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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Likelihood-ratio gradient estimator in linear dynamical system in python (Jax)

TL;DR I am trying to implement the likelihood-ratio gradient estimator in a linear dynamical system (LDS) with Gaussian transition noise and Gaussian observation noise I am currently using python and ...
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Fixed-leg Kalman filter smoother (Rauch–Tung–Striebel) error bounds

Although very intuitive and with plenty of results that talk about the asymptotic convergence of the estimate I wasn't able to track down any paper stating explicitly convergence bounds based on the ...
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On the noiseless Kalman filter

Introduction I've implemented a simple Kalman filter and I'm facing some difficulties into filtering out the noise of the measurements. If I set a small initial state covariance and a null process ...
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Best method(s) to estimate the parameters of a stochastic process with a hybrid (i.e. switching) random input variable?

I'm looking for the best approach to and/or methods of solving the following inference problem. I have tried searching for similar questions but don't have enough knowledge on the various methods (...
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One-step prediction in JAGS of a dynamic model with unknown variance

I have the following problem. I have a linear dynamic model as follows: $$\theta_{0}\sim N(0,10)$$ $$v,w\sim \text{InverseGamma}(0.1,0.1)$$ $$\theta_{t}\sim N(\theta_{t-1},w), \hspace{0.3cm} y_{t}\sim ...
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varying coefficient models parameters and estimation first principles

Please could someone explain in lay terms how the varying coefficient model works? The generalised form looks like eigenvectors. I am unsure why there is Xb(U-u) and K(U - u) and what these are used ...
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Fitting steady state of a recurrent neural network

Our model is $$\underline{\dot{\phi}} = (\mathbf{R}-\tau\mathbf{I} )\underline{\phi} + \mathbf{W}\underline{x}$$ where $\underline{\phi}\in\mathbb{R}^N$, $\mathbf{R}$ is a recurrent weight matrix ...
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Do stochastic chaotic systems decorrelate with time?

Assume I have a dynamical system with additive process noise of the form $$\mathbf{x}_{t} = \mathbf{F}\left(\mathbf{x_{t-1}}\right) + \mathbf{\epsilon}$$ where $\mathbf{x}_{t}$ is the state at time $t$...
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Is there an estimator for the existence of Lyapunov-motivated stability?

Preface: This question is now asking about dynamical stability in a particular sense, and whether its existence can be inferred from data. It is motivated by commentary below the question "Is ...
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Exogenous variable in the state equation in statespace MLEmodel in statsmodels [closed]

I'm trying to fit the following model: $y_t = \left[\begin{matrix} (1-w) & 1 & w \end{matrix}\right] \left[\begin{matrix} d_t \\ \mu_t \\ m_t \end{matrix}\right] + \mathcal{N}(0,\sigma_\eta^2)...
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Choleskly constraint in mlemodel in statsmodel

I want to constraint the off diagonal terms in the covariance matrix in a dynamic linear model. I tried using Cholesky method but it does not seem to converge. I am trying to fit a multivariate CAPM ...
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Approximate a time-discrete linear-dynamical-system using a neural network, when only partial measurements are available

I want to use a simple neural network to approximate a linear time-discrete state-space model, given by the equation: $$\boldsymbol{x}_{k+1} = \mathbf{A} \: \boldsymbol{x}_k$$ with $$\boldsymbol{x} = (...
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Getting started with Bayesian Dynamic Networks?

Dagum developed DBNs to unify and extend traditional linear state-space models such as Kalman filters, linear and normal forecasting models such as ARMA and simple dependency models such as hidden ...
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initial conditions for statespace mlemodel in statsmodel

I am a bit puzzled by very sensitive dependence on the initial conditions in the statespace mlemodels in statsmodel. Let me take a concrete example here. I am trying to fit this Dynamic Linear model ...
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Estimation of Linear Dynamical System with diagonality constraints

I am trying to estimate the parameters of the following linear dynamical system \begin{align} X_t &= \phi X_{t-1}+\varepsilon_t, \quad \varepsilon_t\sim N(0, \Sigma_\varepsilon)\\ Y_t & = h^...
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Model evaluation using Akaike's Information Criterion, Bayesian Information Criterion and Future Prediction Error Criterion

I have come up with 5 different models for a dynamical process which has 3 parameters. In order to decide which model is the best, I am using these criterions from information theory: Akaike's ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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3 votes
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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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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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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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2 votes
2 answers
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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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1 vote
1 answer
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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 ...
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