Md. Rafiqul Islam, Associate Professor

Correspondence to:
Dept. of Population Science and Human Resource Development, Rajshahi University, Bangladesh.
E-mail: rafique_pops@yahoo.com

It is to be mentioned here that mathematical modeling in Population Studies especially in Demography (Fertility, Mortality, Migration) in Bangladesh have been worked very limited scale. In the era of globalization, mathematical models are very realistic and sophisticated mechanisms to express data in Mathematics. Mathematical models are of great useful to demographers in realizing the process in differentiating among various variables to find out the functional relationships and their dynamic behaviors among various demographic phenomena. Finally, model is important for prediction purposes Mathematical models in demography are mainly two groups: stochastic and deterministic.

Deterministic model has only been discussed in the present study. Deterministic models are used to describe the functional relationship between variables that take definite values. Traditionally, one can draw graphs of the demographic parameters but very few of us know in the context of Bangladesh which models are more appropriate for the parameters.

Islam and Ali (2004) found that age specific fertility rates (ASFRs) follows slightly modified biquadratic polynomial model where as forward and backward cumulative ASFRs follow quadratic and cubic polynomial model, respectively in the rural community of Bangladesh. To observe the distribution or pattern of marriage migration associated with distance in Bangladesh, India and other countries of the world a number of models have been fitted to the data set (Libbee and Sopher, 1975; Morril and Pitts, 1967; Perry, 1969a and 1969b; Samuel, 1994; Sharma, 1984; Yadava et. al. 1988). Hossain gave an atention to build up the model of Sharma (1984) and Yadava et. al (1988).

But these models did not supply good fit and then Hossain used the Pareto-Exponential model proposed by Morril and Pitts (1967) to present the marriage migration related to distance for his data of Bangladesh. Although Pareto-Exponential model supplied better approximation than the models of Sharma (1984) and Yadava et. al. (1988) but not significantly fit to the utilised data set. It is to be noted that proposed models of Sharma (1984) and Yadava et. al (1988) are suitable for Hinda community in India. For this, Yadava et. al. (2002) tried to show that exponential distribution provides a better fit to the distribution of marriage migration associated with distance than pareto-Exponential function as applied by Hossain (2000). Also Yadava et. al (2002) compared with pareto-exponential functin applied by Yadava et. al (1998).

In this study an effort has been given attention to build mathematical model to total marriage migration associated with distance, that is, the same data aggregate which was already used by Yadava et. al. (2002). For this purpose, a polynomial model is chosen to applied here. A brief discussion about polynomial model is given below:

A general expression of the form
(Waerden, 1948)
where a₀ is the constant term ;a_i is the coefficient of xⁱ (i =1, 2, 3, ..., n) but a₁, a₂,..., a_n are also constants but these belong to a field (field means a nonempty set in which group for addition, group for multiplication and left as well as right distributive law hold) and n is the positive integer,
is called a polynomial of degree n and the symbol x is called an indeterminate.

An effort has been made here to find out what types of models are more appropriate to total marriage migration by distance in Comilla of Bangladesh. Thus, the fundamental objectives of this study are briefly mentioned below:
i) to build up mathematical models to total marriage migration by distance and
ii) to apply cross-validity prediction power (CVPP), , to the model to verify how much the model is valid or not.

Sources of Data
The data on total marriage migration associated with distance of Comilla district in Bangladesh have been taken from Yadava et. al. (2002). This data was also available in Hossain (2000) and prohibited in Table 1.

Mathematical Model Fitting
Using the scattered plot of marriage migration associated with distance of Bangladesh (Fig. 1), it is observed that marriage migration can be fitted by polynomial model with respect to distance. Therefore, an nth degree polynomial model is considered and the form of the model is

(Gupta and Kapoor, 1997)
where, x is distance; y is marriage migration; is the constant; is the coefficient of (i =1, 2, 3, ..., n) and u is the stochastic error term of the model. Here a suitable n has been selected for which the error sum of square is minimum.

The software STATISTICA was used to fit the mathematical model.

Checking Model Validation
To check how much the model is stable, the cross validity prediction power
(CVPP), , is applied. Here, ; where, n is the number of cases, k = the number of regressors in the model and the cross-validated R is the correlation between observed and predicted values of the dependent variable (Stevens, 1996). The shrinkage of the model is the absolute value of the difference of and R². The stability of R² of this model is equal to 1- shrinkage.

F-test
To verify the measure of the overall significance of the fitted model as well as the significance of R2, the F-test is employed to this model. The formula for F-test in mathematics is as follows:

where k is the number of parameters to be estimated, n is the number of cases and R² is the coefficient of determination in the model (Gujarati, 1998).

The polynomial model is assumed for marriage migration due to distance in Comilla of Bangladesh and the fitted equation is
y = 1025.557-169.5126x+9.613215x²-0.182286x³
t-stats- (105.562) (-47.561) (27.80423) (-19.2117)
p-value- (0.000) (0.000) (0.00001) (0.0000)
providing R²=0.999714324 and =0.998875. This is the polynomial of degree three i.e. cubic polynomial.

From this statistics we see that the fitted model is highly cross-validated and its shrinkage is 0.000839. These imply that the fitted model is 99.8875% stable. Moreover, all the parameters of the fitted model are also highly statistically significant with 99.9714324% of variance explained.

Moreover, the stability of R2 of this model is also more than 99%.
In this study the calculated value of F-test is 4665.96, that is, large quantity which means that the fitted model is overall highly significant at 1% level of significance. Therefore, from these statistics we see that the fitted model and corresponding R² are highly statistically significance. As a result, the model is good fit. Thereafter, the prediction is dine and the predicted values of the model are also demonstrated in the last.

Fig. 1 Observed and Fitted Marriage Migration Associated with Distance of Comilla in Bangladesh

In this paper it is found that third degree polynomial model is fitted to the distribution of marriage migration associated with distance of Muslim community in Bangladesh. The results show that this model is also applicable or suitable even if Hossain fitted pareto exponential model and Yadava et. al. showed that exponeutial distribution provided better approximation than Hossain. Hence it is concluded that the pattern of marriage migration due to distance follow 3rd degree polynomial model.

Gujarati, Damodar N. (1998). Basic Econometric, Third Edition, McGraw Hill, Inc., New York.

Gupta, S. C. & Kapoor, V. K. (1997). Fundamentals of Mathematical Statistics, Ninth Extensively Revised Edition, Sultan Chand & Sons, Educational Publishers, New Delhi.

Hossain, M.Z. (2000). Some Demographic Models and their Applications with Special Reference to Bangladesh. An Unpublished Ph. D. Thesis in Statistics, Banaras Hindu University, Varanasi, India.

Islam, Md. Rafiqul & Ali, M. Korban. (2004). Mathematical Modeling of Age Specific Fertility Rates and Study the Reproductivity in the Rural Area of Bangladesh During 1980-1998, Pakistan Journal of Statistics, Pakistan Vol. 20(3), Page-381-394.

Libbee, M.J. and Sopher, D. E. (1975). Marriage and Migration in Rural India. in Kosinski, L.A. and Prothero, R.M. (eds.) People on the Move: Studies on International, Migration, London: Methuen and Company.

Morril, R. L. and Pitts, F.R. (1967). Marriage, Migration and the Mean Information Field. Annals, Association of American Geographers, 57, 401-22.

Perry, P. (1969a). Marriage Distance Relationsship in North Otago 1975-1914. New Zealand's Geographers, 25(1), 36-43.

Perry, P. (1969b). Working class Isolation and Mobility in Rural Dorset 1837-1936: A Study of Marriage Distance, Trans. Inst. Brit. Geographers, 47, 121-140.

Samuel, M.J. (1994). Patterns of Female Migration, In Maithili Vishwanathan (eds). Women and Society, IV, Printwell, Jaipur, India.

Sharma, L. (1984). A Study of the Pattern of Out-migration from Rural Areas. Unpublished Ph. D. Thesis in Statistics, Banaras Hindu University. India.

Stevens, J. (1996). Applied Multivariate Statistics for the Social Sciences, Third Edition, Lawrence Erlbaum Associates, Inc., Publishers, New Jersey.

Waerden, B.L.Van Der. (1948). Modern Algebra, Vol. 1, ICK Ungar Publishing Co. New York.

Yadavs, K.N.S., Srivastava, Soni and Islam, Sabina. (2002). Distribution of Distance Associated with Marriage Migration, International Journal of Statistical Sciences, Vol. 1, Dept. of Statistics, University of Rajshahi, Bangladesh, Page 49-54.

Yadavs, K.N.S., Hossain, M.Z and Islam, Sabina (1998). Distribution of Distance Associated with Marriage Migration in Rural Areas of Bangladesh. The Bangladesh Journal of Scientific Research, 16(2), 201-207.

Yadava. K.N. S. Raju, K.N.M. and Yadava, G.S. (1988). On the Distribution of Distance Associated with Marriage Migration in Rural Areas of Uttar Pradesh. India. Rural Demography, XV (1), 7-18.