American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: http://www.sciepub.com/journal/ajams Editor-in-chief: Mohamed Seddeek
Open Access
Journal Browser
Go
American Journal of Applied Mathematics and Statistics. 2015, 3(5), 199-205
DOI: 10.12691/ajams-3-5-4
Open AccessArticle

Imputation of Missing Values for Pure Bilinear Time Series Models with Normally Distributed Innovations

Poti Owili Abaja1, , Dankit Nassiuma2 and Luke Orawo3

1Mathematics and Computer Science Department, Laikipia University, Nyahururu, Kenya

2Mathematics Department, Africa International University, Nairobi

3Mathematics Department, Egerton University, Private Bag, Egerton-Njoro, Nakuru, Kenya

Pub. Date: October 21, 2015

Cite this paper:
Poti Owili Abaja, Dankit Nassiuma and Luke Orawo. Imputation of Missing Values for Pure Bilinear Time Series Models with Normally Distributed Innovations. American Journal of Applied Mathematics and Statistics. 2015; 3(5):199-205. doi: 10.12691/ajams-3-5-4

Abstract

In this study, estimates of missing values for bilinear time series models with normally distributed innovations were derived 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 different pure bilinear time series models 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. Further, the optimal linear estimates were equivalent to one step-ahead forecast of the missing value. The study recommends OLE estimates for estimating missing values for pure bilinear time series data with normally distributed innovations.

Keywords:
optimal linear interpolation simulation MAD innovations ANN exponential smoothing

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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 UniversityPress.
 
[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).ThresholdModels 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.