Page - 389 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 with1Uk thecharacteristic functionof thesetUk. It seemsnatural toextendthedensityobtainedfrom samples toanotheronebasedonthe trajectories themselvesusinganintegral form: dk=λ−1 N ∑ i=1 ∫ 1 0 1Uk (γi(t))dt (2) where thenormalizingconstantλ is chosensothatdk isadiscreteprobabilitydistribution: λ= P ∑ k=1 N ∑ i=1 ∫ 1 0 1Uk (γi(t))dt= N ∑ i=1 ∫ 1 0 P ∑ k=1 1Uk (γi(t))dt andsinceUk,k=1,. . . ,P isapartition: P ∑ k=1 1Uk (γi(t))=1 (3) so thatλ=N. Densitycanbeviewedasanempiricalprobabilitydistributionwith theUk consideredasbins inanhistogram. It is thusnatural toextendtheabovecomputationsoas togiverise toacontinuous distributiononΩ. For thatpurpose, localweighting techniques, suchaskerneldensityestimation methods, arewell known in nonparametric statistics, because they are a useful data-drivenway toyield continuousdensity estimation. Many referencesmaybe found in the literature as in [5,6]. Giventheobservations, theresultingestimationwillbe thesumofweights taking intoaccount the distancebetweentheobservationsandthelocationxatwhichthedensityhastobeestimated; themore an observation is close to x, the greater is theweighting. Theweights are definedby selecting a summable functioncenteredontheobservations, calledakernel,usuallydenotedbyK :R→R+ in theunivariatecase,andasmoothedversionof theParzen–Rosenblattdensityestimator [7,8] isused. Standardchoices for theK functionare theonesusedfornonparametrickernelestimation, like the Epanechnikovfunction[9]: K : x → ( 1−x2 ) 1[−1,1](x). There exists a large variety of kernel functions, and any density function satisfying the normalizationconditioncanbeconsidered, so that theestimation isaprobabilitydensity.Moreover, thekernel function isasymmetricpositive function,with thefirstmomentequal tozeroandafinite secondordermoment. In themultivariatecase,amultivariatekernel functionK :Rq→R+ is selected that can be expressed bymeans of a real kernelK associatedwith a norm, denoted by ‖.‖, inRq as follows: K(x)=K(‖x‖), x∈Rq. Thenormalizationconditionbecomes:∫ Rq K(x)dx= ∫ Rq K(‖x‖)dx=1. Akernelversionof thedensity is thendefinedasamappingd fromΩ to [0,1]: d: x → ∑ N i=1 ∫1 0 K(‖x−γi(t)‖)dt ∑Ni=1 ∫ Ω ∫1 0 K(‖x−γi(t)‖)dtdx . (4) Normalizingthekernel isnotmandatory,as thenormalizationoccurswiththedefinitionofd. It is neverthelesseasier toconsider thesekindsofkernels, as isdone innonparametricdensityestimation. 389