Monotone Regression and Correction for Order Relation Deviations in Indicator Kriging

INTRODUCTION

It is well-known that the indicator kriging (IK) proposed by Journel (1983) is one of the most efficient nonparametric methods in geo-statistics. It has been applied successfully in many fields. It has become the basis of some estimate algorithms and sequential indicator simulations (Journel, 1989). The important function of IK is to estimate the conditional cumulative distribution function (ccdf) at a series of increasing cutoffs z_k

where (n) represents the conditional information available in the neighborhood of location u; z_k is the cutoff; E is mathematical expectation, and * is the estimation. From these estimates the local uncertainty of a regionalized variable Z(u) can be assessed.

As the estimates of ccdf, the)] F ^*[u; z_k | (n)] must satisfy the following order relations

However, because these estimates are obtained independently, the above-mentioned order relations may not be ensured. To obtain a legitimate distribution, the order relation deviations have to be corrected by resetting some estimated values F ^*[u; z_k | (n)] to the new ones.

Deutsch and Journel (1998) systematically researched the problem about the order relation deviations. They associated the correction with quadratic programming or quadratic optimization to obtain a new distribution, which minimizes departure from the original quantile values, although it ensures all order relations. However, their practical correction algorithm considers the average of an upward and downward correction and cannot ensure the optimization of the corrected results.

In this article we suggest using monotone regression to realize the correction for order relation deviation. It is essentially a method of quadratic optimization, and can ensure that the result is optimal.

MONOTONE REGRESSION

The model of monotone regression was proposed by Kruskal(1964a, b), for the method of multidimensional scaling. Its theory and computation procedure are summarized in this section, taking the notation used by Zhou and Xia (1993).

Let f₁, f₂, ..., f_K be real numbers K. The aim is to find K numbers , which ensure the order relation

and minimize the departure

Here the method to estimate the K numbers is introduced.

It is easy to find a monotone array related to the original one. Let the array {f₁, f₂, …, f_K} be divided into m blocks and presume the ith one is {f_i1, ..., f_iαi}. The average of the ith block is

A suitable division (to be given later) can yield these averages, satisfying the order relation

The worst situation is m=1, therefore, only one block and one average are present. With each number of original array replaced by the average in its block, a monotone array is obtained

To obtain the solution of monotone regression, the following procedure can be implemented successively

(i) Let temporarily.

(ii) Suppose is already present, then forego and go to step (iv).

(iii) If , then is corrected as follows: among the subscripts satisfying 1≤i≤k－ 2 and

The minimum one is found. If it exists, it is reset as

else it follows that

and reset

Therefore, this can follow

(iv) If k < K then replace k by k + 1 and go to (ii), else go to (v).

(v) Finally, a monotone array is obtained

which minimizes (4).

Kruskal(1964a, b) proved that the solution minimizes the departure D in equation (4).

In the earlier discussion, let

Then the solution of the monotone regression , denoted by (k = 1, 2, …, K) as a new estimate of ccdf, satisfies relation (2).

If there are some of regression values outside the interval [0, 1], then reset them to the nearest bounds, 0 or 1, the constraint (1) will be met easily.

CASE STUDY AND DISCUSSION

To show the algorithm procedure of this method and compare it with Deutsch and Journel's method, two numerical examples have been introduced.

Figure IV.1 in Deutsch and Journel's book (1998) shows a concrete example computed by means of the average of an upward and downward correction. The figure is digitized and computed again by the method mentioned earlier, to demonstrate the application of the monotone regression method. The steps of the procedure and result of the two methods are listed in Table 1 and Fig. 1.

Table  1.  The algorithm steps and results of example 1

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Figure  1.  Correction for order relation deviation of example 1. The dots represent IK-returned ccdf values. The thick line 1 represents the corrected ccdf obtained by Deutsch and Journel's original method, and the thin line 2 by the new method in this article.

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Figure 1 shows the corrected results obtained by two methods. Comparing the two results, the following conclusions are drawn.

(1) Figure 1 shows that the thin line 2 is nearer to the original dots than the thick line 1. It means that the corrected results of the new method are better than those of former method.

(2) If departure (4) is used as a measure of difference between the corrected ccdf and the original quantile values, the new results (represented by line 2) are nearer to the original values than former results (represented by line 1), because, D₂= 0.092 8 < 0.106 8=D₁.

(3) The new corrected ccdf has more equal values

From these equal ccdf values, the estimated probabilities are obtained

The original data, algorithm steps of this method, and corrected results of two methods are listed in Table 2 and represented in Fig. 2. From the configuration of two lines expressing corrected results of two methods and D₂=0.032 4 < 0.040 1=D₁, it can be concluded that the results of the new method are better than those of the former methods.

Table  2.  The algorithm steps and results of example 2

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Figure  2.  Correction for order relation deviation of example 2. The dots represent IK-returned ccdf values. The thick line 1 represents the corrected ccdf obtained by Deutsch and Journel's original method, and the thin line 2 by the new method in this article.

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CONCLUSIONS

A new method of correcting the order relation deviation in indicator kriging is proposed by means of monotone regression. The corrected results can minimize departure from original quantile values, although they ensure all order relations. Two concrete examples show its simple algorithm procedure and efficiency.