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Citation: | Yan Han, Yiheng Yang. Monotone Regression and Correction for Order Relation Deviations in Indicator Kriging. Journal of Earth Science, 2008, 19(1): 93-96. |
The indicator kriging (IK) is one of the most efficient nonparametric methods in geo-statistics. The order relation problem in the conditional cumulative distribution values obtained by IK is the most severe drawback of it. The correction of order relation deviations is an essential and important part of IK approach. A monotone regression was proposed as a new correction method which could minimize the deviation from original quintiles value, although, ensuring all order relations.
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 zk
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where (n) represents the conditional information available in the neighborhood of location u; zk 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; zk | (n)] must satisfy the following order relations
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(1) |
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(2) |
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; zk | (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.
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, ..., fK be real numbers K. The aim is to find K numbers
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(3) |
and minimize the departure
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(4) |
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, …, fK} be divided into m blocks and presume the ith one is {fi1, ..., fiαi}. The average of the ith block is
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A suitable division (to be given later) can yield these averages, satisfying the order relation
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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
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To obtain the solution of monotone regression, the following procedure can be implemented successively
(i) Let
(ii) Suppose
(iii) If
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(5) |
The minimum one is found. If it exists, it is reset as
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else it follows that
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and reset
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Therefore, this can follow
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(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
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which minimizes (4).
Kruskal(1964a, b) proved that the solution minimizes the departure D in equation (4).
In the earlier discussion, let
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Then the solution of the monotone regression
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.
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.
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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
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(6) |
From these equal ccdf values, the estimated probabilities are obtained
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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.
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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.
ACKNOWLEDGMENTS: The authors would like to thank Prof. Q. Cheng for his encouragement, many good suggestions, and valuable comments on earlier drafts.Deutsch, C. V., Journel, A. G., 1998. GSLIB Geostatistical Software Library and User's Guide. Oxford University Press, New York. 369 |
Journel, A. G., 1983. Non-parametric Estimation of Spatial Distributions. Math. Geol. , 15(3): 445–468 |
Journel, A. G., 1989. Fundamentals of Geostatistics in Five Lessons. In: Crawford, M. L., Padovani, E., eds., Volume 8 Short Course in Geology. American Geophysical Union, Washington, D. C. . 40 |
Kruskal, J. B., 1964a. Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis. Psychometrika, 29(1): 1–28 |
Kruskal, J. B., 1964b. Nonmetric Multidimensional Scaling: A Numerical Method. Psychometrika, 29(2): 115–129 doi: 10.1007/BF02289694 |
Zhou, G., Xia, L., 1993. Analysis of Categorical Data and Its Application. Science Press, Beijing. 295 (in Chinese) |
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