Point pattern analysis and classication on compact twopoint homogeneous spaces evolving time

2025-05-02 0 0 2.26MB 22 页 10玖币
侵权投诉
Point pattern analysis and classification on
compact two–point homogeneous spaces
evolving time
M. P. Fr´ıas1, A. Torres2and M. D. Ruiz–Medina3
Abstract
This paper introduces a new modeling framework for the statistical
analysis of point patterns on a manifold Md,defined by a connected
and compact two–point homogeneous space, including the special case
of the sphere. The presented approach is based on temporal Cox pro-
cesses driven by a L2(Md)–valued log–intensity. Different aggregation
schemes on the manifold of the spatiotemporal point–referenced data
are implemented in terms of the time–varying discrete Jacobi polyno-
mial transform of the log–risk process. The n–dimensional microscale
point pattern evolution in time at different manifold spatial scales is
then characterized from such a transform. The simulation study un-
dertaken illustrates the construction of spherical point process mod-
els displaying aggregation at low Legendre polynomial transform fre-
quencies (large scale), while regularity is observed at high frequencies
(small scale). K–function analysis supports these results under tem-
poral short–, intermediate– and long–range dependence of the log–risk
process.
Keywords: Connected and compact two–point homogeneous spaces; Cox
processes; discrete Jacobi polynomial transform; K–function; Md–supported
random fields; point pattern analysis; statistical distances.
1 Introduction
Several statistical approaches arise for processing spatial areally-
aggregated or/and misalignment data in several environmental disciplines
requiring, for example, the application of Geophysical, Ecological and Epi-
demiological models. The approach presented in this paper goes beyond the
1
arXiv:2210.11134v1 [stat.ME] 20 Oct 2022
Euclidean setting, analyzing count models on a manifold defined by a con-
nected and compact two–point homogeneous space. Under spatial isotropy
we consider weighted aggregation schemes adapted to the geometry of the
manifold, in terms of the elements of the Jacobi polynomial basis (see The-
orems 4 and 5 in [22], and [23] for the special case of the sphere). The
application of harmonic analysis in this more general context leads to the
characterization of the evolution of point patterns at different spatial scales
in the manifold.
Markov random field (MRF) models, particularly, Conditional Autore-
gressive (CAR) models have been widely applied to represent the dynamics
of the log–intensity process, interpreted as a log–risk process in the context
of double stochastic Poisson processes, also named Cox processes (see [5]).
In disease mapping, areal disease counts have been usually analyzed under
this Markovian log–risk process framework (see, e.g., [32]; [33]; [34]). Par-
ticularly, different parametric, semiparametric and nonparametric statistical
approaches have been adopted in the estimation of deterministic and random
intensities (see [4]; [13]; [16]; [17], and the references therein). In point pat-
tern analysis, special attention has been paid to functional summary statistics
like the nearest neighbour, empty space, and Kfunctions (see, e.g., [11];[19]).
Recently, LASSO estimation based on spherical autoregressive processes has
been proposed in [8] beyond the Euclidean setting.
Alternatively, in the functional data analysis (FDA) framework, condi-
tional autoregressive Hilbertian process (CARH process) models were con-
sidered by [9], [10] and [18], developing projection estimation methods for
prediction. In [28], an Autoregressive Hilbertian process (ARH(1) process)
framework was adopted to represent the dynamics of the spatiotemporal
log–risk process. This framework has also been adopted in [31] for COVID–
19 mortality prediction by applying multivariate curve regression and ma-
chine learning. As an alternative, to analyze the spatial interaction be-
tween log–risk curves at different regions, in [15], a Spatial Autoregressive
Hilbertian process (SARH(1) process) based modeling was applied. Recently,
wavelet–based projection methods are implemented in [30] to developing an
infinite–dimensional spatial multiresolution point pattern analysis, based on
spatiotemporal Log-Gaussian Cox processes in the Euclidean setting. The
present paper goes beyond this Euclidean setting. At each spatial resolution
level on the manifold, defined in terms of time–varying discrete Jacobi trans-
form, temporal point pattern analysis is achieved from the latent random
intensity process in time, and its higher order moments. In the particular
case of the sphere, suitable log–intensity models can be found in [7], where
spherical functional autoregressive (SPHAR) processes are introduced, and
2
their asymptotically analysis is derived. Additionally, spherical functional
autoregressive–moving average (SPHARMA) processes are considered in [6],
extending SPHAR processes, for suitable approximation of isotropic and sta-
tionary sphere–cross–time random fields. Here, functional spectral analysis
tools are applied, and Wold–like decomposition results are derived.
A growing interest on spherical point processes, and its functional sum-
mary statistics is observed in recent contributions (see, e.g., [26]; [27]). In this
paper, our interest relies on point patterns analysis in compact two–point ho-
mogeneous spaces evolving time. The framework of temporal Cox processes
driven by log–intensities, evaluated in the space L2(Md, dν) of square in-
tegrable functions on a compact two–point homogeneous space Mdis then
considered. Particularly, Mdis a manifold with ddenoting its topological di-
mension, and dν denotes its measure, induced by the probabilistic invariant
measure on the connected component of the group of isometries of Md.The
associated infinite–dimensional n–order product density is identified with
the infinite product of temporal n–order product densities. A spatial multi–
scale analysis of the point process evolution is achieved from these temporal
n–order product densities, and the usual functional summary statistics con-
structed from them.
The interest of the extended family of Cox processes analyzed here re-
lies on well–known examples of compact two–point homogeneous spaces like
the sphere Sd⊂Rd+1,and the projective spaces over different algebras (see
Section 2 in [22] for more details). Recent advances on modeling, analysis
and simulation of Gaussian spherical isotropic random fields, including ran-
dom fields obeying a fractional stochastic partial differential equation on the
sphere, can be exploited in our more general L2(Md, dν)–valued Gaussian
log–risk process framework (see [1]; [3]; [14]; [21], among others). Particu-
larly, [3] and [21] focalize on Cosmic Microwave Background (CMB) evolution
modeling and data analysis. The approach presented here can contribute to
this modeling framework to approximate the distribution of CMB hot and
cold spots.
In point pattern analysis on a d–dimensional manifold Md,embedded into
Rd+1,one can apply the isometric identification of (Sd, dSd) with (Md, dMd)
via the identity dSd(x1,x2) = arccos xT
1x2,for x1,x2∈Sd.This geodesic
distance dMdis involved in the definition of functional summary statistics
characterizing the aggregation, regularity or inhibition of the point pattern.
In particular, point pattern classification is achieved in terms of this geodesic
distance. This paper presents a new manifold spatial–scale–dependent point
pattern classification analysis over time, via time–varying discrete Jacobi
transform, achieved in terms of different statistical distances. Kfunction
3
analysis is also performed describing the cumulative counting properties of
pair correlation function in time through different spatial scales. In the
simulation study undertaken on the sphere, temporal short–, intermediate–
and long–range dependence models are tested, the statistical distance based
methods implemented reflect a departure from complete randomness of the
point pattern at coarser (large) scales in the manifold (low frequencies of the
time–varying discrete Legendre polynomial transform). While their small
scale (high–frequency) behavior shows regularity. K–function based analy-
sis supports the same classification results, independently of the underlying
dependence range of the log–intensity. At coarser spatial scales, stronger
departure from point pattern regularity is observed when long–range depen-
dence log–intensity models are tested. As mentioned above, the approach
presented in this paper then provides a framework to detect non–uniformity
of the spherical distribution of CMB hot and cold spots, since these deviations
from uniformity are usually geometrically described in terms of clustering,
girdling or ring structures (see, e.g., [20]; [29]).
The outline of the paper is the following. Preliminaries on connected and
compact two–point homogeneous spaces are given in Section 2. The new
class of Cox processes analyzed in a metric space framework is introduced in
Section 3. The proposed statistical distance based classification methodology
through spatial scales, involving n–order product density, is formulated in
Section 4. Kfunction is also explicitly computed from the time–varying
discrete Jacobi transform of the second–order structure of the L2(Md)–valued
temporal log–intensity. The results of the simulation study undertaken are
displayed in Section 5. Some final remarks and discussion can be found in
Section 6 to ending the paper.
2 Preliminaries
Let {Xt(·), t ∈ T ⊆ R}be an infinite–dimensional random process such that,
for each t∈ T ⊆ R,almost surely log(Xt)∈L2(Md),and
E[log(Xt)] =
L2(Md)0,with log(Xt) having characteristic functional
flog(Xt)(h) = ZL2(Md)
exp ihh, log(xt)iL2(Md)µlog(Xt)(dlog(xt))
= exp −hR0(h), hiL2(Md)
2!, h ∈L2(Md),(1)
where R0=E[log(Xt)⊗log(Xt)] ∈ L1(L2(Md)) denotes the covariance op-
erator of log(Xt),and L1(L2(Md)) is the space of trace or nuclear operators
4
on L2(Md).Here, µlog(Xt)is the induced Gaussian measure by log(Xt) on
(L2(Md),B(L2(Md))) ,with B(L2(Md)) being the σ–algebra generated by all
cylindrical subsets of L2(Md).In the subsequent development, we will also
assume that, for any t, s ∈ T ,
E[log(Xt)(z) log(Xs)(y)] = rt−s(dMd(z,y)) = er(dMd(z,y), t −s),z,y∈Md,
(2)
i.e., stationarity in time and isotropy over Mdin the weak sense are assumed.
Note that the covariance operator Rt−swith kernel rt−s(·,·) is a nuclear
operator, and its kernel rt−s(dMd(z,y)) is assumed to be continuous.
For the special case rt−s(·,·) = rs−t(·,·),the following series expansion is
obtained from Theorems 4 and 5 in [22]:
log(Xt)(z) =
∞
X
n=0
Vn(t)P(α,β)
n(cos (dMd(z,U)) ,z∈Md, t ∈R,(3)
where P(α,β)
nis a Jacobi polynomial of degree ndepending on parameter
vector (α, β) (see, e.g., [2]). Here, {Vn(t), n ∈N0}is a sequence of inde-
pendent stationary random processes on T ⊆ R,satisfying E[Vn(t)] = 0
and E[Vn(t1)Vn(t2)] = a2
nbn(t1−t2), n ∈N0.The random variable Uis
uniformly distributed on Md,and is independent of {Vn(t), n ∈N0},and
P∞
n=0 bn(0)P(α,β)
n(1) converges. Also,
cov Vn(t)P(α,β)
n(cos (dMd(z,U)) , Vm(t)P(α,β)
m(cos (dMd(z,U))= 0,
for m6=n, z∈Md,and t∈ T .
3 Cox processes family
Let now consider the measure dν(z) induced on the homogeneous space
Md=G/K, by the probabilistic invariant measure on G, with Gbeing
the connected component of the group of isometries of Md,and Kbe the
stationary subgroup of a fixed point o∈Md.As before, H=L2(Md, dν(z)) .
Our spatiotemporal count data model {Nt(·), t ∈ T } characterizes the
behavior of the temporal family Y={Yt, t ∈ T ⊆ R}of finite point sets of
Md,randomly arising at different times in the interval family {[0, t], t ∈ T }.
Specifically, for every t∈ T ,and any Borel set A⊆Md, Nt(A) denotes the
number of points in the pattern Ytfalling in the region A⊆Md,randomly
arising in the interval [0, t].Here, we consider the σ–algebra Fgenerated by
the events {Nt(A) = n}indicating that npoints in Ytare falling in a region
5
摘要：

Pointpatternanalysisandclassi cationoncompacttwo{pointhomogeneousspacesevolvingtimeM.P.Fras1,A.Torres2andM.D.Ruiz{Medina3AbstractThispaperintroducesanewmodelingframeworkforthestatisticalanalysisofpointpatternsonamanifoldMd;de nedbyaconnectedandcompacttwo{pointhomogeneousspace,includingthespecialca...

展开>> 收起<<
Point pattern analysis and classication on compact twopoint homogeneous spaces evolving time.pdf

共22页,预览5页

还剩页未读， 继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

相关推荐

更多
立即下载
分类：图书资源 价格：10玖币 属性：22 页 大小：2.26MB 格式：PDF 时间：2025-05-02

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务

作者详情

相关内容

更多

热门标签

人际关系 动力学连接体 力的合成 配电装置 高考理综 全宋诗作者索引 公务员考试
/ 22
评分 收藏
分享
立即下载
客服
关注