Change-point Detection for Sparse and Dense Functional Data in General Dimensions
Anonymous Author(s)
AffiliationAddress email
Abstract
We study the problem of change-point detection and localisation for functionaldata sequentially observed on a general d -dimensional space, where we allowthe functional curves to be either sparsely or densely sampled.
Data of this formnaturally arise in a wide range of applications such as biology, neuroscience, climatology and finance.
To achieve such a task, we propose a kernel-based algorithmnamely functional seeded binary segmentation (FSBS).
FSBS is computationallyefficient, can handle discretely observed functional data, and is theoretically soundfor heavy-tailed and temporally-dependent observations.
Moreover, FSBS worksfor a general d -dimensional domain, which is the first in the literature of changepoint estimation for functional data.
We show the consistency of FSBS for multiplechange-point estimation and further provide a sharp localisation error rate, whichreveals an interesting phase transition phenomenon depending on the number offunctional curves observed and the sampling frequency for each curve.
Extensivenumerical experiments illustrate the effectiveness of FSBS and its advantage overexisting methods in the literature under various settings.
A real data application isfurther conducted, where FSBS localises change-points of sea surface temperaturepatterns in the south Pacific attributed to El Niño.
Introduction
Recent technological advancement has boosted the emergence of functional data in various applicationareas, including neuroscience [e g. 11, 23], finance [e g. 13], transportation [e.g. 10], climatology[e.g. 7, 14] and others.
We refer the readers to [29] - a comprehensive review, for recent developmentof statistical research in functional data analysis.
In this paper we study the problem of change-point detection and localisation for functional data,where the data are observed sequentially as a time series and the mean functions are piecewisestationary, with abrupt changes occurring at unknown time points.
To be specific, denote D as ageneral d -dimensional space that is homeomorphic to [0 , 1] d , where d ∈ N + is considered as arbitrarybut fixed.
We assume that the observations { ( x t,i , y t,i ) } T,nt =1 ,i =1 ⊆
D ×
R are generated based ony t,i = f
[Equation]
t
[Equation]
x t
[Equation]
i )
+
[Equation]
t
[Equation]
x t
[Equation]
i )
+
[Equation]
t,i , for t =
[Equation] [Equation]
T and i =
[Equation] [Equation]
[Equation]
In this model, { x t,i } T,nt =1 ,i =1 ⊆
D denotes the discrete grids where the (noisy) functional data{ y t,i } T,nt =1 ,i =1 ⊆ R are observed, { f ∗ t : D → R } Tt =1 denotes the deterministic mean functions,{ ξ t : D → R } Tt =1 denotes the functional noise and { δ t,i } T,nt =1 ,i =1 ⊆
R denotes the measurement error.
We refer to Assumption 1 below for detailed technical conditions on the model.
To model the unstationarity of sequentially observed functional data which commonly exists in realworld applications, we assume that there exist K ∈ N change-points, namely
=
η 0 < η 1 < · · · < η K < η K +1 =
T , satisfying that f ∗ t ̸
= f ∗ t +1 , if and only if t ∈ { η k } Kk =1 .
Our primary interest is toaccurately estimate { η k } Kk =1 .
Due to the importance of modelling unstationary functional data in various scientific fields, thisproblem has received extensive attention in the statistical change-point literature, see e.g. [3], [6],[17], [31], [4] and [12].
Despite the popularity, we identify a few limitations in the existing works.
Firstly, both the methodological validity and theoretical guarantees of all these papers require fullyobserved functional data without measurement error, which may not be realistic in practice.
Secondly,most existing works focus on the single change-point setting and to our best knowledge, thereis no consistency result of multiple change-point estimation for functional data.
Lastly but mostimportantly, existing algorithms only consider functional data with support on [0 , 1] and thus are notapplicable to functional data with multi-dimensional domain, a type of data frequently encounteredin neuroscience and climatology.
In view of the aforementioned three limitations, in this paper, we make several theoretical andmethodological contributions, summarized below.
• In terms of methodology, our proposed kernel-based change-point detection algorithm, functionalseeded binary segmentation (FSBS), is computationally efficient, can handle discretely observedfunctional data contaminated with measurement error, and allows for temporally-dependent andheavy-tailed data.
FSBS, in particular, works for a general d -dimensional domain with arbitrary butfixed d ∈ N + .
This level of generality is the first time seen in the literature.• In terms of theory, we show that under standard regularity conditions, FSBS is consistent indetecting and localising multiple change-points.
We also provide a sharp localisation error rate,which reveals an interesting phase transition phenomenon depending on the number of functional
curves observed T and the sampling frequency for each curve n .
To the best of our knowledge, the
theoretical results we provide in this paper are the sharpest in the existing literature.•
A striking case we handle in this paper is that each curve is only sampled at one point, i.e. n = 1 .To
the best of our knowledge, all the existing functional data change-point analysis papers assumefull curves are observed.
We not only allow for discrete observation, but carefully study this mostextreme sparse case n = 1 and provide consistent localisation of the change-points.•
We conduct extensive numerical experiments on simulated and real data.
The result further supportsour theoretical findings, showcases the advantages of FSBS over existing methods and illustrates thepracticality of FSBS.• A byproduct of our theoretical analysis is new theoretical results on kernel estimation for functionaldata under temporal dependence and heavy-tailedness.
This set of new results per se are novel,enlarging the toolboxes of functional data analysis.
Notation and definition.
For any function f : [0 , 1] d → R and for 1 ≤ p < ∞ , define ∥ f
∥ p =( (cid:82) [0 , 1] d | f ( x ) | p d x ) 1 /p and for p = ∞ , define ∥ f ∥ ∞ = sup x ∈ [0 , 1] d | f ( x ) | .
Define L p = { f :[0 , 1] d → R , ∥ f ∥ p < ∞} .
For any vector s =
( s 1 , . . .
, s d ) ⊤ ∈ N d , define | s | = (cid:80) di =1 s i ,s !
= s 1 ! · · · s d !
and the associated partial differential operator D s = ∂
| s | ∂x s 11 ··· ∂x sdd .
For α > 0 , denote⌊ α ⌋ to be the largest integer smaller than α .
For any function f : [0 , 1] d
→ R that is ⌊ α ⌋ -timescontinuously differentiable at point x 0 , denote by f αx 0 its Taylor polynomial of degree ⌊ α ⌋ at x 0 ,which is defined as f αx 0 ( x ) = (cid:80) | s |≤⌊
α ⌋ ( x − x 0 ) s s !
D s f ( x 0 ) .
For a constant L > 0 , let H α ( L ) be theset of functions f : [0 , 1] d → R such that f is ⌊ α ⌋ -times differentiable for all x ∈ [0 , 1] d and satisfy| f ( x ) − f αx 0 ( x ) | ≤ L | x − x 0 | α , for all x, x 0 ∈ [0 , 1] d .
Here | x − x 0 | is the Euclidean distancebetween x, x 0 ∈ R d .
In non-parametric statistical literature, H α ( L ) are often referred to as the classof Hölder smooth functions.
We refer the interested readers to [25] for more detailed discussion on
Hölder smooth functions.
For two positive sequences { a n } n ∈ N + and { b n } n ∈ N + , we write a n = O ( b n ) or a n ≲ b n if a n ≤
Cb nwith some constant C > 0 that does not depend on n , and a n = Θ( b n ) or a n ≍ b n if a n = O ( b n )and b n = O ( a n ) .
Functional seeded binary segmentation2.1 Problem formulation
Detailed model assumptions imposed on model (1) are collected in Assumption 1.
For notationalsimplicity, without loss of generality, we set the general d -dimensional domain D to be [0 , 1] d , as theresults apply to any D that is homeomorphic to [0 , 1] d .
Assumption 1.
The data { ( x t,i , y t,i ) } T,nt =1 ,i =1 ⊆
[0 , 1] d × R are generated based on model (1) .a.
(Discrete grids)
The grids { x t,i } T,nt =1 ,i =1 ⊆
[0 , 1] d are independently sampled from a commondensity function u : [0 , 1] d → R .
In addition, there exist constants r > 0 and L > 0 such thatu ∈ H r ( L ) and that inf x ∈ [0 , 1] d u ( x ) ≥ ˜ c with an absolute constant ˜ c > 0 .b.
(Mean functions)
For r > 0 and L > 0 , we have f ∗ t ∈
H r ( L ) .
The minimal spacing between twoconsecutive change-points ∆ =
min K +1 k =1 ( η k − η k − 1 ) satisfies that ∆ =
Θ( T ) .c.
(Functional noise) Let { ε i , ε ′ 0 } i ∈
Z be i.i.d.
random elements taking values in a measurable space
S ξ and g be a measurable function g : S ∞ ξ
→ L 2 .
The functional noise { ξ t } Tt =1 ⊆
L 2 takes the formξ t =
g ( G t ) , with G t
= ( . . .
, ε − 1 , ε 0 , ε 1 , . .
. , ε t − 1 , ε t ) .
There exists an absolute constant q ≥ 3 , such that E ( ∥ ξ t ∥ q ∞ ) <
C ξ, 1 for some absolute constant C ξ, 1 .
Define a coupled processξ ∗ t =
g ( G ∗ t ) , with G ∗ t = ( . . .
, ε − 1 , ε ′ 0 , ε 1 , . .
. , ε t − 1 , ε t ) .
We have (cid:80) ∞ t =1 t 1 / 2 − 1 /q { E ∥ ξ t − ξ ∗ t ∥ q ∞ } 1 /q < C ξ, 2 for some absolute constant C ξ, 2 > 0 .d.
(Measurement error)
Let { ϵ i , ϵ ′ 0 }
i ∈
Z be i.i.d.
random elements taking values in a measurablespace S δ and ˜ g
n be a measurable function ˜ g n : S ∞ δ
→ R n .
The measurement error { δ t } Tt =1 ⊆
R ntakes the form
[Equation]
t =
[Equation]
n
[Equation]
F t ) , with F t
= ( . . .
, ϵ − 1 , ϵ 0 , ϵ 1 , . .
, ϵ t − 1 , ϵ t ) .
There exists an absolute constant q ≥ 3 , such that max ni =1 E ( | δ t,i | q ) <
C δ, 1 for some absoluteconstant C δ, 1 .
Define a coupled process
[Equation]
with F ∗ t = ( . . .
, ϵ − 1 , ϵ ′ 0 , ϵ 1 , . .
, ϵ t − 1 , ϵ t ) .
We have max ni =1 (cid:80) ∞ t =1 t 1 / 2 − 1 /q { E | δ t,i − δ ∗ t,i | q } 1 /q <
C δ, 2 for some absolute constant C δ, 2 > 0 .
Assumption 1 a allows the functional data to be observed on discrete grids and moreover, we allow fordifferent grids at different time points.
The sampling distribution µ is required to be lower boundedon the support [0 , 1] d , which is a standard assumption widely used in the nonparametric literature[e.g. 26].
Here, different functional curves are assumed to have the same number of grid points n .
We remark that this is made for presentation simplicity only.
It can indeed be further relaxed and themain results below will then depend on both the minimum and maximum numbers of grid points.
Note that Assumption 1 a does not impose any restriction between the sampling frequency n and thenumber of functional curves T , and indeed our method can handle both the dense case where n ≫ Tand the sparse case where n can be upper bounded by a constant.
Besides the random samplingscheme studied here, another commonly studied scenario is the fixed design, where it usually assumesthat the sampling locations { x i } ni =1 are common to all functional curves across time.
We remark thatwhile we focus on the random design here, our proposed algorithm can be directly applied to thefixed design case without any modification.
Furthermore, its theoretical justification under the fixeddesign case can be established similarly with minor modifications, which is omitted.
The observed functional data have mean functions { f ∗ t } Tt =1 , which are assumed to be Hölder continuous in Assumption 1 b .
Note that the Hölder parameters in Assumption 1 a and b are both denotedby r .
We remark that different smoothness are allowed and we use the same r here for notationalsimplicity.
This sequence of mean functions is our primary interest and is assumed to possess a piecewise constant pattern, with the minimal spacing ∆ being of the same order as T .
This assumptionessentially requires that the number of change-points is upper bounded.
It can also be further relaxedand we will have more elaborated discussions on this matter in Section 5.
Our model allows for two sources of noise - functional noise and measurement error, which aredetailed in Assumption 1 c and d , respectively.
Both the functional noise and the measurement errorare allowed to possess temporal dependence and heavy-tailedness.
For temporal dependence, weadopt the physical dependence framework by [30], which covers a wide range of time series models,such as ARMA and vector AR models.
It further covers popular functional time series models such asfunctional AR and MA models [17].
We also remark that Assumption 1 c and d impose a short rangedependence, which is characterized by the absolute upper bounds C ξ, 2 and C δ, 2 .
Further relaxation ispossible by allowing the upper bounds C ξ, 2 and C δ, 2 to vary with the sample size T .
The heavy-tail behavior is encoded in the parameter q .
In Assumption 1 c and d , we adopt thesame quantity q for presentational simplicity and remark that different heavy-tailedness levelsare allowed.
An extreme example is that when q = ∞ , the noise is essentially sub-Gaussian.
Importantly, Assumption 1 d does not impose any restriction on the cross-sectional dependenceamong measurement errors observed on the same time t , which can be even perfectly correlated.
2.2 Kernel-based change-point detection
To estimate the change-point { η k } Kk =1 in the mean functions { f ∗ t } Tt =1 , we propose a kernel-basedcumulative sum (CUSUM) statistic, which is simple, intuitive and computationally efficient.
Thekey idea is to recover the unobserved { f ∗ t } Tt =1 from the observations { ( x t,i , y t,i ) } T,n t =1 ,i =1 based onkernel estimation.
Given a kernel function K ( · ) : R d → R + and a bandwidth parameter h > 0 , we define K h ( x )
=
h − d K ( x/h ) for x ∈ R d .
Given the random grids { x t,i } T,nt =1 ,i =1 and a bandwidth parameter ¯ h , wedefine the density estimator of the sampling distribution u ( x ) as
[Equation]
Given ˆ p ( x ) and a bandwidth parameter h > 0 , for any time t = 1 , 2 , · · · , T , we define the kernelbased estimation for f ∗ t ( x ) as
[Equation]
Based on the kernel estimation F t,h ( x ) , for any integer pair 0 ≤ s < e ≤ T , we define the CUSUMstatistic as
[Equation]
The CUSUM statistic defined in (3) is the cornerstone of our algorithm and is based on two kernelestimators ˆ p ( · ) and F t,h ( · ) .
At a high level, the CUSUM statistic (cid:101) F ( s,e ] t,h ( · ) estimates the difference inmean between the functional data in the time intervals ( s, t ] and ( t, e ] .
In the functional data analysisliterature, other popular approaches for mean function estimation are reproducing kernel Hilbert spacebased methods and local polynomial regression.
However, to our best knowledge, existing worksbased on the two approaches typically require that the functional data are temporally independentand it is not obvious how to extend their theoretical guarantees to the temporal dependence case.
Wetherefore choose the kernel estimation method owing to its flexibility in terms of both methodologyand theory and we derive new theoretical results on kernel estimation for functional data undertemporal dependence and heavy-tailedness.
We would like to point out that in the existing literature,kernel-based change-point estimation methods are used in detecting change-points in nonparametricmodels [e.g. 20, 1, 21, 22].
For multiple change-point estimation, a key ingredient is to isolate each single change-point withwell-designed intervals in [0 , T ] .
To achieve this, we combine the CUSUM statistic in (3) with amodified version of the seeded binary segmentation (SBS) proposed in [19].
SBS is based on acollection of deterministic intervals defined in Definition 1.
Definition 1 (Seeded intervals) .
Let K =
⌈ C K log log( T ) ⌉ , with some sufficiently large absolute constant C K > 0 .
For k ∈ { 1 , . . .
, K} , let J k be the collection of 2 k − 1 intervals of length l k
= T 2 − k +1 that are evenly shifted by l k /
= T 2 − k , i.e.
[Equation]
The overall collection of seeded intervals is denoted as J = ∪ K k
=1 J k .
The essential idea of the seeded intervals defined in Definition 1 is to provide a multi-scale system ofsearching regions for multiple change-points.
SBS is computationally efficient with a computationalcost of the order O ( T log( T ))
Based on the CUSUM statistic and seeded intervals, Algorithm 1 summarises the proposed functionalseeded binary segmentation algorithm (FSBS) for multiple change-point estimation in sequentiallyobserved functional data.
There are three main tuning parameters involved in Algorithm 1, the kernelbandwidth ¯ h in the estimation of the sampling distribution, the kernel bandwidth h in the estimationof the mean function and the threshold parameter τ for declaring change-points.
Their theoreticaland numerical guidance will be presented in Sections 3.1 and 4, respectively.
, tuning parameters
Initialization : If ( s, e ] =
(0 , n ] , set S ← ∅ and set ρ ← log( T ) n − 1 h − d .
Furthermore, sample ⌈ log( T ) ⌉ points from { x t,i } T,nt
=1 ,i =1 uniformly at random without replacement and denote them as { u m } ⌈ log( T ) ⌉ m = .
Estimate the sampling distribution evaluated at { ˆ p ¯ h ( u m ) } ⌈ log( T ) ⌉ m = .
for I =
( α, β ] ∈ J and m ∈ { 1 , . .
. , ⌈ log( T ) ⌉ } do if I =
( α, β ] ⊆
( s, e ] and β − α > 2 ρ then
A I m ← max α + ρ ≤ t ≤ β − ρ | (cid:101) F ( α,β ] t,h ( u m ) | , D I m ← arg max α + ρ ≤ t ≤ β − ρ | (cid:101) F ( α,β ] t,h ( u m ) | else ( A I m , D I m ) ← ( − 1 , 0) end if end for ( m ∗ , I ∗ ) ← arg max m =1 ,..., ⌈ log( T ) ⌉ , I∈J A I m .
if A I ∗ m ∗ > τ then
S ← S ∪ D I ∗ m ∗ FSBS (
[Equation]
[Equation]
FSBS (( D I ∗ m ∗ , e ] , ¯ h, h, τ ) end if OUTPUT: The set of estimated change-points S . Algorithm 1 is conducted in an iterative way, starting with the whole time course, using the multi-scaleseeded intervals to search for the point according to the largest CUSUM value.
A change-point isdeclared if the corresponding maximum CUSUM value exceeds a pre-specified threshold τ and thewhole sequence is then split into two with the procedure being carried on in the sub-intervals.
Algorithm 1 utilizes a collection of random grid points { u m } ⌈ log( T ) ⌉ m = ⊆
{ x t,i } T,nt
=1 ,i =1 to detectchanges in the functional data.
For a change of mean functions at the time point η with ∥ f ∗ η +1 −f ∗ η ∥ ∞ > 0 , we show in the appendix that, as long as ⌈ log( T ) ⌉ grid points are sampled, with highprobability, there is at least one point u m ′ ∈ { u m } ⌈ log( T )
⌉ m = such that | f ∗ η +1 ( u m ′ )
− f ∗ η ( u m ′ )
| ≍∥ f ∗ η +1 − f ∗ η ∥ ∞ .
Thus, this procedure allows FSBS to detect changes in the mean functions withoutevaluating functions on a dense lattice grid and thus improves computational efficiency.
Main Results
3.1 Assumptions and theory
We begin by imposing assumptions on the kernel function K ( · ) used in FSBS.
Assumption 2 (Kernel function) .
Let K ( · ) :
R d → R + be compactly supported and satisfy thefollowing conditions.
a.
The kernel function K ( · ) is adaptive to the Hölder class H r ( L ) ,
i
e. for any f ∈ H r ( L ) , it holdsthat sup x ∈ [0 , 1] d (cid:12)(cid:12) (cid:82) [0 , 1] d K h ( x − z )
f ( z ) d z − f ( x ) (cid:12)(cid:12) ≤ ˜
Ch r , where ˜ C > 0 is a constant that onlydepends on L .
b.
The class of functions F K = { K ( x − · ) /h :
R d → R + , h > 0 } is separable in L ∞ ( R d ) and is auniformly bounded VC-class.
This means that there exist constants A, ν > 0 such that for every probability measure Q on R d and every u ∈ (0 , ∥ K ∥ ∞ ) , it holds that N ( F K , L 2 ( Q ) , u ) ≤ ( A ∥ K ∥ ∞ /u ) v ,where N ( F K , L 2 ( Q ) , u ) denotes the u -covering number of the metric space ( F K , L 2 ( Q ))
Assumption 2 is a standard assumption in the nonparametric literature, see [15, 16], [18], [27]among many others.
These assumptions hold for most commonly used kernels, including uniform,polynomial and Gaussian kernels.
Recall the minimal spacing ∆ =
min K +1 k =1 ( η k − η k − 1 ) defined in Assumption 1 b .
We further definethe jump size at the k th change-point as κ k = ∥
f ∗ η k +1 − f ∗ η k ∥ ∞ and define κ = min
Kk =1 as theminimal jump size.
Assumption 3 below details how strong the signal needs to be in terms of κ and∆ , given the grid size n , the number of functional curves T , smoothness parameter r , dimensionalityd and moment condition q .
Assumption 3 (Signal-to-noise ratio, SNR) .
There exists an arbitrarily-slow diverging sequence
C SNR =
C SNR ( T ) such that
[Equation]
We are now ready to present the main theorem, showing the consistency of FSBS.
Theorem 1.
Under Assumptions 1, 2 and 3, let { (cid:98) η k } (cid:98)
Kk
=1 be the estimated change-points by FSBSdetailed in Algorithm 1 with data { x t,i , y t,i } T,nt =1 ,i =1 , bandwidth parameters ¯ h
= C ¯ h ( Tn ) − 1 2 r + d ,h = C h ( Tn ) − 1 2 r + d and threshold parameter
τ
=
C τ log
[Equation]
[Equation]
[Equation] q } [Equation] T )
[Equation]
+ T
[Equation]
n [Equation] d (cid:17) 1 / 2 , forsome absolute constants C ¯ h , C h , C τ > 0 .
It holds that
[Equation]
where C FSBS > 0 is an absolute constant.
In view of Assumption 3 and Theorem 1, we see that with properly chosen tuning parameters andwith probability tending to one as the sample size T grows, the output of FSBS estimates the correctnumber of change-points and
[Equation]
where the last inequality follows from Assumption 3.
The above inequality shows that there exists aone-to-one mapping from { (cid:98) η k } Kk =1 to { η k } Kk =1 , assigning by the smallest distance.
3.2 Discussions on functional seeded binary segmentation (FSBS)
From sparse to dense regimes.
In our setup, each curve is only observed at n discrete points and weallow the full range of choices of n , representing from sparse to dense scenarios, all accompaniedwith consistency results.
In the most sparse case n = 1 , Assumption 3 reads as κ √ ∆ ≳
T d/ (4 r +2 d ) ×a logarithmic factor, under which the localisation error is upper bounded by T d/ (2 r + d ) κ − 2 , up toa logarithmic factor.
To the best of our knowledge, this challenging case has not been dealt inthe existing change-point detection literature for functional data.
In the most dense case, we canheuristically let n = ∞ and for simplicity let q = ∞ representing the sub-Gaussian noise case.
Assumption 3 reads as κ √ ∆ ≍ log 1 / 2 ( T ) and the localisation error is upper bounded by κ − 2 log( T ) .
Both the SNR ratio and localisation error are the optimal rate in the univariate mean change-pointlocalisation problem [28], implying the optimality of FSBS in the dense situation.
Tuning parameters.
There are three tuning parameters involved.
In the CUSUM statistic (3), wespecify that the density estimator of the sampling distribution is a kernel estimator with bandwidth¯ h ≍ ( Tn ) − 1 / (2 r + d ) .
Due to the independence of the observation grids, such a choice of the bandwidthfollows from the classical nonparametric literature [e.g. 26] and is minimax-rate optimal in termsof the estimation error.
For completeness, we include the study of ˆ p ( · ) ’s theoretical properties in
Appendix B.
In practice, there exist different default methods for the selection of ¯ h , see for examplethe function Hpi from the R package ks ([9]).
The other bandwidth tuning parameter h is also required to be h ≍ ( Tn ) − 1 / (2 r + d ) .
Despite that weallow for physical dependence in both functional noise and measurement error, we show that thesame order of bandwidth (as ¯ h ) is required under Assumption 1.
This is an interesting finding, if notsurprising.
This particular choice of h is due to the fact that the physical dependence put forward by[30] is a short range dependence condition and does not change the rate of the sample size.
The threshold tuning parameter τ is set to be a high-probability upper bound on the CUSUM statisticswhen there is no change-point and is in fact of the form τ = C τ log max { 1 / 2 , 5 /q } ( T ) √ n − 1 h − d + 1 .
This also reflects the requirement on the SNR detailed in Assumption 3, that κ √ ∆ ≳
τ .
Phase transition.
Recall that the number of curves is T and the number of observations on eachcurve is n .
The asymptotic regime we discuss is to let T diverge, while allowing all other parameters,including n , to be functions of T .
In Theorem 1, we allow a full range of cases in terms of therelationship between n and T .
As a concrete example, when the smooth parameter r = 2 , the jumpsize κ ≍ 1 and in the one-dimensional case d = 1 , with high probability (ignoring logarithmic factorsfor simplicity),
[Equation]
This relationship between n and T was previously demonstrated in the mean function estimation literature [e.g. 8, 32], where the observations are discretely sampled from independently and identicallydistributed functional data.
It is shown that the minimax estimation error rate also possesses the samephase transition between n and T , i.e. with the transition boundary n ≍ T 1 / 4 , which agrees with ourfinding under the change-point setting.
Physical dependence and heavy-tailedness
In Assumption 1 c and d , we allow for physical dependence type temporal dependence and heavy-tailed additive noise.
As we have discussed, since thephysical dependence is in fact a short range dependence, all the rates involved are the same as those inthe independence cases, up to logarithmic factors.
Having said this, the technical details required indealing with this short range dependence are fundamentally different from those in the independencecases.
From the result, it might be more interesting to discuss the effect of the heavy-tail behaviours,which are characterised by the parameter q .
It can be seen from the rates in Assumption 3 and
Theorem 1 that the effect of q disappears and it behaves the same as if the noise is sub-Gaussianwhen q ≥ 10 .
Numerical Experiments
4.1 Simulated data analysis
We compare the proposed FSBS with state-of-the-art methods for change-point detection in functionaldata across a wide range of simulation settings.
Specifically, we compare with three competitors:
BGHK in [6], HK in [17] and SN in [31].
All three methods estimate change-points via examiningmean change in the leading functional principal components of the observed functional data.
BGHKis designed for temporally independent data while HK and SN can handle temporal dependence viathe estimation of long-run variance and the use of self-normalization principle, respectively.
All threemethods require fully observed functional data.
In practice, they convert discrete data to functionalobservations by using B-splines with 20 basis functions.
For the implementation of FSBS, we adopt the Gaussian kernel.
Following the standard practice inkernel density estimation, the bandwidth ¯ h is selected by the function Hpi in the R package ks ([9]).
The tuning parameter τ and the bandwidth h are chosen by cross-validation, with evenly-indexeddata being the training set and oddly-indexed data being the validation set.
For each pair of candidate( h, τ ) , we obtain change-point estimators { (cid:98) η k } (cid:98)
Kk =1 on the training set and compute the validationloss (cid:80) (cid:98)
Kk =1 (cid:80) t ∈ [ (cid:98) η k , (cid:98)
η k +1 ) (cid:80) ni
=1 { ( (cid:98)
η k +1 − (cid:98) η k ) − 1 (cid:80) (cid:98)
η k +1 t = (cid:98)
η k +1 F t,h ( x t,i ) − y t,i } 2 .
The pair ( h, τ ) is thenchosen to be the one corresponding to the lowest validation loss.
We consider five different scenarios for the observations { x ti , y ti } T,nt =1 ,i =1 .
For all scenarios 1-5,we set T = 200 .
Given the dimensionality d , denote a generic grid point as x = ( x (1) , · · · , x ( d ) ) .
Scenarios 1 to 4 are generated based on model (1).
The basic setting is as follows
• Scenario 1 (S1) Let ( n, d ) =
(1 , 1) , the unevenly-spaced change-points be ( η 1 , η 2 ) =
(30 , 130)and the three distinct mean functions be 6 cos( · ) , 6 sin( · ) and 6 cos( · ) .
• Scenario 2 (S2) Let ( n, d ) =
(10 , 1) , the unevenly-spaced change-points be ( η 1 , η 2 ) =
(30 , 130)and the three distinct mean functions be 2 cos( · ) , 2 sin( · ) and 2 cos( · ) .•
Scenario 3 (S3) Let ( n, d ) =
(50 , 1) , the unevenly-spaced change-points be ( η 1 , η 2 ) =
(30 , 130)and the three distinct mean functions be cos( · ) , sin( · ) and cos( · ) .•
Scenario 4 (S4) Let ( n, d ) =
(10 , 2) , the unevenly-spaced change-points be ( η 1 , η 2 ) =
(100 , 150)and the three distinct mean functions be 0 , 3 x (1) x (2) and 0 .For S1 - S4 , the functional noise is generated as ξ t ( x ) = 0 .
5 ξ t − 1 ( x ) + (cid:80) 50 i =1 i − 1 b t,i h
i ( x ) , where{ h i ( x ) = (cid:81) dj
=1 (1 / √ 2) π sin( ix ( j ) ) } 50 i =1 are basis functions and { b t,i } T, 50 t =1 ,i =1 are i.i.d.
standardnormal random variables.
The measurement error is generated as δ t = 0 .
3 δ t − 1 + ϵ t , where { ϵ t } Tt =1are i.i.d.
N (0 , 0 .
5 I n ) .
We observe the noisy functional data { y ti } T,nt =1 ,i =1 at grid points { x ti } T,nt =1 ,i =1independently sampled from Unif([0 , 1] d ) .Scenario
5 is adopted from [31] for densely-sampled functional data without measurement error.• Scenario 5 (S5) Let ( n, d ) =
(50 , 1) , the evenly-spaced change-points be ( η 1 , η 2 ) =
(68 , 134) andthe three distinct mean functions be 0, sin( · ) and 2 sin( · ) .
The grid points { x ti } 50 i =1 are 50 evenly-spaced points in for all
t = · , T . The
functional noise is generated as ξ t ( · ) = (cid:82) [0 , 1] ψ ( · , u ) ξ t − 1 ( u ) d u + ϵ t ( · ) , where { ϵ t ( · ) } Tt =1 are independent  standard Brownian motions and ψ ( v, u ) = 1 / 3 exp(( v 2 + u 2 ) / 2) is a bivariate Gaussian kernel.
S1-S5 represent a wide range of simulation settings including the extreme sparse case S1 , sparse case
S2 , the two-dimensional domain case S4 , and the densely sampled cases S3 and S5 .
Note that S1 and
S4 can only be handled by FSBS as for S1 it is impossible to estimate a function via B-spline basedon one point and for S4 , the domain is of dimension 2.
Evaluation result :
For a given set of true change-points C = { η k } Kk =1 , we evaluate the accuracyof the estimator { (cid:98) η k } (cid:98) Kk =1 by the difference | (cid:98) K − K | and the Hausdorff distance d ( ˆ C , C ) , definedby d ( ˆ C , C ) = max { max x ∈ ˆ C min y ∈C {| x − y |} ,
max y ∈ ˆ C min x ∈C {| x − y |}} .
For ˆ C = ∅ , we use theconvention that | (cid:98) K − K | =
K and d ( ˆ C , C ) =
T .
For each scenario, we repeat the experiments 100 times and Figure 1 summarizes the performanceof FSBS, BGHK, HK and SN.
Tabulated results can be found in Appendix A.1. As can be seen, FSBS consistently outperforms the competing methods by a wide margin and demonstrates robustbehaviour across the board for both sparsely and densely sampled functional data.
4.2 Real data application
We consider the COBE-SSTE dataset [24], which consists of monthly average sea surface temperature(SST) from 1940 to 2019, on a 1 degree latitude by 1 degree longitude grid (48 × 30) covering
Australia.
The specific coordinates are latitude 10 S- 39 S and longitude 110 E- 157 E.
We apply FSBS to detect potential change-points in the two-dimensional SST.
The implementation of
FSBS is the same as the one described in Section 4.1.
To avoid seasonality, we apply
For both the June and July data, two change-points are identified by FSBS, Year 1981 and 1996,suggesting the robustness of the finding.
The two change-points might be associated with years whenboth the Indian Ocean Dipole and Oceanic Niño Index had extreme events [2].
The El Niño/Southern
Oscillation has been recognized as an important manifestation of the tropical ocean-atmosphere-landcoupled system.
It is an irregular periodic variation in winds and sea surface temperatures over thetropical eastern Pacific Ocean.
Much of the variability in the climate of Australia is connected withthis phenomenon [5].
To visualize the estimated change, Figure 2 depicts the average SST before the first change-point
Year 1981, between the two change-points, and after the second change-point Year 1996.
The tworows correspond to the June and July data, respectively.
As we can see, the top left corners exhibitdifferent patterns in the three periods, suggesting the existence of change-points.
Conclusion
In this paper, we study change-point detection for sparse and dense functional data in generaldimensions.
We show that our algorithm FSBS can consistently estimate the change-points even inthe extreme sparse setting with n = 1 .
Our theoretical analysis reveals an interesting phase transitionbetween n and T , which has not been discovered in the existing literature for functional change-pointdetection.
The consistency of FSBS relies on the assumption that the minimal spacing ∆ ≍ T .
Torelax this assumption, we may consider increasing K in Definition 1 to enlarge the coverage of theseeded intervals in FSBS and apply the narrowest over threshold selection method [Theorem 3 in19].
With minor modifications of the current theoretical analysis, the consistency of FSBS can beestablished for the case of ∆ ≪ T .
Since such a relaxation does not add much more methodologicalinsights to our paper, we omit this additional technical discussion for conciseness.
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