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Functional Analysis 04-03-2020 Field,Vector Space,Linear Independence and Dependence,Spanning Set,Basis of Vector Space and Dementions Part 2 by Muhammad Hussain lecturer at Govt.Post Graduate College Asghar Mall Rawalpindi

Functional Analysis 04-03-2020 Field,Vector Space,Linear Independence and Dependence,Spanning Set,Basis of Vector Space and Dementions Part 2 by Muhammad Hussain lecturer at Govt.Post Graduate College Asghar Mall Rawalpindi

Functional Analysis 04-03-2020 Field,Vector Space,Linear Independence and Dependence,Spanning Set,Basis of Vector Space and Dementions Part 1 by Muhammad Hussain lecturer at Govt.Post Graduate College Asghar Mall Rawalpindi

We propose a prediction procedure for the functional linear quantile regression model by using partial quantile covariance techniques and develop a simple partial quantile regression (SIMPQR) algorithm to efficiently extract partial quantile regression (PQR) basis for estimating functional coefficients. We further extend our partial quantile covariance techniques to functional composite quantile regression (CQR) defining partial composite quantile covariance. There are three major contributions. (1) We define partial quantile covariance between two scalar variables through linear quantile regression. We compute PQR basis by sequentially maximizing the partial quantile covariance between the response and projections of functional covariates. (2) In order to efficiently extract PQR basis, we develop a SIMPQR algorithm analogous to simple partial least squares (SIMPLS). (3) Under the homoscedasticity assumption, we extend our techniques to partial composite quantile covariance and use it to find the partial composite quantile regression (PCQR) basis. The SIMPQR algorithm is then modified to obtain the SIMPCQR algorithm. Two simulation studies show the superiority of our proposed methods. Two real data from ADHD-200 sample and ADNI are analyzed using our proposed methods.

We propose a prediction procedure for the functional linear quantile regression model by using partial quantile covariance techniques and develop a simple partial quantile regression (SIMPQR) algorithm to efficiently extract partial quantile regression (PQR) basis for estimating functional coefficients. We further extend our partial quantile covariance techniques to functional composite quantile regression (CQR) defining partial composite quantile covariance. There are three major contributions. (1) We define partial quantile covariance between two scalar variables through linear quantile regression. We compute PQR basis by sequentially maximizing the partial quantile covariance between the response and projections of functional covariates. (2) In order to efficiently extract PQR basis, we develop a SIMPQR algorithm analogous to simple partial least squares (SIMPLS). (3) Under the homoscedasticity assumption, we extend our techniques to partial composite quantile covariance and use it to find the partial composite quantile regression (PCQR) basis. The SIMPQR algorithm is then modified to obtain the SIMPCQR algorithm. Two simulation studies show the superiority of our proposed methods. Two real data from ADHD-200 sample and ADNI are analyzed using our proposed methods.

Carry out �standard� constructions in (linear) functional analysis: Abstract Hilbert space � one in each dimension, Concrete Hilbert space � Many, such as L2([0, 1])

Carry out �standard� constructions in (linear) functional analysis: Abstract Hilbert space � one in each dimension, Concrete Hilbert space � Many, such as L2([0, 1])

Carry out �standard� constructions in (linear) functional analysis: Abstract Hilbert space � one in each dimension, Concrete Hilbert space � Many, such as L2([0, 1])

De�nition 4. A set E _ R is of measure zero if there exists an absolutely summable series of step functions fn such that ...

Performing a stability analysis during the design of any electronic circuit is critical to guarantee its correct operation. A closed-loop stability analysis can be performed by analysing the impedance presented by the circuit at a well-chosen node without internal access to the simulator. If any of the poles of this impedance lie in the complex right half-plane, the circuit is unstable. The classic way to detect unstable poles is to fit a rational model on the impedance. This rational approximation has to deal with model order selection, which is difficult in circuits with transmission lines. In this paper, a projection-based method is proposed which splits the impedance into a stable and an unstable part by projecting on an orthogonal basis of stable and unstable functions. Working with a projection instead of a rational approximation greatly simplifies the stability analysis. When the projection is mapped from the complex plane to the unit disc, the projection boils down to calculating a Fourier series. If a significant part of the impedance is projected on the unstable part, a low-order rational approximation is fitted on this unstable part to find the location of the unstable poles.

We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\to \infty$, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.

The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in case of homogeneous linear functional equations. The foundation of the theory can be found in M. Laczkovich and G. Kiss \cite{KL}, see also G. Kiss and A. Varga \cite{KV}. We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to T. Szostok \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}. They are motivated by quadrature rules of approximate integration.

A new functional-based approach is developed for the stability analysis of linear impulsive systems. The new method, which introduces looped-functionals, considers non-monotonic Lyapunov functions and leads to LMIs conditions devoid of exponential terms. This allows one to easily formulate dwell-times results, for both certain and uncertain systems. It is also shown that this approach may be applied to a wider class of impulsive systems than existing methods. Some examples, notably on sampled-data systems, illustrate the efficiency of the approach.

We study regression models for the situation where both dependent and independent variables are square-integrable stochastic processes. Questions concerning the definition and existence of the corresponding functional linear regression models and some basic properties are explored for this situation. We derive a representation of the regression parameter function in terms of the canonical components of the processes involved. This representation establishes a connection between functional regression and functional canonical analysis and suggests alternative approaches for the implementation of functional linear regression analysis. A specific procedure for the estimation of the regression parameter function using canonical expansions is proposed and compared with an established functional principal component regression approach. As an example of an application, we present an analysis of mortality data for cohorts of medflies, obtained in experimental studies of aging and longevity.

We study regression models for the situation where both dependent and independent variables are square-integrable stochastic processes. Questions concerning the definition and existence of the corresponding functional linear regression models and some basic properties are explored for this situation. We derive a representation of the regression parameter function in terms of the canonical components of the processes involved. This representation establishes a connection between functional regression and functional canonical analysis and suggests alternative approaches for the implementation of functional linear regression analysis. A specific procedure for the estimation of the regression parameter function using canonical expansions is proposed and compared with an established functional principal component regression approach. As an example of an application, we present an analysis of mortality data for cohorts of medflies, obtained in experimental studies of aging and longevity.

The article provides an introduction to infinite-dimensional differential calculus over topological fields and surveys some of its applications, notably in the areas of infinite-dimensional Lie groups and dynamical systems.