On the utility of asymptotic bandwidth selectors for spatially adaptive kernel density estimation

Surveillance Routing of COVID-19 Infection Spread Using an Intelligent Infectious Diseases Algorithm

“…The objective of this phase was to identify the hot spots in each of the clusters generated in Phase 1. The KDE function is a non-parametric method to estimate the probability density function of a random variable [22]. A popular version of this type of methodology is the sample point adaptive density estimator, defined bŷ…”

“…The objective of this phase was to identify the hot spots in each of the clusters generated in Phase 1. The KDE function is a non-parametric method to estimate the probability density function of a random variable [22] . A popular version of this type of methodology is the sample point adaptive density estimator, defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{equation*} \hat {f}_{h}(x)=\frac {1}{n} \sum _{i=1}^{n} K_{h} (x-x_{i})=\frac {1}{nh} \sum _{i=1}^{n} K \frac {x-x_{i}}{h} \tag{3}\end{equation*} \end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$x_{1},\ldots,x_{n}$ \end{document} are the bivariate coordinates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$n$ \end{document} independent, identically distributed observations; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$K$ \end{document} is the kernel (a non-negative function); and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$h$ \end{document} , a smoothing parameter called the bandwidth, is greater than 0.…”

Tutorial on kernel estimation of continuous spatial and spatiotemporal relative risk

“…Use of fixed-bandwidth scale rules such as h NS and h OS can, as rough guesses, be used to set h 0 in an adaptive kernel estimate as per (3); in recent work, it has been shown that specific expressions analogous to (10) for the adaptive estimator are not possible for theoretical reasons. 43 Furthermore, combining a 2D and 1D version of either of the above, we may also use the resulting bandwidths as h and in a spatiotemporal density estimate given by (6). The cautionary notes regarding the approximate and exploratory nature of such bandwidths and the densities that result remain an important consideration in these settings.…”

“…We clarify this remark in Section 4. Use of fixed bandwidth scale rules such as h NS and h OS can, as rough guesses, be used to set h 0 in an adaptive kernel estimate as per (3); in recent work it has been shown that specific expressions analagous to (10) for the adaptive estimator are not possible for theoretical reasons [15]. Furthermore, combining a 2D and 1D version of either of the above, we may also use the resulting bandwidths as h and λ in a spatiotemporal density estimate given by (6).…”

“…Furthermore, it can be applied with little modification to the spatially adaptive and spatiotemporal estimators. By replacing fh in (15) with fh0 , optimisation can be performed with respect to the global bandwidth h 0 . In the spatiotemporal case, we replace fh with fh,λ , and optimisation is 2-dimensional over both h and λ.…”