Artificial Neural Network Model Based Estimation of Finite Population Total

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1 International Journal of Science and Researc (IJSR), India Online ISSN: Artificial Neural Network Model Based Estimation of Finite Population Total Robert Kasisi 1, Romanus O. Odiambo 2, Antony G. Waititu 3 1, 2, 3 Jomo Kenyatta University, Scool of Matematical Sciences, Department of Statistics and Actuarial Sciences P.O Box 62000, Nairobi-Kenya Abstract: In a finite population denoted U1, 2, 3,..., N of N identifiable units, let X be te survey variable. Te survey variables are observations from a super population. It is possible to get total information about tese survey variables suc as teir total population, mean or teir variance. In most cases auxiliary information about X is provided. A simple approac to using tis auxiliary information is to assume a working model tat describes te connection between te study variable of interest and te auxiliary variables. Estimators are ten derived on te basis of tis model. Te best estimators are te ones tat ave good efficiency if te model is true, and are consistent if te model is inappropriate. In tis study, we derive a nonparametric artificial neural network estimator of finite population total. Te estimator is design unbiased, design consistent and asymptotic normal. Keywords: super population, auxiliary information, artificial neural networks, sample, survey sampling. 1. Introduction A finite population is a list or a frame denoted U1, 2, 3,..., N of N identifiable units. In tis case N is usually known. E.g. we can ave a finite population of size N1000.Te need to get certain information about te finite population wic cannot be ceaply obtained by involving every individual calls for a sample (a selection of te population) to be taken. Finite populations are of interest to government for policy making. Total information about a population can be obtained from census data were every individual is involved in giving information. A simple way to incorporate known population totals of auxiliary variables is troug ratio and regression estimation. More general situations are andled by means of generalized regression estimation (Sarndal, 1980) and calibration estimation (Deville and Sarndal, 1992). Estimation procedures ave been employed in getting information from te census data, administrative registers and oter surveys. However, in most cases tese are callenging due to cost, time, literacy levels and oter geograpical factors. In tese metods, part of te population referred to as te sample is used and te information about te population is inferred into te sample. For estimating te finite population total we suggest an alternative estimation procedure using artificial neural networks. 2. A Neural Network Model Based Estimator Te goal is to estimate te population mean of te survey variable so tat we can get te population total, tat is 1 From Deville and S arndal (1992) using te notion of a calibration estimators, we can define our artificial neural network estimator to be a linear combination of te observations Wit weigts cosen to minimize an average distance measure from te basic design weigts 1 Minimization is constrained to satisfy, were is te known vector of population means for te auxiliary variables. Altoug alternative distance measures are available in Deville and S arndal (1992), all resulting estimators are asymptotically equivalent to te one obtained from minimizing te ci-squared distance ( ) Were te are known positive weigts unrelated to, i.e. Y +( ) Were Y and are te Horvitz-Tompson estimators of and, respectively, and (d ) Consider te following superpopulation model ( ) ( ) 1,2,, ( ) ( ) 1,2, 0 i j Were E and denote expectation and variance, respectively, wit respect to ; ( ) takes te form of a feed forward neural network wit skip-layer connections and (. ) is a known function of. Hence, ( ) (1) Were M is te number of neurons at te idden layer (Ripley, 1996, Capter 5). Since we consider M as fixed, we can denote by te set of all parameters of te network, and write() in (1) as (, ),,,,,,,,,,, Following te approac of Wu and Sitter (2001) to estimate,te first step is to obtain a design-based metod for estimating te model parameters and terefore obtain estimates of te regression function at, for i1,...,n, troug te resulting fitted values. In oter words, we first seek for an estimate of te model parameters based on Paper ID:

2 International Journal of Science and Researc (IJSR), India Online ISSN: te data from te entire finite population. We ten obtain a design-based estimate of based on te sampled data only. Te population parameter is defined by weigted least squares wit a weigt decay penalty term, i.e. Ø ( ( ) (, )) + (2) were is a tuning parameter and p is te dimension of te parameters vector. Te estimate is defined as te solution of te design-based sample version of (2), tat is Ø ( ( ) (, ))2+ (3) Once te estimates are obtained, te available auxiliary information is included in te estimator troug te fitted values (, ), 1,2,.Ten, we can define te neural network estimator as Y were te calibrated weigts are sougt to minimize te distance measure subject to 1 and f (, ) Using te tecnique of Deville and S arndal (1992) to derive te optimal weigts, We can propose tat Y Y + f f (4) Were And We wis to combine te kernel tecnique to our neural network estimation. Terefore we briefly describe kernel smooting. A continuous kernel is denoted as k (.) and te bandwidt as. Te conditional regression estimator ()is te solution to a natural weigted least squares problem being te minimizer of (0) ( ) (5) (6) ( ) Were By differentiating equation (6) wit respect to and equating to zero we get ( ) ( ) 0 2 ( ) 0 ( ) 0 Similarly So tat i.e is an approximation of ( )wit a constant weigting value of Y corresponding to 's closest to more eavily. Alternatively, let [ ] be te n vector of 's obtained in te sample. Define te 1 matrix [1] and define te matrix 1 Ten a sample based estimator of ( ) is given by X W X X W X W as long as X W X is invertible. It follows tat X W X X W X W We note tat we can use te neural network package (nnet) metod to obtain te mean function of te fitted values. From te kernel tecnique, X W X X W X Te weigts W are subjected to te network and learnt. Ten te network adjusts te weigts until tey are optimal. Now te mean function of te fitted values will be. nnmodel$fitted.values*maximum( ). ( ) () For a target, 1,2,, we ave () Terefore X.X X. y In oter words. 3. Asymptotic P roperties of Ar tificial Neu ral Network Estimator 3.1 Unbiasness For a design expectation and model expectation ᶓ ten, lim ( ) Next, lim ( ) lim + lim ( ) lim + E (I ) Paper ID:

3 International Journal of Science and Researc (IJSR), India Online ISSN: But, ( ) Hence 3.2 Consistency lim + lim By Cebcev tecnique [ X >] Y Ten [ Y >] Y We know Y is unbiased, terefore as a bias 0. From, MSE(Y ) Var(Y )+(( )) Var(Y ) Next [ Y >] ( ) ( lim [ Y ] > ] lim ) but Var(Y ) lim 0 Due to convergence in probability ten, lim [ Y ] > 0 Hence Y is consistent. 3.3 Asymptotic Normality Ten ( ) ( ) + + (0,1) As implies (0,1) 4. Some Results Displayed ( ) ( ) Using R statistical package we simulate two populations of x as independent and identically distributed uniform (0,1) and gamma (1,1) random variables. Te populations are of size N300.Samples of size n30 are generated by simple random sampling. Te population size is considered large enoug for several samples and te sample size is 10 percent of population size. For eac population of x,mean function, and bandwidt,100 replicate samples are generated and te estimates calculated.te population is kept fixed during tese 100 replicates in order to be able to evaluate te design averaged performance of te estimators. We consider four mean functions: Te Horvitz Tompson estimator is a design based estimator wile te oters are nonparametric estimators wic are model based. Te Epanecnikov kernel () 3 4 (1 ), 1 is used for all four nonparametric estimators. Several bandwidts are considered (0.1, 0.25, 0.5, 0.75, 1 and 2) to elp see ow efficiency of te estimators vary wit bandwidt. Te second bandwidt is based on te ad oc rule of te data range. Te bandwidts 1 and 2 are large bandwidts relative to te data range,[0,1]. For te linear mean function, and te results sow equal performance evident from equal mean squared errors for bot uniform and gamma s. We terefore examine ow muc efficiency is lost if we used te oter estimators. Te oter mean functions represent departures from te linear model. For quadratic function performs better followed by (linear), except for a small portion for te range of x i.e for (0.1, 0.5, and 0.75 (linear) performs better under te gamma. Te biases at tese turning points for (linear) are seen to be less compared to tose of.for te exponential mean function under uniform, performs better followed by (linear).it is interesting to see te cycle and exponential mean functions yield similar MSE values under gamma. Te performance of any estimator, in,,, is evaluated using its relative bias and MSEs.Te relative bias is defined as were R is te replicate number of samples. We evaluate te actual design variance and estimated te mean squared error as +( ) We also consider an estimate of te mean square error Were is calculated from te simulated sample. 5. Table of Results Table 1: Comparative MSEs for te nonparametric estimators for a quadratic mean function under uniform uniform MSE of nn MSE of local MSE of Local linear MSE of t Linear Quadratic (2 + 5) 3. Exponential ( 8) (2) We report on some performance of several estimators. Paper ID:

4 International Journal of Science and Researc (IJSR), India Online ISSN: Table 2: Comparative biases for te nonparametric estimators for a quadratic mean function under uniform uniform Bias. nn Bias. local Bias. Local linear Bias.t Table 3: Comparative MSEs for te nonparametric estimators for a quadratic mean function under gamma gamma MSE of nn MSE of local MSE of local linear MSE of t Table 4: Comparative Absolute biases for te nonparametric estimators for a quadratic mean function under gamma gamma Bias nn Bias. Local Bias. Local Bias.t linear About Efficiency Considering te MSEs of te various estimators, we make several observations. Performs exceptionally well under linear and quadratic functions. also performs well since its itself linear, and ence is almost a true model for te linear function. For most of te oter mean functions, retained consistent efficiencies. Terefore from our results we are able to meet our objective tat te artificial neural network outperforms te kernel and local estimators. 7. Conclusions We ave derived an artificial neural network estimator for finite population total. Te properties of tis estimator outclass te existing nonparametric estimators suc te kernel and local estimators. We also note tat tis estimator remains invariant (i.e. gives same result) under different bandwidts. Te only closest competitor of tis estimator is te linear local estimator. However our estimator is more applicable since we do not ave to determine te degrees to use. We ave also found tat if te mean ()of a sample is known, ten we can use tis information to find te mean of te non-sampled elements wic leads to overall population estimation. Our objectives ave been acieved tat te artificial neural network estimator outperforms kernel estimators and also local estimators. We ave also successfully derived te asymptotic properties of te estimator. Tese are asymptotic unbiasness, design consistency and asymptotic normality. Te overall remark is tat an artificial neural network estimator is an improvement of existing nonparametric and parametric estimators. References [1] Breidt,F.J Kim,J.Y and Opsormer,J.D.(2005) Nonparametric regression of finite population totals under Two-stage sampling. Annals of Statistics.25, [2] Breidt,F.J and Opsormer,J.D.(2000).Local regression estimators in survey sampling. Annals of Statistics.28, [3] Cassel,et al (1976).Some results on generalized different estimation and generalized regression estimation for finite populations.biometrika 63, [4] Firt,D.and Bennet,K.E(2006).Robust models in probability sampling.journal of Royal Statistical Society.B,17, [5] K. Deb, S. Agrawal, A. Pratab, T. Meyarivan, A Fast Elitist Non-dominated Sorting Genetic Algoritms for Multiobjective Optimization: NSGA II, KanGAL report , Indian Institute of Tecnology, Kanpur, India, (tecnical report style) [6] Godambe,V.P and Tompson, M.E(1973).Estimation in sampling Teory wit excangeable prior s. Annals of Statistics.1, [7] B He,t Oki,F Sun,D Komon,S Kanae,Y Wang Estimating montly total nitrogen concentration in streams by using artificial neural networks, journal of environmental,2011 [8] Wu,C. and Sitter,R.R(2001).Efficient estimation of quadratic finite population functions in te presence of auxiliary information. Journal of American Statistical Association, 97, [9] William G.Cocran(1992),Sampling Tecniques, tird edition,44-49, [10] Godambe,V.P(1995)A Unified Teory of Sampling from finite populations.journal of Royal Statistical Society.B,17, [11] MA Mazurowski,PA Habas,JM Zurada JY L O Training neural network classifiers for medical decision making Neural networks,2008 [12] Isaki,C.T.and Fuller,W.A.(1982),Survey design under te regression super population model. Autor Profile Robert K asisi received a BSc in Matematics and Computer Science from Jomo Kenyatta University of Agriculture and Tecnology in te year 2011; He is currently pursuing a Master of Science in Applied Statistics at Jomo Kenyatta University of Agriculture and Tecnology. Romanus O. Odiambo is a professor of Statistics. He is currently working as a professor at Jomo Kenyatta University of Agriculture and Tecnology in te college of Pure and Applied Science, Department of Statistics and Actuarial Science. He olds PD (Statistics) degree conferred to im at Kenyatta University, a Master s of Science (Statistics) degree from Kenyatta University Paper ID:

5 International Journal of Science and Researc (IJSR), India Online ISSN: and Executive Master of Organizational Development (EMOD) from te United States International University (USIU). Antony G. Waititu olds a PD in Applied Statistics. He is currently a senior lecturer in te department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Tecnology. Paper ID: