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Seasonal functional autoregressive models
Article in Journal of Time Series Analysis · June 2021
DOI: 10.1111/jtsa.12608
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Seasonal functional autoregressive models
Atefeh Zamani1 , Hossein Haghbin2 , Maryam Hashemi3 , and Rob J Hyndman4
27 May 2020
Abstract
Functional autoregressive models are popular for functional time series analysis, but
the standard formulation fails to address seasonal behavior in functional time series data.
To overcome this shortcoming, we introduce seasonal functional autoregressive time series
models. For the model of order one, we derive sufficient stationarity conditions and limiting behavior, and provide estimation and prediction methods. Moreover, we consider a
portmanteau test for testing the adequacy of this model and we derive its asymptotic distribution. The merits of this model is demonstrated using simulation studies and via an
application to real data.
Keywords: Functional time series analysis, seasonal functional autoregressive model, central
limit theorem, prediction, estimation.
1
Introduction
Improved acquisition techniques make it possible to collect a large amount of high-dimensional
data, including data that can be considered functional (Ramsay, 1982). Functional time series
arise when such data are collected over time. We are interested in functional time series which
exhibit seasonality. The underlying functional process is denoted by f t ( x ), where t = 1, . . . , T
indexes regularly spaced time and x is a continuous variable.
A seasonal pattern exists when f t ( x ) is influenced by seasonal factors (e.g., the quarter of the
year, the month, the day of the week, etc.). Usually seasonality is considered to have a fixed
and known period. For example, consider satellite observations measuring the normalized
difference vegetation index (NDVI) (He, 2018). These are often averaged to obtain monthly
observations over the land surface. Here x denotes the two spatial dimensions, while t denotes
the month. Seasonal patterns are present due to the natural annual patterns of vegetation
variation.
1 Department
of Statistics, Faculty of Science, Shiraz University, Shiraz, Iran.
of Statistics, Faculty of Science, Persian Gulf University, Boushehr, Iran.
3 Department of Statistics,University of Khansar, Khansar, Iran.
4 Department of Econometrics & Business Statistics, Monash University, Clayton VIC 3800, Australia.
Email: Rob.Hyndman@monash.edu. Corresponding author.
2 Department
1
Another example arises in demography where f t ( x ) is the mortality rate for people aged x
at time t (Hyndman and Ullah, 2007). When such data are collected more frequently than
annually, seasonality arises due to deaths being influenced by weather.
In other applications, x may denote a second time variable. For example, Hörmann, Kokoszka,
Nisol, et al., 2018 study pollution data observed every 30 minutes. The long time series is sliced
into separate functions, where x denotes the time of day, and t denotes the day. A similar
approach has been applied to the El Nino-Southern Oscillation (ENSO) (Besse, Cardot, and
Stephenson, 2000), Eurodollar futures rates (Kargin and Onatski, 2008; Horváth, Kokoszka,
and Reeder, 2013), electricity demand (Shang, 2013), and many other applications.
Although, the term ”functional data analysis” was coined by Ramsay (1982), the history of this
area is much older and dates back to Grenander (1950) and Rao (1958). Functional data cannot
be analyzed using classical statistical tools and need appropriate new techniques in order to
be studied theoretically and computationally.
A popular functional time series model is the functional autoregressive (FAR) process introduced by Bosq (2000), and further studied by Hörmann, Kokoszka, et al. (2010), Horváth,
Hušková, and Kokoszka (2010), Horváth and Kokoszka (2012), Berkes, Horváth, and Rice
(2013), Hörmann, Horváth, and Reeder (2013) and Aue, Norinho, and Hörmann (2015).
Although these models are applied in the analysis of various functional time series, they cannot
handle seasonality adequately. For example, although it seems that the traffic flow follows a
weakly pattern, Klepsch, Klüppelberg, and Wei (2017) applied functional ARMA processes for
modeling highway traffic data.
A popular model for seasonal univariate time series, { X1 , . . . , XT }, is the class of seasonal
autoregressive processes denoted by SAR(P)S , where S is the seasonal period and P is the
order of the autoregression. These models satisfy the following equation:
Xt = φ1 Xt−S + φ2 Xt−2S + · · · + φP Xt− PS + ε t ,
where φ( x ) = 1 − φ1 x S − φ2 x2S − · · · − φP x PS is the seasonal characteristic polynomial and ε t
is independent of Xt−1 , Xt−2 , . . . . For stationarity, the roots of φ( x ) = 0 must be greater than 1
in absolute value. This model is a special case of the AR(p) model, which is of order p = PS,
with nonzero φ-coefficients only at the seasonal lags S, 2S, 3S, . . . , PS.
In this paper, we propose a class of seasonal functional AR models, which are analogous to seasonal autoregressive models. We present some notation and definitions in Section 2. Section 3
introduces the seasonal functional AR(1) model and discusses some of its properties. Estimation of the parameters of this model is studied in Section 4. In Section 5, the portmanteau
test for seasonal functional AR(1) is presented and its asymptotic distribution is derived. The
prediction problem is considered in Section 6 and Sections 7 and 8 are devoted to simulation
studies and real data analysis. We conclude in Section 9.
2
2
Preliminary notations and definitions
Let H = L2 ([0, 1]) be the separable real Hilbert space of square integrable functions x : [0, 1] →
R
1/2
R1
1 2
. Let B
R with the inner product h x, yi = 0 x (t)y(t)dt and the norm k x k =
x
(
t
)
dt
0
denote the Borel field in H, and (Ω, F , P) stand for a probability space. A random function in
H is an F /B measurable mapping from Ω into H.
Additionally, let L( H ) denote the space of all continuous linear operators from H to H, with
operatorial norm k · kL . An important subspace of L( H ) is the space of Hilbert-Schmidt operators, HS( H ), which forms a Hilbert space equipped with the inner product h A, BiHS =
n
o1/2
2
∞
and
the
norm
A
=
Aφ
, where {φk } is any orthonormal
Aφ
,
Bφ
k
i
k
k
k
h
∑
∑∞
k
k
k
k =1
k =1
HS
basis on H. The space of nuclear or trace class operators, N ( H ), is another notable subclass of
HS ( H ) and the associated norm is defined as
∞
k AkN =
∑ h| A| φk , φk i =
k =1
∞
∑
D
E
( A∗ A)1/2 φk , φk ,
(2.1)
k =1
where A∗ is the adjoint of A (Conway, 2000). If A is a self-adjoint nuclear operator with
associated eigenvalues λk , then k AkN = ∑∞
k =1 | λk |. If, in addition, A is non-negative, then
∞
∞
k AkN = ∑k=1 h Aφk , φk i = ∑k=1 λk . Note that k · kL ≤ k · kHS ≤ k · kN (Hsing and Eubank,
2015).
For x and y in H, the tensorial product of x and y, x ⊗ y, is a nuclear operator and is defined as
( x ⊗ y)z := hy, zi x, z ∈ H. Here, we point out some relations which are simple consequences
of the definition of x ⊗ y and will be applied in the following sections:
( x ⊗ y)∗ = y ⊗ x,
( Ax ) ⊗ y = A ( x ⊗ y) ,
x ⊗ ( Ay) = ( x ⊗ y) A∗ .
(Conway, 2000).
Let Z denote the set of integers. Then, we define a functional discrete time stochastic process
as a sequence of random functions in H, namely X = { Xt , t ∈ Z}. A random function X
in H is said to be strongly second order if Ek X k2 < ∞. Similarly, a functional discrete time
stochastic process X is said to be strongly second order if every Xt is strongly second order.
Let L2H (Ω, F , P) stand for the Hilbert space of all strongly second order random function on
the probability space (Ω, F , P).
For the random function Xt , the mean function is denoted by µt := E( Xt ) and is defined
in terms of Bochner integral. For any t, t0 ∈ Z, the auto-covariance operator is defined as
3
X : = E X − µ ) ⊗ ( X 0 − µ 0 )]. Besides, as an integral operator, C X can be represented as
Ct,t
[( t
0
t
t
t
t,t0
X
Ct,t
0 h(s) =
Z 1
0
X
0
0
0
Ct,t
0 ( s, s ) h ( s ) ds ,
s, s0 ∈ [0, 1]
and
t, t0 ∈ Z,
X s, s0 : = E X ( s ) − µ ( s ))( X 0 ( s0 ) − µ 0 ( s0 ))] is the corresponding auto-covariance
where Ct,t
)
[( t
0 (
t
t
t
X as C X and if no confusion arises,
kernel. When the process is stationary, we will denote Ct,t
0
t−t0
we will drop superscript X.
As in any time series analysis, functional white noise processes are of great importance in
functional time series analysis.
Definition 2.1. A sequence ε = {ε t , t ∈ Z} of random functions is called functional white noise if
ε : = C ε do not depend on t,
(i) 0 < E kε t k2 = σ2 < ∞, E (ε t ) = 0 and Ct,t
0
ε = 0.
(ii) ε t is orthogonal to ε t0 , t, t0 ∈ Z, t 6= t0 ; i.e., Ct,t
0
The gaussian random functions are of great importance in the analysis of function data. The
next definition is devoted to introducing these random functions, Lord, Powell, and Shardlow
(2014).
Definition 2.2. The random function X is called gaussian with mean µ and covariance operator C,
N (µ, C ) in abbreviation, if for all x ∈ H, h X, x i is a real-valued gaussian random variable with mean
hµ, x i and variance hC ( x ) , x i .
The following definitions will be applied in the subsequent sections.
Definition 2.3. Let Xn be a sequence of random functions. We say that Xn converges to X in L2H (Ω, F , P)
if Ek Xn − X k2 → 0, as n goes to infinity.
Definition 2.4. G is said to be an L-closed subspace (or hermetically closed subspace) of L2H (Ω, F , P)
if G is a Hilbertian subspace of L2H (Ω, F , P) and, if X ∈ G and ` ∈ L, then `( X ) ∈ G . A zero-mean
LCS is an L-closed subspace which contains only zero-mean random functions.
p
Moreover, let H p = L2 ([0, 1]) denote the productqHilbert space equipped with the inner
p
product hx, yi p = ∑i=1 h xi , yi i and the norm kxk p = hx, xi p . We denote by L( H p ) the space
of bounded linear operators on H p .
3
The SFAR(1)S model
Following the development of univariate seasonal autoregressions in Harrison (1965), Chatfield and Prothero (1973) and Box et al. (2015), we define the seasonal functional autoregressive
process of order one as follows.
4
Definition 3.1. A sequence X = { Xt ; t ∈ Z} of random functions is said to be a pure seasonal
functional autoregressive process of order 1 with seasonality S (SFAR(1)S ) associated with (µ, ε, φ) if
Xt − µ = φ ( Xt − S − µ ) + ε t ,
(3.1)
where ε = {ε t ; t ∈ Z} is a functional white noise process, µ ∈ H and φ ∈ L( H ), with φ 6= 0.
The SFAR(1)S processes can be studied from two different perspectives. As the first viewpoint,
a SFAR(1)S model is an FAR(S) model with most coefficients equal to zero. This point of view
will be applied when dealing with basic properties of these processes in the next subsection.
In the other perspective, SFAR(1)S processes are studied as a special case of an autoregressive
time series of order one with values in the product Hilbert space H S , which will be used while
studying the limit theorems of such processes.
For simplicity of notation, let us consider µ as zero. In this case, as there is no intercept, the
unconditional mean of the process regardless of the season is equal to zero. However, the
conditional mean on the past of Xt depends in S, since E ( Xt | Xt−1 , . . . ) = φXt−S (Ghysels and
Osborn, 2001).
3.1
Basic properties
In order to study the existence of the process X, the following assumption is required:
Assumption 1: There exists an integer j0 ≥ 1 such that φ j0
L
< 1.
In the following, we will call an SFAR(1)S a standard time series if µ = 0 and Assumption 1
holds.
The following theorem, which deals with the existence and uniqueness of SFAR(1)S process, is
a reformulation of Theorem 3.1, Bosq (2000).
Theorem 3.1. Let Xt be a standard SFAR(1)S time series. Then,
Xt = φXt−S + ε t ,
(3.2)
has a unique stationary solution given by
∞
Xt =
∑ φ j ε t− jS ,
t ∈ Z,
(3.3)
j =0
where the series converges in L2H (Ω, F , P) and with probability 1 and ε is the functional white noise
process of Xt .
Proof. The proof is an extension of the proof of Theorem 3.1 of Bosq (2000) for the functional
AR(1) and consequently is omitted.
5
In time series analysis, the auto-covariance operator is of great importance and plays a crucial
role in the analysis of the data. The following theorem demonstrates the structure of autocovariance operator for the SFAR(1)S processes.
Theorem 3.2. If X is a standard SFAR(1)S , the following relations hold:
C0X = CSX φ∗ + C ε = φC0X φ∗ + C0ε ,
(3.4)
∞
C0X =
∑ φ j C0ε φ∗ j ,
(3.5)
j =0
where CSX is the auto-covariance operator of X at lag S and the series converges in the k · kN sense.
Besides, for t > t0 ,
(
X
Ct,t
0 =
φk C0X
if t − t0 = kS
0
Otherwise
,
(3.6)
,
(3.7)
and
(
CtX0 ,t
=
C0X φk
∗
if t − t0 = kS
Otherwise
0
where k is some positive integer.
Proof. For each x in H, we have:
C0X = E ( Xt ⊗ Xt )
= E ( Xt ⊗ (φXt−S + ε t )) ,
which, using the properties of tensorial product, results in that C0X = CSX φ∗ + C0ε . Similarly,
C0X = E ((φXt−S + ε t ) ⊗ (φXt−S + ε t )) , which implies that C0X = φC0X φ∗ + C0ε .
As demonstrated in the proof of Theorem 3.1, Xt = ∑kj=0 φ j ε t−Sj + φk+1 Xt−(k+1)S , which results
in that:
C0X = E ( Xt ⊗ Xt )
=E
k
∑ φ j ε t−Sj + φk+1 Xt−(k+1)S
!
j =0
k
⊗
∑ φ j ε t−Sj + φk+1 Xt−(k+1)S
!!
.
j =0
Consequently, C0X = ∑kj=1 φ j C0ε φ∗ j + φ(k+1) C0X φ∗ (k+1) . Since φ(k+1) C0X φ∗ (k+1) is the auto-covariance
operator of φ(k+1) X0 , it is a nuclear operator and
φ(k+1) C0X φ∗ (k+1)
N
= φ(k+1) A1/2 A∗ 1/2 φ∗ (k+1)
= φ(k+1) A1/2
6
2
HS
N
≤ φ ( k +1)
2
L
k AkHS ,
where C0X = A1/2 A∗ 1/2 . Therefore, C0X − ∑kj=1 φ j C ε φ∗ j
N
≤ φ ( k +1)
2
L
k AkHS , which tends
j ε ∗j
to zero as k goes to infinity, and results in that C0X = ∑∞
j =0 φ C φ .
X =
Let t − t0 = kS, for some positive integer k. Using recursive calculation, we have Ct,t
0
X = φk C X . Finally, Concerning (3.7), it
E φk Xt0 + ε t0 +S ⊗ Xt0 , which demonstrates that Ct,t
0
0
suffices to write
∗ h
i∗
∗
X
= φk C0X = C0X φk ,
CtX0 ,t = Ct,t
0
and the proof is completed.
Note 3.1. The auto-covariance operators characterize all the second-order dynamical properties of a
time series and are the focus of time domain analysis of time series data. However, sometimes studying
time series in the frequency domain can open new avenues in time series research. Here, we are going
to formulate the spectral density operator of { Xt }, based on Panaretos and Tavakoli (2013). If the
auto-covariance operators satisfy ∑t∈Z CtX N < ∞, then the spectral density operator of a stationary
functional time series, { Xt }, at frequency λ (which is nuclear), is defined as
Fλ =
1
2π
∑ e−iλt CtX ,
(3.8)
t ∈Z
where λ ∈ R and the convergence holds in nuclear norm. Similarly, based on Theorem 3.2, for SFAR(1)S
time series, if ∑t∈Z CtX
Fλ =
N
1
2π
1
=
2π
< ∞, then the spectral density operator at frequency λ, will be
∑ e−iλt CtX ,
t ∈Z
∞
(
φC0X φ∗
ε
+C +
∑e
−iλkS k
φ
C0X
k =1
−1
+
∑
k =−∞
e
−iλkS
∗
C0X φk
)
.
(3.9)
Note 3.2. Let ε = Xt − φXt−S . It can be shown that
 X
X
X ∗
X ∗
h=0

 C0 − φC−S − CS φ + φC0 φ
ε
X
X
X
∗
X
∗
CkS − φC(k−1)S − C(k+1)S φ + φCkS φ h = kS .
Ch =


0
h 6= kS
(3.10)
This result will be used in Section 5.
Consider φ be a symmetric compact operator. In this case, φ admits the spectral decomposition
∞
φ=
∑ αj ej ⊗ ej,
j =1
7
(3.11)
where e j is an orthonormal basis for H and α j is a sequence of real numbers, such that
2
α1 ≥ α2 ≥ · · · ≥ 0 and lim j→∞ α j = 0. If there exists j such that E ε 0 , e j
> 0, we can define
the operator φε , which connects ε and φ, as follows:
∞
φε =
∑
αj ej ⊗ ej,
(3.12)
j= jmin
where jmin is the smallest j satisfying E
ε 0, ej
2
> 0. If such a jmin does not exists, we set
φε = 0.
Theorem 3.3. Let φ be a symmetric compact operator over H and ε be a functional white noise process.
Then, equation Xt = φXt−S + ε t has a stationary solution with innovation ε if and only if kφε kL < 1.
Proof. The proof is an easy consequence of Theorem 3.5, Page 83, Bosq (2000).
The following theorem is of great importance when dealing with portmanteau tests for SFAR(1)S .
Theorem 3.4. Let φ = ∑ j=1 α j e j ⊗ e j be a symmetric compact operator on H. Then, Xt is a SFAR(1)S
model if and only if h Xt , ek i is a SAR(1)S model.
Proof. Let Xt be a pure SFAR(1)S process, where ε = {ε t ; t ∈ Z} is a functional white noise
and φ ∈ L( H ). Then, using spectral decomposition and symmetric property of φ, it can be
shown that
h Xt , e k i = α k h Xt − S , e k i + h ε t , e k i ,
j ≥ 1.
(3.13)
If αk 6= 0 and E hε t , ek i2 > 0, then h Xt , ek i is a SAR(1) model. If αk = 0, h Xt , ek i = hε t , ek i
and h Xt , ek i is a degenerate AR(1). Conversely, by substituting equation (3.13) in h Xt , x i =
∑k=1 h Xt , ek i h x, ek i and applying spectral decomposition, we have h Xt , x i = hφXt−S + ε t , x i .
Consequently, Xt = φXt−S + ε t .
3.2
Limit theorems
In this subsection, we will focus on some limiting properties of the SFAR(1)S model. Let us set
Yt = ( Xt , . . . , Xt−S+1 )0 and εt = (ε t , 0, . . . , 0)0 , where 0 appears S − 1 times. Define the operator
ρ on H S as:

0 0 ···

I 0 · · ·


ρ = 0 I · · ·
. .
 .. .. · · ·

0 0 ···
8
0 φ


0 0

0 0
,
.. .. 
. .

I 0
(3.14)
where I denotes the identity operator on H. Consequently, we have the following simple but
crucial lemma.
Lemma 3.1. If X is a SFAR(1)S process, associated with (ε, φ), then Y is an autoregressive time series
of order one with values in the product Hilbert space H S associated with (ε, ρ), i.e.,
Yt = ρYt−1 + εt .
(3.15)
Proof. This lemma is an immediate consequence of Lemma 5.1, Bosq (2000).
The next theorem demonstrates the required condition for existence and uniqueness of the
process X based on the operator ρ.
Theorem 3.5. Let L H S denote the space of bounded linear operators on H S equipped with the norm
k · kLS . If
ρ j0
< 1,
LS
for some j0 ≥ 1,
(3.16)
t ∈ Z,
(3.17)
then (3.1) has a unique stationary solution given by
∞
Xt =
∑
πρ j
εt− j ,
j =0
where the series converges in L2H S (Ω, F , P) and with probability 1 and π is the projector of H S onto H,
defined as π ( x1 , . . . , xS ) = x1 , ( x1 , . . . , xS ) ∈ H S .
By the structure of ρ, it can be demonstrated that ρS = φI, where I denotes the identity operator on H S . Consequently, the following lemma is clear.
Lemma 3.2. If kφk < 1, then (3.16) holds.
Considering Sn = ∑nj=1 X j , we can state the law of large number for the process X.
Theorem 3.6. If X is a standard SFAR(1)S then, as n → ∞,
n0.25 Sn
→ 0,
(log n) β n
β > 0.5.
(3.18)
Proof. The proof is an easy consequence of Theorem 5.6 of Bosq (2000).
The central limit theorem for SFAR(1)S processes is established in the next theorem, which can
be proved using Theorem 5.9 of Bosq (2000).
Theorem 3.7. Let X be a standard SFAR(1)S associated with a functional white noise ε and such that
I − φ is invertible. Then
S
√n → N (0, Γ),
n
9
(3.19)
where
Γ = ( I − φ ) −1 C ε ( I − φ ∗ ) −1 .
4
(3.20)
Parameter estimation
In this section, three methods are applied to estimate the parameter φ in model (3.1). These
methods, namely method of moments, unconditional least squares estimation and the KarginOnatski method, are studied in brief in the following subsections.
4.1
Method of moments
Identification of SFAR(1)S takes place via estimation of the parameter φ. This parameter can be
estimated using the classical method of moments. For this purpose, it can be shown that
E ( Xt ⊗ Xt−S ) = E (φXt−S ⊗ Xt−S ) + E (ε t ⊗ Xt−S ) ,
or equivalently, by stationarity of SFAR(1)S ,
ε,X
CSX = φC0X + Ct,t
−S .
Since CSX,ε = 0, it can be concluded that
CSX = φC0X .
(4.1)
The operator C0X is compact and, consequently, it has the following spectral decomposition:
C0X =
∑
λm νm ⊗ νm ,
m ∈N
∑ λm < ∞,
(4.2)
where (λm )m>1 is the sequence of the non-negative eigenvalues of C0X , such that, λ1 ≥ λ2 ≥
· · · ≥ 0. Besides, (νm )m>1 is the corresponding set of eigenfunctions, which forms a complete
orthogonal system in H. Note that, if, for some k ∈ Z, λk > 0 and λk+1 = 0, then Xt belongs
to the k-dimensional space spanned by νm , m = 1, 2, · · · , k, which is isometric to Rk . Therefore,
in the following and in order to reflect the infinite-dimensional nature of functional data, we
consider the case where λm > 0, for all m ≥ 1, Zhang (2016).
From equation (4.1), we have
CSX νj = φC0X νj = λ j φ νj .
Then, for any x ∈ H, the derived equation leads to the following representation:
∞
φ( x ) = φ
∑
j =1
10
!
x, νj νj
∞
=
∑
x, νj φ νj
j =1
∞
C X νj
=∑ S
λj
j =1
x, νj ,
(4.3)
which gives a core idea for the estimation of φ.
Define the empirical auto-covariance operators at lags 0 and S, respectively, by Ĉ0X = n1 ∑nj=1 X j ⊗
X j and ĈSX = n1 ∑nj=S X j ⊗ X j−S . Let λ̂ j , ν̂j 1≤ j≤n be the eigenvalues and the corresponding
eigenfunctions of Ĉ0X , where λ̂1 ≥ λ̂2 ≥ · · · λ̂n ≥ 0. Using Equation (4.3), φ can be estimated
by λ̂ j , ν̂j 1≤ j≤k , which leads to the following truncated equation:
n
kn
φ̂( x ) =
∑
j =1
=
ĈSX ν̂j
λ̂ j
x, ν̂j
1 n k n −1
∑ λ̂ j Xi−S , ν̂j
n i∑
= S j =1
x, ν̂j Xi ,
(4.4)
where k n ≤ max j| 1 ≤ j ≤ n, λ̂ j > 0 . The choice of k n can be based on the cumulative percentage of total variance (CPV).
Apart from the truncation method which is used in equation (4.4), there are other regularization methods that can be applied in the estimation of the parameter φ. For instance, one may
alternatively consider Tikhonov regularization given by
φ̃( x ) =
λ̂ j
1 n n
Xi−S , ν̂j
∑
∑
2
n i=S j=1 λ̂ j + an
x, ν̂j Xij ,
1 n n
1
Xi−S , ν̂j
∑
∑
n i=S j=1 λ̂ j + an
x, ν̂j Xi ,
or the penalization given by
φ̌( x ) =
with ( an )n∈Z being a positive sequence that converges to zero (Tikhonov and Arsenin, 1977;
Groetsch, 1993; Zhang, 2016).
Following the same steps as Theorem 8.7 of Bosq (2000), it can be proved that under certain
technical assumptions, φ̂ is a strongly consistent estimator of φ. Hörmann and Kidziński (2015)
showed that, even if some of the assumed conditions in Bosq (2000) are violated, φ̂ stays consistent. Moreover, Zhang (2016) presented assumptions which slightly generalizes the results of
Bosq (2000) by allowing the errors to be dependent. These assumption are stated in Section 5.
11
Note 4.1. For a sequence of zero mean SFAR(1)S processes, the Yule-Walker equations can be obtained
as:
(
ChX =
φChX−S
if h = kS
0
Otherwise
,
(4.5)
and
C0X = CSX φ∗ + C ε ,
(4.6)
where k is some positive integer. If the auto-covariance operators are estimated using their empirical
counterparts, the Yule-Walker equations can be applied for estimating the unknown parameters of the
model.
4.2
Unconditional least square estimation method
In this section we obtain the least square estimation of the model parameter, φ. Let C0X be the
auto-covariance operator of Xt , with the sequence of λ j , νj as its eigen-values and eigenfunctions. The idea is that the functional data can be represented by their coordinates with
respect to the functional principal components (FPC) of the Xt , e.g. Xtk = h Xt , νk i, which is the
projection of the tth observation onto the kth largest FPC. Therefore,
h Xt , νk i = hφXt−S , νk i + hε t , νk i
∞
=
∑
Xt−S , νj
φ(νj ), νk + hε t , νk i
Xt−S , νj
φ(νj ), νk + δtk ,
j =1
p
=
∑
(4.7)
j =1
where δtk = hε t , νk i + ∑∞
j= p+1 h Xt−S , νk i φ ( νj ), νk . Consequently, for any 1 ≤ k ≤ p, we have
p
Xtk =
∑ φkj Xt−S,j + δtk
j =1
where φkj = φ(νj ), νk . Note that the δtk are not i.i.d. Setting
Xt = ( Xt1 , . . . , Xtp )0 , δt = (δt1 , . . . , δtp )0 ,
φ = (φ11 , . . . , φ1p , φ21 , . . . , φ2p , . . . , φ p1 , . . . , φ pp )0 ,
we rewrite (4.8) as
Xt = Zt−S φ + δt ,
12
t = 1, 2, . . . , n,
(4.8)
where each Zt is a p × p2 matrix

Xt0
 0
 0p
Zt = 
 ..
 .
00p
00p
...
Xt0 . . .
..
. ...
00p
...
00p


00p 
.. 
,
. 
(4.9)
Xt0
with 0 p = (0, . . . , 0)0 .
Finally, defining the N p × 1 vectors X and δ and the N p × p2 matrix Z by

X1


δ1


Z 1− S

 
 


 X2 
 δ2 
 Z 2− S 





X =  . , δ =  . , Z =  . 
,
 .. 
 .. 
 .. 
Xn
δn
Zn−S
we obtain the following linear model
X = Zφ + δ
(4.10)
Representation (4.10) leads to the formal least square estimator for φ :
φ̃ = ( Z 0 Z )−1 Z 0 X
(4.11)
which can be computed if the eigenvalues and eigenfunctions of C0X are known. However, in
practical problems, since C0X is estimated by its empirical estimator, the νj must be replaced
by its empirical counterpart, called ν̂j . Using φ̃, the unconditional least square estimator of φ,
called φ̃, can be reconstructed. Following the approach of Kokoszka and Reimherr (2013), it
can be proved that φ̃ is a consitent estimator of φ.
4.3
The Kargin-Onatski method
The Kargin-Onatski method, which is also known as the predictive factors, is introduced
by Kargin and Onatski (2008). The main idea of this methods is to focus on the estimation of those linear functionals of the data that can most reduce the expected square error
of prediction. In fact, we would like to find an operator A, approximating φ, that minimizes
Ek Xt+1 − A( Xt−S+1 )k2 . Using this method, we obtain a consistent estimator as follows.
Let Ĉ0 and ĈS be the empirical auto-covariance and empirical auto-covariance of lags 0 and S,
respectively, i.e.,
Ĉ0 ( X ) =
1 n
h Xt , x i Xt ,
n t∑
=1
ĈS ( X ) =
13
n
1
h Xt , x i Xt − S .
n − S t∑
=S
Define Ĉα as Ĉ0 + αI, where α is a positive real number. Let {υα,i , i = 1, 2, . . . , k n } be the k n
eigenfunctions of the operator Ĉα−1/2 ĈS∗ ĈS Ĉα−1/2 , corresponding to the first largest eigenvalues
τ̂α,1 > · · · > τ̂α,kn . Based on these eigenfunctions, we construct the empirical estimator of φ,
φ̂α,kn , as follows:
kn
φ̂α,kn =
∑ Ĉα−1/2 υα,i ⊗ ĈS Ĉα−1/2 υα,i .
(4.12)
i =1
It can be demonstrated that that if {αn } is a sequence of positive real numbers such that
n1/6 αn → 0, as n goes to infinity, and {k n } is any sequence of positive integers such that
K ≤ n−1/4 k n ≤ n, for some K > 0, then φ̂α,kn is a consistent estimator of φ.
Based on Kargin and Onatski (2008) and Horváth and Kokoszka (2012), in practical implementation, k n can be considered to be equal to the number of the EFPC’s and αn = 0.75.
5
Diagnostic Checking
Based on the Box-Jenkins methodology in time series analysis, an appropriate method for a
long-term forecasting is consisted of model identification, parameter estimation, diagnostic
checking and forecasting. In the functional time series, the portmanteau tests for testing the
independence of a sequence of functional observations are proposed by Gabrys and Kokoszka
(2007) and Horváth, Hušková, and Rice (2013). They applied the independent and identically
distributed (i.i.d) assumption to validity the distribution of the proposed test statistics. Zhang
(2016) proposed a Cramérvon-Mises type test based on the functional periodogram to test
the uncorrelatedness of the residuals. In functional time series, the effect of estimating the
model parameters and replacing the unobservable innovations with their estimates are not
asymptotically negligible due to the dependence structure of these models. To overcome this
problem, Zhang (2016) introduced a modified block bootstrap that takes the estimation effect
into account.
In order to check the adequacy of seasonal time series models, it is recommended to use tests
in seasonal and nonseasonal lags, since implementing tests in just nonseasonal lags results
in ignoring the possibility of autocorrelations at seasonal lags S, 2S, · · · , mS, Mahdi (2016).
Therefore, in this section, we presented the seasonal version of the test introduced by Zhang
(2016), which can be applied to check the adequacy of the fit of the SFAR(1)S model in seasonal
lags.
Let Rt1 ,t2 ,··· ,t2k−1 (s1 , · · · , s2k−1 , s2k ) = cum Xt1 (s1 ) , · · · , Xt2k−1 (s2k−1 ) , X0 (s2k ) be the cumulant
function and the 2kth order cumulant operator be defined as
Rt1 ,t2 ,··· ,t2k−1 g (s1 , · · · , s2k−1 , s2k ) =
Z
[0,1]k
Rt1 ,t2 ,··· ,t2k−1 (s1 , · · · , s2k−1 , s2k ) g (sk+1 , · · · , s2k ) dsk+1 · · · ds2k .
(5.1)
14
To establish the test statistic for diagnostic checking, we make the following assumption.
ε
Assumption 5.1: Assume that kφkHS < 1 and ∑∞
t1 ,t2 ,t3 Rt1 ,t2 ,t3
Besides, considering the autocovariance operator
C0X
< ∞.
and its spectral decomposition presented
HS
as equation (4.2), we assume that
Assumption 5.2: For all j, λ j > 0.
n
√
√
−1
−1 o
Assumption 5.3: Let a1 = 2 2 (λ1 − λ2 )−1 and a j = 2 2 max λ j−1 − λ j
, λ j − λ j +1
2
2
= o (1) , and ∑ j≥kn φ νj
=
for j ≥ 2. Assume that n−1/4 ∑kj=n 1 a j /λ j = o (1) , λkn n−1/2 ∑kj=n 1 λ−
j
√ o 1/ n
It can be shown that under Assumptions 5.1-5.3, φ̂, defined in Section 4.1, is a consistent estiε̂ , with
mator for φ. Let ε̂ t = Xt − φ̂Xt−S , t = S, S + 1, · · · , n, and X0 = 0. Define V̂hε̂ = Ĉhε̂ + Ĉ−
h
the corresponding kernel
1 n V̂hε̂ s, s0 =
ε̂ t (s) ε̂ t−h s0 + ε̂ t s0 ε̂ t−h (s)
∑
n t = h +1
(5.2)
As can be seen in Equation (3.10), if the model fits the data adequately, we may expect zero
autocovariance in non-seasonal lags. Consequently, the test statistic defined by Zhang (2016)
is redefined as
[n/S] Z 1 Z 1
n
Qε̂ =
8π 2
∑
k =1
Let ψh (λ) = sin (hφλ) /hπ and C0X
−1/2
0
0
ε̂
V̂kS
s, s0
2
ds ds0 .
(5.3)
−1/2
νj ⊗ νj . The following theorem deter= ∑∞
j =1 λ j
mine the asymptotic behavior of Qε̂ .
Theorem 5.1. Consider the standard SFAR(1)S model with ε t being a sequence of i.i.d random function
in L2H (Ω, F , P) . If Assumptions 5.1-5.3 are fulfilled and
C0X
uniformly for all k = 1, · · · ,
−1/2
n
S
√
−1/2
φkS−1 (C0ε )2 φ∗ kS−1 C0X
< ∞,
(5.4)
N
, we have
n
2
[n/S]
∑
ε̂
V̂kS
s, s0 ψkS (λ) ⇒ g λ, s, s0 ,
(5.5)
k =1
and
Qε̂ →
Z 1Z 1Z 1
0
0
0
g λ, s, s0 ds ds0 dλ
15
(5.6)
where ⇒ denotes the weak convergence and g (λ, s, s0 ) stands for a zero mean Gaussian random process
such that
1 ∞
Cov g λ, s, s0 , g λ0 , s1 , s1 0 =
ψkS (λ) ψk0 S λ0
∑
4 k,k0 =1
× Cov ξ 0,kS s, s0 + ξ 0,kS s0 , s , ξ 0,k0 S s, s0 + ξ 0,k0 S s0 , s
(5.7)
and ξ i,j (s, s0 ) = ε i (s) ε i− j (s0 ) − ε i (s) C0ε φ∗ j−1 C0X
6
−1
( Xi − 1 ) ( s 0 ) .
Prediction
In this section, we present the h-step functional best linear predictor (FBLP) X̂n+h of Xn+h based
on X1 , . . . , Xn . For this purpose, we will follow the method of Bosq (2014). In the last section,
this prediction method will be compared with the Hyndman-Ullah method and multivariate
predictors.
Let us define Xn = ( X1 , X2 , . . . , Xn )0 . It can be demonstrated that the L−closed subspace
generated by Xn , called G, is the closure of {`0 Xn ; `0 ∈ L ( H n , H )}, (see Bosq, 2000, Theorem
1.8 for the proof). The h-step FBLP of Xn+h is defined as the projection of Xn+h on G; i.e.,
X̂n+h = PG Xn+h . The following proposition, which is a modified version of (see Bosq, 2014,
Proposition 2.2), presents the necessary and sufficient condition for the existence of FBLP in
terms of bounded linear operators and determines its form.
Proposition 6.1. For h ∈ N the following statements are equivalent:
(i) There exists `0 ∈ L ( H n , H ) such that CXn ,Xn+h = `0 CXn .
(ii) PG Xn+h = `0 Xn for some `0 ∈ L ( H n , H ).
Based on this proposition, to determine the FBLP of Xn+h , it is required to find `0 ∈ L ( H n , H )
such that CXn ,Xn+h = `0 CXn . Let h = aS + c > 0, a ≥ 0 and 0 < c ≤ S. Then , for any
x = ( x1 , . . . , xn ) ∈ H n , we have:
CXn ,Xn+h ( x) = E (h Xn , xi H n Xn+h )
= E h Xn , x i H n
φ
a +1
a
Xn − S + c +
∑ φ ε n−(k−a)S+c
k =0
= E h X n , x i H n φ a +1 X n − S + c
1
= E h Xn , xi H n Φna+
X
n
−S+c
1
= Φna+
−S+c E (h Xn , x i H n Xn )
1
= Φna+
− S + c CX n ( x ) ,
16
k
!!
where Φij is in L ( H n , H ) and is defined as an n × 1 vector of zeros with φi in the jth position.
1
a +1 X
Consequently, X̂n+h = PG Xn+h = Φna+
n−S+c .
−S+c Xn = φ
Note 6.1. Based on The Kargin-Onatski estimation method, the 1-step ahead predictor of Xn+1 is:
kn
X̂n+1 =
∑ < Xn−S+1 , ẑα,i > ĈS (ẑα,i ),
(6.1)
i =1
where
q
ẑα,i =
∑ τ̂j−1/2
υα,i , ν̂j ν̂j + αυα,i ,
(6.2)
j =1
The method depends on a selection of q and k n . We selected q by the cumulative variance method and
set k n = q.
7
Simulation results
Let { Xt } follows a SFAR(1)S model, i.e.,
Xt (τ ) = φXt−S (τ ) + ε t (τ ) , t = 1, . . . , n,
(7.1)
where φ is an integral operator with parabolic kernel
k φ (τ, u) = γ0 2 − (2τ − 1)2 − (2u − 1)2 .
The value of γ0 is chosen such that kφkS =
qR R
1 1
0
0
2
k φ (τ, u) dsdu = 0.1, 0.5, 0.9. Moreover,
the white noise terms ε t (τ ) are considered as independent standard Brownian motion process
on [0, 1] with variance 0.05. Then B = 1000 trajectories are simulated from the Model (7.1) and
the model parameter is estimated using the method of moments (MME), Unconditional Least
Square (ULSE) and Kargin-Onatski (KOE) estimators. As a measure to check the adequacy of
the fit of the model, the root mean-squared error (RMSE), defined as
v
u B
u1
RMSE = t ∑ kφ̂i − φk2S ,
B i =1
(7.2)
is applied. The MSE results are reported in Table 1. By comparing the results in this table, it can
be seen that MME outperform other methods when kφkS = 0.5, 0.9, but ULSE and KOE have
better MSE when kφkS = 0.1. Generally and as expected, the MSE of the methods improves as
the sample size (n) get larger. To better understand these methods, a trajectory path of length
n = 200 from the model (7.1) is generated by considering kφkS = 0.9 and k n = 1. Then
three estimation methods are compared using the generated path and the results are shown
in Figure 1. Moreover, the figures related to MME are shown in Figure 2 for k n = 1, 2, 3. As
17
can be seen in these figures, the method of moments and unconditional least squares have a
better performance than the Kargin-Onatski method. Besides, increasing the number of k n in
the estimation methods, decreases the accuracy of estimation.
kφkS = 0.1
kφkS = 0.5
kφkS = 0.9
n
kn
MME
ULSE
KOE
MME
ULSE
KOE
MME
ULSE
KOE
50
1
2
3
4
0.1750
0.5484
1.0239
1.5573
0.1645
0.5189
0.9657
1.4934
0.0951
0.0959
0.0961
0.0962
0.2403
0.4931
0.9988
1.5340
0.2838
0.7000
1.1478
1.6513
0.3716
0.3720
0.3721
0.3721
0.1986
0.4387
1.0435
1.6382
0.2323
1.0381
1.7282
2.4725
0.5096
0.5099
0.5099
0.5099
100
1
2
3
4
0.1222
0.3662
0.6830
1.0645
0.1183
0.3598
0.6798
1.0243
0.0861
0.0866
0.0868
0.0868
0.2050
0.3325
0.6661
1.0245
0.2579
0.6087
0.8723
1.1973
0.3539
0.3541
0.3541
0.3542
0.1387
0.2743
0.6694
1.0193
0.1709
0.9728
1.3925
1.9377
0.4134
0.4136
0.4136
0.4136
150
1
2
3
4
0.1033
0.2903
0.5533
0.8560
0.1027
0.2900
0.5449
0.8237
0.0825
0.0830
0.0831
0.0831
0.1946
0.2666
0.5387
0.8256
0.2505
0.5704
0.7601
1.0040
0.3460
0.3462
0.3462
0.3462
0.1205
0.2149
0.5272
0.8106
0.1517
0.9478
1.2493
1.6683
0.3735
0.3737
0.3736
0.3736
200
1
2
3
4
0.0917
0.2490
0.4790
0.7438
0.0935
0.2610
0.4745
0.7127
0.0798
0.0803
0.0804
0.0804
0.1879
0.2285
0.4684
0.7199
0.2457
0.5568
0.7047
0.9042
0.3393
0.3394
0.3394
0.3394
0.1114
0.1818
0.4542
0.7040
0.1419
0.9411
1.1896
1.5134
0.3496
0.3497
0.3497
0.3497
Table 1: The NMSE of the parameter estimators.
Figure 1: Comparison of all methods of estimations when n = 200, kφkS = 0.9, and k n = 1.
18
Figure 2: Method of moment estimation of the model parameter kernel by considering n =
200, kφkS = 0.9, and k n = 1, 2, 3.
60
20
20
0
40
60
Sunday
0
40
60
40
Saturday
20
20
0
Friday
0
20
40
60
Thursday
0
40
60
Wednesday
20
40
60
Tuesday
0
0
20
40
60
Monday
Figure 3: Square root of functional pedestrian hourly counts at Flagstaff Street Station, by day
of the week.
8
Application to pedestrian traffic
In this section, we study pedestrian behavior in Melbourne, Australia. These data are counts
of pedestrians captured at hourly intervals by 43 sensors scattered around the city. The dataset
can shed light on people’s daily rhythms, and assist the city administration and local businesses with event planning and operational management. Our dataset consists of measurements at Flagstaff Street Station from 00:00 1 Jan 2016 to 23:00 31 Dec 2016.1 In Figure 3, we
show the pedestrian behavior by day of the week, which are converted to functional objects
using 11 Fourier basis functions.
Although Figure 3 shows the existence of a weekly pattern in the data, the different pattern
in weekends can put stationarity of data in doubt. Therefore, we do not consider weekends
in data analysis and SFAR(1)7 model is applied to the rest of the data. The number of bases
is chosen using a generalized cross-validation criteria. Figure 4 shows the estimation of the
autocorrelation kernel using different estimation methods, as described in Section 4.
A rolling forecast origin for one day ahead throughout the data set was computed. The Root
Mean Squared Error (RMSE) and Mean Absolute Error (MAE) of the forecasts are tabulated in
Table 2, comparing the results when using MME and ULSE estimation methods. The table also
1 The
data are available using the rwalkr package (Wang, 2019) in R.
19
MME(Kn=1)
ULSE(Kn=1)
3.5
3.0
2.5
2.0
z
z
1.5
1.0
y
0.5
y
x
x
0.0
Figure 4: The estimated kernel of the autocorrelation operator using MME and ULSE methods.
include the seasonal naive (SN) forecast for comparison. As can be seen in this table, the best
predictions come with small values of k n (similar to the results in the last section) and using
MME. Using this method of prediction, the one-day-ahead forecasts are shown in Figure ??,
along with the functional pedestrian data.
MAE
RMSE
kn
MME
ULSE
MME
ULSE
1
2
3
4
5
6
3.04
135.19
139.46
155.52
165.34
180.86
3.025
24.55
27.65
30.07
32.60
33.20
6.08
138.09
142.09
157.62
168.06
186.92
6.073
24.70
27.88
30.41
33.23
34.01
SN
3.67
6.46
Table 2: The MAE and RMSE of the 1-step predictor when the autocorrelation operator is
estimated by MME and ULSE methods.
9
Conclusion
In this paper, we have focused on seasonal functional autoregressive time series. We have
presented conditions for the existence of a unique stationary and causal solution. Furthermore, we have derived some basic properties and limiting behavior. In FSAR(1)S , we proposed three estimation methods, namely methods of moment, unconditional least square and
Kargin-Onatski. Furthermore, we have presented a test for diagnostic checking and, for arbitrary h ∈ N, we have investigated the h-step predictor. The performance of the estimation
methods are compared using a simulation study, which demonstrates that the MME has the
20
best performance among mentioned estimation methods. Finally, a real data set is analyzed
using the SFAR(1)S model.
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