is related to control theory, dynamic optimization and statistical methods to build mathematical models based on some measured data.

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Efficient estimator: best choice for the weight matrix in the Weighted Least Squares Estimation

I am facing the linear regression problem in the form: $$y = \Phi\theta+\eta$$ where $y\in\mathbb{R}^N$ is the vector of the measurements (the available data), $\Phi\in\mathbb{R}^{N\times n}$ the ...
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32 views

Properties of Injective Operator on Hilbert Space

I am new to functional analysis and have the following issue: Given an infinite dimensional Hilbert space $H$ and an operator $f: H \times \Omega \to H$, where $\Omega$ is some finite dimensional ...
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55 views

What is the name of this formula??

Let formula_a = lambda array: sum(array) / len(array). We may rename this formula to be average or ...
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42 views

Linear Map of an ellipsoid in $\mathbb{R}^N$ into another ellipsoid in $\mathbb{R}^n$, with $n<N$

Starting from the closed set describing an ellipsoid in $\mathbb{R}^N$: $$\Omega_x = \{ x \in \mathbb{R}^N : (x-x_0)^T\Sigma_x^{-1}(x-x_0) \leq \varepsilon^2 \}$$ where $\Sigma_x \in \mathbb{R}^{N\...
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Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
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Surface fitting: Where to start from?

Often, we deal with identification problems such as identifying the parameters $\alpha_i$ where $z(x) = f_{\alpha_i}(x,y)$, which means simply $z$ is a function of $(x,y)$ and the parameters $\alpha_i$...
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34 views

Identification of non-linear functions:polynomial+exponential

Is there a way to perform a non linear least square to identify the following function: $$\alpha_2\cdot x^2 + \alpha_1\cdot x + \alpha_0 + \beta e^{\frac{\gamma}{x}}=Y$$ I aim at identifying the ...
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Support of a Random Variable and Linear Subspaces in $\mathbb{R}^d$

I am reading a Text about Single Index Models where a theorem is given for the identification in case all covariates are continuous. The theorem states these four conditions: $G$ is differentiable ...
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38 views

How to treat non-identifiable states in Kalman filtering/dynamic linear models?

Let $x_t = G_tx_{t-1}+\omega_t$ with $\omega_t \sim \mathrm{N}(\mathbf{0}, \mathbf{W}_t)$ be a state equation and $y_t = F_tx_t+\nu_t$ with $\nu_t \sim \mathrm{N}(\mathbf{0}, \mathbf{V}_t)$ be a ...
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85 views

Identification of real functions

this my second question, so I'm still new... thanks in advance for any help! Basically, I'm looking for some references and tools to study the following problem. Consider the following function $f(\...
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29 views

Automatic component identification/extraction of a system

This may probably be a nonsensical question, but here goes… A definition of a system could be: “a set of interacting components, which give structure and behaviour to the system”. E.g. say, a car (a ...
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140 views

Identifiability of a state space system

I'm trying to solve assignment 4E.5 from this sheet (ship steering dynamics). My question are: Do I need to perform the Laplace Transform in order to check for identifiability? The state space model ...
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281 views

Explain the existence of limit in Persistence Excitation — mostly zero and non-existent?

Definitions Persistence Excitation on page 121 here or shortly here and here. A signal is PE if this limit exists $$r_u(\tau)=\lim_{N\rightarrow\infty}\frac 1 N \sum_{t=1}^{N} u(t+\tau)u^T(t)$$...