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1. There is a Difference in terms of optimality criteria Kalman filter is a Linear estimator. It is a linear optimal estimator - i.e. infers model parameters of interest from indirect, inaccurate and uncertain observations. But optimal in what sense? If all noise is Gaussian, the Kalman filter minimizes the mean square error of the estimated parameters. ...


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Maximum likelihood (ML) estimator Here will be derived a maximum-likelihood estimator of the power of the clean signal, but it doesn't seem to be improving things in terms of root mean square error, for any SNR, compared to spectral power subtraction. Introduction Let's introduce the normalized clean amplitude $a$ and normalized noisy magnitude $m$ ...


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A Gaussian process is completely specified by its mean and variance. A Kalman Filter updates the process mean which is the state, and its variance. These are the sufficient statistics. The measurement noise is reduced but the process noise is part of the recursive state history and is tracked. Your heading question and subsequent paragraph aren't ...


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The difference is quite huge, since they are two completely different models which can be used to tackle the same problem. Let's do a quick recap. Polynomial regression is a way of function approximation. We have a data set of the form $\lbrace x_i, z_i \rbrace$ and wish to determine the functional relationship, which is often expressed by estimating the ...


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*STOP! If you only want a hint and not the complete solution please see Stanley P.'s or Peter K.'s answers. * Since you do not specify if there is model for the temperature evolving over time $n$, I will derive an estimator which is a combination of $Y_1$ and $Y_2$ for each fixed $n$. Let $\alpha \in (0,1)$ and suppose the estimate of $X$ can be written as $...


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Not an expert on kalman filters, however I believe traditional Kalman filtering presumes a linear relationship between the observable data, and data you wish to infer, in contrast to more intricate ones like the Extended Kalman filters that can assume non-linear relationships. With that in mind, I believe that for a traditional Kalman filter, on-line linear ...


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Kalman filters are useful when your input signal consists of noisy observations of some linear dynamical system's state. Given a series of observations of the system state, the Kalman filter aims to recursively provide better and better estimates of the underlying system's state. In order to apply it successfully, you need to have a model for the dynamics of ...


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Kalman filtering gives multiple predictions for the next state, where an extrapolation of a regression would not. The Kalman filters are also focused on including noise factors (based on Gaussian distributions).


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as the first answer (with the most votes) says, the kalman filter is better in Any case when signal is changing. Notice the problem statement These use the algorithm to estimate some constant voltage. How could using a Kalman filter for this be better than just keeping a running average? Are these examples just oversimplified use cases of the filter? using a ...


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The covariance matrix of a Kalman filter is a function of the $ Q $ and $ R $ matrices of the model. If you use a model where $ R $ and $ Q $ are time invariant or known in prior then the calculation of the covariance matrix $ P $ can be done offline and isn't a function of the measurements. In some cases, advanced implementations of Kalman Filter estimate ...


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First of all let us assure that a Kalman filter (estimator) does not only remove Gaussian noise, but can remove (with certain success) any other type of noise as long as it's designed accordingly. However, what lies at the heart of the standard Kalman filter is the linear estimator; and that linear estimator will be the optimum minimum mean square error ...


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Update: I'm sorry to have to say that testing shows the following argument seems to break down under heavy noise. This is not what I expected, so I have definitely learned something new. My prior testing had all been in the high SNR range as my focus has been on finding exact solutions in the noiseless case. Olli, If your goal is to find the parameters ...


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Assumption of a Gaussian process allows us to obtain optimality. This uses the facts A linear( or better affine) map takes a Gaussian random variable and maps to another Gaussian random variable. A linear combination of two jointly Gaussian random variables is again a Gaussian random variable. So we don't have to track the mean and the variance. If we ...


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Another take: The Kalman Filter lets you add more information about how the system you're filtering works. In other words, you can use a signal model to improve the output of the filter. Sure, a moving average filter can give very good results when you're expecting a close-to-constant output. But as soon as the signal you're modelling is dynamic (think ...


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I won't add any equations, I will just add some intuition. I will also limit my self for Additive Gaussian White Noise. Now, in that case the Kalman filter can written as a Least Squares problem to solve. I'd say even more, the Kalman Filter is linear, if you have the samples up to certain time $ T $, you can write the Kalman filter as weighted sum of all ...


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In ideal world you'd have the correct model and use it. In your case, the model isn't perfect. Yet the steps you're suggesting are based on a knowledge you have about the process - which you should incorporate into your process equation using your dynamic model matrix: The classic and correct way given F matrix is built correctly according to your knowledge....


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A lot has been said already, allow me to add some comments: Kalman filters are an application of Bayesian probability theory, which means that "a priori information" or "prior uncertainty" can (and must) be specified. As I understand, this is not the case with traditional least-squares fitting. While observations (data) can be weighted with probabilities in ...


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KF is actually a mixture of a deterministic state propagator and a statistical estimator. Despite it's name including the term filter, Kalman filter is not a simple frequency selective one. It's indeed a statistical recursive estimator of a state of a (linear) dynamic system. Yet on a broader sense it's called as a filter as it will separate a desired ...


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If I understood it correctly, you have something that is moving and you can observe the speed and this speed is noisy. From your measurements, you observe 2 kinds of variations.\ Variations caused by the noise Variations because the object is truly changing the speed (e.g. turning) The reason your Kalman gain goes to zero is that you have implicitly ...


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You need a dynamic system to use a Kalman Filter. I would suggest $$y = \sum\limits_{i =0}^n a_i\, x^i $$ $$a[k+1] = a[k] + w $$ $$ cov(w) = Q$$ Measurement: $$z = \sum\limits_{i =0}^n a_i\, x^i = y$$ So instead of using $x$ as states, introduce the coefficient ($a$) as states


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HINT: Since in your model $X[n]$ is independent of $X[n-1]$ or $X[n+1]$ or any other shifts greater than zero, the linear estimate will have form: $$ \hat{X}[n]= \alpha Y_1[n] + \beta Y_2[n] $$ One might consider how you would weight $\alpha$ and $\beta$ given $\sigma_v$ and $\sigma_w$. One might also make a connection for Gaussian models, linear ...


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I'm going to assume that we have no information about how $X[n]$ varies with time, so we can just do one-at-time estimation of $X[n]$ using $Y_1[n]$ and $Y_2[n]$. One way to get an estimate from $Y_1$ and $Y_2$ is to average the measurements: $$ \hat{X}[n] = \frac{Y_1[n] + Y_2[n]}{2} $$ Using the fact that the sum of two independent Gaussian variables, $A$...


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The covariance decreases to a steady state regardless of how much error I introduce into the measurement. Yes, as @Drazick notes, if the $Q$ and $R$ matrices are time invariant, then the $P$ matrix will converge to a steady state that does not depend on the data (measurements). The variance for x and y are exactly the same even though I introduce more ...


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I will tell you something, even if it is differntiable, use Unscented Kalman Filter for any non linear case. This flavor of Kalman Filter, based on the Unscented Transform, is almost always superior to the Extended Kalman due to its properties. The main reason is it is able to better predict the mean and variance (Which all Kalman Filter needs) of the ...


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The Kalman Filter properties allows is to be the best linear estimator (What you call removing noise) for any White Noise (Not only Gaussian White Noise). The idea of Kalman Filter is estimating the Mean and Covariance of the State Vector at each iteration. Since optimal linear estimator (For the MMSE criteria) are based on the Mean and Covariance, as long ...


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Update If I understood your model, you have a model of Constant Velocity in 2D (Cartesian Coordinate System). While your measurement are in Polar Coordinate System. Pay attention that your measurement function is: $$ h \left( x, y, {v}_{x}, {v}_{y} \right) = \begin{bmatrix} \sqrt{ {x}^{2} + {y}^{2} } \\ {\tan}^{-1} \left( \frac{y}{x} \right ) \end{...


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An interesting approximative solution of the maximum likelihood (ML) estimation problem is obtained by using the asymptotic formula $$I_0(x)\approx \frac{e^x}{\sqrt{2\pi x}},\qquad x\gg 1\tag{1}$$ Using the notation and formulas from Olli's answer, the optimum ML estimate of the normalized clean signal amplitude satisfies $$\hat{a}=m\frac{I_1(2m\hat{a})}{...


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Scale-invariant minimum mean square error (MMSE) improper uniform prior estimators of transformed amplitude This answer presents a family scale-invariant estimators, parameterized by a single parameter which controls both the Bayesian prior distribution of amplitude and the transformation of amplitude to another scale. The estimators are minimum mean square ...


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Perhaps an analogy might be constructive. Consider a submarine commander with a fat tanker in the cross hairs of his periscope. He needs to shoot his torpedoes, not at where the target is now, but at some place where the torpedo will intersect with the target. A skilled commander will have knowledge about how fast or slow the tanker can go. knowledge ...


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UPDATE It looks like we are all stumbling forward This section is wrong, and I appreciate being corrected. First your Jacobian is incorrect for the polar to cartesian transform, from https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant $$ \begin{align} x &= r \cos \varphi ; \\ y &= r \sin \varphi . \end{align}$$ $$\mathbf J_{\mathbf ...


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