Imputation of Missing Values for BL (P,0,P,P) Models with Normally Distributed Innovations
Science Journal of Applied Mathematics and Statistics
Volume 3, Issue 6, December 2015, Pages: 234-242
Received: Oct. 4, 2015; Accepted: Oct. 21, 2015; Published: Oct. 30, 2015
Views 3659      Downloads 114
Author
Poti Abaja Owili, Mathematics and Computer Science Department, Laikipia University, Nyahururu, Kenya
Article Tools
Follow on us
Abstract
This study derived estimates of missing values for bilinear time series models BL (p, 0, p, p) with normally distributed innovations by minimizing the h-steps-ahead dispersion error. For comparison purposes, missing value estimates based on artificial neural network (ANN) and exponential smoothing (EXP) techniques were also obtained. Simulated data was used in the study. 100 samples of size 500 each were generated for bilinear time series models BL (1, 0, 1, 1) using the R-statistical software. In each sample, artificial missing observations were created at data positions 48, 293 and 496 and estimated using these methods. The performance criteria used to ascertain the efficiency of these estimates were the mean absolute deviation (MAD) and mean squared error (MSE). The study found that optimal linear estimates were the most efficient estimates for estimating missing values for BL (p, 0, p, p). The study recommends OLE estimates for estimating missing values for bilinear time series data with normally distributed innovations.
Keywords
Optimal Linear Interpolation, Simulation, MSE, Innovations, ANN, Exponential Smoothing
To cite this article
Poti Abaja Owili, Imputation of Missing Values for BL (P,0,P,P) Models with Normally Distributed Innovations, Science Journal of Applied Mathematics and Statistics. Vol. 3, No. 6, 2015, pp. 234-242. doi: 10.11648/j.sjams.20150306.12
Copyright
Copyright © 2015 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Abdalla, M.; Marwalla, T.(2005). The use of Genetic Algorithms and neural networks to approximate missing data. Computing and Informatics vol. 24, 5571-589.
[2]
Abraham, B. (1981). Missing observations in time series. Comm. Statist. A-Theory Methods.
[3]
Abraham, B.; Thavaneswaeran, A. (1991). A Nonlinear Time Series and Estimation of missing observations. Ann. Inst. Statist. Math. Vol. 43, 493-504.
[4]
Bishop, C. M.(1995). Neural Networks for pattern recognition. Oxford: Oxford University Press.
[5]
Cao, Y; Poh K, L and Wen Juan Cui, W, J.(2008). A non-parametric regression approach for missing value imputation in microarray. Intelligent Information Systems. pages 25–34.
[6]
Cheng and D. M. Titterington D M(1994): Neural networks: review from a statistical perspective.
[7]
Cheng, P.(1994). Nonparametric estimation of mean of functionals with data missing at random. Journal of the American statistical association, 89, 81-87.
[8]
Cortiñas J, A.; Sotto, C; Molenberghs, G; Vromman, G.(2011). A comparison of various software tools for dealing with missing data via imputation. Bart Bierinckx pages 1653-1675.
[9]
De Gooijer, J.C.(1989) Testing Nonlinearities in World Stock Market Prices, Economics Letters v31, 31-35
[10]
Granger, C. W; Anderson, A. P.(1978). An Introduction to Bilinear Time Series model. Vandenhoeck and Ruprecht: Guttingen.
[11]
Hannan, E J. (1982). "A Note on Bilinear Time Series Models", Stochastic Processes and their Applications, vol. 12, p. 221-24.
[12]
Hellem, B T (2004). Lsimpute accurate estimation of missing values in micro Array data with least squares method. Nucleic Acids, 32, e34.
[13]
Hirano, K; Imbens, G, W.; Ridder (2002). Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71, 1161 - 1189.
[14]
Howitt, P. (1988), "Business Cycles with Costly Search and Recruiting", Quarterly Journal of Economics, vol.103 (1), p. 147-65.
[15]
Kim, J K and Fuller. W. (2004). Fractional hot deck imputations. Biometrika 91, 559-578.
[16]
Ledolter, J. (2008). Time Series Analysis Smoothing Time Series with Local Polynomial Regression on Time series. Communications in Statistics—Theory and Methods, 37: 959–971.
[17]
Liu J. and Brockwell P. J. 1988. “On the general bilinear time series model.” Journal of Applied probability, 25, 553–564.
[18]
Liu, J. (1989). A simple condition for the existence of some stationary bilinear time series.
[19]
Ljung, G. M. (1989). A note on the Estimation of missing Values in Time Series. Communications in statistics simulation 18(2), 459-465.
[20]
Luceno, A.(1997). Estimation of Missing Values in Possibly Partially Nonstationary Vector Time Series. Biometrika Vol. 84, No. 2 (Jun., 1997), pp. 495-499 .Oxford University Press.
[21]
Maravall, A. (1983), "An application of nonlinear time series forecasting", Journal of Businesa 6 Econamic Statistics, 1, 66-74.
[22]
Mcknight, E, P; McKnight, M, K; Sidani, S.; Figueredo, A.(2007). Missing data. Guiford New York.
[23]
Nassiuma, D. K.(1994). A Note on Interpolation of Stable Processes. Handbook of statistics, Vol. 5 Journal of Agriculture, Science and Technology Vol.3(1) 2001: 81-8
[24]
Nassiuma, D. K.(1994). Symmetric stable sequence with missing observations. J.T.S.A. volume 15, page 317.
[25]
Nassiuma, D.K and Thavaneswaran, A. (1992). Smoothed estimates for nonlinear time series models with irregular data. Communications in Statistics-Theory and Methods 21 (8), 2247–2259.
[26]
Norazian, M. N., Shukri, Y. A., Azam, R. N., & Al Bakri, A. M. M. (2008). Estimation of missing values in air observations. Lecture notes in Statistics Vol. 25. Sprínger verlag. New York.
[27]
Oba S, Sato MA, Takemasa I, et al. A Bayesian missing value estimation method for gene expressioprofile data. Bioinformatics 2003; 19(16):2088-2096.
[28]
Pascal, B. (2005): Influence of Missing Values on the Prediction of a Stationary Time Series. Journal of Time Series Analysis. Volume 26, Issue 4, pages 519–525.
[29]
Pena, D., & Tiao, G. C. (1991). A Note on Likelihood Estimation of Missing Values in perspective. Multivariate Behavioral Research, 33, 545−571.
[30]
Pourahmadi, M. (1989) Estimation and interpolation of missing values of a stationary time series. Journal of Time Series Analysis 10(2), 149–69.
[31]
Priestley, M.B. (1980). State dependent models: A general approach to time series analysis. profile data. Bioinformatics 2003; 19(16):2088-2096.
[32]
Ripley, B. (1996).Pattern recognition and neural networks. Cambridge: Cambridge University Press.
[33]
Smith, K.W and Aretxabaleta, A.L (2007). Expectation–maximization analysis of spatial time series. Nonlinear Process Geophys 14(1):73–77.
[34]
Subba Rao, T. and Gabr, M.M. (1980). A test for non-linearity of stationary time series. Time Series Analysis, 1,145-158.
[35]
Subba, R.T.; Gabr, M.M.(1984). An Introduction to Bispectral Analysis and Bilinear Time Series Models. Lecture notes in statistics, 24.New York. Springer.
[36]
Thavaneswaran, A.; Abraham (1987). Recursive estimation of Nonlinear Time series models. Institute of statistical Mimeo series No 1835.Time Series. The American statistician, 45(3), 212-213.
[37]
Tong, H. (1983).Threshold Models in Non-Linear Time Series analysis. Springer Verlag, Berlin.
[38]
Troyanskaya, O, Cantor, M, Sherlock G, Brown, P, Hastie, T, Tibshirani R, Botstein D and Russ B. Altman1 (2001). Missing value estimation methods for DNA microarrays BIOINFORMATICS Vol. 17 no. 6 2001Pages 520–525.
[39]
Sesay, S.A and Subba Rao, T (1988): Yule Walker type difference equations for higher order moments and cumulants for bilinear time series models. J. Time Ser. Anal.9, 385-401.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186