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Slope from all samples obtained To summarize the question's problem, you want to calculate the slope based on all samples obtained thus far, and as new samples are obtained, update the slope without going through all the samples again. On the page you cite is the equation for calculation of the slope $m_n$ that together with $b_n$ minimizes the sum of ...


5

The Jacobian is not computed numerically but analytically and then just evaluated. The frequency response of the IIR filter is $$H(e^{j\omega})=\frac{b_0+b_1e^{-j\omega}+\ldots+b_Me^{-jM\omega}}{1+a_1e^{-j\omega}+\ldots+a_Ne^{-jN\omega}}=\frac{B(e^{j\omega})}{A(e^{j\omega})}\tag{1}$$ Now you need the derivative with respect to the filter coefficients: $$\...


3

There are really great answers. I will try to give the Sequential Least Squares approach which generalizes to any Linear Model. Sequential Least Squares Model We're after solving the Linear Least Squares model: $$ \arg \min_{\boldsymbol{\theta}} {\left\| H \boldsymbol{\theta} - \boldsymbol{x} \right\|}_{2}^{2} $$ Now imagine that we have new measurement ...


2

It is not clear what are you asking but I will try answer both things. Deriving the Matrix Inversion Lemma The Matrix Inversion Lemma goes as: $$ {\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-1} U {\left( {C}^{-1} + V {A}^{-1} U \right)}^{-1} V {A}^{-1} $$ Deriving it is by utilizing these useful identities: $$\begin{align} U + U C V {A}^{-1} U &...


2

I'm not sure what's you model is. Let's say it is something like: $$ y = H x + n $$ Now, using the Least Squares model is optimal (In the MSE sense) when $ n $ is AWGN (It is the linear optimal estimator if the noise is white). So unless the noise in your model is colored, no gain by filtering the data before applying the Least Squares method. Now, what ...


2

The notation $ \hat{y} \left( k \mid k - 1 \right) $ usually means this is an estimated value of $ y \left( k \right) $ given all the available data up to time index $ k - 1 $. So generally speaking, this is a prediction of one step in time of the data. The case above also suggests linear estimation. Namely, $ \hat{y} \left( k \right) $ is built using ...


2

I understand your question like this You have new points $(x_i,y_i)$ coming in constantly, and would like to update the estimate of your slope $m$, analogously as you would with a running average (ie without computing the whole sums again for all the values). Suggestion Why don't you simply take the formula that is given in your link and split the terms ...


2

A resistance isn’t a particularly dynamic state, should be an unknown constant, but you might have a bin of resistors where they might vary. Taking one “randomly” out the bin makes it a random variable. The bin of resistors will have a mean value, so perhaps that mean constitutes a state variable. Perhaps someone starts putting resistors in the bin from a ...


1

The problem that recursive least squares (RLS) can solve can be formulated as recursively solving for $\hat{\theta}$, such that it is the least squares solution to $$ \hat{\theta}_n = \arg\min_x \sum_{k=0}^n w[k]\,\|z[k] - \phi[k]^\top x\|_2^2, $$ where $w[k]$ are weights, $z[k]$ and $\phi[k]$ are known and $z[k]$ is assumed to be generated by using $\phi[...


1

Without loss of generality, let's define your anchor positions to be on the $x$ axis with $x_0=0$ and the gap between each position be $g$. Thus $$ x[n] = n \cdot g $$ Obviously, this can be exactly interpolated with the equivalent linear function. $$ x(n) = n \cdot g $$ Let $(x_u,y_u)$ be the position of the user. Now the distance from any position ...


1

Maximum Likelihood under the assumption of Additive White Gaussian Noise (AWGN) is always equivalent to finding the hypothesis with the minimum distance to given data. Since minimizing distance is equivalent (In the euclidean Space) of maximizing the correlation you can always build the idea of Match Filter for parameter estimation in the settings of ML ...


1

Now I wanted to show you how to get those minimum linear mean square estimator coefficients $a$ and $b$ for your given problem setup. The procedure is summarised from the book Statistical Digital Signal Processing_MonsonHayes. Given two random variables $X$ and $Y$, we observe $X$ and want to estimate $Y$ using a linear estimator : $$ \hat{Y} = a\cdot X + ...


1

So in your case doesn't the relation $x = n+y$ help ? I mean, assuming your derivation for the mean square estimtor is right, then to compute $E\{xn\}$ you would look for $E\{ (y+n)n\}$ and using properties of $x$ and $n$ you would get $$E\{xn\} = E\{(y+n)n\} = E\{yn\} + E\{n^2\} = 0.5 + 1 = 1.5 $$


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I am joining the party, as fitting lines (and polynomials) remains a current topic when it come to huge numbers of points $N$. Indeed, in a recent work, I had to extrapolate data from cyber-physical systems, in a causal and real-time manner, with low-degree $D$ polynomials. With uniform sampling, numerical instabilities were observed with $N\gtrapprox 1.000....


1

In general, for position-related Kalman filters, you want your state vector to contain $x$, and $\dot{x}$ (location and velocity) components. See, for example, the Wikipedia page or this question and answer here. If you have a measure of velocity, then it can certainly also be an input to the Kalman filter.


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Indeed the Linear Least Squares uses the Covariance (Which resembles Cross Correlation) and the Variance (Which resembles the Auto Correlation) for parameter estimation. Let's see that using simple example - Linear Function fitting. Assume our model is given by (Simple Polynomial Model of Order 2): $$ {y}_{i} = a {x}_{i} + b, \; i = 1, 2, \cdots, n $$ In ...


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