Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Make your voice heard. Take the 2019 Developer Survey now

Questions tagged [kalman-filter]

The Kalman filter is an algorithm for estimating the mean vector and variance-covariance matrix of the unknown state in a state space model.

0
votes
0answers
16 views

Observation Operator - Data Assimilation

In data assimilation, one assumes the existence of a observation operator $\mathcal{H}$ that maps the model-state vector $\bf{x_b}$ to $ \bf{y_b}$ (the model-equivalent of the observations $\bf{y_o}$) ...
0
votes
0answers
29 views

Fixed-Delay Kalman smoother with/without augmented measurements

There are several algorithms regarding fixed-lag Kalman smoothing. In most cases, an augmented state vector is defined in which the elements are the current and delays of the original state vector. ...
0
votes
1answer
27 views

Is a Kalman filter ever the optimal way to estimate a dynamic value given a full history of measurements?

I'm trying to get some intuition for Kalman filtering, and I conceived this toy example: Say that I have a sensor that tracks a moving 1-dimensional target. Say that the measurements from the sensor ...
2
votes
0answers
41 views

ARMA process forecasts and maximum likelihood parameters

I have some trouble understanding the forecasting/inference process of ARMA models. From Hamilton (which I am reading now), we can obtain forecasts at $Y$ from any linear process with r.v. values $X$...
0
votes
0answers
16 views

Trouble replicating the experiments in “On-line Novelty Detection Using the Kalman Filter and Extreme Value Theory”

I'm trying to replicate the online novelty detection algorithm from "On-line Novelty Detection Using the Kalman Filter and Extreme Value Theory" by Hyoung-joo Lee and Stephen J. Roberts. In the first ...
2
votes
1answer
64 views

Math questions in Kalman filter equation derivation

I am interested in data analysis. While my working data (actually it's shopping mall's daily sale) is accumlating, I wish to find some statistical laws underlying business phenomena. I left school for ...
1
vote
1answer
65 views

Kalman Filter vs. Regression

I'm an economics undergraduate with a fundamental understanding of regression and some experience with machine learning models (e.g. regression trees, boosting). To my knowledge, Kalman Filter is ...
0
votes
1answer
36 views

(Online) intuitive explanation of state space models

I have a similar question to the one in the link below: Intuitive explanation of state space models In the link they recommend the book by Commandeur and Koopman. I have this book already. I was ...
1
vote
0answers
19 views

How to create the initial ensemble samples for EnKF

As we know, for the ensemble Kalman filter (EnKF), we need to create a set of samples in the beginning and then to run the predict and analysis step. But for now I have a question of how to create the ...
0
votes
0answers
51 views

Examples of state space models where the filtering problem can be solved analytically

Background A discrete-time, Markovian state space model takes the form \begin{align} \mathbf{y}_t&\sim p(\mathbf{y}_t\,|\,\mathbf{s}_t,\,\boldsymbol{\theta})\\ \mathbf{s}_t&\sim p(\mathbf{s}...
0
votes
0answers
17 views

How can I not show the initialization of the estimation in the Extended Kalman Filter?

I'm making estimates through the Extended Kalman Filter and I have a problem related to the vertical axis of my figure, it's too big, so I can not see population dynamics. However, I wish it did not ...
0
votes
1answer
48 views

Tracking Moving Objects with Kalman Filters— Over-fitting over time?

I've been learning about Kalman Filters, and the classic example given is tracking an object via radar/gps. My issue here is that each time you get a new data point, you update the error in the ...
0
votes
0answers
34 views

How can one use Kalman filtering to estimate stochastic volatility models?

Assume that we have returns modelled by a stochastic volatility model with parameters that are unknown. Say we want to estimate the parameters with Quasi-Maximum Likelihood estimation and the ...
1
vote
0answers
14 views

Are there any R code examples for estimating the state space vector in this case?

I couldn't make sure Whether the model I'm using is a local level model with multiplicative components (state vector $\times$ regressor vector) or a linear gaussian state-space model. And couldn't ...
0
votes
1answer
42 views

How to sample an unobserved Markov process using the forward-backward algorithm?

The setup Let $X = (x_1, \ldots, x_T)$ denote a state variable that follows a Markov process, where $x_t \in S$. The transition distribution is denoted by \begin{equation} p(x_{t}|x_{t-1}) . \end{...
0
votes
0answers
19 views

How to interpret log-likelihood score as compared to mse

Say one has a linear dynamic system as follows: $x_k = Fx_{k-1} + v_k$ $y_k = Hx_{k-1} + w_k$ with $v \sim (0, Q)$ and $w \sim (0, R)$. I am estimating $(x)_k$ using a normal Kalman Filter and ...
1
vote
1answer
66 views

State-space model with contemporaneous effects

I have the following system of equations: $$ \begin{align} y_t^{(1)}&=y_t^{(2)}-x_t+\epsilon_t\\ y_t^{(2)}&=x_t+\nu_t\\ x_t&=\alpha x_{t-1}+u_t \end{align} $$ where $y_t^{(1)}, y_t^{(2)}$ ...
1
vote
2answers
36 views

Probability of a measurement with uncertainty covariance being generated by a normal distribution

I have the following situation: A set of Kalman filters with the same model, each with its own current estimated state and state covariance. A measurement with a covariance matrix expressing its ...
0
votes
0answers
36 views

Kalman-filter different sampling rates

I'm trying to implement a simple Kalman filer on GPS and Linear Acceleration in Python. I'm having trouble because I get acceleration data 100 times a second, but GPS data just once a second (so 100 ...
0
votes
1answer
35 views

Proper Imputation and bias-correction on degrading signal with Kalman Filtering?

A signal degrades in its quality. Some signals are far more robust to degradation while others are not. We will simulate degradation by randomly removing values from a function and then applying ...
2
votes
1answer
81 views

State Space Model Form for Equations

I have a set of equations which I have to write in state space model form but unfortunately I'm having a bit of difficulty doing so. They are given as: $y_{t} = x_{t} + z_{t}$ $x_{t} = x_{t-1} + w_{...
2
votes
1answer
106 views

Deriving a filter like a Kalman filter from a non-Gaussian state space model

Assume we specify a state space model as $$Y_t = a X_t + W_t$$ and $$X_{t+1} = b X_t + V_t$$ where $b,a \in R$, $E[W_t] = E[V_t] = 0 \quad \forall{t }$ and $W_t $ and $V_t$ are indipendent for ...
1
vote
1answer
30 views

Student doubts about maximmization

I am an economics student and I am having doubts about optimization. For example, at some point in my course I will estimate a state space model via kalman filter and I will need to find parameters ...
2
votes
0answers
43 views

Using Kalman Filters with different dimensionality in an Interacting Multiple Model Algorithm

I am currently reading a lot about Kalman Filtering and am especially interested in the IMM - Interactive Multiple Model Algorithm. In the literature (e.g. here), IMM is used for Kalman Filters with ...
1
vote
1answer
31 views

Why is Qk not included in the cost function that is optimised by the Kalman filter?

Assume the following linear discrete system: $x_k = Fx_{k-1} + w_{k-1}$ where $w_{k} \sim N(0, Q)$ $y_k = Hx_k + v_{k}$ where $v_{k} \sim N(0, R)$ One way to prove that the Kalman filter is optimal ...
0
votes
0answers
30 views

Chi^2 Test: Alpha and it's Relation to Sigma

I'm using an Extended Kalman Filter and the $Chi^2$ test to test a part of it's residual during the update. Here I want to determine if the residual lays within 3 Sigma and reject outliers that are ...
0
votes
0answers
59 views

How to approach SSM models for time series forecasting in general?

I have worked on SSM model using KFAS package (https://cran.r-project.org/web/packages/KFAS/KFAS.pdf) in R. Package suggests me to use one of the Box_Jenkins method to implement SSM. So we convert ...
2
votes
0answers
41 views

What is the difference between Noise, error and residuals?

I was reading about Kalman filter. http://web.mit.edu/kirtley/kirtley/binlustuff/literature/control/Kalman%20filter.pdf They talk about additive noise and error. I need to understand difference ...
0
votes
1answer
48 views

Does the MLE-Kalman prediction maximize the likelyhood of the prediction?

The question is the following. Say I have observations of a Gaussian stochastic process ($\{x_i\}_{i=1}^n$) for which is convenient to use the state space formalism (and Kalman recursions) to describe ...
0
votes
0answers
27 views

Interpreting the standard deviation in this setting

I am having trouble relating the estimate of the standard deviation to the precision of the estimate of the $y$-variable. For example: $$\tilde{y}_t=a_{y,1}\,\tilde{y}_{t-1}+a_{y,2}\,\tilde{y}_{t-2}+\...
0
votes
0answers
37 views

Intuition behind definition of one-step-ahead prediction errors

So in this note, https://www.bankofengland.co.uk/-/media/boe/files/archive/discussion-paper/a-note-on-the-estimation-of-grach-m-models-using-the-kalman-filter the author defines the discrete system ...
0
votes
1answer
68 views

Which is the random variable in a Kalman filter?

When estimating a hidden state $x$ with a Kalman filter, there is the posterior and prior estimate. There are also covariances associated with those estimates. Some authors call these the covariances ...
1
vote
0answers
40 views

Extended kalman filter vs online passive-aggressive

I was wondering, what are the advantages and disadvantages of extended Kalman filter and online passive-aggressive algorithm when we use them to train our networks. I have RBF neural network and I'm ...
1
vote
0answers
26 views

Bayes filter with delayed measurements

I have some straight and curve pieces with numbers, they are used to build tracks (of $5$ lanes) for my cars (figure $1$), I can send commands to the cars using an SDK on the Raspberry (set the speed ...
1
vote
0answers
20 views

best statistical approach to study the time evolution of clustering in a data set

I am using a stochastic method for the clustering of a data set. The number of clusters that this approach returns, can differ in each iteration. On the other hand, I would like to study the evolution ...
2
votes
0answers
151 views

traditional state-space models and LSTMs

I am trying to understand the nature of LSTMs in relation to intuitions from traditional state-space models (e.g., Kalman filtering). The code below aims to simulate a simple univariate linear state-...
0
votes
0answers
83 views

Initialization of the Kalman Filter

I would like to try out different initialization procedures of the Kalman Filter in order to see if it effects the estimation paths of the state variables. One way of initializing is to use the first ...
0
votes
0answers
82 views

Implementation of kalman filter with inner ARIMA non seasonal model

I am trying to write an application which impute some missing values on one time series signal. I have done it similarly in R using ImputeTS package but now need to do it similarly in Java. I just ...
1
vote
0answers
42 views

unscented kalman filter for non-linear state-space

I intend to use unscented kalman filter to estimate a non-linear state -space problem. latent factor $X_t$ in the formulation has usual VAR(1) specification $$X_t = \phi X_{t-1} +\epsilon_t$$ ...
3
votes
1answer
94 views

Negative variances in Kalman smoother (FFBS)

I have implemented the forward-filtering-backwards-sampling (ffbs) algorithm. It consists of kalman filtering forward in time (to obtain mean and sigma). Then it uses these values and the Kalman ...
0
votes
0answers
137 views

ARIMAX and Kalman filter to impute missing data

Following this post How to use auto.arima to impute missing values, and the really comprehensive answers there: Is it possible to implement this gap filling method with covariates, e.g. using climatic ...
0
votes
0answers
151 views

Transaction and Observation matrices in Kalman Filtering for univariate data

I am working on Kalman Filtering and I have questions about observation and transition matrices. So, I have a variable and I want to clean the noise from my variable by using Kalman Filtering. What ...
3
votes
0answers
112 views

Is a Kalman Filter applicable for irregular, infrequent measurement?

I have taken on a project previously approached by someone else, looking at sensor data. Each sensor produces about three days of data (sampling about once a second), and each day a calibration is ...
0
votes
0answers
55 views

Why are Kalman Filters nontrivial?

I'm starting to learn about Kalman Filters and its variants but am having a hard time understanding why they're interesting or nontrivial. As far as I understand, giving some observations $O_{0:t}$ ...
1
vote
1answer
37 views

MLE derivation of the Recursive Least Squares estimator

I think I'm able to derive the RLS estimate using simple properties of the likelihood/score function, assuming standard normal errors. If the model is $$Y_t = X_t\beta + W_t$$ then the likelihood ...
0
votes
1answer
220 views

Bayesian Filtering for linear but non-Gaussian estimation problems

It seems that most optimal estimation literature is divided into either linear Gaussian problems, for which you use Kalman Filter, or non linear and non Gaussian problems for which you use EKF, UKF or ...
1
vote
0answers
65 views

Using xreg function in R while using kalman filter [closed]

I am using a kalman filter on my data to both impute missing values and predict 20 samples ahead. However, I would like to use a ARIMAX model instead of Arima model using XREG as well... this is my ...
0
votes
0answers
52 views

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 ...
1
vote
1answer
249 views

Kalman Filter prediction using different time step

Typically Kalman Filter or any other time series forecasting methods use a single step prediction - update step. For eg: Let us say I have sensor data collected at every 1ms. Let z denote ...
1
vote
0answers
11 views

Including features in a Dirichlet model with Markov dynamics

I have a fantasizing about a model here, so please keep in mind that this is not even half-baked: I have categorical time series data $y_t\sim\text{Cat}(y\ |\ \lambda_t)$ with a hidden variable $\...