Robust Estimation Based on Lognormal Kriging Technique for Some Soil Data

Spatial data obtained from applications of environment, agriculture, and earth sciences are defined as regional spatial variables that differ from ordinary variables in statistics, as each spatial variable represents that it has a real value for the phenomenon or observations (such as the degree of raw metal, air pollution, or the level of gases and groundwater). Each spatial variable has its location (two or three dimensions).

Suppose it represents the spatial variable T(v) at location (v) within region D. In Euclid space.

And number of regionalized spatial variables defied as T(v_i) at the location, i=1, 2, …, h.

The hypotheses of intrinsically stationary and isotropic for all v and v+h the region D.

$\begin{gathered}E[T(v)]=\mu \\ \operatorname{Var}[T(v+h)-T(v)]=2 \gamma(h) \\ \gamma(h)=\gamma(\|h\|) \text { Isotropy }\end{gathered}$

Let we have a finite number of points $\left\{T\left(v_i\right), v_i \in D\right\}$, for $i=1, \ldots$, n. and let $v_0 \in D$ and T(v₀) is spatial variable at location v₀.

2.1 The empirical variorum function

The analysis of the empirical variorum function is calculated from the real spatial data, The um function is defined by averaging one-half the difference squared of spatial variable values over all pairs of observations with the serrated distance and direction. Spatial variable expressed a function of spatial location T(v) where v is a location of real variable T the properties an intrinsically stationary random function T(v) with known the semi-variogram function: $\gamma(h)=\frac{1}{2} \operatorname{var}[T(v+h)-T(v)], \gamma(h)=\frac{1}{2} E\left[(T(v+h)-T(v))^2\right]$. where h is the log along for direction. Then we can write:

$\gamma(h)=\frac{1}{2 m(h)} \sum_{i=1}^{m(h)}\left[\left(T\left(v_i+h\right)-T\left(v_i\right)\right)\right]^2$                     (1)

where, γ(h) is the variogram function n lag(h) of the sample points and (v_i+h), (v_i), where i=1, 2, …, h, m(h) is the number of pairs.

The theoretical semi-variogram function has some basic properties. Let T(v) be variable be an interracially stationary, then the semi-variogram function denoted as γ(h) satisfies the following conditions:

·γ(0)=0

·γ(h)≥0

·γ(-h)=γ(h)

·$\lim _{h \rightarrow \infty} \frac{\gamma(h)}{|h|^2} \neq 0$

·Semi-variogram defined as negatives

i.e., for any finite sequence of points (v_i), i=1, 2, …, h and for any finite of real numbers (λ_i), i=1, 2, …, h such that $\sum_{i=1}^n \lambda_i=0$, then $\sum_{i=1}^n \sum_{j=1}^n \lambda_i \lambda_j \gamma\left(v_i-v_j\right) \leq 0$.

2.2 Kriging techniques

The kriging technique is used to estimate a value at a point of real spatial data of the study area. The variables satisfy the second-order stationarity. In ordinary kriging, we want to estimate a value of (v₀) using the data values from neighboring sample point (v₀). The predictor of ordinary kriging linearly with weights can be written as:

$\hat{T}_{o k}(v)=\sum_{i=1}^n \lambda_i * T\left(v_0\right)$                   (2)

where, λ_i, i=1, 2, …, h is the weights, the estimate variance is defined by $\sigma_E^2$:

$\sigma_E{ }^2=E(\hat{T}(v)-T(v))^2=-\gamma\left(v_i-v_j\right) \sum_{i=1}^n \sum_{j=1}^n \lambda_i \lambda_j \gamma\left(v_i-v_j\right)+2 \sum_{i=1}^n \lambda_i \gamma\left(v_i-v_0\right)$                 (3)

By minimizing the estimate variance with the condition on the weight, the ordinary kriging system:

$\left(\begin{array}{cccc}\gamma_{\left(v_1-v_1\right)} & \cdots & \gamma_{\left(v_1-v_n\right)} & \cdots 1 \\ & \cdot & & \cdot \\ & \cdot & & \cdot \\ \gamma_{\left(v_n-v_1\right)} & \cdots & \gamma_{\left(v_2-v_n\right)} & \cdots 1 \\ 1 & & \cdots & 0\end{array}\right)\left(\begin{array}{c}\lambda_1 \\ \cdot \\ \cdot \\ \lambda_n \\ \mu_{0 k}\end{array}\right)=\left(\begin{array}{c}\gamma_{\left(v_1-v_0\right)} \\ \cdot \\ \cdot \\ \gamma_{\left(v_n-v_0\right)} \\ 1\end{array}\right)$                     (4)

where, μ_0k is the Lagrange parameter and λ_i are the weights the ordinary kriging system can be defined in the form:

$\left\{\begin{array}{c}\sum_{\alpha=1}^n \lambda_i \gamma_{\left(v_i-v_\alpha\right)}+\mu_{0 k}=\gamma_{\left(v_1-v_0\right)}, i=1,2, \ldots n \\ \sum_{\alpha=1}^n \lambda_\alpha=1\end{array}\right\}$                      (5)

The estimated variance is defined as:

$\sigma_{0 k}^2=\mu_{0 k}-\gamma_{\left(v_0-v_0\right)}+\sum_{\alpha=1}^n \lambda_\alpha \gamma_{\left(v_\alpha-v_0\right)}$                       (6)

where, v₀ is the location of the real data, then $\hat{T}\left(v_0\right)=T\left(v_\alpha\right)$  if v₀=v_α [15].

2.3 Simple kriging with an estimated mean

Let T(v) a second-order stationery with o(h) known covariance function the mean of kriging using the linear combination:

$\left\{\sum_{i=1}^n \lambda_i{ }^{k m} T\left(v_i\right), \quad\right.$ with $\left.\sum_{i=1}^n \lambda_i^{k m}=1\right\}$                      (7)

where, $\lambda_i^{k m}$ are the weight of kriging of the mean and T(v_i) the data at location (v_i) then:

$\hat{T}_{s k m}\left(v_0\right)=\sum_{i=1}^n \lambda_i^{k m} T\left(v_i\right)+\sum_{i=1}^n \lambda_i^{s k} T\left(v_i\right)-\sum_{i=1}^n \lambda_i^{s k} \sum_{j=1}^n \lambda_j^{k m} T\left(v_j\right)=\sum_{i=1}^n\left\lceil\lambda_i^{s k}+\lambda_i^{k m}\left(1-\sum_{j=1}^n \lambda_j^{s k}\right)\right] T\left(v_i\right)$                       (8)

Then the weight means:

$\begin{gathered}\bar{w}=1-\sum_{i=1}^n \lambda_i^{s k} \\ \hat{T}_{s k m}\left(v_0\right)=\sum_{i=1}^n\left[\lambda_i{ }^{s k}+\bar{w} \lambda_i{ }^{k m}\right] T\left(v_i\right)=\sum_{i=1}^n \lambda_i{ }^{\prime} T\left(v_i\right)\end{gathered}$                  (9)

where, $\sum_{i=1}^n \lambda_i{ }^{\prime}=1$ from an ordinary known system by the weight $\lambda_i{ }^{\prime}$:

$\sum_{j=1}^n \lambda_j{ }^{\prime} C\left(v_i-v_j\right)=\sum_{j=1}^n \lambda_i{ }^{s k} C\left(v_i-v_j\right)+\bar{w} \sum_{j=1}^n \lambda_i{ }^{k m} C\left(v_i-v_j\right)=C\left(v_i-v_j\right)+\bar{w} \mu_{m k}$

We denote μ' as a product of the mean weight with the $\lambda_i{ }^{\prime}$, Then the ordinary known system:

$\left\{\begin{array}{c}\sum_{j=1}^n \lambda_j{ }^{\prime} C\left(v_i-v_j\right)=C\left(v_i-v_0\right)+\mu^{\prime} \,\, \text { for } i=1,2, \ldots n \\ \sum_{j=1}^n \lambda_j{ }^{\prime}=1\end{array}\right\}$                     (10)

Then the estimated mean is $\hat{T}_{s k m}\left(v_0\right)=\hat{T}_{o k}\left(v_0\right)$ and the variance of the ordinary known can be written as:

$\sigma_{0 k}{ }^2=\sigma_{s k}{ }^2-\bar{w} \sigma_{k m}{ }^2$                  (11)

When the mean weight is small the sum of simple weights is close to one, also the variance of kriging near is small. Predictor of ordinarily known we denote $\hat{T}_{s k m}\left(v_0\right)$ to ordinary Predictor of the value at (v₀) is the linear combination of T(v) at each sample v_i, i=1, 2, …, n.

$\hat{T}_\lambda\left(v_0\right)=\sum_{i=1}^n \lambda_i T\left(v_i\right)=\lambda^T T$

where, $\lambda=\left(\lambda_1, \ldots \lambda_n\right)^T \in R^n$.

The unbiasedness combination of linear Predictor $\hat{T}_\lambda$.

$\sum_{i=1}^n \lambda_i=1 \leftrightarrow \lambda^T{ }_1=1$

And the express over definer as:

$E\left(\hat{T}_\lambda\left(v_0\right)-T\left(v_0\right)\right)=E\left(\sum_{i=1}^n \lambda_i T\left(v_i\right)-T\left(v_0\right) \sum_{T=1}^n \lambda_i\right)=\sum_{i=1}^n \lambda_i E\left(T\left(v_i\right)-T\left(v_0\right)\right)=0$

Spatiality the value o $E\left(\hat{T}_\lambda\left(v_0\right)-T\left(v_0\right)\right)$ equal to zero, where, T(v) to be intrinsically stationary Cressie according to Cressie [14]. (Since any second-order stationary random process is automatically intrinsically stationary, i.e. the set of all second-order stationary random functions is a subset of the set of all intrinsically stationary functions) [16].

Then variance

$\begin{gathered}\sigma_E{ }^2=\operatorname{Var}\left(\hat{T}_\lambda\left(v_0\right)-T\left(v_0\right)\right)=E\left(\hat{T}_\lambda\left(v_0\right)-T\left(v_0\right)\right)^2=-\lambda^T \Gamma \lambda+2 \lambda^T \gamma_0 \\ \sigma_E^2=-\lambda^T\left(2 \gamma_0-\Gamma \lambda\right) \geq 0\end{gathered}$                     (12)

where, γ is the variance of T, Γ is the variance matrix and γ₀=γ(v₁- v₁), …γ(v_n-v₀)^T [15].

2.4 Variogram estimates

Experimental variogram function or so-called variogram estimate to compare between the following estimators.

2.4.1 Median classical variogram estimates

Matheson proposed the estimates based on the moments by defined:

$2 \hat{\gamma}(h)=\frac{1}{N(h)} \sum_{i=1}^n\left(v_i-v_j\right)^2$                (13)

is the best linear unbiased estimator of real observations?

2.4.2 Cressie-Hawkins robustness

Cressie proposed the formula:

$2 \hat{\gamma}(h)=\frac{1}{p_h}\left[\frac{1}{N(h)} \sum_{N=n}^n\left(T v_i-T v_j\right)^{\frac{1}{2}}\right]$                    (14)

where, $p_h=0.457+\frac{0.457}{N(h)}+\frac{0.457}{N(h)^2}$.

This estimation is robust given to others of non-normality.

2.4.3 Median variogram estimator

Defined the median variorum as:

$2 \hat{\gamma}(h)=\frac{1}{B_h}\left[\operatorname{med}\left(T v_i-T v_j\right)^{\frac{1}{2}},\left(v_i, v_j\right) \in N(h)\right]^4$                    (15)

where, med is median and B_h=0. To 457 is according to factor where the distribution is normal [17].

2.5 Kriging on transformed data

The transformation of data is used to normalize the data, to define the outliers, and to know the statesman of data and impose it. Also, transform data gives clearer of data and more stationary variorum by using the log others become the skewers of observations lognormal order of log transform of data i.e.

$y\left(v_i\right)=\log \left(T\left(v_i\right)\right)$                      (16)

where, T(v_i) is the original data at the location v_i, and y(v_i) is data taking the logarithms.

The transformed data have the uniform dis tⁿ on [0, 1] is a practical side, and the transformed [18].

1. Simple agreement in according to order, T(r) is called the r^th order statistics.

$T^{(1)} \leq \cdots r \leq T^{(r)} \leq \cdots \leq T^{(n)}$                  (17)

2. Calculate the value y^(r) of the sample

$y^{(r)}=\frac{r}{n}$                  (18)

3. The value of y^(r) is between $\frac{1}{n}$ and 1.

4. Calculate kriging on the values

$T^*(v)=F\left(y^*(v)\right)$                   (19)

where, T(.) is the random variable and y(.) is the lognormal of T. The value for y^*(v) usually falls between two adjacent ranks, then T^*(v) will be between T(γ) and T(r+1), i.e., T is put off mid-point [19].

$2 \hat{\gamma}(h)=\frac{1}{2}\left[T^{(r)}+T^{(r+1)}\right]$                   (20)

$\begin{aligned} & \text { If } y^*(v)=\frac{r}{n} \\ & (v)=T^{(r)}\end{aligned}$                  (21)

Sometimes the value y^*(v) by kriging estimation. Falls outside the range of the minimum and the maximum acceptable limits, in case we reassigned $\left(\frac{b_1}{n}\right)$ to equal $\left(\frac{1}{n}\right)$ and N₁≠0 equal 1, then back than function and normal order kriging are similarity, and the denotes Gaussian G(y) the normal store transform steps as the following.

1. Similar to order statistics, the n simple data in a section order

$T^{(1)} \leq \cdots r \leq T^{(k)} \leq \cdots \leq T^{(n)}$               (22)

k is rank of data T^(k).

The simple frequency is computed as: $p^k=\frac{k}{n}$.

2. The normal transform of T^(k) is matched to p^k and

$y^k=G^{-1}\left[F_z{ }^{(K)}\right]=G^{-1}\left(p^k\right)$                   (23)

3. Kriging is performed on the transform data.

$T^*(v)=F^{-1}\left(G\left(y^*(v)\right)\right)$                    (24)

where, F(.) is the CDF of original data, let T(v₁), T(v₂) …, T(v_n) a sample of variable and $N(h)=\left(v_i, v_j\right):\left\|v_i-v_j\right\|=h$ [20].

Because of the small effect of extreme values on the estimates, it is clear from this effect that pollution may be large by properly knowing the weight restrictions for the level of pollution, and reducing the level of pollution depends practically and for a long period whenever some estimates lose their stability or constancy. The characteristics of robustness methods are studied using a prediction study and the results indicate several factors that affect the efficiency of the analyzed methods. In particular, the methods depend on: whether the data set is multivariate normal or not; the Dimension of the data set; Type of outliers. The proportion of outliers in the data set; and the degree of contamination of outliers (dimension). The study prompted the authors to recommend the use of “a combination of multivariate methods” on the data set to detect potential outliers. We fully adopt this recommendation and consider that the combination of methods should depend, but also on other factors such as the dimension and size of the data structure. Masking of one outlier by another outlier occurs if the second outlier can be considered an outlier in itself only, but not in the presence of the first outlier. Thus, because of this masking, a set of outlying observations skews the mean and covariance estimates toward it. the results showed the outliers Matheron median of variogram predict as corresponding to the theoretical model. For data without outliers, Matheson's and Haslett's robust estimators had better performance than Cressie Hawkins's robust soil data comparison with the empirical variogram. Soil data showed the spatial dependence of the variation at different scales.