Fast same-step forecast in SUTSE model and its theoretical properties Wataru Yoshida and Kei Hirose

2025-05-06 0 0 1.59MB 29 页 10玖币

侵权投诉

Fast same-step forecast in SUTSE model and its

theoretical properties

Wataru Yoshida and Kei Hirose

Kyushu University

Abstract

We consider the problem of forecasting multivariate time series by a Seemingly Unrelated

Time Series Equations (SUTSE) model. The SUTSE model usually assumes that error variables

are correlated. A crucial issue is that the model estimation requires heavy computational loads

because of a large matrix computation, especially for high-dimensional data. To alleviate the

computational issue, we propose a two-stage procedure for forecasting. First, we perform the

Kalman ﬁlter as if error variables are uncorrelated; that is, univariate time-series analyses are

conducted separately to avoid a large matrix computation. Next, the forecast value is computed

by using a distribution of forecast error. The proposed algorithm is much faster than the

ordinary SUTSE model because we do not require a large matrix computation. Some theoretical

properties of our proposed estimator are presented. Monte Carlo simulation is performed to

investigate the eﬀectiveness of our proposed method. The usefulness of our proposed procedure

is illustrated through a bus congestion data application.

Keywords: Kalman ﬁlter, state-space model, SUTSE model, limiting Kalman ﬁlter

1 Introduction

Multivariate time series data analysis has been recently developed in various ﬁelds to achieve high-

quality forecasts and investigate the correlations among time series: for example, forecast of energy

consumption and crowd-ﬂow [Gong et al. 2020]. In particular, a state-space model is applied widely

for forecasting time series, and a number of model estimation procedures have been proposed. One

of the most famous methods is the Kalman ﬁlter [Kalman 1960b] and its extensions: for example,

the Adaptive Kalman ﬁlter [Mohamed and Schwarz 1999] and the Robust Kalman ﬁlter [Koch and

Yang 1998]. In recent times, the deep Kalman ﬁltering network which fuses deep neural networks

and the Kalman ﬁlter, has been proposed as well [Lu et al. 2018]. The Kalman ﬁlter and its

arXiv:2210.09578v2 [math.ST] 17 Jan 2023

extensions have been used in various ﬁelds of research, such as tracking [Jondhale and Deshpande

2018], photovoltaic forecasting [Pelland et al. 2013], and traﬃc volume forecasting [Xie et al. 2007].

In this study, we employ a Seemingly Unrelated Time Series Equations (SUTSE) model [Fern´andez

and Harvey 1990, Antoniou and Yannis 2013], a special case of the state-space model. In the SUTSE

model, multiple univariate time series equations are combined to express a single multivariate linear

Gaussian state-space model. The SUTSE model usually assumes that components of a noise vector

are correlated. In this case, the components of an observation vector are also correlated. As an

example of an analysis using such a correlation structure, Moauro and Savio [2005] employed tem-

poral disaggregation; that is, a low-frequency time series is transformed into a high-frequency series

by interpolation. They performed the interpolation by making use of the correlation structure of a

multivariate time series with diﬀerent frequencies.

The SUTSE model can be applied not only to interpolation but also to forecasting. Suppose

we have nobservations of d-dimensional time series data, {{y1, . . . , yn}:yt= (y1,t, . . . , yd,t)T, t =

1, . . . , n}, and some components of yn+1, say yA,n+1. Here, A ⊂ {1, . . . , d}and yA,n+1 indicates a

subvector of yn+1 whose indices consist of A. We may consider two types of forecasts: the one-

step-ahead forecast and the same-step forecast. The one-step-ahead forecast calculates the forecast

value yn+1 by using the y1, ..., yn. In the same-step forecast, we forecast the value of yk,n+1 by

using {y1, . . . , yn}and yA,n+1, where k /∈ A.

Figure 1.1 : Example: bus congestion same-step forecast

Figure 1.1 shows one example of the application of the same-step forecast. In this example, yi,t

indicates the congestion of ith bus on tth day. We forecast the congestion using past congestion

data; in other words, A={1, . . . , j}when we forecast yk,n+1, where ksatisﬁes k > j. The same-

step forecast can be used not only for a bus congestion forecast but also for a wide variety of

practical applications, including electricity demand and price forecasting.

The same-step forecast is performed by modifying the one-step forecast based on the covariance

matrix estimation of the one-step-ahead forecast error. This estimation is conducted with the

Kalman Filter. However, the Kalman ﬁlter requires heavy computational loads because of a large

matrix computation, especially for high-dimensional data. The computational complexity is about

O(max{d3, p3}) [Willner et al. 1976], where pdenotes the dimension of the state vector. In fact, we

attempted to analyze 32-dimensional bus congestion data with the SUTSE model. Despite using

the package FKF [Luethi et al. 2022], which can perform the Kalman ﬁlter very fast because of the

implementation in C [Tusell 2011], it took approximately 11 hours to complete the analysis. The

details of this experiment are given in Section 6.2.

One way to handle this computational issue would be to apply a method that accelerates the

Kalman Filter algorithm. For a general linear Gaussian state-space model, a method to accelerate

the Kalman ﬁlter is proposed by Koopman and Durbin [2000]. This method transforms the obser-

vation vectors into a univariate time series and then applies the Kalman ﬁlter to this univariate

time series. A transformed univariate time series results in larger sample sizes than the original

observation vectors, and its state vectors have the same dimension as the original vectors. Thus,

this method reduces the cost of matrix calculations related to the original observation vectors.

However, the matrix calculations related to the state vectors of transformed univariate time series

must be done more times than the ordinary Kalman ﬁlter. Empirically, this method works well

when p≤d. Meanwhile, in the SUTSE model, it is mostly p>d. Therefore, this method may not

speed up the Kalman ﬁlter adequately.

As seen above, the general methods for accelerating the Kalman ﬁlter in state-space models

may not always speed up the SUTSE model. Thus, in this study, we propose a simple and faster

method specialized for the same-step forecast in the SUTSE model. Speciﬁcally, the following

two-stage procedure is proposed. First, a model estimation with the Kalman ﬁlter is performed

separately for each dimension of observation vectors, as if components of the observation vector

are uncorrelated, and the one-step-ahead forecast value is computed. With this procedure, the

cost of matrix calculations involving both the observation and state vectors in the Kalman ﬁlter is

signiﬁcantly reduced. In addition, parallel computing can be applied. Next, the mean vector and

the covariance matrix of the one-step-ahead forecast error are estimated using the results obtained

from the Kalman ﬁlter in the ﬁrst step. The same-step forecast value is computed by using the one-

step-ahead forecast error distribution. Note that the mean vector and covariance matrix of the one-

step-ahead forecast error are time-varying, and thus they are viewed as time-varying parameters.

We show that these time-varying parameters converge under some assumptions. In particular,

the mean vector converges to 0; consequently, we estimate the mean vector as 0. The sample

covariance matrix is shown to be the consistent estimator of the limiting value; hence, we use the

sample covariance matrix as an estimator of the covariance matrix of the one-step-ahead forecast

error. Another possible covariance estimation method is applying the graphical lasso [Friedman

et al. 2008] to the sample covariance matrix. When the sample size is not large enough compared

with d, this estimation would be more suitable. Regardless of which covariance estimation method

is chosen, the second step does not take long. A Monte Carlo simulation shows that the proposed

algorithm slightly sacriﬁces the forecast accuracy, while the computational time is greatly improved,

resulting in a practical method of the same-step forecast for high-dimensional time series data. We

also apply our proposed method to the bus congestion data. The result shows that the proposed

method took about 8 seconds to analyze, while the existing method took approximately 11 hours.

The rest of this paper is structured as follows. In Section 2, the forecast methods using the

SUTSE model are presented. We also review the Kalman ﬁlter in this section. In Section 3, we

propose the fast same-step forecasting method and provide its theoretical properties. In Section

4, we prove the convergence of the mean vector and the covariance matrix of the one-step-ahead

forecast error. Using this convergence, in Section 5, we prove the consistency of the estimator

in the second step of the fast method. In Section 6, we conduct a Monte Carlo simulation and

bus congestion forecasting to investigate the computational time and the forecast accuracy of our

proposed algorithm. Section 7 presents the conclusion, and the Appendix contains proofs of lemmas

and other supplemental work.

2 Forecasts using the SUTSE model

2.1 SUTSE model

Let y1, y2, . . . , ynbe d×1 observation vectors with yt:= (y1,t, . . . , yd,t)T. Assume that yj,t follow

the linear-Gaussian state-space model:

(yj,t =Z(j)

tα(j)

t+εj,t

α(j)

t+1 =T(j)

tα(j)

t+η(j)

(t= 1,2, . . . , n, j = 1,2, . . . , d),(2.1)

where α(j)

tis a p(j)×1 state vector, εj,t is an observation noise, η(j)

tis a p(j)×1 state noise vector, Z(j)

is a 1×p(j)design matrix, and T(j)

tis p(j)×p(j)a transition matrix. Let Zt:= Diag(Z(1)

t, . . . , Z(d)

t),

αt:= (α(1)T

t, . . . , α(d)T

t)T,Tt:= Diag(T(1)

t, . . . , T (d)

t), ηt:= (η(1)T

t, . . . , η(d)T

t)T, and p:= p(1) +

. . . p(d). Then model ( 2.1 ) can be rewritten as the form of the multivariate linear-Gaussian state-

space model:

(yt=Ztαt+εt

αt+1 =Ttαt+ηt

(t= 1,2, . . . , n),(2.2)

where αtis a p×1 state vector, εtis a d×1 observation noise vector, ηtis a p×1 state noise

vector, Ztis a d×pdesign matrix, and Ttis a p×ptransition matrix. We assume εt∼N(0,Σε),

ηt∼N(0,Ση), α1∼N(a1, P1), and εt,εk(t6=k), ηt, and ηkare mutually uncorrelated. This

model is called Seemingly Unrelated Time Series Equations model (SUTSE model) [Fern´andez and

Harvey 1990].

2.2 One-step-ahead forecast

We now consider the forecast of yn+1 when y1, y2, . . . , ynare given. The Kalman ﬁlter is well

known to be used for this forecast. Let at:= E(αt|Yt−1), Pt:= V(αt|Yt−1), vt:= yt−Ztat, and

Ft:= V(vt|Yt−1), where Ytdenotes {y1, y2, . . . , yt}. These conditional expectations and covariance

matrices can be computed with the Kalman ﬁlter as follows:











vt=yt−Ztat

Ft=ZtPtZT

t+ Σε

at+1 =Ttat+TtKtvt

Pt+1 =TtPtLT

tTT

t+ Ση

(t= 1,2, . . . , n),(2.3)

where Kt:= PtZT

tF−1

tand Lt:= Ip−KtZt. Note that a1and P1are the mean vector and

the covariance matrix of the initial state vector α1respectively. Then, we deﬁne “one-step-ahead

forecast” of ytas

¯yt:= Ztat,(2.4)

and call vt“one-step-ahead forecast error” in this paper. In particular, we can get the one-step-

ahead forecast of yn+1 as ¯yn+1 =Zn+1an+1. Note that by the deﬁnition of at, we have ¯yt=

2.3 Same-step forecast

We now consider the forecast of yk,n+1 when y1, y2, . . . , ynand also y1,n+1, . . . , yj,n+1 are given with

k > j. The one-step-ahead forecast of yk,n+1 can be modiﬁed by using the conditional expectation

of the one-step-ahead forecast error vk,n+1 as follows:

ˆyk,n+1 = [Zn+1an+1](k) + Evk,n+1|Yn, vn+1(1 : j),(2.5)

where [Zn+1an+1](k) denotes the kth component of Zn+1an+1 and c(i:j) denotes (ci, . . . , cj)T. We

call this forecast “same-step forecast”. To compute the modiﬁcation term (the second term) of the

right-hand side of ( 2.5 ), we use the formula for a conditional expectation of a multivariate normal

distribution (e.g., see [Eaton 1983]):

E(x1|x2) = E(x1) + Cov(x1, x2)V(x2)−1(x2−E(x2)),

where x1and x2follow multivariate normal distribution. (2.6)

Hence, the modiﬁcation term can be expressed as

Evk,n+1|Yn, vn+1(1 : j)

=E(vk,n+1|Yn) + Cov(vk,n+1, vn+1(1 : j)|Yn)V(vn+1(1 : j)|Yn)−1nvn+1(1 : j)−E(vn+1(1 : j)|Yn)o.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Fastsame-stepforecastinSUTSEmodelanditstheoreticalpropertiesWataruYoshidaandKeiHiroseKyushuUniversityAbstractWeconsidertheproblemofforecastingmultivariatetimeseriesbyaSeeminglyUnrelatedTimeSeriesEquations(SUTSE)model.TheSUTSEmodelusuallyassumesthaterrorvariablesarecorrelated.Acrucialissueisthatthemo...

展开>> 收起<<

Fast same-step forecast in SUTSE model and its theoretical properties Wataru Yoshida and Kei Hirose.pdf

共29页,预览5页

还剩页未读，继续阅读

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

Fast same-step forecast in SUTSE model and its theoretical properties Wataru Yoshida and Kei Hirose

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: