APPLICATION TREND SURFACE MODELS WITH ESTIMATION

T his research deal with estimation of trend surface analysis and with spatial data with three models to real spatial data represents a rising ground water. The first method in assessment is to estimation trend surface model parameters by (ml), the second method requires decision on the maximum time difference to be calculated (s). while, the third method need a resolution, and the residuals r of dots to take for the conjecture the f(Dij) parameters. The first and second methods require resolution principle of "neighbor" from determines of "W". These three approaches are applied to real data which represent the ground water levels in 47 wells in mountain region in Sin jar district in Nineveh governorate.


Introduction
Trend -surface analysis is used for describing a certain style to represent real desktop data or highlights to represent changes that occur the spatial stochastic process.Base model is usually in the trend surface analysis in a way that ordinary least-squares and that assumes that the residuals are independent but if least-squares method is not efficient the standard errors and test morale be biased.The link between residuum can appear in the graphs, for example of groundwater location in a certain area .Statistical models are often used to represent views in terms of random variables and these models can be used to estimate the based on Probability theory.[1].attribute actual data trend -surface using of duet dimension polynomial equation An observation at a dot i to enable to writs Y i = Y(u i , v i ), where u and v represent a rectangular coordinate system for the area.Trend -surface analysis assume that Y i consists of two components, a trend component T i represented by the polynomial equation and a residual component (R i ) so that Y i = T i + R i T(u, v) = β 00 + β 10 u + β 01 v + β 20 u 2 + β 11 uv + β 02 v 2 + ⋯ +β pq μ p v q ……(1) Then term p + q represents the order of the trend surface,first order or linear p + q = 1 second order or quadratic p + q = 2, third order or cubic p + q = 3,and so on.thetechnique has been documented in a number of textbooks .
To a multidimensional research due to the finder [2], and circulated earth science by [3].The basic model is often installed by the normal minimum squares (ols) assuming that residues separate, assumption residues are spatially autocorelated, the estimates of the lower squares ineffective ,the standard errors and important test of the biased.Auto correlated can be mistakenly generated by error in determining the order of the form (the auto correlated of the first-order shape residue will be linked real shape quadrature) or measurement error or both.However, of the localized effects of spatial operations that process on a medium scale between the regional trend component and the sit or residual component so that an attempting to force this element into the trend component of the model may be inappropriate or to give rise unnecessary complicated model.The study examines alternative methods for mounting trend-model to accommodate spatial autocorrelation.

Regression analysis
Defines the concept of a general regression mathematical representation for the average relationship between one variable and other variables called independent variables.Specialized Regression analysis to describe the relationship between variables in the form of have from that contains one independent variable is then called (simple linear regression model) write the following formula [4,5] Y=β 0 + β 1 X + e Also, if the from contains several independent variables it is called (multiple linear regression model ).General linear model takes a written form and takes more than independent variables and the formula be:   =  0 +  1  1 +  2  2 + ⋯ +     +   j = 1,2,3 ….,  ;   =  0 + ∑     +    =1 ‫ﱠ‬  = 1,2,3 ….,  Using arrays can formulate the following general form . [6], [5].The vector of observation, denoted by Y and matrix notation the model is ( and That E(e) = 0 and E(ee ꞌ ) = Ω, That Ω matrix of nxn identical , [7].
In most practical applications  be unknown data shall be the standard case appreciation when independent data, to find a linear model parameters estimation indicated in equation ( 2) using( ols) follow: If there is a Spatial Autocorrelation between subjective random errors   because of random error in each site depends on nieghbouring sites and the error in this case we (    ) ≠ 0 ∀  ≠  From the simplest Autocorrelation is that appears in the first order autoregressive scheme .[8] a i = ρa i−1 + e i That  autocorrelation between   and  −1 and random error for this model takes the same assumptions used in some way (OLS) and e i~ N (0, σ e 2 ) The relationship is found by ( 4 That j=2,3,4,…….,n and i=1,2,3,…...,n-1Not that () and () represent the standard deviation of the data and (ℎ) represents the heterogeneity and which can be found from the relationship:

Estimating the usual method model parameters
Model (1), rate first of the case where the directional orderly arrangement of the trend surface is known .[9] proposed a regression pattern for the regression models with the remaining spatial autocorrelation by following these steps; [10] First Step.Appreciation β in (1) (call this β ̂ by (ols) assuming that the Residuals is in depended β ̂= (X ꞌ X) −1 X ꞌ Y Second Step.Gain the ( ols) residuals , e ̂= Y − Xβ ̂ Third Step.Use the residuals to achieve Ω ̂ (or Ω ̂−1 ,if this can be gained immediately.Fourth Step.Appreciation  by generalized (gls). ̃= (X ꞌ Ω −1 X) −1 X ꞌ Ω −1 Y (6) Fifth Step .get the (gls) residuals e ̂= Y − Xβ ̂ ,and return to third step.Iterate until convergence.The critical step is three, because the specification of Ω could be approached in several different methods.I contrast a maximum likelihood (ml) approach (method 1) with two other approaches, direct estimation of the residual autocovariances (way 2) and estimation of an autocovariance function for the residue (way3).Throughout, I shall write  for residuals, it being understood that at the first cycles the (ols) residuals ̂ are used, but in subsequent cycles (gls) residuals e ̂ are used.

Method 1 : estimation maximum likelihood
[11]studied the ML estimators for spatial regression model parameters where errors are automatically linked.trend-surface shape trend is a special case of regression modeling so that a natural start.point is to study this approach.
The auto covariance among Y values is stated by a parametric model function, a position two values (usually only dependent space between sites) and a set of unknown parameters (θ).The autocovariances matrix must be positive, clear, and twice latent in relation to unknown parameters.Assuming a Gaussian process (ml) of β ,θ are achieved by maximizelikelihood Where || denotes determinat of .the second model is the mulataneous autoregressive scheme can be written as: e = ρWe + a, where a~N(0, σ 2 I) and W is an  ×  connectivity or describing linkage matrix the proximity or neighbor relationship among the sites i j is an neighbor of To improve the previous models suggested following methods.

Method 2: lag order (autocovariance estimates).
residuals Autocovriance the makes direct estimated .Let C(s) indicate residuals Autocovriance at lag s. state [10] estimated several approaches of autocovariances.Comparative selection coefficient adopted  2 as well as mean square error MSE .
Method 3: autocovariance function discussed [12] the Auto covariance function are parameters are estimated of a continuous in a different context He assumed that the structure of dependence among the residuals .the function can be described byu i = F(D ij )u i + e i , F(D ij ) refer to function of the distance among j ,i .It is more a summed , E[e i ] = 0 u i u j = F(D ij )u j 2 + e i e j providing u i and e i are independent, that Where (ρ ij ) is the autocorrelation separated among two sites by a distance   .Although this seems like a approach a reasonable, e and u processes cannot be independent , and in general the expected value will be an underestimate of the true autocorrelation function.
Suggested approximating the distance function by the expression, and discuss an (ols) procedure for estimated the parameters A,B, and C .f A=1, then Procedure so that F 1 (D ij ) can be interpreted as an autocorrelation function.depends the way to define constant radius (r) around each point within which pairs of location captivated for the interpret of A, B, and C. past interim autocorrelation is equal 0. r > If D ij F ̂(D ij ) w ̂ij = 0 otherwise.= w ̂ij The way, the previous way unlike, allows the values in Ω to be adgusted for distance betwwen location, it may be suitable for the case of irregular site distributions.For this reason.For this reason.I representing evaluating.When you make up w ̂ij in equation σ 2 [(I − ρw ̂)(I − ρw ̂)] −1 = Ω * * estimate (gls) get by Comparative selection coefficient adopted  2 as well as mean square error

Conclusion
The present study has been considered as a repeat pattern to estimating the parameters of the trend surface models with residuals autocorrelation.The procedures discussed here would seem appropriate in in those situations in which three levels of variation are suspects in the data, but they may also be useful in avoiding order misspecification trend-surface analysis.The importance of fine tests does not seem to be available, but sufficient information can be taken from the variance of the common contrast matrix to design approximate tests.Of conclusion supply, the (ml) way to be the best, Method 3 works well, and special suitable irregular Method 2 requires the evaluation of multiple autocorrelation requests and shows the progress of the method 1 in the current context, however, perhaps to a lot of autocorrelation in the data sites, from all this we can conclude that it is better on the third method way depending on the comparison criteria.

1 Y
Then find the estimated values trend surface model Y ̃ That, Y ̃= X ̃ Comparative selection coefficient adopted  2 as well as maen square error MSE MSE