CN104834938A - Hyper-spectral information extraction method based on main component and cluster analysis - Google Patents

Abstract

The invention discloses a hyper-spectral information extraction method based on main component and cluster analysis. The method comprises the steps of: determining a standardized matrix Z of data of n samples; determining a correlation coefficient matrix R of the standardized matrix Z, calculating the characteristic roots of the correlation coefficient matrix R and characteristic vectors respectively corresponding to each characteristic root, and converting standardized sample data variables into main components, wherein a variable U1 highest in contribution rate is a number 1 main component, a variable U2 the second highest in contribution rate is a number 2 main component, ..., and a variable whose ranking is p is the number p main component; carrying out weighted summation on m main components which are highest in contribution rate to obtain an accumulated contribution rate, and taking the m main components whose accumulated contribution rate exceeds a threshold as cluster analysis main components; and carrying out classification scale calculation on the cluster analysis main components to determine the similarity of the samples, and classifying the samples according to the determined similarity of the samples. According to the invention, the main component analysis method and the cluster analysis method are combined to realize effective extraction of hyper-spectral image information.

Description

Based on the hyperspectral information extracting method of principle component and cluster analysis

Technical field

The present invention relates to spectral data analysis, particularly a kind of hyperspectral information extracting method based on principle component and cluster analysis.

Background technology

Hyper-spectral image technique combines target optical spectrum information and spatial characters, and the image cube got comprises abundant space and spectral information, thus makes the observation of the mankind and information obtaining ability stride forward major step.But hyper-spectral image technique also exists a lot of problem and needs solution badly, such as, eliminate in class and the impact of class inherited change, realize real-time magnanimity information processing etc.

Traditional analytical approach comprises principal component analysis (PCA).Principal component analysis (PCA) is a kind of important statistical method, its objective is and original variable is carried out linear transformation, to select a kind of Multielement statistical analysis method of several significant variable.Its main object carries out the dimensionality reduction of data under being not lose the prerequisite of raw data important information.

Cluster analysis is classified according to similarity degree at one group of physics or abstract object.Wherein more similar object forms one group, and the object differed greatly is classified as different groups.Difference between group and group can represent with similarity distance, and similarity distance is shorter, and illustrate that between two groups, difference is less, similarity distance is larger, illustrates that between two groups, difference is larger.The result that cluster analysis finally obtains can represent with the tree-like figure of classification.

Traditional analytical approach, all there are some drawbacks, such as principal component analysis (PCA), although can play dimensionality reduction and in vector space to the effect that different features is roughly distinguished, but the information recoverable amount after dimensionality reduction is often unstable, can not classify to the target of different characteristic.Traditional single statistical method has been difficult to the requirement meeting current high spectrum image information extraction, therefore needs integrated statistical methods.

Summary of the invention

In view of this, the invention provides a kind of hyperspectral information extraction method method based on principle component and cluster analysis, the method comprises: the normalized matrix Z determining the data of n sample, the data of each sample be xi=(xi1, xi2 ..., xip) ^t, wherein i=1,2 ..., n, p are the integer being less than n; The correlation matrix R of confirmed standardization matrix Z, calculates p the characteristic root of correlation matrix R and vectorial with each characteristic root characteristic of correspondence wherein, j=1,2 ..., n; Based on be major component by the sample data variable transitions after standardization, wherein, j=1,2,3 ..., p, the variable U1 that contribution rate is the highest is first principal component, and the variable U2 that contribution rate second is high is Second principal component, ..., contribution rate rank is the variable of p is p major component; Contribution rate m factor weighted method is from high to low sued for peace to obtain contribution rate of accumulative total, and m major component contribution rate of accumulative total being exceeded predetermined threshold is as cluster analysis major component, wherein, m is less than p, and the weight of each major component in m major component is and this major component characteristic of correspondence root; Carry out classificatory scale to described cluster analysis major component to calculate with the similarity determining sample, and according to the sample similarity determined, sample is classified.

Wherein, determine that the normalized matrix Z of the data of n sample is determined by formula 1 and formula 2:

$Z_{\mathrm{ij}}=\frac{x_{\mathrm{ij}}-{\stackrel{\‾}{x}}_j}{s_j},i=\mathrm{1,2},...,n;j=\mathrm{1,2},...,p$ (formula 1)

${\stackrel{\‾}{x}}_j=\frac{{\mathrm{\Σ}}_{i=1}^n{x}_{\mathrm{ij}}}{n},s_j^2=\frac{{\mathrm{\Σ}}_{i=1}^n{(x_{\mathrm{ij}}-{\stackrel{\‾}{x}}_j)}^2}{n-1}$ (formula 2).

Wherein, according to the correlation matrix of formula 3 confirmed standardization matrix Z:

$R={\left[r_{\mathrm{ij}}\right]}_p\mathrm{xp}=\frac{Z^{T}Z}{n-1}$ (formula 3)

$r_{\mathrm{ij}}=\frac{\mathrm{\Σ}{z}_{\mathrm{kj}}\·z_{\mathrm{kj}}}{n-1},i,j=\mathrm{1,2},...,p$ (formula 4);

P the characteristic root of calculating correlation matrix R is the secular equation by the correlation matrix R to sample | R-λ I _p|=0 carries out solving and obtains.

Preferably, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the related coefficient of m major component determine:

$R_{\mathrm{ij}}=\frac{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}(Z_{\mathrm{ik}}-\stackrel{\‾}{Z_i})(Z_{\mathrm{jk}}-\stackrel{\‾}{Z_j})}{\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{(Z_{\mathrm{ik}}-\stackrel{\‾}{Z_i})}^2{(Z_{\mathrm{jk}}-\stackrel{\‾}{Z_j})}^2}},i,j=\mathrm{1,2},...,m;$

$\stackrel{\‾}{Z_i}=\frac{1}{m}\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z}_{\mathrm{ik}};\stackrel{\‾}{Z_j}=\frac{1}{n}\underset{k=1}{\overset{n}{\mathrm{\Σ}}}{Z}_{\mathrm{jk}}$

Wherein, one 1≤R _ij≤ 1 and R _ijmore close to 1 time, represent that two samples are more close, R _ijmore close-1, then the relation between two samples is more become estranged.

Preferably, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the similarity coefficient of m major component determine:

$S_{\mathrm{ij}}=\cos Q_{\mathrm{ij}}=\frac{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z}_{\mathrm{ik}}-Z_{\mathrm{jk}}}{\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z_{\mathrm{ik}}}^2\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z_{\mathrm{jk}}}^2}},i,j=\mathrm{1,2},...,m;$

Wherein, one 1≤S _ij≤ 1, and S _ijvalue larger, more close to 1, represent that two sample relations are more similar.

Preferably, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the Euclidean distance of m major component determine:

$D_{\mathrm{ij}}=\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{(Z_{\mathrm{ik}}-Z_{\mathrm{jk}})}^2},i,j=\mathrm{1,2},...,m;$

Wherein, 0≤D _ij≤ 1, distance D _ijless, represent that two samples are more similar.

Preferably, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the oblique space length of m major component determine:

$D_{\mathrm{lij}}=\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}\underset{L=1}{\overset{m}{\mathrm{\Σ}}}(Z_{\mathrm{ik}}-Z_{\mathrm{jk}})(Z_{\mathrm{iL}}-Z_{\mathrm{jL}})r_{\mathrm{kL}}},i,j=\mathrm{1,2},...,m;$

Wherein

$r_{\mathrm{kL}}=\frac{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}(X_{\mathrm{ik}}-\stackrel{\‾}{Z_k})(Z_{\mathrm{iL}}-\stackrel{\‾}{Z_L})}{\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(X_{\mathrm{ik}}-\stackrel{\‾}{X_k})}^2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(X_{\mathrm{iL}}-\stackrel{\‾}{X_L})}^2}},k,L=\mathrm{1,2},...,m;$

Wherein, 0≤D _lij≤ 1, distance D _lijless expression two samples are more similar.

The result of principal component analysis (PCA), compared with independent statistical analysis technique in the past, is carried out cluster analysis, principal component analysis (PCA) and cluster analysis is combined by method that the present invention analyzes high-spectral data again.Be provided with the advantage of principal component analysis (PCA) and cluster analysis simultaneously.

The process of methods analyst data of the present invention is clear, and physical meaning is clear and definite, and effect is fairly obvious, effectively can be distinguished by different materials, carry out correct classification.

Principal component analysis (PCA) combines with clustering methodology by method of the present invention, can realize the effective dimensionality reduction to data; Spectral signature is transformed into feature space, draws the distinctive points of different characteristic target clearly.Further, the similarity between target is drawn by cluster analysis, thus the classification realized target and identification, realize the effective extraction to high spectrum image information.

Accompanying drawing explanation

Fig. 1 shows the process flow diagram of the hyperspectral information extraction method method based on principle component and cluster analysis of the present invention.

Fig. 2 shows the curve of spectrum and their difference curve of different material.

Fig. 3 shows the major component scatter diagram obtained through principal component transform by each curve of spectrum.

Fig. 4 shows the first principal component curve of load figure by each curve of spectrum.

Fig. 5 shows according to cluster analysis result example of the present invention.

Embodiment

For making object of the present invention, technical scheme and advantage clearly understand, to develop simultaneously embodiment referring to accompanying drawing, the present invention is described in more detail.

The present invention is mainly used in high-spectral data analysis.Different materials has the different curves of spectrum.As shown in Figure 2, the curve of spectrum and their difference curve of different material is shown.In Fig. 2, ordinate represents intensity (Intensity).According to the present invention, the spectroscopic data of different objects is carried out feature extraction, thus effectively classifies, identify.

According to the present invention, using the curve of spectrum of each pixel of high-spectral data of multiple sample as research object, first principal component analysis (PCA) is carried out to the every bar curve of spectrum corresponding with different material, again major component the highest for the contribution rate of predetermined number is carried out cluster analysis, according to similarity system design, thus different objects difference is come, realize classification and identify.According to the result of principal component analysis (PCA), can also obtain principal component analysis (PCA) scatter diagram and the major component curve of load, scatter diagram shows the difference degree between the different material curve of spectrum, and the curve of load shows the difference wave band that the different material curve of spectrum is concrete.

See Fig. 1, in step 100, determine the normalized matrix Z of the data of n sample.At the Articles detecting of reality with in identifying, the change between the data of the sample obtained may be very large, needs to carry out conversion process to carry out follow-up operation to original sample data.The data of each sample can be expressed as xi=(xi1, xi2 ..., xip) ^t, wherein i=1,2 ..., n, p are the integer being less than n.

Preferably, normalized matrix Z can be obtained as follows by publicity 1 and 2:

$Z_{\mathrm{ij}}=\frac{x_{\mathrm{ij}}-{\stackrel{\‾}{x}}_j}{s_j},i=\mathrm{1,2},...,n;j=\mathrm{1,2},...,p$ (formula 1)

Wherein, ${\stackrel{\‾}{x}}_j=\frac{{\mathrm{\Σ}}_{i=1}^n{x}_{\mathrm{ij}}}{n},s_j^2=\frac{{\mathrm{\Σ}}_{i=1}^n{(x_{\mathrm{ij}}-{\stackrel{\‾}{x}}_j)}^2}{n-1},$ (formula 2).

In step 102, the correlation matrix R of confirmed standardization matrix Z, calculates p the characteristic root of correlation matrix R and vectorial with each characteristic root characteristic of correspondence wherein, j=1,2 ..., n.

$R={\left[r_{\mathrm{ij}}\right]}_p\mathrm{xp}=\frac{Z^{T}Z}{n-1},$ (formula 3)

Wherein, $r_{\mathrm{ij}}=\frac{\mathrm{\Σ}{z}_{\mathrm{kj}}\·z_{\mathrm{kj}}}{n-1},i,j=\mathrm{1,2},...,p,$ (formula 4)

Wherein, p the characteristic root of calculating correlation matrix R is the secular equation by the correlation matrix R to sample | R mono-λ I _p|=0 carries out solving and obtains.

In step 104, based on be major component by the sample data variable transitions after standardization, wherein, j=1,2,3 ..., p, the variable U1 that contribution rate is the highest is first principal component, and the variable U2 that contribution rate second is high is Second principal component, ..., contribution rate rank is the variable of p is p major component;

In step 106, contribution rate m factor weighted method is from high to low sued for peace to obtain contribution rate of accumulative total, and m major component contribution rate of accumulative total being exceeded predetermined threshold is as cluster analysis major component, wherein, m is less than p, and the weight of each major component in m major component is and this major component characteristic of correspondence root.Preferably, predetermined threshold can be set make it possible to choose 7 major components as cluster molecule major component.

In step 108, classificatory scale is carried out to described cluster analysis major component and calculates with the similarity determining sample, and according to the sample similarity determined, sample is classified.

Preferably, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the related coefficient of m major component determine:

$R_{\mathrm{ij}}=\frac{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}(Z_{\mathrm{ik}}-\stackrel{\‾}{Z_i})(Z_{\mathrm{jk}}-\stackrel{\‾}{Z_j})}{\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{(Z_{\mathrm{ik}}-\stackrel{\‾}{Z_i})}^2{(Z_{\mathrm{jk}}-\stackrel{\‾}{Z_j})}^2}},i,j=\mathrm{1,2},...,m;$

$\stackrel{\‾}{Z_i}=\frac{1}{m}\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z}_{\mathrm{ik}};\stackrel{\‾}{Z_j}=\frac{1}{n}\underset{k=1}{\overset{n}{\mathrm{\Σ}}}{Z}_{\mathrm{jk}}$

Wherein, one 1≤R _ij≤ 1 and R _ijmore close to 1 time, represent that two samples are more close, R _ijmore close-1, then the relation between two samples is more become estranged.

Alternatively, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the similarity coefficient of m major component determine:

$S_{\mathrm{ij}}=\cos Q_{\mathrm{ij}}=\frac{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z}_{\mathrm{ik}}-Z_{\mathrm{jk}}}{\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z_{\mathrm{ik}}}^2\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z_{\mathrm{jk}}}^2}},i,j=\mathrm{1,2},...,m;$

Wherein, one 1≤S _ij≤ 1, and S _ijvalue larger, more close to 1, represent that two sample relations are more similar.

Alternatively, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the Euclidean distance of m major component determine:

$D_{\mathrm{ij}}=\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{(Z_{\mathrm{ik}}-Z_{\mathrm{jk}})}^2},i,j=\mathrm{1,2},...,m;$

Wherein, 0≤D _ij≤ 1, distance D _ijless, represent that two samples are more similar.

Inventor notices, sometimes calculate distance with Euclidean distance and also there will be some deviations, owing to often there is correlationship between variable, relevant degree is had nothing in common with each other, this can make result that the deviation not wishing to occur occurs, to sample k, the distance of 1 can with more broadly oblique space length as classificatory scale.Therefore, alternatively, to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the oblique space length of m major component determine:

$D_{\mathrm{lij}}=\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}\underset{L=1}{\overset{m}{\mathrm{\Σ}}}(Z_{\mathrm{ik}}-Z_{\mathrm{jk}})(Z_{\mathrm{iL}}-Z_{\mathrm{jL}})r_{\mathrm{kL}}},i,j=\mathrm{1,2},...,m;$

Wherein

$r_{\mathrm{kL}}=\frac{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}(X_{\mathrm{ik}}-\stackrel{\‾}{Z_k})(Z_{\mathrm{iL}}-\stackrel{\‾}{Z_L})}{\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(X_{\mathrm{ik}}-\stackrel{\‾}{X_k})}^2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(X_{\mathrm{iL}}-\stackrel{\‾}{X_L})}^2}},k,L=\mathrm{1,2},...,m;$

Wherein, 0≤D _lij≤ 1, distance D _lijless expression two samples are more similar.

The principle of cluster analysis can see Fig. 5.Cluster analysis is classified to spectrum according to the similarity degree of spectrum.In the example of hgure 5, compare 28 spectrum, the most similar two spectral lines be classified as a class, then this class and 26 remaining spectral lines are compared again, the most similar again and be a class.The rest may be inferred.Thus carry out correct classification and identification.

According to the present invention, principal component scores figure can be obtained according to the result of principal component analysis (PCA).Fig. 3 is principal component scores figure.In Fig. 3, each point is that each point represents a spectral line by a curve of spectrum by major component algorithmic transformation, and some position in the drawings represents the feature of spectral line.If two points are very close, then illustrate that two spectral line characteristics are very similar, otherwise illustrate that spectral line characteristic differs greatly.

Equally, major component curve of load figure can also be obtained according to the result of principal component analysis (PCA).For multiple sample, after determining m major component, the curve of load of each major component can be obtained.Such as, Fig. 4 is first principal component (PC1) curve of load.Obviously, other major component also can have corresponding curve of load figure.In Fig. 4, transverse axis is wave number, the difference degree between the absolute value representation of the longitudinal axis 20 spectral lines.The larger expression of absolute value is larger in the difference of this this group curve of spectrum of wave number place.

The foregoing is only preferred embodiment of the present invention, not in order to limit the present invention, within the spirit and principles in the present invention all, any amendment made, equivalent replacement, improvement etc., all should be included within the scope of protection of the invention.

Claims (7)

1., based on a hyperspectral information extraction method method for principle component and cluster analysis, the method comprises:

Determine the normalized matrix Z of the data of n sample, the data of each sample be xi=(xi1, xi2 ..., xip) ^t, wherein i=1,2 ..., n, p are the integer being less than n;

The correlation matrix R of confirmed standardization matrix Z, calculates p the characteristic root of correlation matrix R and vectorial with each characteristic root characteristic of correspondence wherein, j=1,2 ..., n;

Based on be major component by the sample data variable transitions after standardization, wherein, j=1,2,3 ..., p, the variable U1 that contribution rate is the highest is first principal component, and the variable U2 that contribution rate second is high is Second principal component, ..., contribution rate rank is the variable of p is p major component;

Contribution rate m factor weighted method is from high to low sued for peace to obtain contribution rate of accumulative total, and m major component contribution rate of accumulative total being exceeded predetermined threshold is as cluster analysis major component, wherein, m is less than p, and the weight of each major component in m major component is and this major component characteristic of correspondence root;

Carry out classificatory scale to described cluster analysis major component to calculate with the similarity determining sample, and according to the sample similarity determined, sample is classified.

2. the method for claim 1, wherein determine that the normalized matrix Z of the data of n sample is determined by formula 1 and formula 2:

$Z_{\mathrm{ij}}=\frac{x_{\mathrm{ij}}-{\stackrel{\‾}{x}}_j}{s_j},i=\mathrm{1,2},...,n;j=\mathrm{1,2},...,p$ (formula 1)

${\stackrel{\‾}{x}}_j=\frac{{\mathrm{\Σ}}_{i=1}^n{x}_{\mathrm{ij}}}{n},s_j^2=\frac{{\mathrm{\Σ}}_{i=1}^n{(x_{\mathrm{ij}}-{\stackrel{\‾}{x}}_j)}^2}{n-1}$ (formula 2).

3. the method for claim 1, wherein according to the correlation matrix of formula 3 confirmed standardization matrix Z:

$R={\left[r_{\mathrm{ij}}\right]}_p\mathrm{xp}=\frac{Z^{T}Z}{n-1}$ (formula 3)

$r_{\mathrm{ij}}=\frac{\mathrm{\Σ}{z}_{\mathrm{kj}}\·z_{\mathrm{kj}}}{n-1},i,j=\mathrm{1,2},...,p$ (formula 4);

P the characteristic root of calculating correlation matrix R is the secular equation by the correlation matrix R to sample | R-λ I _p|=0 carries out solving and obtains.

4. the method for claim 1, wherein to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the related coefficient of m major component determine:

$R_{\mathrm{ij}}=\frac{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}(Z_{\mathrm{ik}}-\stackrel{\‾}{Z_i})(Z_{\mathrm{jk}}-\stackrel{\‾}{Z_j})}{\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{(Z_{\mathrm{ik}}-\stackrel{\‾}{Z_i})}^2{(Z_{\mathrm{jk}}-\stackrel{\‾}{Z_j})}^2}},i,j=\mathrm{1,2},...,m$

$\stackrel{\‾}{Z_i}=\frac{1}{m}\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z}_{\mathrm{ik}};\stackrel{\‾}{Z_j}=\frac{1}{n}\underset{k=1}{\overset{n}{\mathrm{\Σ}}}{Z}_{\mathrm{jk}}$

Wherein, one 1≤R _ij≤ 1 and R _ijmore close to 1 time, represent that two samples are more close, R _ijmore close-1, then the relation between two samples is more become estranged.

5. the method for claim 1, wherein to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the similarity coefficient of m major component determine:

$S_{\mathrm{ij}}=\cos Q_{\mathrm{ij}}=\frac{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z}_{\mathrm{ik}}-Z_{\mathrm{jk}}}{\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z_{\mathrm{ik}}}^2\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{Z_{\mathrm{jk}}}^2}},i,j=\mathrm{1,2},...,m;$

Wherein, one 1≤S _ij≤ 1, and S _ijvalue larger, more close to 1, represent that two sample relations are more similar.

6. the method for claim 1, wherein to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the Euclidean distance of m major component determine:

$D_{\mathrm{ij}}=\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{(Z_{\mathrm{ik}}-Z_{\mathrm{jk}})}^2},i,j=\mathrm{1,2},...,m;$

Wherein, 0≤D _ij≤ 1, distance D _ijless, represent that two samples are more similar.

7. the method for claim 1, wherein to described cluster analysis major component carry out classificatory scale calculate with determine the similarity of sample comprise by determine according to following formula different sample the oblique space length of m major component determine:

$D_{1\mathrm{ij}}=\sqrt{\underset{k=1}{\overset{m}{\mathrm{\Σ}}}\underset{L=1}{\overset{m}{\mathrm{\Σ}}}(Z_{\mathrm{ik}}-Z_{\mathrm{jk}})(Z_{\mathrm{iL}}-Z_{\mathrm{jL}})r_{\mathrm{KL}}},i,j=\mathrm{1,2},...,m;$

Wherein

$r_{\mathrm{kL}}=\frac{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}(X_{\mathrm{ik}}-\stackrel{\‾}{Z_k})(Z_{\mathrm{iL}}-\stackrel{\‾}{Z_L})}{\sqrt{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(X_{\mathrm{ik}}-\stackrel{\‾}{X_k})}^2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(X_{\mathrm{iL}}-\stackrel{\‾}{X_L})}^2}},k,L=\mathrm{1,2},...,m;$

Wherein, 0≤D _1ij≤ 1, distance D _1ijless expression two samples are more similar.