Multiset Canonical Correlations Analysis and Multispectral ... · In Section II, ordinary two-set...

Multiset Canonical Correlations Analysisand Multispectral, Truly Multitemporal

Remote Sensing DataAllan Aasbjerg Nielsen

Abstract—This paper describes two- and multiset canonicalcorrelations analysis (CCA) for data fusion, multisource, multiset,or multitemporal exploratory data analysis. These techniquestransform multivariate multiset data into new orthogonal vari-ables called canonical variates (CVs) which, when applied inremote sensing, exhibit ever-decreasing similarity (as expressed bycorrelation measures) over sets consisting of 1) spectral variablesat fixed points in time (R-mode analysis), or 2) temporal variableswith fixed wavelengths (T-mode analysis). The CVs are invariantto linear and affine transformations of the original variableswithin sets which means, for example, that the R-mode CVs areinsensitive to changes over time in offset and gain in a measuringdevice. In a case study, CVs are calculated from Landsat TMdata with six spectral bands over six consecutive years. Both R-and T-mode CVs clearly exhibit the desired characteristic: theyshow maximum similarity for the low-order canonical variatesand minimum similarity for the high-order canonical variates.These characteristics are seen both visually and in objectivemeasures. The results from the multiset CCA R- and T-modeanalyses are very different. This difference is ascribed to the noisestructure in the data. The CCA methods are related to partialleast squares (PLS) methods. This paper very briefly describesmultiset CCA-based multiset PLS. Also, the CCA methods canbe applied as multivariate extensions to empirical orthogonalfunctions (EOF) techniques. (Multiset) CCA is well-suited forinclusion in geographical information systems (GIS).

Index Terms—Geographical information systems (GIS), min-imum and maximum similarity variates, multiset partial leastsquares (PLS), multisource data fusion, multivariate empiricalorthogonal functions (EOF).

I. INTRODUCTION

T HIS paper deals with multiset canonical correlationsanalysis (MCCA) for data fusion, multisource, multiset,

or multitemporal exploratory data analysis. MCCA deals withdata that naturally splits up into more (than two) groups ofvariables, e.g., multispectral satellite data covering the samegeographical region over several points in time.

In Section II, ordinary two-set canonical analysis is de-scribed. Two-set canonical correlations analysis investigatesthe relationship between two groups of variables. It findscorresponding sets of linear combinations of the two groups

Manuscript received July 23, 1999; revised June 22, 2001. This work wassupported by the European Union under Contract BRE2-CT92-0201. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Michael R. Frater.

The author is with Informatics and Mathematical Modelling, TechnicalUniversity of Denmark, DK-2800 Lyngby, Denmark (e-mail: [email protected];http://www.imm.dtu.dk/~aa).

Publisher Item Identifier S 1057-7149(02)01323-4.

of original variables with maximum correlation. Partial leastsquares regression is mentioned in Section II also.

In Section III, this analysis is generalized to deal with morethan two sets of variables. The idea is to optimize characteris-tics of the dispersion matrix of the transformed variables to ob-tain high correlations between all new variables simultaneously.These characteristics include

• maximization of the sum of the elements;• maximization of the sum of the squared elements;• maximization of the largest eigenvalue;• minimization of the smallest eigenvalue;• minimization of the determinant.

These measures are not confined and the optimizations takeplace subject to different chosen constraints and orthogonalitycriteria. The latter are briefly mentioned in Section IV, whichalso describes computer implementations of the techniques.

Results from such analyses are linear combinations termedcanonical variates (CVs) that when used with remote sensingdata transform the original data into new orthogonal variablesthat show decreasing similarity over sets consisting of

1) spectral variables at fixed points in time (R-modeanalysis);

2) temporal variables with fixed wavelengths (T-modeanalysis).

The higher order canonical variates exhibit minimum similarityand they are therefore measures of differences in all variablessimultaneously.

Multiset partial least squares methods emerge from this typeof description with a special choice of optimization criteria, con-straints, and orthogonality criteria leading to an optimization ofcovariance rather than correlation measures.

If applied to several variables that change over time, this typeof analysis constitutes a multivariate extension to the techniqueof empirical orthogonal functions (EOF) [1] often applied ingeophysical data analysis.

In Section V, a Landsat TM case with data from 1984 to 1989covering a small forested region in northern Sweden is usedto illustrate the technique. The purpose of the case study is todemonstrate the method and to suggest a possible way of inter-preting the resulting transformed variables. It is not the purposeto assess the similarity (or lack of similarity, i.e., change) overtime on the ground or in the atmosphere.

Multiset or multisource data analysis techniques such as theapplication of the Mahalanobis distance in joint distributionsof multiset data to point out potential mineralization areas [2]

1057–7149/02$17.00 © 2002 IEEE

294 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 11, NO. 3, MARCH 2002

or Markov random field methods to fuse image data with largespatial resolution differences [3] are not dealt with here.

The methods described here are well suited for integration ina geographical information system (GIS).

II. TWO-SET CANONICAL CORRELATIONS

Canonical correlations analysis was first introduced in [4] toanalyze linear relations between two sets of variables. The tech-nique is described in most standard textbooks on multivariatestatistics, e.g., [5] and [6]. Work on nonlinear canonical corre-lations analysis is dealt with in [7]–[11]. This type of analysiswill not be pursued here.

Two-set canonical correlations analysis investigates the rela-tionship between two groups of variables. It finds correspondingsets of linear combinations of the original two groups of vari-ables. The first set of linear combinations are the ones with thelargest correlation. This correlation is called the first canonicalcorrelation and the two linear combinations are called the firstcanonical variates. The second set of linear combinations arethe ones with the largest correlation subject to the condition thatthey are orthogonal to the first canonical variates. This correla-tion is called the second canonical correlation and the two linearcombinations are called the second canonical variates. Higherorder canonical correlations and canonical variates are definedsimilarly.

We consider a -dimensional random variable ( )ideally following a Gaussian distribution split into two groupsof dimensions and , respectively, (without loss of generalitywe assume that , where denotesexpectation)

and we assume that the relevant dispersion matrices are nonsin-gular. Of course .

We are searching for linear combinations ofand

V

V

(where V denotes variance) with maximum correlation

CorrCov

V V

Let Cov denote the covariance betweenand .To maximize we set and get

Without loss of generality, we choose [ ] so thatwhich leads to

i.e., we find the desired projections for by considering theconjugate eigenvectors corresponding to the eigen-values of with respect to . Sim-ilarly, we may find the desired projections forby consideringthe conjugate eigenvectors of with re-spect to corresponding to the same eigenvalues. Ifthis will be all the eigenvalues and -vectors of . If

the last eigenvalue will be 0 with multiplicity . (Asthe solutions and are interrelated we only need to find oneof them.)

A. Partial Least Squares (PLS)

Canonical corelations analysis (CCA) is closely re-lated to the method of partial least squares, PLS, in which

Cov (often with as a scalarresponse variable) is maximized with another choice of con-straints, namely leading to

(see [12]). We see that in this case matrix inversion is not neededwhich is good if we have many variables and few observations.Only the first pair of canonical variates (or latent variables) cor-responding to the largest eigenvalue are calculated and the re-sponse CV is regressed on the predictor CV

If more information is present in the residuals, their projec-tions replace the original response variables (i.e., replaceby

), the predictor variables are projected into a subspaceorthogonal to the solution found (i.e., replaceby )with ), and we iterate; see also [13]–[16].

B. The MAD Transformation

The above CCA technique is used in [17]–[22] to find linearcombinations that give maximal multivariate differences. Thename chosen for the transformation, multivariate alteration de-tection (MAD), is due to the application to change detectionin remote sensing (and the acronym). Although it is presentedas a change detection technique in remote sensing, the tech-nique applies to nonspatial multivariate differences also. TheMAD transformation has been used in an attempt to differen-tiate between geogenic and anthropogenic influences on soils ina mining processing area, see [23]. References [17] and [19] alsosuggest the use of the maximum autocorrelation factor (MAF)transformation, [24], to postprocess the MAD variates. Refer-ence [25] uses MAF to process the simple differences. Refer-ence [26] uses a hybrid canonical correlation/principal compo-nents technique to enhance uncorrelated parts of Landsat TMequivalents of ATM data in a gold exploration study. Changedetection techniques based on canonical variates are also de-scribed in [27] and [28].

NIELSEN: MULTISET CANONICAL CORRELATIONS ANALYSIS 295

III. M ULTISET CANONICAL CORRELATIONS

Multiset canonical correlations analysis (MCCA) is a tech-nique for analyzing linear relations between more (than two)sets of variables. Earlier work in this field comprise [29]–[32].Reference [33] gives an interesting example using satellite dataand two types of geochemical data. Work on nonlinear MCCAis reported in [48].

We consider an dimensionalrandom variable ideally following a Gaussian distributionsplit into groups of dimensions , to

, respectively, (without loss of generality we assumethat )

...

......

.... . .

...

and we assume that the relevant dispersion matrices are nonsin-gular. Of course .

An obvious extension from the two-set case is tosearch for linear combinations of

...

with dispersion matrix

......

.. ....

or for short. As in the two-set casethere is one , and for each

, .In the two-set case we obtain new variables with a

high measure of similarity by maximizing the scalarCorr . Here, we must maximize all correla-tions/covariances between the new variables simultaneously.To do this, the following measures of can be optimized:

1) maximize sum of elements ( );2) maximize sum of squared elements (

);3) maximize largest eigenvalue ();4) minimize smallest eigenvalue ();5) minimize determinant (det ).

Reference [32] lists all these possibilities and names them

1) SUMCOR;2) SSQCOR;

Fig. 1. Sketch of R-mode multiset canonical correlations analysis. Variablesindicated in top row are transformed into CVs in bottom row.

Fig. 2. Sketch of T-mode multiset canonical correlations analysis. Variablesindicated in top row are transformed into CVs in bottom row.

3) MAXVAR ;4) MINVAR;5) GENVAR.These measures are not constrained, but several natural

choices for constraints under which to carry out the optimiza-tions come to mind:

1) the projection vectors are unit vectors within each set( );

2) the sum of the projection vectors is a unit vector( );

3) the new variables have unit variance ( );4) the sum of the variances of the new variables is unity

( tr );

where constraint 3 (causing to be correlations rather thancovariances) is the natural extension from the two-set case. Ifthe sets analyzed are different variables measured over timethis type of analysis constitutes a multivariate extension to thetechnique of empirical orthogonal functions (EOF) [1].


TABLE ICORRELATIONSBETWEEN R-MODE CANONICAL VARIATES 1 FOR ALL

FIVE METHODS

In the two-set case all of these methods with constraints 3 and4 reduce to the standard Hotelling case described in Section II(except for a scaling factor for constraint 4).

Reference [31] examines the SUMCOR method and [32] ex-amines all the above methods using constraint 3. Reference [17]examines all the above methods using all constraints. As an il-lustration, we consider the SUMCOR method with constraints 3and 4 shown in the following.

A. Maximize Sum of Covariances

To maximize the sum of covariances under constraints we usea Lagrange multiplier technique.

1) Constraint 3: : Introduce

TABLE IICORRELATIONS BETWEEN T-MODE CANONICAL VARIATES 1 FOR ALL

FIVE METHODS

and maximize without constraints. By settingwe get

or

......

.. ....

...

......

. . ....

...(1)

Setting merely reproduces the constraints. Be-cause the s are not equal this system of equations is more gen-eral than a generalized eigensystem. Invariance of the solutionto linear transformations within sets is easily shown (after thetransformation the s will be the same, the s will not).


Fig. 3. Landsat TM data. Rows are years 1984 to 1989 and columns are TM bands 1 to 5 and 7.

2) Constraint 4: : Introduce

and maximize without constraints. By settingwe get

or

......

.. ....

...

......

. . ....

...

This is a (real, symmetric) generalized eigensystem, i.e., wefind the desired projections for by computing the conjugateeigenvectors corresponding to the first


Fig. 4. R-mode CVs. Rows are years 1984 to 1989 and columns are CV1–CV6.

eigenvalues ofthe above eigensystem.

B. Multiset Canonical Variates

We are now able to define themultiset canonical variates

...

where the s come from either of the above solutions. With anobvious choice of notation we get

...

where

is

is...

is


Fig. 5. T-mode CVs. Rows are TM bands 1 to 5 and 7 and columns are CV1–CV6.

C. Multiset Partial Least Squares

We can base a true multiset or multiblock partial least squares(PLS) method on MCCA with modified optimization criteria 1and 2 mentioned above if we use constraint 1, , with

replaced by the null matrix since in this case we do notwant to include the diagonal terms of . To see this, considerfor example the maximization of the sum of all nondiagonalelements in

To maximize under constraint 1, again use Lagrange multi-pliers and maximize

without constraints. By setting we get


Fig. 6. Landsat TM bands 4, 5, and 3 as red, green, and blue (1984–1989).

Fig. 7. R-mode CVs 1, 2, and 3 as red, green, and blue (1984–1989).

Fig. 8. R-mode CVs 6, 5, and 4 as red, green, and blue (1984–1989).

Fig. 9. T-mode CVs 1, 2, and 3 as red, green, and blue (TM bands 1–5 and 7).

Fig. 10. T-mode CVs 6, 5, and 4 as red, green, and blue (TM bands 1–5 and 7).

or

......

. . ....

......

Now, calculate all first sets of latent variables, i.e., the firstcanonical variates , . Without loss of generality

we place the response variables in the first set and perform amultiple regression analysis

If more information is present in the residuals their projectionsreplace the original response variables (i.e., replaceby

), the predictor variables are projected into a sub-space orthogonal to the solution found (i.e., forreplace by with ) and weiterate. For ordinary two-set PLS, see [13]–[16].


IV. COMPUTERIMPLEMENTATIONS

Two-set canonical correlations analysis is implemented in acomputer program, , which is a general orthogonalizationprogram that also finds principal components, (rotated) prin-cipal factors, maximum autocorrelation factors (MAF), [24],scaled MAFs [34], minimum noise fractions (MNF), [35], [36],multivariate alteration detection (MAD) variates, [17]–[19]and canonical discriminant functions, etc. Also the multisetcanonical correlations analysis methods of maximizing thesum of covariances under constraints 2 ( ) and4 ( ) are implemented in .

All dispersion (variance–covariance) matrices are found bythe method of provisional means, [37]. The eigenvalue problemsassociated with the analysis are solved by means of LAPACKroutines, [38]. Good general descriptions of the methods usedare given in, e.g., [39]–[41].

The remaining optimization problems concerning MCCA (infact, all of them, including the eigenvalue problems) are solvedby means of the general algebraic modeling system (GAMS),[42], NLP solver CONOPT, [43]. A computer programthat calls GAMS to perform the analysis is implemented. Asa parallel to the solution of the eigensystem involved in thetwo-set case the orthogonality criteria chosen in aresimilar to the normalization criteria, for example for constraint3 , where if and if notis Kronecker’s delta.

V. CASE: LANDSAT TM DATA IN FORESTRY

The utility of multiset canonical correlations analysis to mul-tivariate and truly multi-temporal data is demonstrated in a casestudy using Landsat-5 Thematic Mapper (TM) data covering asmall forested area approximately 20 kilometers north of Umeåin northern Sweden. The data consist of six by six spectral bandswith 512 512 20-m pixels from the summers 1984–1989 rec-tified to the Swedish national grid. The acquisition dates are1 August 1984, 26 June 1985, 6 June 1986, 12 August 1987,27 June 1988, and 21 June 1989. These data are also analyzedin [44]–[46].

As an illustration, all results reported here (except the correla-tions in Tables I and II) relate to the SUMCOR method with con-straint and orthogonality criterion 3, i.e., the CVs have unit vari-ance. In R-mode analysis, we consider Landsat TM bands 1 to 5and 7 for 1984 as one set of variables and similarly for 1985,etc. In T-mode analysis we consider TM bands 1 for all years1984–1989 as one set of variables, TM bands 2 for all years1984–1989 as another set of variables, etc. [1]. For a sketch ofR- and T-mode analysis setup; see Figs. 1 and 2. In both figuresthe six sets of variables indicated on the top are transformed intosix sets of new variables on the bottom. For example, in T-modeanalysis the variables 1984 TM1, 1985 TM1,, 1989 TM1 aretransformed into TM1 CVs and similarly for TM2, etc.

Fig. 3 shows the original TM data. Column one is TM1,column two is TM2, etc. Row one is 1984, row two is 1985, etc.Fig. 4 shows the R-mode CVs. Column one is CV1, columntwo is CV2, etc. Row one is 1984, row two is 1985, etc. Fig. 5shows the T-mode CVs. Column one is CV1, column two isCV2, etc. Row one is TM1, row two is TM2, etc.

Fig. 11. Correlations between R-mode CVs 1 and original data.

Fig. 6 shows Landsat TM bands 4, 5, and 3 as red, green,and blue, respectively. Fig. 7 shows R-mode canonical variates1, 2, and 3 as red, green, and blue, respectively. We see thatwe have indeed obtained a high degree of similarity over years.Fig. 8 shows R-mode canonical variates 6, 5, and 4 as red, green,and blue, respectively. This is the RGB combination that showsminimum similarity over years. We see that noise (striping anddropouts) is depicted well as is to be expected: if data from oneyear is noisy and data from another year is not (or if the noisepatterns are different) then certainly the largest difference couldbe that noise (or that difference). This observation inspires aniterative use of the procedure: first identify noise, restore dataor exclude areas with noise from further analysis and carry outthe analysis once more. This iterative use is not illustrated here.

Fig. 9 shows T-mode canonical variates 1, 2, and 3 as red,green, and blue, respectively. Again, we see that we have ob-tained a high degree of similarity, this time over TM bands.Fig. 10 shows T-mode canonical variates 6, 5, and 4 as red,green, and blue, respectively. We see that striping is stronglypresent in TM bands 1 and 2.

The transformation matrices containing the weights appliedto the original variables to obtain the CVs are not shown as theseweights are difficult to interpret because of inter-correlation be-tween the original variables. Instead we show correlations be-tween the original variables and the CVs. (It is often seen that


Fig. 12. Correlations between R-mode CVs 6 and original data.

an original variable that has a, say, negative weight in the cal-culation of some transformed variable has a positive correlationwith that transformed variable.) Correlations between R-modeCVs 6 and the original data given in Fig. 12 show that dissimilar-ities (differences between years) are associated with TM bands1 especially from 1984 to 1987. This is probably because of dif-ferences in atmospheric conditions. Therefore analysis of atmo-spherically corrected data would be interesting. Correlations be-tween T-mode CVs 6 and the original variables given in Fig. 14,for TM bands 1, 2, 3, 5, and 7 reveal a pattern of positive cor-relation with 1984, negative correlation with 1985 and againpositive correlation with 1986 (but not as high as with 1984)combined with (nearly) no correlation with 1987–1989. T-modeCV6 for TM4 is positively correlated with TM4 in 1984–1986,uncorrelated with TM4 in 1987 and negatively correlated withTM4 in 1988 and 1989. This could indicate that vegetation re-lated changes occurred from 1986 to 1988. This finding is con-firmed by an observation in [45]: “Several stands with ScotsPine (Pinus Sylvestris) had been damaged by the snow-breakin the winter 1987/1988.” Correlations between T-mode CVs 1and TM4 given in Fig. 13 are (except for TM4 CV1) lower thancorrelations between T-mode CVs 1 and the other bands. Again,this indicates changes that are related with TM4, possibly vege-tation changes. For completeness Fig. 11 gives correlations be-tween R-mode CVs 1 and the original data.

Fig. 13. Correlations between T-mode CVs 1 and original data.

Table I shows correlations between R-mode canonical vari-ates 1 ( ) for all five methods investigated. The same correla-tions for T-mode analysis are shown in Table II. Again, we seea special behavior for TM4 indicating vegetation changes.

Table III shows the values of objective function, i.e., thequantity which is maximized, namely the sum of the elementsin , under the constraint that

. We see that although R-mode analysis obtainsthe highest objective function value (i.e., level for all correla-tions simultaneously), T-mode analysis maintains a high levelfor this measure for higher order CVs than does R-mode. Also,for R-mode the highest difference between objective functionvalues occurs between CVs 3 and 4, whereas for T-mode itoccurs between CVs 5 and 6. Also, there is a big reduction inthe objective function values between T-mode CVs 1 and 2.This difference is clearly visible in the imagery also (Fig. 5).

In the comparisons performed in Tables I and II, SUMCOR,SSQCOR, and MAXVAR seem to perform similarly. MINVAR andGENVAR seem to perform differently and not in the same fashion.[47] observes a similar different behavior for MINVAR. This isunderstandable when contemplating the design criteria behindthe individual methods. SUMCOR and SSQCORboth focus on allcorrelations between CVs, i.e., all elements in . MAXVAR

maximizes the largest eigenvalue, again a focus onall elements in. MINVAR relies heavily on the smallest eigenvalue, whereas


Fig. 14. Correlations between T-mode CVs 6 and original data.

TABLE IIIVALUE OF OBJECTIVE FUNCTION: SUM OF ELEMENTS IN�

GENVAR minimizes the determinant of and therefore relieson several small eigenvalues. Due to lack of ground truth data,it has not been possible to determine empirically which of thefive methods (if any) perform best in this context.

Tables IV and V show comparisons of the actual values ofthe optimization criteria for the five methods discussed for R-and T-mode canonical variates 1. The optimization criteria arenot contradicted, e.g., for MINVAR is smaller than for theother methods. Also in this comparison, SUMCOR, SSQCOR, andMAXVAR seem to perform similarly and MINVAR and GENVAR

seem to perform differently and not in the same fashion.Fig. 15 shows for both R- and T-mode CV1–CV6 [see (1)].

Some differences within individual CVs are seen especially forT-mode.

The difference between the results of the R- and T-modeanalyses is believed to be due to the expected correlation in noiseover bands in the same year and the expected lack of correlationin noise over years in the same bands.

TABLE IVVALUE OF OBJECTIVEFUNCTION FORALL FIVE METHODS, R-MODE CV1

TABLE VVALUE OF OBJECTIVEFUNCTION FORALL FIVE METHODS, T-MODE CV1

Fig. 15. � for both R-mode (top) and T-mode (bottom) CV1 (indexes 1–6),CV2 (indexes 7–12),. . ., CV6 (indexes 31–26) [see (1)].

If we assign variables to (Section III) in an appropriatefashion the methods described can be used for simultaneous op-timization of more objectives such as interset correlation andspatial correlation.

VI. CONCLUSIONS

Two- and multiset canonical correlations analysis for data fu-sion, multisource, multiset, or multitemporal exploratory dataanalysis is described and applied to six spectral bands fromLandsat TM summer data from 1984 to 1989. The resultingcanonical variates are invariant to linear and affine transforma-tions of the original data within sets. This means, for example,that the R-mode CVs are insensitive to changes over time inoffset and gain in a measuring device. The CVs show the de-sired characteristic, namely that they exhibit ever decreasingsimilarity (as measured by correlation) with increasing order ofthe CVs. There is a big (visual) difference between the results ofthe R- and T-mode analyses. This difference is ascribed to thenoise structure of the data.

Choosing other optimization criteria and constraints thanthe ones usually chosen for canonical correlation analysis, themethods described also form a basis for true multiset PLS.

If the data analyzed are variables measured over time this typeof analysis constitutes a multivariate extension to the techniqueof EOF.


Although applied to Landsat TM data here, the methods aresuitable to the analysis of any data that are naturally divided intoseveral multivariate groups.

The methods described are well suited for integration in aGIS.

ACKNOWLEDGMENT

The author would like to thank Prof. K. Conradsen, IMM, forhis cooperation in the development of the MAD concept and forestablishing a professional contact to Dr. A. Drud whom the au-thor thanks for the immediate interest he took in the noneigen-value type optimization problems. Dr. Drud wrote the GAMSNLP solver CONOPT applied and he also wrote the genericGAMS code that is the heart of the computer program .The author would like to thank Dr. R. Larsen, IMM, for gooddiscussions on many aspects of multivariate statistics and forprogramming in close cooperation with him. The accessto six years of geometrically corrected Landsat TM data fromnorthern Sweden given by the Swedish Space Corporation isacknowledged.

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Allan Aasbjerg Nielsenreceived the M.Sc. degree in1978 from the Department of Electrophysics, Tech-nical University of Denmark, Lyngby, and the Ph.D.degree in 1994 from Informatics and MathematicalModelling (IMM), Technical University of Denmark.

He is an Associate Professor with IMM. He waswith the Danish Defense Research Establishmentin 1977–1978. He worked on energy conservationin housing with the Thermal Insulation Laboratory,Technical University of Denmark, from 1978 to1985. Since 1985, he has been with the Section for

Image Analysis, IMM. He has worked on several national and internationalprojects on development, implementation, and application of statisticalmethods and remote sensing in mineral exploration, mapping, geology,environment, oceanography, geodesy, and agriculture funded by industry, theEuropean Union, Danida (the Danish International Development Agency), andthe Danish National Research Councils.

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