Questions tagged [state-space-models]
It describes the probabilistic dependence between the latent state variable and the observed measurement.
367 questions
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On the equivalence of ARIMA models with ETS
It is often said that exponential smoothing models can, in some cases, be seen as ARIMA models. For instance, if we have a simple exponential smoothing model:
$$X_t = l_{t-1} + \varepsilon_t$$
$$l_t = ...
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Why is exponential smoothing forecast the median of the forecast density?
I am reading Hyndman & Athanasopoulos "Forecasting: Principles and Practice" 2nd edition (FPP2). (I am aware that 3rd edition exists.) In the chapter about exponential smoothing, section ...
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Dynamic factor analysis/Kalman filter using fixed scores/state and dynamic loadings/observation model?
I have to following data setting. There's $n$ genomes whose expression for a trait of interest is measured in $p$ locations. This is repeated for $\tau$ years so the total number of datapoints is $n\...
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Dynamic Factors with Statsmodels in Python
I am fitting a linear gaussian state space model in python using statsmodels.DynamicFactorMQ. It is giving me back only the model summary, but I want to extract the estimated AR(1) transition matrix ...
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Coupled Multivariate Regression and AR(1) Process for the Covariates
I am wondering how to fit the following model:
I have a standard multivariate regression, and a set of covariates X. I want to couple my regression equation with an AR(1) process for the dynamics of ...
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State Space Model with All Observed Factors
I have seen state space models implemented when we have latent/unobserved factors we want to uncover that jointly explain the dynamics of a time series of dependent variables. I have also seen state ...
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State space models and Kalman Filter
I have the following model specification:
$y_t = \mu_t + v_t,$
$\mu_{t+1|t} = \phi \, \mu_{t|t-1} + k\, v_t $
where v_t= y_t - mu_{t|t-1}, v_t|F_{t-1} ~ tv(0, sigma^2).
I was asked to provide the ...
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Ways to parametrise a positive parameter
I am working with a differentiable state-space model involving a noise variance term $\sigma^2$ which I want to parametrise based on some features, e.g. $\sigma^2 = g(X\beta) > 0$, wherer $\beta$ ...
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What if there is only one measurement equation containing two (or more) state variables while there are two unobservable state variables in a model?
I am learning Kalman Filter and ran into a question about the case in which only one signal is available.
It is commonly assumed that the number of states equals the number of observations (signals) ...
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Errors and residuals in simple exponential smoothing (state space form) in FPP textbook
I am reading Hyndman & Athanasopoulos "Forecasting: Principles and Practice" 2nd edition (FPP2). (I am aware that 3rd edition exists.) In the chapter about exponential smoothing, section ...
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How to incorporate sources of observational error in state space model?
I’m learning about state space models. I understand the concept of a latent process that is unobserved, and a noisy set of observed data that we can use to estimate the latent process. I am trying to ...
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Regressing time series of continuous proportions Y(invested in different buckets) against X (prices) variables
I am not sure of what model need to be applied in my case. So my goal is to model the below data.
I have the monthly deposits data from 2001 till 2024 for this analysis. Each customer can invest in ...
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Does this state space model make sense?
I'm working on a problem,
Consider that a times series $\{y_t\}$ is generated from an $\text{ARIMA}(1,1,1)$ model, so that $$y_t-y_{t-1}=\alpha(y_{t-1}-y_{t-2})+\epsilon_t+\gamma\epsilon_{t-1},$$ ...
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model averaging with MARSS
I have a large number of MARSS dynamic factor analysis models that, based on AICc, are all competitive for the 'best' model. Is there a way to implement a model averaging process so that I can extract ...
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Repeated measures of multiple time series processes
I am struggling with a comparison of temporal processes, which are observed in several time series. The problem is as follows:
Suppose there are some semi-experimental conditions, with several ...
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Bayesian estimation of nonlinear state space model using Gibbs Sampler
I want to estimate a state space model that looks like this:
$y_t = exp(\beta_t)x_t + \varepsilon_t \hspace{1cm} \varepsilon_t \sim \mathcal{N}(0, \sigma^2_{\varepsilon})$
$\beta_t = \rho\beta_{t-1} +...
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What kind of series is this? Time series Model Choice
I am trying to model a series (mainly as self-education, as a way of deepening my time series knowledge). The series is percentage growth in GDP per capita for the United States, taken from the World ...
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Looking for books and/or other resources for Multivariate Statistics for Optimal Estimation
I was trying to familiarize myself with state estimation theory by going through Optimal State Estimation and I realized I don't have the required background in Multiple Random Variables and ...
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Local linear trend-exponential smoothing duality with non-Gaussian likelihood?
Say we have the following state space/structural time series model:
$$
y_t \mid \mu, \sigma \sim \text{Normal(} \mu = \mu_t, \sigma^2_\varepsilon) \\
\mu_{t+1} = \mu_t + \eta_t
$$
where
$$
\eta_t \sim ...
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Kalman filter + MCMC vs pure MCMC for bayesian dynamic linear(state space) models
I have heard that you can use both kalman filter integrated with MCMC to estimate bayesian state space models, i have also heard that you can use pure MCMC to estimate bayesian state space models.
Why ...
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timevarying effects in regression vs state space models
Consider the following regression model:
$z_t = \alpha + trend_t + seasonality_t + \epsilon$
$\alpha \sim HalfNormal(1)$,
$trend_t = \mu t$ where as $\mu \sim Normal(0, 1)$
$seasonality_t = \sum_k^K [\...
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Determine the parameters of a particle filter that best fit observations
I am wondering is there any established framework to optimize the parameter $\lambda$ of a particle filter such that $p(O|\lambda)$ is maximized, where $O$ is the observation sequence. For HMM and ...
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What does it mean to have an observation variance of zero when fitting a model using StructTS in R?
I'm using the StructTS() function in R to fit 50 structural time series models (state-space models) on 50 univariate time series. I'm using the "trend" ...
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Block sampling hidden state using forward algorithm only
In a hidden Markov model, I can't get my mind around why I can't sample the full hidden state $\vec x$ using only a forward sampling algorithm.
Let $\vec y$ be the observed data and $\theta$ the model ...
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Why should we need the acf of a series to define its joint distribution?
I'm working on an unassessed course problem,
Consider the time series $y_t$ generated by the state space model with $x_t=1$, $F_t=\lambda$, $\sigma_2$, $Z_t=Z$, where the variances $\sigma^2,Z$ and ...
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Identification of Non-Gaussian State Space Model
The following paper details necessary assumptions in order to have a non-gaussian state-space model be identifiable (see A1-A5); 'A General Linear Non-Gaussian State-Space Model: Identifiability, ...
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What can you do with quantified uncertainty in latent variable time-series models?
Uncertainty quantification in latent variable models is a topic I am interested in, but I am struggling to grasp the difference between what you can do with quantified parameter uncertainty and ...
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How to handle exact diffuse initialization of a Kalman filter?
This is partially a coding question so I hope I'm on the right platform for this.
I am fitting a dynamic factor model using the state space framework.
I don't know the initial distribution of the ...
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ARIMA, VAR and State Space Model (SSM) forecasting comparison
I am trying to compare the asset price forecasting abilities of SSMs with ARIMA and VAR models. To keep it brief, this is the plan that I am following:
Collect multivariate data
Perform ADF ...
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Problem from Chatfield and Xing on State-Space Model
Chapter 10, Problem 2 from The Analysis of Time Series by Chatfield and Xing:
The problem says, "Consider the following special case of the linear growth model:"
$$ X_t = \mu_t + n_t $$
$$ \...
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Comparison of two models with different number of parameters
I want to compare two models, which has different number of parameters. The first model is Arbitrage free Nelson-Siegel model, which has the following equation:
$y_{t}(\tau )=X_{1,t}+X_{2,t}(\frac{1-e^...
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Do we need to propagate state covariance matrix 'P' during missing observations in the Extended Kalman Filter?
Basically as the title says: in a scenario, we have missing observations where the entire state vector is unknown for consecutive time steps. Do we just run through the prediction section of the ...
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BSTS package cannot capture seasons in simulated data
I am self-learning about structural time series, and for me the best way to understand topic is to simulate the data myself. I want to simulate a time series of local level model with seasonal ...
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Normality of data used to estimate significance of ARIMA parameters?
Does significance testing (p-value, 95% prediction interval of a forecast, etc.) of an ARIMA model require time series data to be normally distributed? As in, according to the tests used to assess ...
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Recurrent neural networks vs. State space models
I'm trying to understand the differences between RNNs and State Space Models (SSMs). I know that SSMs can take on different definitions depending on who you ask, but here I define it as in Learning ...
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How can I re-write the expectation for a linear quadratic Gaussian problem?
Let $x_0 \sim N(\mu_{x_0},\Sigma_{x_0})$, where $\mu_{x_0} \in \mathbb R^n$ and $\Sigma_{x_0} \in \mathbb R^{n \times n}$. Then, let
$$
y_0 = Cx_0 + v_0
$$
where $C \in \mathbb R^{n \times n}, v_0 \...
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State Space with Space Lags in R (dlm, MARSS or anything else)
** Edited to reflect on some first comments **
I am trying to estimate a state space model which does a kind of disaggregation. In particular, I am interested in estimating high-frequency unobserved ...
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How to use Forwards algorithm for HMM with Continuous Observation model $P(y|z_t=k)$
I have implemented a Forwards-Backwards algorithm for discrete latents HMM given the observed distribution matrix $B$.
Now if the observed distribution matrix is a Gaussian instead of finite ...
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Relationship Between "ARIMA" and "State Space"
I attended a "Lunch Seminar" hosted by the Engineering Faculty in my school where they showed how Time Series Models are being used in different Electrical Engineering projects.
Some of the ...
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How to smoothen state probabilistic time series?
I have an array of where each columns represents the probability of being in a certain state and rows represent time indexes. Each one of the rows sum up to one (we always are in one state).
\begin{...
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State space model parameterized by a neural network
For the sake of argument, assume the following simple state-space model:
Transition model: $x_{t}=a\cdot x_{t-1}+\epsilon_t$
Measurement model: $y_t=f(x_t)+\delta_t$
Assume $x_t$ and $y_t$ are 1D, ...
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Parameters in mlemodel in statsmodel
I am trying to run a TVP-VAR on statsmodel for a big data, but seems to run in a problem when I am trying to validate the vector matrix and the vector shape. Particularly, in the start and the update ...
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How do we identify parameters in this simple model?
Consider the following model:
$$ y_{it}=\nu_{it}+\epsilon_{it}$$
$$\nu_{it}=\rho \nu_{it-1}+\zeta_{it}$$
Where $y_{it}$ is the income for $i$ at time $t$. $\epsilon_{it}$ is the idiosyncratic income ...
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State space models in python statsmodels: including lag state innovation in observation equation
I am trying to fit the following state space model via the (excellent and highly useful) statsmodels state-space module. The model is a standard local level model but where last period's innovation ...
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Fitting TVP-VAR statespace mlemodel in statsmodel
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Thanks to everyone in advance for their time!
I am trying to run a TVP-VAR for a panel in the statespace mlemodels in statsmodel. I get an error while trying to fit the model
...
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Slice sampling in Particle Gibbs with Ancestral Sampling
Bear with me as I am not from statistical background. My question is about the implementation of PGAS algorithm as given in Lindsten et. al 2014 concerning sampling in state-space models. The two ...
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Subspace method fails at identifying parameters in a state space system
I am trying to infer the parameters of a linear multivariate time-invariant state space system using a subspace method. However, the inferred parameters do not match the ground-truth parameters used ...
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statespace mlemodel in statsmodel does not converge
I'm trying to estimate the following state-space model:
$$
y_{t} = \begin{bmatrix}1& 1 \end{bmatrix}\begin{bmatrix}\mu_{t}\\v_{t}\end{bmatrix} + \epsilon_{t},\quad \epsilon_{t}\sim N(0,\sigma_{\...
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TVP-VAR fails statespace.MLEModel
I am trying to run a TVP-VAR for a panel in python using statsmodels. I am using the site example, trying to adopt it in my model. Data are from 1945-2020 for 50 countries Furthermore, I am getting ...
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What is the difference between regression and state-space models?
I would like to know the differences between a regression model with autocorrelated errors and state space models (time series). When should each be used?
According to this lecture, regression (linear ...
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