationdata in a state space model with a stochastic level and stochastic seasonal. But now I recognized that I need a stationary time series because I have to compare it ...
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19 views

a general approach to derive state space representation from ARMA?

I see that the likes of this question has been asked many times but I'm just wondering whether there is a general approach to write ARMA models in state space representation? I have an exam in a few ...
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1answer
77 views

Linearisation of Kalman filter

Assume we have the following state-space model: $$ z_{k} = \theta_{k} z_{k-1} + v_{k}\\ \theta_{k} = \phi \theta_{k-1} + w_{k}, $$ where $v_{k}$ and $w_{k}$ are independent and normal for all $k$. The ...
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1answer
54 views

Kalman filter: asymptotic of state estimate

Assume we have a linear state-space model: $$ z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1} + Bu + w_{k}, $$ where $u$ is some control variable (constant intercept is the simplest case). Kalman filter, ...
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72 views

Kalman filter for AR(1) plus noise

I am working the following AR(1) plus noise state-space model $$ z_{t} = x_{t} + v_{t}\\ x_{t} = \phi x_{t-1} + c + w_{t} $$ Therefore, the transition matrix is $[\phi]$, the observation matrix is $[1]...
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1answer
76 views

Parameter space restriction in random walk + noise model

Suppose we have a random walk + noise model so \begin{align} y_t & = \mu_{t-1} + \epsilon_t\\ \mu_t & = \mu_{t-1} + \eta_t \end{align} Then, it's straightforward to show that $$\...
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1answer
44 views

Variance of AR(1) plus noise and its “equivalent” ARMA(1,1)

Let us consider the following state-space model $$ z_{t} = x_{t} + v_{t}\\ x_{t} = \phi x_{t-1} + w_{t} $$ where $ \phi< 1$, the errors $v_{t}\sim \mathcal{N}(0,V^{2})$ and $w_{t}\sim \mathcal{N}(0,...
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1answer
235 views

Estimation of ARMA from state-space generated data

There is a simple book-problem: the following state-space model $$ z_{t} = x_{t} + v_{t}\\ x_{t} = \phi x_{t-1} + w_{t} $$ where $v_{t}\sim \mathcal{N}(0,\sigma^{2}_{v})$ and $w_{t}\sim \mathcal{N}(0,\...
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1answer
86 views

Superposition of random walk and autoregressive process

Let us consider the following model: $$ y_{t} = c_{t} + \alpha y_{t-1} + v_{t} \\ c_{t+1} = c_{t} + w_{t} $$ where $v_{t} \in \mathcal{N}(0, \sigma^{2}_{v})$ and $w_{t} \in \mathcal{N}(0, \sigma^{2}...
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1answer
35 views

RegARMA in state space representation

I am attempting to fit a state space regression model of the form: $Y_{t} = i^* + \beta_{1}Y_{t-1} + \beta_{2}X_{t} + \epsilon_{1,t}$ $i^* = i^*_{t-1} + \epsilon_{2,t}$ How could I represent the ...
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95 views

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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19 views

covariances in Kalman Filter

I am confused with the Kalman filter. Could you, please, explain the solution here https://stackoverflow.com/questions/46198246/em-algorithm-with-pykalman/58560992#58560992 In the simulations ...
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32 views

Kalman EM estimation of observation variance

Let us consider a simple AR(1) process: $$ y_{t} = \mu + \beta y_{t-1} + \varepsilon_{t}, $$ with $t = 0, \dots, N$. Assume that the parameters $\mu$ and $\beta$ slowly change in time and let's ...
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1answer
55 views

Stationary Kalman Filter

Ecercise 4.5 from Bayesian Filtering & Smoothing by Simo Särkkä: Derive the stationary Kalman filter for the Gaussian random walk model. That is, compute the limiting Kalman filter gain when $...
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1answer
53 views

Finite grid approximation to the Bayesian filtering problem

I need some hints for solving Ecercise 4.4 from Bayesian Filtering & Smoothing by Simo Särkkä: Select a finite interval in the state space, say, $x \in [-10, 10]$ and discretize it evenly to N ...
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1answer
60 views

ARMA and HMM equivalence

The question might be trivial, but I am confused. Assume, one has $$ x_{t} = s_{t} + v_{t},\\ s_{t} = \phi s_{t-1} + w_{t}, $$ where $v_{t}$ and $w_{t}$ are two independent white noise sequences with ...
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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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Questions about the stability (and stationarity) of a system and state space representations

I'm pretty new to the topic and I'm trying to understand how to determine the stability of a process. I'm giving this discrete-time stochastic system: $$ \cases{ s_t = 2s_{t-2} + 3w_{t-2} \\ y_t = ...
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39 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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33 views

Outliers Kalman Filtering

This might not be the right place to ask this questions, but I figured it's more of a machine learning question. I am also asking on the pyro forum for brevity. I'm working with the simple extended ...
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16 views

What is behind “forecast” in Eviews?

I have been trying to use state space models in order to represent some gestural data. Until now I have been using Eviews to to do all the dynamic forecasting part, so I was curious what is behind ...
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11 views

Transition Matrix definition in State Space models using gestural data

I am trying to represent some gestural data (x,y,z from right hand & x,y,z from left hand) I am getting from sensors in a state space form, so as to predict the next x,y,z. Since statistics &...
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1answer
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What are the differences between Bayesian filters and adaptive filters?

I am learning about state estimation and I am having difficulty understanding the difference between Bayesian filters such as Kalman filter and particle filters compared to adaptive filters. According ...
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10 views

Online learning from a Bayesian Perspective in a State-Space Model

I'm trying to learn how to do online learning from a Bayesian Perspective. My main interest is to use it for a State-Space model. However, any explanation/reference in a different context, which may ...
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74 views

Dynamic factor model (DFM) with R, please help

I'm interested doing a dynamic factor model (DLM) similar to Doz, Giannone and Reichlin (2011) and Giannone, Reichlin and Small (2008). Moreover, I'm trying doing macroeconomic nowcasting model. In ...
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1answer
19 views

how to model a state space with GARCH noise

I'm trying to model a state space model with GARCH noise and get stuck by the complexity of the equation. so the first equation is a observation equation and second one is a state equation where both ...
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2answers
98 views

Forecasting in a state-space model from a Bayesian perspective

We have the following state-space model(or linear dynamical model): \begin{align} x_t&\sim N(Ax_{t-1},Q)\\ y_t&\sim N(Bx_{t},\Sigma) \end{align} I want to obtain a sample from $p(y_{T+1}\mid ...
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16 views

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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1answer
156 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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1answer
81 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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65 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
29 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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31 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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54 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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44 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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3answers
295 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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0answers
110 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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0answers
27 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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0answers
78 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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0answers
20 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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0answers
82 views

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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1answer
77 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
36 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 ...