Sains Malaysiana 43(10)(2014): 1599–1607

Imputing Missing Values in Modelling the PM₁₀ Concentrations

(Mengganti Nilai Hilang dalam Pemodelan Kepekatan PM₁₀)

NURADHIATHY ABD RAZAK¹, YONG ZULINA ZUBAIRI²* & ROSSITA M. YUNUS³

¹Institute of Graduate Studies, University of Malaya, 50603 Kuala Lumpur, Malaysia

²Centre for Foundation Studies in Science, University of Malaya

50603 Kuala Lumpur, Malaysia

³Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

Diserahkan: 30 Julai 2013/Diterima: 13 Februari 2014

ABSTRACT

Missing values have always been a problem in analysis. Most exclude the missing values from the analyses which may lead to biased parameter estimates. Some imputations methods are considered in this paper in which simulation study is conducted to compare three methods of imputation namely mean substitution, hot deck and expectation maximization (EM) imputation. The EM imputation is found to be superior especially when the percentage of missing values is high as it constantly gives low RMSE as compared with other two methods. The EM imputation method is then applied to the PM₁₀ concentrations data set for the southwest and northeast monsoons in Petaling Jaya and Seberang Perai, Malaysia which has missing values. Four types of distributions, namely the Weibull, lognormal, gamma and Gumbel distribution are considered to describe the PM₁₀ concentrations. The Weibull distribution gives the best fit for the southwest monsoon data for Petaling Jaya. The lognormal distribution outperformed the others in describing the southwest monsoon in Seberang Perai. Meanwhile, for the northeast monsoon in both locations, gamma distribution is the best distribution to describe the data.

Keywords: Expectation maximization; mean imputation; missing value; PM₁₀; Weibull

ABSTRAK

Nilai hilang selalu menjadi masalah dalam analisis. Kebanyakan mengabaikan nilai hilang ini daripada analisis yang mungkin menyebabkan kepincangan dalam anggaran parameter. Beberapa kaedah gantian dipertimbangkan dalam kertas kerja ini dengan kaedah simulasi telah dijalankan untuk membandingkan kaedah-kaedah gantian tersebut iaitu penggantian menggunakan min, geladak panas dan jangkaan pemaksimuman (EM). Gantian EM didapati yang terbaik terutama apabila peratus nilai hilang adalah tinggi kerana ia berterusan memberi RMSE yang rendah berbanding dua kaedah yang lain. Kaedah gantian EM ini kemudiannya diaplikasikan pada set data kepekatan PM₁₀ bagi monsun barat daya dan timur laut di Petaling Jaya dan Seberang Perai, Malaysia yang mempunyai nilai hilang. Empat jenis taburan, iaitu taburan Weibull, lognormal, gama dan Gumbel dipertimbangkan untuk menggambarkan kepekatan-kepekatan PM₁₀. Taburan Weibull memberi kesesuaian terbaik untuk data monsun barat daya bagi Petaling Jaya. Taburan lognormal pula mengatasi yang lain dalam menggambarkan monsun barat daya di Seberang Perai. Manakala bagi monsun timur laut di kedua-dua kawasan, taburan gama adalah taburan yang terbaik yang menggambarkan data tersebut.

Kata kunci: Jangkaan pemaksimuman; min gantian; nilai hilang; PM₁₀; Weibull

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*Pengarang untuk surat-menyurat; email: yzulina@um.edu.my