Questions tagged [linear-dynamical-system]

Dynamic linear models refers to modeling problems where coefficients (as in regression) are allowed to vary with time. This is the so called state-space approach.

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Compute initial value in Kalman Smoother

Suppose we observe data $y_t$ and $X_t$ from $t=1,...,T$ and want to estimate a dynamic linear model of the form $y_t = X_t\beta_t + \epsilon_t$ $\beta_t = \beta_{t-1} + \omega_t$ where $\epsilon_t$...
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26 views

Examples of Real Applications for Time-series with Continuous-valued Targets and Continuous-valued Observations

Suppose that we are interested in estimating continuous-valued targets $y_t$ from continuous-valued observations $x_t$ over discrete time steps $t = \{1,2,3,\dots,T\}$. Could you give me some ...
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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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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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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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20 views

How to model this as a POMDP?

I would like to fit a DLM to a dataset in R but I don't know the underlying transition matrices between states nor do I have a guess for the emission matrix (given a state, what responses should I see)...
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39 views

Understanding the DLM

The DLM model in my notes is described as: $f_k(\theta,u)=F_k\theta+u$ and $h_k(\theta,v)=H_k\theta+v$, where $F_k$ is a $d\times d$ matrix and $H_k$ is a $d'\times d$ matrix, respectively called ...
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94 views

How to numerically solve a matrix differential equation in R? [closed]

I have interest in using the R language and environment to numerically solve a system of linear ordinary differential equations. The numerical solver, deSolve, ...
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27 views

How important is research on model selection methods in Statistics?

My question is nothing technical. I just wanted your opinion on how important is the model selection problem in the field of Statistics considering the age of big data. Are the current methods such as ...
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1answer
241 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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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 ...
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360 views

How to ensure covariance matrix is positive semi definite in linear dynamical model learning?

I am trying to learn a linear dynamical model for a data using expectation-maximization algorithm. The model is defined as follows: $$x_0 \sim \mathcal{N}(\mu_0 ,\Sigma_0)$$ $$ x_{t+1} = Fx_t + w_t, \...
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53 views

Help on statistical modeling of pedestrian flow in subways

I'm a New Yorker and take the subways every day. I have a growing interest in understanding the distribution of paths people take on the subways to work every day. I.e. if there are $n$ subway ...
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35 views

Determining state space for a dynamic linear model

Are there any techniques for determining a good state space to use for a dynamic linear model? I'm trying to model ad-clicks with observed values being whether a user clicked on an ad and I'm curious ...
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60 views

What should be the termination criteria for my problem with a closed loop system identification?

I have modelled a dynamic system which needs to be validated against test data. A closed loop system identification process is used for the validation. In this process, the time domain simulation of ...
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191 views

How to add stochastic drift in dynamic linear model?

As I'm not able to comment (yet), my question follows the one raised by @mzuba here I would like to use the DLM R package to model the local linear trend model, which unlike mzuba specified, has a ...
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35 views

Predictions in a control loop like airconditions

I wonder if there are special things to consider with predictions in a control loop, e.g. An airconditioner trys to hold the target temperature at 20 degrees. I want to predict the energy consumption,...
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1answer
414 views

How to estimate coefficients of a state space when relevant data is provided?

I have a state space system $\dot{x}$ = $Ax$ + $Bu$ $y$ = $Cx$ I know C matrix exactly. And A matrix looks something like this, and some of the $x_{ij}$ in A are known as well. Same goes with B. \...
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1answer
124 views

Help in CRLB for linear model

The model is an FIR (MA) filter $$x(t) = h_1 u(t-1) + h_2 u(t-2) + u(t) \tag{1}$$ $$ y(t) = h^T x(t) + v(t) \tag{2}$$ $u(t)$ is a pseudo-random binary signal (PRBS) that excites/ drives the ...
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93 views

Simulating a dynamical system

Basically I need to replicate Hartley's 'A User's Guide to Solving Real Business Cycle Models' . Specifically (to make question relevant to stats.stackexchange), I want to simulate the dynamical ...
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156 views

State Space model question [closed]

I am looking for some help with estimating Space state model of this form: $r_{t} = r^{*}_{t} + \pi + \varepsilon_{1}$ $R_{t}= r^{*}_{t} + \alpha + \pi + \varepsilon_{2}$ $r^{*}_{t} = r^{*}_{t-...
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Forward Filtering Backwards Sampling (FFBS) and Look-Ahead Bias

Assumptions / Context: Let's assume that I have data that can be modeled as a dynamic linear model. To estimate the parameters (e.g., covariance matrix of the state/system equation), I use a Gibbs ...
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1answer
353 views

Model selection and parameter estimation in forecasting with a Dynamic Linear Model

I am implementing a general purpose prediction tool for time series. I want to tolerate missing values, so I decided to settle for DLMs. To make it as relevant as possible on a large number of ...
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52 views

Estimation from two observations [closed]

Suppose there are two vector signals $x$, $z$. The observer 1 receives a linear version of $x$ plus Gaussian noise. Observer 2 receives a linear sum of both $x$ and $z$ plus Gaussian noise as shown ...
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69 views

How to include prior knowledge that a model might be able to figure out itself

I have a problem where I want to predict the outcome of a sequence given another sequence online. Let $(x_1, x_2, ... x_T)$ be denoted by $x_{1:T}$, then I am estimating: $$ p(y_T|x_{1:T}) $$ where $...
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2answers
289 views

Learning a mapping from one time series to another with a Kalman Filter

I am interested in finding the relation between two (possibly multi dimensional) time series $x_{1:T}$ and $y_{1:T}$. I wonder how I can do that with a linear dynamical system/Kalman filter. My ...
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408 views

Assumption of Gaussian distribution of acceleration

I have a data set consisting of noisy position values of a trajectory of a human hand. I want to estimate a generative model of these trajectories, and the obvious choice is a Kalman Filter/linear ...