# 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.

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### Linearisation of Kalman filter

Assume we have the following state-space model: $$z_{k} = \theta_{k} z_{k-1} + v_{k}\\ \theta_{k} = \phi \theta_{k-1} + w_{k},$$ where $v_{k}$ and $w_{k}$ are independent and normal for all $k$. The ...
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### Kalman filter: asymptotic of state estimate

Assume we have a linear state-space model: $$z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1} + Bu + w_{k},$$ where $u$ is some control variable (constant intercept is the simplest case). Kalman filter, ...
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### covariances in Kalman Filter

I am confused with the Kalman filter. Could you, please, explain the solution here https://stackoverflow.com/questions/46198246/em-algorithm-with-pykalman/58560992#58560992 In the simulations ...
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### Kalman EM estimation of observation variance

Let us consider a simple AR(1) process: $$y_{t} = \mu + \beta y_{t-1} + \varepsilon_{t},$$ with $t = 0, \dots, N$. Assume that the parameters $\mu$ and $\beta$ slowly change in time and let's ...
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### Kalman filter multi-step prediction

I have historical data for five correlated time series, denoted by $\{\mathbf{x}_t\}_{t=-m}^0$, where $$\mathbf{x}_t\equiv[x_{1t}, x_{2t}, x_{3t}, x_{4t}, x_{5t}]^\intercal.$$ I already have ...
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### What is behind “forecast” in Eviews?

I have been trying to use state space models in order to represent some gestural data. Until now I have been using Eviews to to do all the dynamic forecasting part, so I was curious what is behind ...
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### Kalman Filter with heteroscedastic Q (covariance of the transition noise)

I am looking at a generic derivation of the Kalman Filter (like this but you can take any). And I was wondering, checking all the derivation, why are we forced to assume that the covariance matrix Q ...
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### R- Auto Arima with Kalman Filter

In Python auto arima, it is clearly stated that when you set the method parameter as "ml" (maximum likelihood), residuals are obtained via the Kalman Filter. What is its equivalent in R?
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### What are the differences between Bayesian filters and adaptive filters?

I am learning about state estimation and I am having difficulty understanding the difference between Bayesian filters such as Kalman filter and particle filters compared to adaptive filters. According ...
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I'm interested doing a dynamic factor model (DLM) similar to Doz, Giannone and Reichlin (2011) and Giannone, Reichlin and Small (2008). Moreover, I'm trying doing macroeconomic nowcasting model. In ...
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### How to plot results from Kalman filter

I am interested in representing the performance/consistency of my Kalman filter in a single plot. I would like to compare the norm of the estimate error against 3$\sigma$ error. I would also like the ...
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I see some state space models specify their innovation process as log innovations and some squaring the term. For example, the examples in the R package DLM favours the use of log innovations when ...
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### Are Kalman Filter recursions valid when the state noise has a singular covariance matrix?

Consider a Linear Gaussian State-Space Model where the states are denoted by $X_t$ and observations are denoted by $Y_t$: \begin{align} X_t &= A X_{t-1} + \epsilon_t, &&\epsilon_t \sim \...
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### How do I model the chaotic behaviour(like the sequence from Lorenz attractor) in a stochastic sense?

Recently, I encountered a difficulty of prediction Lorenz attractor by using a GRU. (See the code from here.) I think that it's inevitable since the original system, i.e. Lorenz equation, is too ...
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### Two time-varying coefficients in Kalman filter with DLM package [closed]

I am trying to estimate a model that has two time varying coefficients in R using the "DLM" package. My measurement equation would be = Yt = F1tx1t + F2tx2t + v The state equations are: F1t = F1t-...
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### When forecasting, is it better to remove the outliers or just to transform them?

I am forecasting the number of logins. I have a dataset with the number of logins for each hour. First, I use LOF (local outlier factor) to find the outliers and then I remove them. Second, I use ...
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### How to Derive SISO Kalman Filter Update Equation Using only Probability Density Functions

I'm trying to prove to myself that a single state/single measurement kalman update can be derived using bayes theorem (as proof of concept for a more complicated task) only using the PDF. I am able ...
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### Do we need to stationarize a time series signal when using Kalman filter?

I am working on forecasting the number of logins. I know that before using ARIMA, it is important to remove trend and seasonality. But in the case of Kalman filter, I am not sure. After all it is a ...
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### Continuous-time Kalman filter with no observation/measurement noise

The continuous-time (linear) state space model can be written \begin{align*} \text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t + \mathbf{G} \,\text{d} \boldsymbol{\...
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### Books on systematic risk(beta)

I am looking for some reference material on beta (systematic risk in market model regression). I want to calculate time varying beta for stocks using kalman filter. First I would like to go through ...
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### Kalman filter parameter estimation

From what I've known about Kalman filter, it requires all the parameters of the underlying state space model. Say the state space model is: $$\xi_{t+1} = F\xi_t + v_{t+1}$$ $$y_t = H\xi_t + w_{t}$$ ...
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### How do I tell if the sensors that feed a Kalman filter has diverged?

I have a time varying variable $x$ that I want to estimate. I have two sensors A and B that measure $x$. I feed their measurements to a Kalman filter. Sometimes, one of the sensors degrades for a ...
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### State space estimation with state dependent state variance

I am estimating a state space of the following form $$Y_t= A X_t + \epsilon_t$$ $$X_{t+1} = B X_{t} + \sigma \sqrt{( a-X_t)(X_t-b)} \eta_t$$ Considering the variance of the state error is state ...
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### Kalman filter on stock sentiment time series

I was wondering if & how I can use a Kalman filter on my dataset which contains closing prices of stocks + sentiment scores of tweets about that stock for each day in a timeframe of 1 month. e.g....
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### Kalman/HMM for (short) multivariate time series from a sample with missing values

The problem in short: I want to estimate (?) a lag-1 Markovian hidden process for offline multi-variate discrete-time time series with continuous distributions via smoothing, with no dimensionality ...
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### Calculating travel distance from GPS updates

I am learning about Kalman filters but struggle to apply them for the following problem: tracking the distance traveled from GPS data. The GPS provides position updates every second and an estimate of ...