Change-point Detection for  Sparse and Dense Functional Data in General Dimensions
Authors: Carlos Misael Madrid Padilla
Daren Wang
Zifeng Zhao
Yi Yu
Venue: NeurIPS 2022 Accept (2022)

Abstract: We study the problem of change-point detection and localisation for functional data sequentially observed on a general $d$-dimensional space, where we allow the functional curves to be either sparsely or densely sampled. Data of this form naturally arise in a wide range of applications such as biology, neuroscience, climatology and finance. To achieve such a task, we propose a kernel-based algorithm named functional seeded binary segmentation (FSBS). FSBS is computationally efficient, can handle discretely observed functional data, and is theoretically sound for heavy-tailed and temporally-dependent observations. Moreover, FSBS works for a general $d$-dimensional domain, which is the first in the literature of change-point estimation for functional data.  We show the consistency of FSBS for multiple change-point estimation and further provide a sharp localisation error rate, which reveals an interesting phase transition phenomenon depending on the number of functional curves observed and the sampling frequency for each curve. Extensive numerical experiments illustrate the effectiveness of FSBS and its advantage over existing methods in the literature under various settings. A real data application is further conducted, where FSBS localises change-points of sea surface temperature patterns in the south Pacific attributed to El Ni\~{n}o.

Change-point Detection for Sparse and Dense Functional Data in General Dimensions
Carlos Misael Madrid Padilla Department of Mathematics University of Notre Dame cmadridp@nd. .ddu
Daren Wang Department of Statistics University of Notre Dame dwang24@nd @rang24@nd.edu edu
Zifeng Zhao
Yi Yu Department of Statistics University of Warwick yi. 20warwick ac. uk
Mendoza College Business University of Notre Dame zzhao20nd. edu
Abstract
We study the problem of change-point detection and localisation for functional data sequentially observed on general d-dimensional space, where we allow the functional curves to be either sparsely densely sampled. Data of this form naturally arise in a wide range of applications such as biology, neuroscience, clima- tology and finance. To achieve such task, propose kernel-based algorithm namely functional seeded binary segmentation (FSBS). FSBS is computationally efficient, can handle discretely observed functional data, and is theoretically sound for heavy-tailed and temporally-dependee for general d-dimensional domain, which is the first in the literature of change- point estimation for functional data. We show the consistency of FSBS for multiple Change-point estimation and further provide sharp localisation reveals an interesting phase transition phenomenon depending on the number functional curves observed and the sampling frequency for each curve. Extensive numerical experiments illustrate the effectiveness of FSBS and its advantage over existing methods in the literature under various settings. A real data application is further conducted, where FSBS localises change-points of sea surface temperature patterns in the south Pacific attributed to Niño. The code to replicate all of our experiments can be found at https: h/tgttgttttt////gg///////ggghhht com/ cmadridp/FSB.
observations. Moreover, FSBS works
rate, which
Introduction
Recent technological advancement has boosted the emergence of functional data in various application
areas, including neuroscience [e.g. [1 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 development
of 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 observed sequentially stationary, with abrupt changes occurring at unknown time points. To be specific, denote D as general d-dimensional space that is homeomorphic [0, 1]d, where d E N+ is considered arbitrary but fixed. We assume that the observations {(1,,s Mr, 1 C D are generated based on
time series and the mean functions piecewise
V1, f (Fv,) + (2 8 for t 1,..., and I,...,m.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

In this model, (21,) S D denotes the discrete grids where the (noisy) functional data / E R are observed, {f: D - R) denotes the deterministic mean functions, {&r : + T denotes the functional noise and {6 1,ا= S denotes the measurement error. We refer to Assumption Saaiptioonneboooorro below for detailed technical conditions on the model.
To model the unstationarity of sequentially observed functional data which commonly exists in real world applications, we assume that there exist K E N change-points, namely 0 = 7/o < '1 < ..... < чд < + T, satisfying that J + Ji+1, if and only if E {nx) Our primary interest is to accurately estimate { )
Due to the importance of modelling unstationary functional data in various scientific fields, this problem has received extensive attention in the statistical change-point literature, see e.g. B [6].
[7]. [31, [ and [2]. 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 fully observed functional data without measurement error, which may not be realistic in practice. Secondly, most existing works focus the single change-point setting and to our best knowledge, there is no consistency result of multiple change- point estimation for fuunctional data. Lastly but most importantly, existing algorithms only consider functional data with support on [0, 1] and thus are not applicable to functional data with multi-dimensional domain, a type of data frequently encountered in neuroscience and climatology.
In view of the aforementioned three limitations, in this paper, we make several theoretical and methodological contributions, summarized below.
In terms of methodology, our proposed kernel-based change-point detection algorithm, functional seeded binary segmentation (FSBS), is computationally efficient, can handle discretely observed functional data contaminated with measurement error, and allows for eemporally-dependerr and heavy-tailed data. FSBS, in particular, works for general d-dimensional domain with arbitrary but fixed E 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 in detecting and localising multiple change-points. We also provide 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. 1. To the best of our knowledge, all the existing functional data change-point analysis papers assume full curves are observed. We not only allow for discrete observation, but carefully study this most extreme 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 supports our theoretical findings, showcases the advantages of FSBS over existing methods and illustrates the practicality of FSBS.
A byproduct of our theoretical analysis is new theoretical results on kernel estimation for functional data 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, 11 + IR and for 1 S < co, define I| Uf..1a (x)) // and for p xo, define |l = sup (f(x). Define Cp = {f [0, 1] R. Illllp < oo). For any vector
(si...., sa) No. define |s|
Σ Di=1 8,.
s! = sy!... sal and the associated partial differential operator D = da,"" ... 2n T. For a > 0, denote
rl4l
|a| to be the largest integer smaller than For any function f [0, 112 - R that is |a] -times continuously differentiable at point Zn. denote by fa, its Taylor polynomial of degree [a] at Zo.
s! D* (f For a constant L 0, let 7. (L) be the set of functions f [0, 1]d + IR such that f is Itimmm differentiable for all . E [0. 1]d and satisfy [f(x) - 1% (x) < T.|r - zol%, for all x, 20 E [0, Here |x vol is the Fuclidean distance
which is defined as f& (x)
(z-zo)"
[|]]
between 2, To E Rdd In canameraettiic statistical literature, H. H²(L) are often referred to as the class

---

of Hölder smooth functions. We refer the interested readers to 22 for more detailed discussion on Hölder smooth functions.
For two positive sequences (an)NENT MEN+ and (bn MEN++ we write an = Obr,) ) or an S bn if. S Cbp with some constant C > that does not depend and a, (bn.) on = bn if On (b,,)
and 6 = O.,,).
2 Functional seeded binary segmentation
2.1 Problem formulation
Detailed model assumptions imposed on model (I) are collected in Assumption I For notational simplicity, without loss of generality, we set the general d-dimensional domain D to be [0, 1] the results apply to any D that is homeomorphic to [0, 1]
Assumption 1. The data ((.,,, Vc,,) [0, 1) are generated based on model T
S [0, 1] are independently sampled from common density function [0, 11 + R. In addition, there exist constants r and > such that E H. (L) and that inf = u(x) 2 é with an absolute constant ė 0.
(Discrete grids) The grids {x,,))
T.
b. (Mean ) For > and L > 0, we have fi E (L). The minimal spacing between two consecutive change-points = min K+1 (Nw - -Tk-1) satisfies that A = (TT). k+ (Functional noise) Let {ei, (6,,6)ez,, be ii.d. random elements taking values in measurable space
Se and g be a measurable fuensurablefunctionn g Se + L2. The functional noise {E:)== S Lo takes the form
g(G)), with G. (.:==..,.E (...,,-1,EE) E1, (1-...
There exists absolute constant 3. such that B(11Eo112a) Ce.1 some absolute constant Cg,1- Define a coupled process
; = 99];),
with =
We have Eee 1/22 [B]E: 1 1/8 < C2.2 same absolute constant Cg.2 > 0.
d. (Measurement error) Let {ei, Gaeez be i.i.d. random elements taking values in a measurable space Ss and In be measurable function functionn In 53 - R., The measurement error 18.)7- S IR" takes the form
8, (F), with J, (..., 6- 61, ,...............111.
There exists an absolute constant q 2 3, such that max;_ B((d,,19) < Co.1 for some absolute constant Co.1. Define coupled process
8 = -.((FF) with Fi (...,e-1y 66: (1.1,,))).
We have max"
< Cx.2 for some absolute constant C8.2 >> O. E 1122 67, 19/2/
Assumption allows the functional data be observed on discrete grids and moreover, we allow for different grids at different time points. The sampling distribution is required to be lower bounded on the support [o, 1]6, which standard assumption widely used in the nonparametric literature [6... Here, different functional curves are assumed to have the same number of grid points We remark that this is made for presentation simplicity only. can indeed be further relaxed and the main results below will then depend on both the minimum and maximum numbers of grid points.
Note that Assumption M does not impose any restriction between the sampling frequency n and the number of functional curves T, and indeed our method can handle both the dense case where n >> T and the sparse case where n can be upper bounded by a constant. Besides the random sampling scheme studied here, another commonly studied scenario is the fixed design, where it usually assumes that the sampling locations { are common to all functional curves across time. We remark that while we focus on the random design here, our proposed algorithm can be directly applied to the fixed design case without any modification. Furthermore, theoretical justification under the fixed design case can be established similarly with minor modifications, which is omitted.

---

The observed functional data have mean functions {fr which are assumed to be Hölder continu- ous in Assumption [p. Note that the Hölder parameters Assumption Ta and b both denoted by We remark that different smoothness are allowed and we use the same here for notational simplicity. This sequence of mean functions is our primary interest and is assumed to possess a piece- wise constant pattern, with the minimal spacing A being of the same order as T. This assumption essentially requires that the number of change-points is upper bounded. It can also be further relaxed and 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 are detailed in Assumption [ and d. respectively. Both the functional noise and the measurement error are allowed to possess temporal dependence and heavy-tailedness. For temporal dependence, we adopt the physical dependence framework by [30], which covers wide range of time series models, such as ARMA and vector AR models. further covers popular functional time series models such as functional AR and MA models [ We also remark that Assumption I and d impose short range dependence, which characterized by the absolute upper bounds Cg.2 and C2,2- Further relaxation possible by allowing the upper bounds Cg and Cs vary with the sample size T.
The heavy-tail behavior is encoded in the parameter q. In Assumption D and d, we adopt the same quantity for presentational simplicity and remark that different heavy-tailedness levels are allowed. An extreme example is that when q oc, the noise is essentially uub-Gaussian. Importantly, Assumption Dd does not impose any restriction on the cross-sectionss dependence among 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-poie poinge-pointtoee {nx } in the mean functions (fi) we propose a kernel-based cumulative sum (CUSUM) statistic, which simple, intuitive and computationally efficient. The key idea is to recover the unobserved {f² [T from the observations {( (1,,,,: vt,)) based on kernel estimation.
Given a kernel function K(-) : Rd - + and bandwidth parameter h > define Kx(x) h K (</h)) for x E RRd Given the random grids { and bandwidth parameter ht we define the density estimator of the sampling distribution u(x) as
T
p(x) (x) nT
Kg(x -1,,), E [0.1)
Given p(z) and bandwidth parameter h > 0, for any time t = 1,2, ... ,T, we define the kemel- based estimation for f² (x) as
Fz,(()) = Ei Me, Kn xe.e)
x E 1]
(2)
np(xx)
Based on the kernel estimation FL, (x),, for any integer pair 0 S 58<5 e TT, we define the CUSUM statistic
pla.el (x) 1.h
e V (e s1
s (i, V (e s) 1=0+1
Σ Fir()), a x 10.1% (3)
1=++1
The CUSUM statistic defined in (3) is the cornerstone of our algorithm and is based on two kernel estimators p0-) and (.). At high level, the CUSUM statistic Fi (.) estimates the difference in mean between the functional data in the time intervals (s, f] and (t, e]. In the functional data analysis literature, other popular approaches for mean function estimation are reproducing kernel Hilbert space based methods and local polynomial regression. However, to our best knowledge, existing works based on the two approaches typically require that the functional data are temporally independent and it not obvious how to extend their theoretical guarantees to the temporal dependence case. We therefore choose the kernel estimation method owing to its flexibility in terms of both methodology and theory and we derive new theoretical results on kernel estimation for functional data under temporal dependence and heavy-tailedness. We would like to point out that in the existing literature,

---

kernel- based change-pone point estimation methods are used in detecting Shange-points in nonparametric models [e.g. 20 22 22]
For multiple change-point estimatiin, a key ingredient is to isolate each single change-poine with well-designed intervals in [0, T. To achieve this, combine the CUSUM statistic in ( with modified version of the seeded binary segmentation (SBS) proposed in [19]. SBS is based on a collection of deterministic intervals defined in Definition
Definition 1 (Seeded intervals). Let K. = [Cx log log(T)), with some sufficiently large absolute constant Cx > O. For k (1....,K) let Jk be the collection of 2 intervals of length k+1 4.- T2-k+1 that are evenly shifted by 12/2 T2 i.e.
Jk
10-2- + + -21k+111 i=
2 13.
The overall collection of seeded intervals is denoted as J = U Jk.
The essential idea of the seeded intervals defined in Definition is to provide multi-scale system of searching regions for multiple change-points. SBS is computationally efficient with computational cost of the order ('7'log(7) [19].
Based the CUSUM statistic and seeded intervals, Algorithm [ summarises the proposed functional seeded binary segmentation algorithm (FSBS) for multiple change-point estimation in sequentially observed functional data. There are three main tuning parameters involved in Algorithm []] the kernel bandwidth h in the estimation of the sampling distribution, the kernel bandwidth h in the estimation of the mean function and the threshold parameter for declaring srange--pintss. Their theoretical and numerical guidance will be presented in Sections 3.1 and respectively.
Algorithm 1 Functional Seeded Binary Segmentation. FSBS ((s, e), h)
Tmn INPUT: Data {Zi,a, Vt, hr ft=1, seeded intervals J, tuning parameters h, h, > 0.
Initialization: If (s, c] (0, n). set S +- 2 and p log(T)n--he Furthermore, sample [og(T)) points from ( uniformly random without replacement and denote them as [log(T)]
[um) m=1 Estimate the sampling distribution evaluated at (fR(Um)) Dog(T))
for 1 = (ax, I and m E (1...., [log(T)]) do
I (ax, 3 S (x, and a > 2p then
A max FF (3).))
(um) )1 D <- arg m8x ap5ts--8 T/a (um) (um))
else
( (AZ DE m
-- (-1,0)
end if
end for
(m² In) +- arg m Dogt....oooo__1 [og(r) Az.
if AT m > T then
5 S SUDT DT' m
FSBS ((s, DT h T)
FSBS ((DT) n c], h, T)
end if
OUTPUT: The set of estimated change-points S.
Algorithm iss conducted in an iterative way. starting with the whole time course, using the multi-scale sceded intervals search for the point according to the largest CUSUM value. A change-poin is declared if the corresponding maximum CUSUM value exceeds pre-specified threshold 7 and the whole sequence then split into two with the procedure being carried on the sub-intervals.
[lgg(T)] C (2 to detect Algorithm I utilizes a collection of random grid points {um M
changes in the functional data. For a change of mean functions at the time point with [lf
f Ilo > O, we show in the appendix that, as long as [log(T)] grid points are sampled, with high probability, there is at least one point Una E { [lgg(T) such that (ftti(um))) f (um)) x
|| - oo. Thus, this procedure allows FSBS to detect changes in the mean functions without evaluating functions on a dense lattice grid and thus improves computational efficiency.
5

---

3 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-) Rd + R be compactly supported and satisfy the following conditions.
a. The kernel function K-) is adaptive to the Hölder class H'' (L), for any f f (L), it holds that s Jo.1j Ka (x - 2) f(z) - [(x) S ChT, where > O is a constart that only depends on L.
b. The class of functions FR [K(x - -.)h : R - 0} is separable in Lox (RR) and is uniformly bounded VC-class. This means that there exist constants A, v > 0 such that for every proba- bility measure Q on R.d and every (0, (o))) holds that (FK, (2))), u) [A]]K]) ol /oo//// u)" where ([F K; (2()), u) denotes the u-covering number of the metric space (Fx. (Q)).
Assumption 2 is standard assumption in the nonparametric literature, among many others. These assumptions hold for most commonly used kernels, including uniform, polynomial and Gaussian kernels.
15 16 18
Recall the minimal spacing A min K+1 (mh 1) defined in Assumption Ip. We further define Ki. I1 S I and define min K as the
the jump size the kth change-point minimal jump size. Assumption 3 below details how strong the signal needs to be in terms of k and A, given the grid size n1 the number of functional curves T, smoothness parameter r, dimensionality d and moment condition q.
Assumption 3 (Signal-to-noiss ratio, SNR). There exists an arbitrarily-slon diverging sequence CsNn (s" such that
RV A > CSNR log max (1/2.5/0) (T) (1 T a
2r 1/2
We are now ready present the main theorem, showing the consistency of FSBS.
Theorem 1. Under Assumptions and 3 let {fk] be the estimated change-points by FSBS
T.n detailed in Algorithm I with data { Vt,))t=i,s=1- bandwidth parameters h C₂(TT) 2
1/2
= Cr (T') = and threshold parameter T = log max (1/.5//11//)) (T) (1+ T' कスス some absolute constants Cr. Ch. C- > 0. It holds that
for
2x P p = K; 17 mkl S CrsBs log max(1,100m))) (T) (1+ Turtaa
- Ak 1,.... K
Ki
2 1 - 3 log -1 (T),
where CrsBs > 0 an absolute constant.
In view of Assumption and Theorem 1 we see that with properly chosen tuning parameters and with probability tending to one as the sample size grows, the output of FSBS estimates the correct number of change-point and
K max Mm I (1+
[ T
log max (1/10/4) (T) = o(1),
where the last inequality follows from Assumption 3 The above inequality shows that there exists a one-to-one mapping from [mx k
{m ) 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 we allow the full range of choices of n, representing from sparse to dense scenarios, all accompanied with consistency results. In the most sparse case 1, Assumption 3 reads as kV (dd(4++2d) x logarithmic factor, under which the localisation error is upper bounded by (2r+(r)d) x up to

---

a logarithmic factor. To the best of our knowledge, this challenging case has not been dealt in the existing change-point detection literature for functional data. In the most dense case, we can heuristically let n oo and for simplicity let x representing the sub-Gaussian noise case. Assumption 3 reads AV x (TT) and the localisation error is upper bounded by x lTg(T). Both the SNR ratio and localisation error are the optimal rate in the univariate mean change-poing localisation problem [28], implying the optimality of FSBS in the dense situation.
Tuning parameters. There are three tuning parameters involved. In the CUSUM statistic 3 we specify that the density estimator of the sampling distribution is a kernel estimator with bandwidth h x (I') 11(2))) Due to the independence of the observation grids, such a choice of the bandwidth follows from the classical nonparametrie literature [e.g. g. 26 and is minimax-rate optimal in terms of the estimation error. For completeness, we include the study of p(-)'s theoretical properties in ??. In practice, there exist different default methods for the selection of h see for example the function Hpi from the R package ks (19).
The other bandwidth tuning parameter is also required to be h x (Tn) 1/(2r+d)) Despite that we allow for physical dependence in both functional noise and measurement enror, we show that the same order of bandwidth (as h) is required under Assumption [ This is an interesting finding, not surprising. This particular choice of h is due 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 is set to be a high-probability upper bound on the CUSUM statistics when there is Shange-point and is in fact of the form C, log (1/2,5/4) (T) vnvn-hh h-
This also reflects the requirement on the SNR detailed in Assumption that RV ZT.
Phase transition. Recall that the number of curves curve is The asymptotic regime discuss is to let T diverge, while allowing all other parameters, including n, to be functions of T. In Theorem [ we allow a full range of cases in terms of the relationship between n and T' As concrete example, when the smooth parameter 2, the jump size k x 1 and in the one-dimensional case = 1, with high probability (ignoring logarithmic factors for simplicity).
and the number of observations on each
K max Th 17k = Op(Tnn - + +1 = (T
(Op(1)
n 2 TV/ S T/4
This relationship between and T was previously demonstrated in the mean function estimation lit- erature [e.g. 8 (32). where the observations are discretely sampled from independently and identically distributed functional data. It is shown that the minimax estimation error rate also possesses the same phase transition between and T, i.e. with the transition boundary n x 11/4, which agrees with our finding under the change-poin setting.
Physical dependence and heavy-tailedness In Assumption D: and d, we allow for physical depen- dence type temporal dependence and heavy-tailed additive noise. As we have discussed, since the physical dependence is in fact short range dependence, all the rates involved are the same those in the independence cases, up to logarithmic factors. Having said this, the technical details required in dealing with this short range dependence are fundamentally different from those in the independence cases. 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 [ that the effect of q disappears and it behaves the same as if the noise is sub-Gaussian when g 2 10.
Numerical Experiments
4.1 Simulated data analysis
We compare the proposed FSBS with state-of-the-an methods for change point detection in functional data 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 examining mean change in the leading functional principal components of the observed functional data. BGHK is designed for temporally independent data while HK and SN can handle temporal dependence via
7

---

the estimation of long-run variance and the use of self- self-normalization principle, respectively. All three methods require fully observed functional data. In practice, they convert discrete data to functional observations by using B-splines with 20 basis functions.
For the implementation of FSBS, we adopt the Gaussian kernel. Following the standard practice in kernel density estimation, the bandwidth h is selected by the function Hpi in the R package ks ((9). The tuning parameter and the bandwidth / are chosen by cross-validation, with evenly-indexed data being the training set and oddly-indexed data being the validation set. For each pair of candidate
(h, T), obtain change-poin estimators ) k=
the training set and compute the validation
loss Ek=1 Eng(h ( 1) Ex= (m ) Σt=uu Fe,(( )t,3 The pair (h, r) is then chosen be the one corresponding to the lowest validation loss.
R
We consider five different scenarios for the observations
{arts, Dti ) For all scenarios 1-5,
we set T = 200. Given the dimensionality d. denote a generic grid point as x = (x x.()). Scenarios 4 are generated based model (T. The basic setting is as follows.
Scenario 1 (S1) Let (n, d) = (1, 1). the unevenly-spaced change-points be (71,7)) = (30, 130) and the three distinct mean functions be 6 cos(-). 6 sin(-) and 6 cos
Scenario (S2) Let (n, d) (10, 1), the unevenly-spaced change-points be (71, 7722 (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-spacee change-points be (71, 72) (30, 130) and the three distinct mean functions be cos(-), sin(-) and cos(-).
Scenario 4 (S4) Let (n, d) = (10, 2). the srevenly-spaced change-points be (71; = (100, 150) and the three distinct mean functions be 0, (z) and O.
- For S1-S4, the functional noise is generated as {x(2) = 0.5 + 50 1- (x))) where
(l((r) = I (L/v2) sin(iz(())) are basis functions and {br.: T,50 are iid. standard normal random variables. The measurement error is generated as 6 0.36, + cs, where {ee)E= are i.i.d. (0, 0.57,)) We observe the noisy functional data [yri)/,, at grid points (2a) 1=1, independently sampled from Uni 1
Scenario is adopted from 31 for densely-samples functional data without measurement error.
Scenario 5 (S5) Let (n, d)- = (50, 1), the evenly-spaced change-points be (7/,7)) = (68, 134) and the three distinct mean functions be O, sin(-) and 2 sin(.).
The grid points {*+x} 150 are 50 evenly-spaced points in [0, 1] for all ( I,,... T. The functional noise is generated as Ec(-) Jo.. (t (2) du (+))) where {<(())]= are independent standard Brownian motions and (0( u) = 1/3 exp((22 + =2)/2) is bivariate Gaussian kernel.
S1-S5 represent a wide range of simulation settings including the extreme sparse case S1, sparse case S2, the wwo-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 impossible estimate function via B-spline based on one point and for $4, the domain is of dimension 2.
Evaluation result: For given set of true change-poin -oontts-pints C {m M evaluate the accuracy of the estimator {Nx J k= by the difference jK- - K| and the Hausdorff distance d(C,, C), defined by d(C,c) max max,ee minyec {|x v/}. max minxec f v1}}. For Ć 2, the convention that j K K and d(C,c)= = T.
For each scenario, we repeat the experiments 100 times and Figure T summarizes the performance of FSBS, BGHK, HK and SN. Tabulated results can be found in ??. As can be seen, FSBS consistently outperforms the competing methods by a wide margin and demonstrates robust behaviour across the board for both sparsely and densely sampled functional data.
8

---

Hausdorff distance K-R
Hausdorff distance
K-R
25
190
20
Error
S1
100
S4
02
0.1
FBBS 0.0
F88S BOHK
FSBS
Algorithm
Figure I: Bar plots for simulation results of $1-S5. Each bar reports the mean and standard deviation computed based on 100 experiments. From left right, the first two plots correspond to the Hausdorff distance and K| in S2, S3 and S5. The last two plots correspond to S1 and S4.
4.2 Real data application
We consider the COBE-SSTE dataset [24]. which consists of monthly average sea surface temperature (SST) from 1940 2019, 1 degree latitude by degree longitude grid (48 30) covering Australia. The specific coordinates are latitude 10S-39S and longitude 110E-157E.
We apply FSBS to detect potential change-poing in the two-dimensional SST. The implementation of FSBS is the same as the one described in Section 4.1 avoid seasonality, we apply FSBS to SST for the month of June from 1940 to 2019. We further conduct the same analysis separately for the month of July for robustness check.
For both the June and July data, two change-poing are identified by FSBS, Year 1981 and 1996, suggesting the robustness of the finding. The two change-points might be associated with years when both the Indian Ocean Dipole and Oceanic Niño Index had extreme events [2]. The El Niñoo/Suthern Oscillation has been recognized as an important manifestation of the tropical ocean- atrranhreemspeeeeeeeeee coupled system. It is an irregular periodic variation in winds and sea surface temperatures over the tropical eastern Pacific Ocean. Much of the variability the climate of Australia is connected with this phenomenon 5
To visualize the estimated change, Figure 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 two rows correspond to the June and July data, respectively. As we can see, the top left corners exhibit different patterns in the three periods, suggesting the existence of change-points.
-10
-20
30
-30
Latiude
40
40
120
130
140
130
-10
120
130
140
150
130
150
20
20
-20
200
30
30
30
120
150
120
130
140
150
120
130
140
100
Longitude
Figure 2: Average SST. From left to right: average SST from 1940 to 1981, average SST from 1982 1996, and average SST from 1997 to 2019. The top and bottom rows correspond to the June and July data respectively.

---

Conclusion
In this paper, we study change-point detection for sparse and dense functional data in general dimensions. We show that our algorithm FSBS can consistently estimate the change-points even in the extreme sparse setting with 1. Our theoretical analysis reveals an interesting phase transition between n and T, which has not been discovered in the existing literature for functional change-poing detection. The consistency of FSBS relies on the assumption that the minimal spacing A T. To relax this assumption, we may consider increasing I. in Definition to enlarge the coverage of the seeded intervals FSBS and apply the narrowest over threshold selection method [Theorem 3 in 19]. With minor modifications of the current thcoretical analysis, the consistency of FSBS can be established for the case of < T. Since such relaxation does not add much more methodological insights paper, we omit this additional technical discussion for conciseness.
Acknowledgments and Disclosure of Funding
YY's research partially funded by EPSRC EP/VOL 3432//..
10

---

References
[1] Arlot, S., Celisse, A. and Harchaoui, Z. [2019], 'A kernel multiple change-pon poigee algorithm via model selection', Journal of machine learning research 20(162).
[2] Ashok, K., Guan, Z. and Yamagat, T. [2003], 'A look at the relationship between the enso and the indian ocean dipole', Journal of the Meteorological Society of Japan. Ser. II 81(1), 41-56.
[3] Aue, A., Gabrys, R., Horváth, L. and Kokoszka, P. [2009], 'Estimation of change-point in the mean function of functional data', d Journal of Multivariate Analysis 100(10), 2269.
[4] Aue, A., Rice, G. and Sönmez, O. [2018], 'Detecting and dating structural breaks in functional data without dimension reduction' Journal of the Royal Statistical Society: Series B (Statistical Methodology) 80(3), 509-529.
[5] Australian Government [2014], 'What is EI Niño and how does it impact Australia?",htar
//www. bom.
and anw.a-no-nndauttrrllia...
8html
[6] Berkes, Gabrys, R., Horváth, L. and Kokoszka, P. [2009], 'Detecting changes in the mean of functional observations', Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71(5). 927-946.
[7] Bonner, S. J., Newlands, N. K. and Heckman, N. E. [2014], 'Modeling regional impacts of climate teleconnections using functional data analysis', Environmental and Ecological Statistics 21(1), 1-26.
[8] Cai, T. T. and Yuan, M. [2011], 'Optimal estimation of the mean function based on discretely sampled functional data: Phase transition'. The Annals of Statistics 39(5), 2330-2355.
[9] Chacón, and Duong, [2018), Multivariate kernel smoothing and its applications, Chapman and Hall/CRC.
[10] Chiou, J..M., Zhang, Y.-C., Chen, W.-H. and Chang, C.-W. [2014], 'A functional data approach to missing value imputation and outlier detection for traffic flow data', Transportmetrica B: Transport Dynamics 2(2), 106-129.
[11] Dai, X., Müller, H.-G., Wang, J.-L. and Deoni, S. C. [2019], 'Age-dynamic networks and func- tional correlation for early white matter myelination' Brain Structure and Function 224(2), 535- 551.
[12] Dette, H., Kokot, K. and Volgushev, S. [2020]. 'Testing relevant hypotheses in functional time series via saff-normalizationn Journal of the Royal Statistical Society: Series B (Staristical Methodology) 82(3), 629-660.
[13] Fan, Y., Foutz, N., James, G. M. and Jank, W. [2014], 'Functional response additive model estimation with online virtual stock markets', The Annals of Applied Statistics 8(4), 2435-24452 4460.
[14] Fraiman, R., Justel, A., Liu, R. and Llop, P. [2014], 'Detecting trends in time series functional data: A study of antarctic climate change', Canadian Journal of Statistics 42(4), 597-609
[15] Giné, and Guillou, A. [1999], 'Laws of the iterated logarithm for censored data', The Annals of Probability 27(4). 2042-2067.
[16] Giné, E. and Guillou, A. [2001]. "On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervals' Annales de l'HHP Probabilités et statistiques 37(4), 503-522.
[17] Hörmann, S. and Kokoszka, P. [2010], 'Weakly dependent functional data', The Annals of Statistics 38(3), 1845-1884.
[18] Kim, J., Shin, J., Rinaldo, A. and Wasserman, I. [2019], Uniform convergence rate of the kernel density estimator adaptive to intrinsic volume dimension, in 'International Conference on Machine Learning', PMLR, pp. 3398-3407.

---

[19] Kovács, S., Li. H., Bühlmann, P. and Munk, A. [2020]. 'Seeded binary segmentation: A general methodology for fast and optimal change point detection', arXiv preprint arXiv:2002.06633
[20] Li, S., Xie, Y., Dai, H. and Song, L. [2015], 'M-statistic for kernel change-point detection', Advances in Neural Information Processing Systems 28.
[21] Li, S., Xie, Y., Dai, H. and Song. L. [2019), 'Scan b-statistic for kernel change-poing detection', Sequential Analysis 38(4), 503-544.
[22] Padilla, O. H. M., Yu, Y., Wang, D. and Rinaldo, A. [2021], 'Optimal nonparametric multivariate
change point detection and localization', IEEE Transactions on Information Theory.
[23] Petersen, A., Deoni, and Müller, H.-G. [2019]. 'Fréchet estimation of tie-varying covariance matrices from sparse data, with application to the regional co-evolution of myclination in the developing brain', The Annals of Applied Statistics 13(1), 393-419.
[24] Physical Sciences Laboratory [2020]. 'COBE SST2 and Sea-Ice', httts::/psl. noa. gov/ batta/grided//eeee
cobe2 html html
[25] Rigollet, P. and Vert, R. [2009], 'Optimal rates for plug-in estimators of density level sets', Bernoulli 15(4), 1154-1178.
[26] Tsybakov, A. B. [2009], Introduction to Nonparametric Estimation, Springer series in statistics, Springer, Dordrecht.
[27] Wang. D., Lu. X. and Rinaldo. A. [2019], 'Dbscam: Optimal rates for density- -based cluster estimation', Journal of Machine Learning Research.
[28] Wang, D., Yu, Y. and Rinaldo, A. [2020], 'Univariate mean change point detection: Penalization, cusum and optimality', Electronic Journal of Statistics 14(1), 1917-1961.
[29] Wang, J.-L., Chiou, J.-M. and Müller, H.-G. [2016), 'Functional data analysis', Annual Review of Statisties and Its Application 3, 257-295.
[30] Wu, W. B. [2005], 'Nonlinear system theory: Another look at dependence', Proceedings of the National Academy of Sciences 102(40), 14150-14154.
[31] Zhang, X., Shao, X., Hayhoe, K. and Wuebbles, D. J. [2011]. 'Testing the structural stability of temporally dependent functional observations and application to climate projections', Electronic Journal of Statistics 5, 1765-1796.
[32] Zhang, X. and Wang, J.-L. [2016]. 'From sparse to dense functional data and beyond', The Annals of Statistics 44(5), 2281-2321.
12

---

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes] See Sections] 3 and
(b) Did you describe the limitations of your work? [Yes) See Section S
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix ??.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main ex- perimental results (either in the supplemental material as a URL)? [Yes] We have included detailed instructions needed to reproduce the numerical results in Section 4
We also submit our code to the supplementary materials.
(b) Did you specify all the training details (c.g.. data splits, hyperparameters, how they were chosen)? [Yes] See Section 4
(c) Did you report error bars (e.g.. with respect to the random seed after running experi- ments multiple times)? [Yes] See Figure
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [No] The computation is sufficiently fast and computational cost is not a concern.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you cite the creators? [N/A]
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you't 're using/curating? [N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
13


Reviews:
[{"review_id": "sdLmW4C2gkc", "reviewer": "Reviewer_3KgB", "summary": "This paper proposes a new kernal-based algorithm called functional seeded binary segmentation (FSBS) for change point detection of functional data. Theoretical error rate and consistency results are presented for functional data with general sampling density (sparse & dense), temporal dependence, and importantly that are multi-dimensional. Numerial experiments are conducted on simulation data as well as climate change data.", "questions": "1. Since the paper stresses extension to multi-dimensional data as a major contribution, there should be more discussion (in section 3.2) and experiments (in both section 4.1 and 4.2) involving multi-dimensional data. What insight can one gain about the theoretical bounds with respect to parameter $d$? How does the algorithm behave for $d >1$ in more than just a single experiment (scenario 4)?\n\n2. How did the computational cost and runtime of the proposed method compare to the existing methods?\n\n3. It's not clear what the reader is supposed to notice from Figure 2. If there is noticable change in the top left corner, a zoomed-in panel might be helpful.\n", "limitations": "N/A", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper is very clearly written and the presentation is great. While I haven't had a chance to read through the proofs in appendix, assuming they are technically sound, extending changepoint detection to multi-dimensional functional data is a significant contribution to the community.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "oG5TKZNbNk2", "reviewer": "Reviewer_Kngx", "summary": "This paper presented a change-point detection method, namely functional seeded binary segmentation (FSBS), for d-dimensional time-series data. FSBS is claimed to be more suitable for heavy-tailed and temporally dependent data, while being computationally efficient. The proposed method is evaluated on both synthetic data and sea surface temperature data and leads to better results compared to three classic methods: BGHK, HK, and SN.", "questions": "FSBS is claimed to handle temporally-dependent observations better theoretically, however, it is not clearly why it has such a property and other methods do not. In line 128 – 130, authors state that “For temporal dependence, we adopt the physical dependence framework by [26], which covers a wide range of time series models, such as ARMA and vector AR models.” while it is hard to tell how [26] is adopted here exactly.", "limitations": "See “weakness” above.", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "* Strengths\n\n-- The proposed method is easy to implement with fewer (only three) tuning parameters, and the authors explained them well in terms of how to the select them.\n\n-- The authors showed that FSBS is consistent in detecting and localising multiple change-points under standard regularity conditions. Besides, it is claimed to have the sharpest localisation error rate.\n\n* Weaknesses\n\n-- The baseline methods appear to be published quit a while ago, e.g., BGHK (2009), HK (2010) and SN (2011), so I’m not sure if they are state-of-the-art methods for change-point detection in functional data as the authors claimed in Line 267.\n\n-- The experiments seem to be quite lightweight, as it has only one real dataset.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "_kyTms0qfT", "reviewer": "Reviewer_WXXv", "summary": "This paper considers the change-point detection problem for functional data in multi-dimensional space. A kernel-based algorithm is proposed, which allows for handling functional data in multivariate domain with measurement error. The author further presents several important theoretical results about this algorithm, such as the consistency of testing, and sharp localization error rate. Numerical results support their contributions. ", "questions": "See **Weakness** part.", "limitations": "See **Weakness** part.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Originality/Significance**: The proposed kernel-based algorithm elegantly solves main challenges in existing literature: (i) observed funtional data may contain measurement error, (ii) there may exist multiple change points, (iii) domain for functional data can be multivariate.  This work would be very interesting for readers in the testing and detection area.\n\n**Quality**: All important points for functional-data change point detection are discussed, including the algorithm development, solid theoretical analysis, and extensive experiment simulations. From my view point, the quality of this paper is at least above 50% of all accepted nips papers.\n\n**Clarity**: Problem setup, algorithm procedures, theory statements and assumptions are clearly presented. \n\n**Weakness**: \n1. It would be helpful to do a literature review for kernel-based change-point detection (even not under the functional data setting). This comparison will make the reader understand the connection and difference for the proposed algorithm.\n2. The choice of kernel will largely influence the performance of change-point detection. The kernel choice in general is specified by the users that have prior knowledge on the data. Although the author does not demonstrate which kernel choice is better (in numerical simulation only Gaussian kernel is used), I think it is not a big issue and it would be an interesting topic for future study.\n3. Minor wording issue: \n- In line 168, the bracket should be $O(T\\log(T))$.\n- In line 211-212, there is a large box for the equation environment. The author may consider repharse it to be \"with probability at least $1-3\\log^{-1}(T)$, it holds that \\[xxx\\]\".", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "XbVgOZk1hzv", "reviewer": "Reviewer_eNoD", "summary": "This paper studies offline change-point detection and estimation for functional data. A kernel-based algorithm named functional seeded binary segmentation (FSBS) is proposed. This paper also shows the consistency of FSBS for multiple change-point estimation and \t.\tprovides a sharp localization error rate. Numerical demonstrations on synthetic and real data are provided.", "questions": "Some notations are not explained sufficiently. For example, what is the meaning of “arbitrarily-slow” in assumption 3, and what does C_SNR mean here?  \n\nAlso, considering kernel-based change-point detection, it might be worthwhile to compare to or comment on the difference with some existing kernel type methods, such as \"Li, S., Xie, Y., Dai, H., & Song, L. (2015). M-statistic for kernel change-point detection. Advances in Neural Information Processing Systems, 28.\" (although this paper didn't target detection for functional data, its online detection scheme might be helpful if the FSBS could be generalized for online detection).\n\nMinor issues:\nThe equation in Theorem 1 is too long, maybe split into two lines.\n", "limitations": "yes", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\n1. A new multiple change point detection algorithm is proposed. The FSBS procedure, based on kernel density estimation and CUSUM statistics, is computationally efficient. \n2. This paper considers a general detection scheme, where the functional data can be discretely observed (even with n=1 in the very extreme case) and can be contaminated with temporally-dependent and heavy-tailed errors.\n3. Theoretical guarantee on the estimation error is provided. \n4. Extensive numerical experiments \n\n\nWeaknesses\nThe method is not new. Both Binary Segmentation and kernel density estimations are well-studied methods for change detection and nonparametric density estimation, respectively. And the theoretical guarantee on BS and KDE are also well-studied.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "HrFNYEEwd9", "reviewer": "Reviewer_1qwN", "summary": "This paper studies change-point detection and localization for sparsely or densely sampled functional data observed on a general d-dimensional space. A computational efficient kernel based FSBS algorithm is proposed to handle discretely observed functional data and general d-dimensional domain. The supporting theory is developed for heavy-tailed and serially dependent curves and measurement errors.  A phase transition phenomenon is established depending on the order of n relative to T. The finite sample performance is evaluated via both simulations under different settings and one real data example. ", "questions": "1, What is the advantage of the local constant smoother used in the paper instead of the commonly adopted local linear smoother?\n\n2, The phase transition result is interesting, but is different from results in Zhang and Wang, 2016, which considers categories of sparse, dense and ultra-dense functional data. In particular, the optimal bandwidths are different under different cases and as the number of measurement locations becomes sufficiently large, the parametric root-n rate can be attained. A detailed discussion is helpful.\n\n3, What are the truly innovative component of the paper taking the advantages of functional, temporally dependent and non-stationary nature of the data?\n\n4, Why is the extreme sparse case n=1 argued as one selling point of the paper? As far as I know, if n is bounded (treated as sparse functional data), the same rate can be attained.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\n1, The first work in existing literature studying change points detection for partially observed functional data. \n\n2, Can handle low-dimensional multivariate functional data.\n\n3, Theory is developed for serially dependent random functions and measurement errors\n\nWeakness:\n\n1, Instead of a machine learning conference like NeurIPS, I feel this paper fits a traditional Statistics journal better. The proposed methodology seems like a combination of some well-established techniques in change points detection and functional data analysis areas.\n\n2, The measurement locations x_{t,i} are fixed, while in standard functional and longitudinal data analysis literature, these locations are assumed to be random and the number of measurement locations across curves can be different (rather than a fixed n) or even random.\n\n3, The detailed comparison with phase transition phenomenon in existing literature is needed. See my question below.\n\n4, Local constant smoother instead of local linear smoother is used and hence may suffer from bad performance around the boundaries. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept