It describes the probabilistic dependence between the latent state variable and the observed measurement.
3
votes
2answers
41 views
Linear Filtering in High Dimensional State Space
I am working with Gaussian Linear State Space models of the form:
$$y_t=F_t\Theta_t+v_t$$
$$\Theta_t=G_t\Theta_{t-1}+w_t$$
$$v_t \sim N(0, V_t)$$
$$w_t \sim N(0, W_t)$$
Where $y_t$ is my observed ...
1
vote
0answers
27 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(...
1
vote
0answers
18 views
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 ...
0
votes
0answers
78 views
How to estimate parameters of an Extended Kalman Filter
I am slightly familiar with how to estimate the parameters in a linear Kalman Filter, by using maximum likelihood. Do we do the same for the extended Kalman Filter? In particular, I am confused as to ...
0
votes
0answers
13 views
Is there a difference between recursive parameter estimates and time-varying parameters?
As the title indicates, is there a difference between recursive parameter estimates and time-varying parameters. I ask this in the context of time-series.
For example, recursive parameter estimates ...
5
votes
1answer
84 views
State space representation of ARMA(p,q) from Hamilton
I have been reading Hamilton Chapter 13 and he has the following state space representation for an ARMA(p,q). Let $r = \max(p,q+1)$.Then the ARMA (p,q) process is as follows:
$$
\begin{aligned}
y_t -\...
0
votes
1answer
23 views
Particle Filtering with Nonlinear Observation Equation of 2 set of variables
I am stuck with this problem in my research.
I am having a State Space Model like the below mentioned one:
State Equation: $\mathbf{d}_k = \mathbf{d}_{k-1} + \mathbf{u}_k + \boldsymbol{\epsilon}_k$ ...
0
votes
0answers
20 views
Machine learning algorithm for validating a state sequence
I am developing a machine learning algorithm to validate a markov chain sequence.
Currently I have :
1. A ngram predictor which predicts the next state from the state history and will check whether ...
3
votes
0answers
52 views
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 ...
0
votes
0answers
26 views
Validity of Durbin Koopman simulation smoother
In the paper "A Simple and Efficient Simulation Smoother for State Space Time Series Analysis" by Durbin and Koopman an algorithm is proposed to sample from the error terms in the observation and ...
3
votes
1answer
35 views
Likelihood of a state space model with multiplicative noise $p(y_k|x_k) = ? $
Consider the following state space model: the transition distribution of the latent variable $x_k$ along with the observational density is given below. Assume that both sequences of the noise terms, $\...
0
votes
0answers
26 views
How are Dynamic Stochastic General Equilibrium (DSGE) Models state space models?
Reading this right now because someone told me that DSGE models are an instance of state space models. I'm not much of an Economics guy. There are a lot of letters here, so which are the 'state' ...
3
votes
0answers
28 views
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 ...
1
vote
0answers
16 views
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} \...
0
votes
0answers
25 views
Hidden state and hidden sequence in HMM
Wikipedia mentions that "A belief state can be calculated at each time step, but doing this does not, in a strict sense, produce the most likely state sequence, but rather the most likely state at ...
0
votes
0answers
12 views
Right Model for Customer States and Transitions?
Suppose an online customer can be in any one of the following states at a point in time:
free (gets fewer features than full customers)
free_trial (full customer, but free, with auto-renew into paid ...
0
votes
0answers
85 views
Multivariate State Space model in r (dlmodeler)
I'm trying to fit a multivariate dlm using the dlmodeler package. The model is a state space representation of a simplified macroeconomic model, as such:
Observation equations:
$h_t = c + A * h_{t-1}...
1
vote
0answers
32 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 ...
1
vote
0answers
29 views
Model approach for extending Dynamic Linear Models - Nested Regression Relationships
I have three multivariate random variables $X_t$, $Y_t$, and $Z_t$.
I have been very happily modeling the relationship between $Y_t$ and $X_t$ through a dynamic linear model
$$Y_t = X_t\beta_t + ...
0
votes
1answer
132 views
Hidden Markov Model with continuous state-space and emission?
I recently started learning HMM and was wondering how do I go about using a model or similar thereof in which an observation is really a realization of the Gaussian distribution of the corresponding ...
1
vote
1answer
37 views
How can I identify market regimes with a Hidden Markov Model?
I am trying to identify market regimes (2 states: bull or bear) with percent changes in equity returns. Can you help me in the mathematicl modeling of this? So far, I thought that for each day, there ...
1
vote
0answers
18 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 ...
1
vote
1answer
52 views
References on state space models and time series
I would like to study time series modeling and state space modeling for the purpose of modeling a complex physical system (vibrations of an industrial machine). The system is mostly linear (small ...
1
vote
0answers
18 views
Deriving the particle filter with driving-force/inputs/control-signal
Whenever the particle filter is derived (I used a different condition for $u_t$ as a solution to the nonlinear filtering problem;
$x_{t+1} \mid x_t \sim f_{\theta}(x_{t+1} \mid x_t,u_t) \\
y_{t} \...
1
vote
0answers
21 views
Modelling rates or levels in state transition model
Q: Suppose I'm modeling the transition of customers from one state to another. Is it better to model the time series of transition rates directly or implicitly as a time-varying parameter in a model ...
0
votes
0answers
25 views
Derivation of covariance equation for state space model
Given a state space model on the form
$\dot{\boldsymbol{x}} = \boldsymbol{Ax}+\boldsymbol{E}v,\quad v\sim Norm(0, Q) $
My suggestion for the covariance equation would be
$cov(\dot{\boldsymbol{x}})=\...
4
votes
0answers
131 views
Estimation of ARMA: state space vs. alternatives
I am interested in estimation of ARMA models. I understand that a popular approach is to write the model down in the state space form and then maximize the likelihood of the model using some ...
2
votes
1answer
74 views
How to model a stochastic trend in the response variable of a regression?
Each time an individual arrives at random, we register a response variable, many covariates and the day of arrival. The response variable is continuous and unbounded (Reals), there are both ...
1
vote
0answers
29 views
Correlation of state and observation error terms in state space model
One of the key assumptions of the state space model is that correlation of state and observation error terms should be zero.
Does this have to be the expected error terms?
How do I test for ...
2
votes
0answers
22 views
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 ...
0
votes
0answers
23 views
How to determine appropriate lagged features for learning systems with states?
In much of machine learning literature, the systems being modelled are instantaneous. Inputs -> outputs, with no notion of impact from past values.
In some ...
1
vote
0answers
26 views
Time series data with a binary output, current state is correlated with past few states if current “unknown” state is 1
Background: I am performing a pattern matching on some data which is being generated real time. I perform the pattern matching on this newly generated data at every 1 second. Every time a match is ...
3
votes
1answer
48 views
Understanding SMC as approximations to a sequence of measures
I am used to thinking about Sequential Monte Carlo (SMC) methods as a method that discretely approximates a sequence of probability distributions. For example, in a state-space framework, SMC can ...
8
votes
1answer
897 views
What are the properties of a half Cauchy distribution?
I am currently working on a problem, where I need to develop a Markov chain Monte Carlo (MCMC) algorithm for a state space model.
To be able to solve the problem, I have been given the following ...
1
vote
0answers
31 views
Identifying the parameters of a linear state-space-model using Kalman Filter
I have a linear state space model (SSM) that looks like this
\begin{align}
{\dot {x}} & = {\rm \textbf{A}}{x} + {\rm \textbf{B}}{u} \\
{y} & = {\...
0
votes
0answers
42 views
Are Particle Filters stochastic models? [duplicate]
I am a student and I am writing an assignment in the subject of the machine learning. My question is, if it is correct to say that:
"Particle Filters are stochastic models, like Hidden Markov Models"?...
0
votes
0answers
40 views
Having trouble implementing an extremely rudimentary Kalman filter
After spending hour unsuccessfully debugging my Kalman filter code I decided to try the most rudimentary case I can think of and see if my issue persist. They do. Please help me figure out what I'm ...
1
vote
0answers
63 views
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 ...
5
votes
2answers
129 views
How to approximate (log-)likelihood from model specification using particle filters
In Calvet et al., "Robust Filtering" (JASA, 2015), the authors obtain (pseudo?) maximum likelihood estimates of the parameters of a time series model of the trading volume of a futures contract. The ...
1
vote
1answer
25 views
MCMC: why params that are directly sampled from posterior converge slower than that sampled from Metropolis steps
I am fitting a Bayesian model (the core is a linear state space model) using MCMC. Most of the parameters are sampled directly from their analytic posteriors whereas the left are sampled from ...
0
votes
0answers
35 views
state space model with non stationary time series data
I want to estimate a state space model. Do the series need to be transformed into stationary series for a state space estimation?
2
votes
0answers
37 views
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 (...
1
vote
0answers
27 views
Formulation of state equations [closed]
I am estimating a state space model but I am totally confused about specification of the state/transition equations. How should I decide whether the state equation should be a random walk or random ...
1
vote
0answers
19 views
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 ...
1
vote
0answers
26 views
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 ...
1
vote
0answers
20 views
Linear model with AR hidden state
Suppose everyday I want to predict the temperature reading taken somewhere in a city. I have some explanatory variables, say humidity and wind speed. And I have data for $K$ cities. So the data look ...
2
votes
1answer
91 views
Periodicity for Markov chain
I don't understand why the only state with period > 1 is 1
Let's take state 2 for example, what's the period for state 2?
Another question is, does an absorbing state(state 4 in this example) only ...
0
votes
0answers
160 views
Time series model to forecast electricity demand given temperature and electricity price
I currently have half-hourly electricity demand, half-hourly electricity price and hourly temperature from 2012 to 2016, and I would like to do both short-term and long-term forecast of electricity ...
1
vote
0answers
128 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 ...
0
votes
0answers
69 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. ...
