CN101055631A - Space data fuzzy evidence weight analysis method - Google Patents

Abstract

The invention provides a fuzzy evidence weight analytical method and technology for mineral forecasting, mineral resource assessment as well as environmental assessment. The forecasting for the minerals resource is completed by constructing multivariate space multivariate spatial information database, forming the spatial information evidence picture layer of minerals forecasting through data processing and information extraction, obtaining the forecasting weight of evidence picture layer through calculation, drawing the probability picture of minerals resource prediction through comprehensive multilayer evidence picture layer. The invention has the advantages of broad practicality and less information loss, high forecasting precision, and is suitable for multisource geological data, for example geological data, mineral data, geochemistry data, geophysics data and remote sensing data or the like, to carry out the precasting process such as information extracting, data integration, mineral resource assessment, mineral detection, environmental contamination assessment, natural calamity, risk assessment, ect.

Description

A kind of space data fuzzy evidence weight analysis method

Technical field

The invention belongs to the Spatial Information Technology field, specifically, is a kind of space data fuzzy evidence weight analysis method for mineral resource prediction, environmental pollution evaluation and geo-hazard early-warning and technology. Handled multi-source Spatial Data can comprise all kinds of geologic datas (geologic map), geochemistry data, geophysical data, geodata (digital topographic map), data of mineral and remotely-sensed data. The method technology by spatial data process, spatial information obtains, the comprehensive realization of how former spatial information estimated prognosis of mineral resource potentials, environmental pollution evaluation, natural calamity risk.

Technical background

It is the important technical of contemporary appraisal and spatial decision support system that spatial data is processed with informix. The spatial decision support of Develop Data and information-driven has very wide application prospect under the GIS-Geographic Information System environment. Evidence weight technology (weights of evidence) is that the medical worker is according to a kind of mode of the various gained evidence diagnosis state of an illness as the artificial intelligence technology the earliest, in " geologist's GIS-Geographic Information System: GIS-Geographic Information System modeling " book that Oxford publishing house 1994 publishes, show, Bang Hakate (Bonham-Carter) the eighties in 20th century with this technology transfer and be extended to the geology field. The how former figure layer of this technology Main Basis is predicted an event space distribution and is estimated. At present this technology combines with GIS and is widely used in mineral resource prediction, environmental pollution evaluation, disaster alarm, the prevention of infectious disease, traffic accident evaluation etc. and forecasts sex work. The method not only can be processed quantitative data such as geochemical elements content, geophysical survey data etc., and can process qualitative data such as the point-lines such as rock type, structure-face type sign. A kind of method of space correlation relation of analysis different dimensional number space feature of uniqueness is provided.

In traditional evidence weight technology is used, unit and dimension meaning for unified figure layer, often continuous figure layer to be carried out discretization, such as by the horizontal range of distance structural line and be divided into area region in the certain distance with the best dependency relation in the space of research object and be two states, form the binary pattern layer. In this figure layer, a state has positive correlation with the research forecasting object, and another state then has the inverse correlation relation. The method and then can provide quantitatively the weight of tolerance correlation size. Multilayer figure layer is carried out the prediction probability figure that the logic stack then can form research object according to its weight size. This prognostic chart has reflected the probability of research object distribution spatially. Binary and tertiary mode are often expressed the existence of evidence attributes or are not existed, true or false, agree with or disapprove, determine or uncertain etc. For continuous attribute layer, common way is that continuous property value is divided into discrete binary or tertiary mode. For example, in the application of combining geographic information system (GIS, Geographical Information System), continuous property value, such as geochemistry or geophysical measured value, can determine optimum partition value by the space correlation relation of it and ore deposit point. In the evidence weight method, with two weights W+ and W～be assigned to each pattern in the evidence, W+ represents that evidence exists, and W～expression evidence lacks. They have been expressed evidence E (evidence) and have supposed estimating of H (hypothesis) relation, and become among the figure at mineral products potential energy, and they have expressed the probability that certain class ore deposit occurs. And in real work, if convert continuous property to discrete binary or tertiary mode, and want to guarantee that in transfer process it is difficulty relatively that information does not lack. Especially multi-layer data is merged, such as remotely-sensed data, geochemistry data and geophysical data etc. In addition, because the quality that sampled data is obtained and the lack of balance characteristic of quantity must reduce in forecasting process because the uncertainty that missing data produces.

On the basis of evidence weight technology, simultaneously there is above-mentioned weak point in order to overcome it, Cheng and A Gete Burger (Cheng and Agterberg) have been delivered " fuzzy evidence weight method and its application in the mineral products potential energy diagram " at " natural resources research " the 8th volume in 1999, fuzzy evidence weight technology (fuzzy weights of evidence) has further been proposed, this technology has been considered the multi-class of evidence rather than has been only limited to two classes or three classes value (latter comprises missing value (missing data)), utilize fuzzy probability and condition fuzzy probability to distribute fuzzy member function's value to middle classification, replace traditional binary or tertiary mode with the multiclass pattern. The fuzzy member function can determine also to utilize Model fitting to obtain by artificial. Fuzzy probability and conditional probability are based on the fuzzy member function and common product of probability assigns to obtain. Under the hypothesis of condition independence, last posterior probability figure obtains by the linear model match of evidence model. Like this, traditional evidence weight technology has just become special circumstances of fuzzy evidence weight, and the latter is the fuzzy evidence weight technology of more broad sense.

Summary of the invention

In order to overcome the deficiencies in the prior art, the object of the present invention is to provide a kind of space data fuzzy evidence weight analysis method. Like this, in real work, we can convert continuous property to discrete multi-mode and be not only binary or tertiary mode, guarantee that effectively information does not lack in transfer process. Such as, multi-layer data merges (remotely-sensed data, geochemistry data and geophysical data etc.).

In order to finish the foregoing invention purpose, the overall technological scheme that the present invention takes is:

A kind of space data fuzzy evidence weight analysis method, the method may further comprise the steps at least:

The mineral data of step 1, State selective measurements are as the evidence layer of MINERAL PREDICTION, and change continuous evidence layer into discrete classification evidence layer;

Step 2, discrete evidence layer is trained;

The fuzzy evidence weight of the evidence layer behind step 3, the calculation training;

The response variable figure of step 4, calculating estimation range;

Step 5, output MINERAL PREDICTION result.

Described step 2 also comprises training parameter is set, and calculates the step of prior probability.

The step of described calculating prior probability, adopt following formula:

Prior probability W₀=logO (D), wherein $O\left(D\right)=\frac{P\left(D\right)}{1-P\left(D\right)},$ W _μA(x) and W_μB(x) corresponding respectively is the weight of fuzzy set A and B;

${\mathrm{\&mu;}}_A\left(x\right)=\left\{\begin{array}{cc}1,& \mathrm{ifx}\&Element;{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="A20061007276300151.png,A20061007276300162.png"\; state="\{\{state\}\}">A</figure-callout>}}_1,\\ 0.5,& \mathrm{otherwise},\\ 0,& \mathrm{ifx}\&Element;{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="A20061007276300161.png"\; state="\{\{state\}\}">A</figure-callout>}}_2.\end{array}\right.$

A ₁，A ₂The expression subset, A₁＝{x；μ _A(x)＝1}，A ₂＝{x；μ _A(x)＝0}；A ₁∪A ₂X，A ₁∩A ₂＝0。

Described training parameter comprises unit are size, training zone and training sample size at least.

Described step 3 further may further comprise the steps:

Step 31, with the evidence layer chosen input computer system;

Step 32, setting and calculating member functional value;

The fuzzy evidence weight of step 33, calculating evidence layer;

Step 34, result of calculation is tested.

Described step 33 also comprises the step of the surface elemant number, training points surface elemant number, fuzzy evidence weight standard deviation, fiducial value or the test value that calculate classification value.

Estimation range response variable figure in the described step 4 comprises posterior probability figure and posterior probability standard deviation figure.

When calculating posterior probability, W_ACarry out following value:

$W_{\mathrm{\&mu;A}\left(x\right)}=\left\{\begin{array}{cc}{W}_A^{+},& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=1,\\ 0,& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=0.5\left(P\right[A_1]+P[A_2]=1),\\ W_A^{-},& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=0.\end{array}\right.$

Here, W_A ⁺(x) and W_A ^-(x) utilize common evidence weight method assignment;

$W_A^{+}=\log \frac{P\left[A_1\right|D]}{P\left[A_1\right|\stackrel{\&OverBar;}{D}]}$

$W_A^{-}=\log \frac{P\left[A_2\right|D]}{P\left[A_2\right|\stackrel{\&OverBar;}{D}]}.$

Advantage of the present invention and good effect are: the variance that the variance ratio of being derived out by the fuzzy evidence weight method derives out from unknown evidence or wrong evidence is little, and the posterior probability of deriving industry than common evidence weight method derive out accurately. Utilize this patent can make the user obtain more accurate and reliable result in MINERAL PREDICTION, mineral resource assessment and environmental evaluation, be the national resources exploitation, estimate to wait and bring immeasurable Economic Contribution.

Description of drawings

Fig. 1 is main flow chart of the present invention;

Fig. 2 is that evidence layer training parameter arranges surface chart;

Fig. 3 is evidence layer fuzzy evidence weight calculation interface figure;

Fig. 4 is the relevant information schematic diagram of evidence layer;

Fig. 5 is the fuzzy weight of evidence layer, standard deviation and corresponding T test value schematic diagram;

Fig. 6 is that Prediction Parameters arranges surface chart;

Fig. 7 is the final response variable figure surface chart of the calculating in the embodiment of the invention.

The specific embodiment

Below in conjunction with Figure of description the specific embodiment of the present invention is described.

The computational methods of the fuzzy weight that the present invention adopts are as follows: supposition evidence X connects together with the hypothesis of certain ore deposit D " appearance ". X={xi, x2 ..., xn}. In mineral resource assessment, xi can be rock category, singular value classification etc. In common discrete statistical method, make up binary and three metavariables with X. Its step is as follows: a dual mode is made of two subsets, A and A (not A), and A ∪ A=X; A ∩ A=0 is that A and A are mutual exclusions. The binary variable value that falls into set A is 1, and the binary variable value that does not fall into set A or fall into set A is 0. If single dual mode can not be satisfied the demand, can from X, produce more dual mode. Different from traditional evidence weight method is that fuzzy evidence weight replaces a plurality of dual modes with the fuzzy set of X:

Definition 1: suppose that A  X is a fuzzy set that is created. The degree of membership that the x that each element wherein belongs to X is subordinate to set A can define μ by member function_A(x)

1.0≤μ _A(x)≤1，

2.μ _A(x)＝1，if，and only if，xA，

3.μ _A(x)＝0，if，and only if，xA.

μ _A(x) it is just higher that value shows that the closer to 1 x belongs to the possibility of A. Fuzzy set A can be considered to the generalized form of common fragile collection. If A has two values 1 or 0, then A just becomes common fragile collection.

For convenience's sake, make A₁，A ₂The expression subset:

A ₁＝{x；μ _A(x)＝1}，A ₂＝{x；μ _A(x)＝0}

Wherein, A₁∪A ₂X，A ₁∩A ₂=0. Under special circumstances, μ_A(x) three values are arranged: 1,0.5 and 0, A becomes tertiary mode so:

${\mathrm{\&mu;}}_A\left(x\right)=\left\{\begin{array}{cc}1,& \mathrm{ifx}\&Element;A_1,\\ 0.5,& \mathrm{otherwise},\\ 0,& \mathrm{ifx}\&Element;A_2.\end{array}\right.---\left(1\right)$

The fuzzy probability of fuzzy set A and condition can be expressed as P[μ_AAnd P[μ (x)]_A(x) | D]. Similar, the joint probability of fuzzy set A and B is expressed as P[μ_A(x)μ _B(x)]。

In common evidence weight method, if being conditions, A and B be independent of D, the probability logarithm converting expressing of the D on fuzzy set A and B is as follows so:

$\log O\left[D\right|{\mathrm{\&mu;}}_A\left(x\right){\mathrm{\&mu;}}_B\left(x\right)]={\mathrm{<figure-callout\; id="0"\; label="W"\; filenames="A20061007276300161.png"\; state="\{\{state\}\}">W</figure-callout>}}_0+W_{{\mathrm{\&mu;}}_A\left(x\right)}+W_{{\mathrm{\&mu;}}_B\left(y\right)}---\left(2\right)$

Here, the equation left side represent the occurrence probability of certain mineral under the fuzzy member function's of evidence A and B condition to number conversion; W₀That corresponding is a priori probability W₀=logO (D), wherein $O\left(D\right)=\frac{P\left(D\right)}{1-P\left(D\right)},$ W _μA(x) and W_μB(x) corresponding respectively is the weight of fuzzy set A and B.

Posterior probability can be calculated by following formula. W wherein_AFollowing attribute is arranged

$W_{\mathrm{\&mu;A}\left(x\right)}=\left\{\begin{array}{cc}{W}_A^{+},& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=1\\ 0,& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=0.5\left(P\right[A_1]+P[A_2]=1),\\ W_A^{-},& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=0.\end{array}\right.---\left(3\right)$

Here, W_A ⁺(x) and W_A ^-(x) utilize common evidence weight method assignment.

$W_A^{+}=\log \frac{P\left[A_1\right|D]}{P\left[A_1\right|\stackrel{\&OverBar;}{D}]}$

$W_A^{-}=\log \frac{P\left[A_2\right|D]}{P\left[A_2\right|\stackrel{\&OverBar;}{D}]}---\left(4\right)$

Body, the variance that the variance ratio of being derived out by the fuzzy evidence weight method derives out from unknown evidence or wrong evidence is little, and the posterior probability of deriving industry than common evidence weight method derive out accurately.

See also Fig. 1 main flow chart of the present invention. At first carry out data and prepare, the mineral data that selection will be measured are as the evidence layer of MINERAL PREDICTION, and change continuous evidence layer into discrete classification evidence layer.

(1) selects the evidence layer.

In the MINERAL PREDICTION process, the present invention determines that at first which measurement data or knowledge can be used as the evidence layer (evidence) of MINERAL PREDICTION, supposes to have N evidence layer, selects the evidence layer mainly to be participated in and judgement by the expert.

(2) the evidence layer changes discrete classification evidence layer into continuously.

In the present invention, fuzzy evidence weight requires the evidence layer for discrete class label, if user's evidence layer is continuous measured value, such as the geochemical survey value, then need to change it into discrete class label, then, in computational process with these discrete variable substitution fuzzy evidence weight models.

Then, training parameter is set, calculates prior probability, and training evidence layer. Comprising the setting unit size, select training zone and training sample size. Can draw the rectangle basket by mouse drag and select training zone, select training sample with structured query sentence (SQL) etc. Based on above-mentioned parameter the calculating prior probability is set, then prior probability is converted to probability and carries out the logarithm conversion, be the W in the formula (2)₀。

Prior probability, the parameters such as the gross area that participate in the unit of calculating will be used when calculating fuzzy evidence weight, and its computational methods are:

Prior probability is the probability that the some event occurs under the prerequisite of known training points, the number of effective unit graticule mesh in the number/survey region of the shared effective unit of prior probability=sampled point graticule mesh.

Participate in the cellar area that the gross area=(the number * individual unit graticule mesh area of effective unit graticule mesh in the survey region)/user sets of computing unit.

Participate in the cellar area that total number=(the number * individual unit graticule mesh area of the shared effective unit of sampled point graticule mesh)/user sets of the point of calculating.

Design parameter arranges such as table 1 and shown in Figure 2.

Table 1 parameter arranges table

Parameter name	Designate	Parameter interpretation
			The survey region base map	StudyArea map	The user can make a survey region base map in this system or ArcGIS/ArcView the inside, and all carry out in the survey region scope in the calculating of this module.
The study area scope	Extent	The study area scope is a rectangular area, and it can be identical with the scope of survey region base map, also can be in the survey region base map scope or comprises a rectangular area of survey region base map, is combined with the survey region map and determines survey region with this.
			The training zone	TrainArea	The training zone can be identical with survey region, and the user also can do a rectangular area in system, and the part that this rectangle drops in the survey region is trained the zone exactly, training data and the calculating of fuzzy weight on evidence layer all in this zone, carry out.

Surface elemant	areaUnit	The size of surface elemant is set, the area of unit equal the unit length of side square. Surface elemant also represents the area of a training points representative, and it arranges will affect the size of prior probability and posterior probability values.
			The lower-left point coordinates	(Left，Bottom)	The lower-left point coordinates of training area.
Upper right point coordinates	(Right，Top)	The upper right point coordinates of training area.
			The grid image cell size	cellsz	The grid size of training area image, the user can arrange according to the resolution of survey region base map or evidence layer.
OK	rows	The grid line number of training area image.
			Row	cols	The grid columns of training area image.
Training point diagram layer	TrainingPointMap	Known training point diagram
			The training points data set	SampDataset	The point that is used to train set from training point diagram layer. The user can be with have a few in all training point diagram layers as training sampling point, also can be from train the point diagram layer chosen in advance or according to sql like language selection part training points, or choose at random certain point.
The effective unit number relevant with training points	ValidCellCount_fo   rPoint	The grid number that in the training zone, intersects with the training points surface elemant.
			Total effective unit number	ValidCellCount_fo   rStudy	All effective grid numbers after in the training zone, removing missing values.
Training points surface elemant number	TotalPoints	After the training zone is carried out dividing elements, contain the number of training points in the unit.

Afterwards, the fuzzy evidence weight of the evidence layer behind the calculation training.

When calculating the fuzzy evidence weight of evidence layer, advanced line parameter setting is called in system with N the evidence layer of choosing, and calculates the statistical indicator of each classification value of evidence layer again, arranges and calculate member's functional value, calculates fuzzy weight and standard deviation and T test value. The parameter setting sees Table 2

Parameter when table 2 calculates fuzzy evidence weight arranges table

Input parameter	Designate	Note
			The evidence layer	Evidential layer	Selection will be calculated the evidence layer of fuzzy weight

The statistical indicator of each classification value of evidence layer comprises: the surface elemant number of classification value and training points surface elemant number; The common evidence weight W+ and the W-that utilize formula (4) to calculate; Their standard deviation s (W+), s (W-)); And contrast C (C=W+-W-); The standard deviation of C (s (C)); The T test value.

As shown in Figure 3, the user can select any evidence weight layer, and (is noun inconsistent after asking case when calculating fuzzy evidence weight? PLSCONFM) calculates its fuzzy evidence weight. As shown in Figure 4, after selecting certain evidence weight layer, some about the relevant information of this evidence layer by prior information (prior probability whether? calculating PLSCONFM), comprising the gross area (Cumulative Area:C_area) of other area of each discrete class and accumulation participation computing unit, total number (Cumulative number of Points:C_points) of the point of the number of the point that participation is calculated in classification and accumulation participation calculating; The W+ and the W-that utilize formula (4) to calculate; Their standard deviation s (W+), s (W-)); And contrast C (C=W+-W-); The standard deviation of C (s (C)); The T test value.

When arranging and calculate member's functional value, the member function value can be according to statistics and experience indirect assignment, particularly when the training points data deficiencies. When the training points data are many, can be according to classification value and certain statistical indicator (C or T etc.) relation, draw relevant scatter diagram, determine that first member's functional value is part A1 and the A2 of μ A (x)=1 and μ A (x)=0, to member function value 0＜μ A (x)＜1, can be according to the relation of classification value and C or T, indirect assignment or carry out linearity or the nonlinear function match, with match value as corresponding member function μ A (x) value.

As shown in Figure 5. Utilize formula (3) to calculate fuzzy weight (Fuzzy Weight) and fuzzy weight standard deviation and the corresponding T test value of each each classification of evidence layer.

Next, carry out the mineral prediction, when predicting, estimation range and Prediction Parameters are set first, calculate again the response variable figure of estimation range, export at last the MINERAL PREDICTION result

(1) Prediction Parameters setting. The Prediction Parameters setting sees Table 3 and Fig. 6, and in the present invention, the user can be made as the same area with estimation range and training zone, also can arrange in base map.

Table 3 mineral Prediction Parameters arranges table

Input parameter	Designate	Note
			Prediction district base map	PredictLayer	Can be identical with the training center base map, also can reset, but should be in survey region.

Estimation range	PredictExtent	Estimation range is a rectangular area in the survey region, can be in advance delineation of user, also can be the scope of prediction district base map.
			The estimation range	PredictArea	Be to go to the determined effective zone of base map by prediction in the estimation range, predict the outcome and to export in this zone.

(2), calculate the response variable figure of estimation range

As shown in Figure 7, the fuzzy evidence weight of class variable of layer on evidence finishing, then after setting up the estimation range, the user can begin to calculate final response variable figure, comprise posterior probability figure and posterior probability standard deviation figure, and the fuzzy evidence weight form, form comprises the fuzzy evidence weights of each evidence layer heavy and standard deviation and new multinomial mixing (new-omnibus) independence test result. In the present invention, the user can with previous calculations the institute on evidence the layer as variable substitution formula (2), the antilogarithm conversion is done on the right, also can be selected the half-proof layer to calculate final mineral resource prediction figure and its corresponding predicated error figure as variable. It is as follows that the antilogarithm conversion method is done on formula (2) the right:

$P\left(D\right|{\mathrm{\&mu;}}_A\left(x\right){\mathrm{\&mu;}}_B\left(y\right))=\frac{O\left(D\right|{\mathrm{\&mu;}}_A\left(x\right){\mathrm{\&mu;}}_B\left(y\right))}{1+O\left(D\right|{\mathrm{\&mu;}}_A\left(x\right){\mathrm{\&mu;}}_B\left(y\right))}$

At last the MINERAL PREDICTION result is exported.

The above is the preferred embodiments of the present invention only, is not limited to the present invention, and for a person skilled in the art, the present invention can have various modifications and variations. Within the spirit and principles in the present invention all, any modification of doing, be equal to replacement, improvement etc., all should be included within the claim scope of the present invention.

Claims (8)

1, a kind of space data fuzzy evidence weight analysis method is characterized in that, the method may further comprise the steps at least:

The mineral data of step 1, State selective measurements are as the evidence layer of MINERAL PREDICTION, and change continuous evidence layer into discrete classification evidence layer;

Step 2, discrete evidence layer is trained;

The fuzzy evidence weight of the evidence layer behind step 3, the calculation training;

The response variable figure of step 4, calculating estimation range;

Step 5, output MINERAL PREDICTION result.

2, space data fuzzy evidence weight analysis method according to claim 2 is characterized in that, described step 2 also comprises training parameter is set, and calculates the step of prior probability.

3, space data fuzzy evidence weight analysis method as claimed in claim 2 is characterized in that, the step of described calculating prior probability adopts following formula:

Prior probability W₀=logO (D), wherein $O\left(D\right)=\frac{P\left(D\right)}{1-P\left(D\right)},$ W _μA(x) and W_μB(x) corresponding respectively is the weight of fuzzy set A and B;

${\mathrm{\&mu;}}_A\left(x\right)=\left\{\begin{array}{cc}1,& \mathrm{ifx}\&Element;A_1\\ 0.5,& \mathrm{otherwise}\\ 0,& \mathrm{ifx}\&Element;A_2.\end{array}\right.$

A ₁，A ₂The expression subset, A₁＝{x；μ _A(x)＝1}，A ₂＝{x；μ _A(x)＝0}；A ₁∪A ₂X，A ₁∩A ₂＝0。

4, according to claim 2 or 3 described space data fuzzy evidence weight analysis methods, it is characterized in that described training parameter comprises unit are size, training zone and training sample size at least.

5, according to claim 2 or 3 described space data fuzzy evidence weight analysis methods, it is characterized in that described step 3 further may further comprise the steps:

Step 31, with the evidence layer chosen input computer system;

Step 32, setting and calculating member functional value;

The fuzzy evidence weight of step 33, calculating evidence layer;

Step 34, result of calculation is tested.

6, space data fuzzy evidence weight analysis method according to claim 5, it is characterized in that described step 33 also comprises the step of the surface elemant number, training points surface elemant number, fuzzy evidence weight standard deviation, fiducial value or the test value that calculate classification value.

7, space data fuzzy evidence weight analysis method according to claim 6 is characterized in that, the estimation range response variable figure in the described step 4 comprises posterior probability figure and posterior probability standard deviation figure.

8, space data fuzzy evidence weight analysis method as claimed in claim 7 is characterized in that, when calculating posterior probability, and W_ACarry out following value:

$W_{\mathrm{\&mu;A}\left(x\right)}=\left\{\begin{array}{cc}{W}_A^{+},& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=1,\\ 0,& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=0.5\left(P\right[A_1]+P[A_2]=1),\\ W_A^{-},& \mathrm{if}{\mathrm{\&mu;}}_A\left(x\right)=0,\end{array}\right.$

Here, W_A ⁺(x) and W_A ^-(x) utilize common evidence weight method assignment;

$W_A^{+}=\log \frac{P\left[A_1\right|D]}{P\left[A_1\right|\stackrel{\&OverBar;}{D}]}$

$W_A^{-}=\log \frac{P\left[A_2\right|D]}{P[A_2+\stackrel{\&OverBar;}{D}]}.$

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