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Sains
Malaysiana 53(6)(2024): 1463-1476
http://doi.org/10.17576/jsm-2024-5306-18
Spatial Functional Outlier Detection in Multivariate Spatial
Functional Data
(Pengesanan Outlier Fungsian Reruang dalam Data Fungsian
Reruang Multivariat)
NUR FATIHAH MOHD ALI1, ROSSITA MOHAMAD YUNUS1,*,
IBRAHIM MOHAMED1 & FARIDAH OTHMAN2
1Institute of
Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603 Kuala
Lumpur, Malaysia
2Department
of Civil Engineering, Faculty of Engineering, Universiti Malaya, 50603 Kuala
Lumpur, Malaysia
Received: 11
January 2024/Accepted: 23 May 2024
Abstract
Multivariate spatial functional data
consists of multiple functions of time-dependent attributes observed at each
spatial point. This study focuses on detecting spatial outliers in spatial
functional data. Firstly, we develop a new method called Mahalanobis Distance
Spatial Outlier (MDSO) to detect functional outliers in the data. The method
introduces the multivariate functional Mahalanobis semi-distance and
multivariate pairwise functional Mahalanobis semi-distance metrics based on the
multivariate functional principal components analysis to calculate the
dissimilarity between functions at each spatial point. Via simulation, we show
that MDSO performs better than the other competing methods. Secondly, MDSO has
been extended to detect spatial functional outliers as well. The functional
outliers can now be categorized as global or/and local functional outliers. The
appropriate number of neighbors and the cut-off point for the degree of
isolation are determined via simulation. Finally, we demonstrate the
application of the MDSO on a water quality data set obtained from Sungai Klang
basin in Malaysia. The results can be used to support the authority in making
better decisions on the management of the river basin or other spatial data
with time-independent attributes.
Keywords: Functional Mahalanobis
distance; multivariate functional data; spatial outlier; water quality
Abstrak
Data reruang multivariat berfungsi
adalah terdiri daripada pelbagai atribut berfungsi mengikut masa yang dicerap
bagi setiap titik reruang. Kajian ini mengutamakan pengesanan reruang terpencil
dalam data reruang berfungsi. Pertama, kajian ini membangunkan kaedah baharu
yang dikenali sebagai Jarak Mahalanobis Reruang Terpencil (JMRT) untuk mengesan
fungsi terpencil dalam data. Kaedah ini memperkenalkan penganggar separa
multivariat Mahalanobis berfungsi dan penganggar separa multivariat Mahalanobis
berfungsi berpasangan berdasarkan analisis komponen utama multivariat berfungsi
bagi menghitung perbezaan antara fungsi pada setiap titik reruang. Melalui
simulasi, kajian menunjukkan bahawa prestasi JMRT lebih baik berbanding
daripada kaedah lain. Kedua, kaedah JMRT dilanjutkan untuk mengesan reruang
terpencil berfungsi. Fungsi terpencil yang sedia ada boleh dikategorikan kepada
pencilan global dan/atau lokal berfungsi. Bilangan jiran dan titik potong bagi
darjah keberasingan yang sesuai ditentukan melalui simulasi. Akhirnya, kami
mengadaptasi aplikasi kaedah JMRT terhadap data kualiti air yang diambil dari
lembangan Sungai Klang di Malaysia. Hasil keputusan dapat membantu pihak
berwajib dalam membuat keputusan yang lebih baik untuk menguruskan lembangan
sungai dan menguruskan data reruang yang bergantung terhadap masa.
Kata kunci: Data multivariat berfungsi; kualiti air; penganggar Mahalanobis
berfungsi; ruang terpencil
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*Corresponding author; email:
rossita@um.edu.my
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