archetypal analysis is a statistical data representation technique developed by to characterise multivariate data sets of the form , where each is an -vector with variables , i.e. . the algorithm represents every member of the set as a mixture ( a constrained linear combination ) of basis vectors that are _ pure types _ or _ archetypes _ of the data set .the archetypes are themselves mixtures of the data vectors .our exploratory study of archetypal analysis of galaxy spectra is justified by noting that the spectra of many galaxies appear to be a superposition of emissions from various _ populations _ or _ mechanisms_. identification of different archetypes with the spectral signature of these different populations or mechanisms promises to provide a natural representation of galaxy spectra .data representation techniques previously applied to galaxy spectra include principal component analysis ( pca ) ( ronen et al .1999 , and references therein ) , multiple optimised parameter estimation and data compression ( moped ) and the information bottleneck method ( ib ) .an objective of these representation techniques has been to identify the physical relationships between different emission signatures , and to attempt to uncover parameterizations describing these physical relationships .additionally , data representation techniques frequently produce a more compact representation of the data set and thereby compress the computation and data storage load .a critical review of both aspects of pca , moped and ib is given by .the eigenvectors determined by principal components analysis and its relatives generally do not resemble any member of the data and may be difficult to interpret . to overcome this ,cutler & breiman required that the basis vectors in archetypal analysis ( the archetypes ) be _ mixtures _ of the actual data points .in particular , archetypes are _ extreme members _ of the data set chosen in the following sense .consider the -dimensional polytope ( a region of -dimensional space enclosed by a set of hyperplanes ) corresponding to the convex hull ( the hypersurface defined by the minimum number of extreme data points ) of the data points ( hereinafter called the _ data convex hull _ ) .the vertices of this polytope are by definition data points , and as proven by any point on the polytope can be represented as a mixture of the data points .archetypes are chosen from the data mixtures lying on the data convex hull . since the typical number of archetypes used is smaller ( generally much smaller ) that the number of vertices of the data convex hull , the -dimensional polytope corresponding to the convex hull of the archetypes ( hereinafter the _ archetype convex hull _ )encompasses a smaller hypervolume than the data convex hull .data points lying inside the archetype convex hull are exact mixtures of the archetypes , while data points lying outside the archetype convex hull are only approximated .the difference reflects a loss of information arising from the use of a reduced representation , and is similar to the loss arising from truncating the number of eigenvectors used in pca .in section 2 of the paper we present a more extensive description of archetypal analysis . in section 3we demonstrate the application of archetypal analysis to extract the star formation history from model galaxy spectra and also the effects on archetypal analysis in the presence of various types of noise .finally , a discussion of future work is presented in section 4 .here we contextualise the mathematical basis of archetypal analysis using galaxy spectra , based on the development in . from a starting point of principal components analysis, we first present the mathematical methodology of archetypal analysis , and then use a schematic example to illustrate the differences between pca and archetypal analysis .suppose that we are given a set of galaxy spectra , each consisting of a measurement ( e.g. of flux density ) in wavelength bins , i.e. . for any given set of -vectors ( hereinafter the basis vectors ) , the linear combination that best approximates any given galaxy spectrum in the data set may be defined through the coefficients that minimize the error the minimize the squared distance between the data point and the represented point in -dimensional hyperspace .the optimal set of basis vectors to represent the entire data set may then be determined as the minimizer of the sum of all squared distances over the entire data set , suppose that the set of basis vectors is chosen to be an orthonormal set of axes. then expression [ equ : min ] minimizes the sum of the distances from the data points to the axes .this is equivalent to maximizing the sum of the projections onto the axes ( cf .2.3 , murtagh & heck 1987 ) given by , where * s * is the variance - covariance matrix of the data set , the minimizers of expression [ equ : max ] are the eigenvectors of * s * corresponding to the largest eigenvalues .when the data are centred on the mean ( ) , this corresponds to the principal component decomposition of principal component analysis ( pca ) .a basis vector derived by such principal component decomposition does not necessarily correspond to , or even resemble , any member or combination of members of the original data set .furthermore , it is not required that each data point is approximated as a mixture of the basis vectors , only that the representation globally minimizes the residual sum of squares .these features may make the physical interpretation of the principal components difficult .for instance , in applying pca to a set of galaxy spectra , the principal components ( eigenspectra ) may not resemble the spectrum of any observed galaxy , and may contain points that are negative , or which vary rapidly and unphysically with wavelength . furthermore , the representations of individual data points may contain negative or wildly varying weightings of the basis vectors .  as noted by ,these problems can be overcome to some extent .these authors applied pca to a set of galaxy spectra obtained in the 2df galaxy redshift survey and then formulated physically significant parameterizations from linear combinations of the first two eigenspectra . as they stated : ``_ _ in effect what we are doing when we utilise these linear combinations is rotating the axes defined by the pca to make the interpretation of the components more straightforward _the axes defined by pca potentially require rotation because pca itself determines orthogonal ( linearly independent ) eigenspectra , while the physical effects that dominate changes in the appearance of galaxy spectra are not necessarily independent in this way .in contrast to pca , archetypal analysis was designed with two distinct features that address the question of physical interpretation more directly : ( 1 ) the basis vectors are themselves required to be members or mixtures of members of the input data set , making an interpretation of the origin of the basis vectors more straightforward , and ( 2 ) the basis vectors are extreme data points lying on the data convex hull , allowing each member of the data set to be represented either exactly or approximately as a mixture of the basis vectors .these properties are achieved by introducing two additional sets of constraints to the general formalism of pca .begin by selecting a set of basis vectors that are `` linear combinations '' of the data values , with so the basis vectors `` resemble '' the data , and so that they are a `` mixture '' of the data . then determine coefficients that allow the set of data points to be `` well represented '' by the basis vectors via minimizing finally , use the constraints so that each data point is a `` physically meaningful '' combination of basis vectors , and so that data points are `` mixtures '' of basis vectors .the set of basis vectors that optimally describes the data set are then minimizers of the _ residual sum of squares _ , or basis vectors determined by these sets of constraints are called _an archetypal representation involves the determination of the two sets of coefficients and .cutler & breiman prove that archetypes lie on the data convex hull , and show how the coefficients can be determined iteratively using an alternating constrained least - square algorithm .as they note , unlike pca , archetypal analysis does not nest , nor are archetypes orthonormal , so that existing archetypes change to capture in progressively better ways ( subject to the advent of noise ) the shape of the data set as a larger set of archetypes is determined .it is instructive to visualise the difference between archetypal analysis and pca using an artificial two - dimensional ( i.e. ) data set .such a data set is exhibited as the points in fig .1 . using pca ,the data are represented by their projections ( coordinates ) referenced to principal components pointing in the two orthogonal directions of maximum data variance , illustrated as the dot - dashed axes .because there are only two independent variables , two principal components will provide an exact reconstruction of every data point .archetypal analysis is a representation using basis vectors lying on the data convex hull . in fig .1 the dashed hexagon traces the data convex hull , like a rubber band stretched around the data set .the archetypal algorithm locates archetypes on this data convex hull that form ( for the given number of archetypes ) an optimal model of the convex hull .for example , if we represent the artificial data set with three archetypes , the archetypal algorithm determines 3 points on the data convex hull that outline a triangle , which is the archetype convex hull .every data point within the triangle is exactly described ( zero residual ) by mixtures of the archetypes .data points lying outside the triangle are represented as the nearest point on the triangle , as illustrated in fig .1 . the distance between the original ( external ) data point and the represented data point is the _ residual_. the sum of these residuals is minimized ( expression [ equ : aaeq ] ) in the search for the optimal set of archetypes . for this particular data set ,the data convex hull is a hexagon and six archetypes would provide an exact reconstruction of every data point .there are three generic issues associated with archetypal analysis that are addressed in the following sections :  1 .the quality of an archetypal representation of multi - dimensional data depends on the number of archetypes adopted .we demonstrate the outcome of using different numbers of archetypes .archetypes are located on the data convex hull and archetypal analysis is potentially sensitive to outliers .we investigate the effects of noise - generated outliers on archetypal analysis .3 .  the input data set sometimes needs to be standardised before archetypal analysis .we discuss our experience in this respect .we apply archetypal analysis to study evolving star formation in model galaxies .we construct a set of spectra of model galaxies by assuming that each galaxy has experienced episodes of star formation with quasi - random amplitudes at past epochs .motivated by the analysis of reichardt et al .( 2001 ) , we model the galaxies as superpositions of stellar populations formed instantaneously , and now observed at ages of 0.007 , 0.045 , 0.3 , 2 and 14 gyr .the _ fiducial spectra _ at these ages are derived using the pgase - ii code of , adopting a initial mass function in the range [ 0.1,120 ] and the stellar library of .spectra were synthesised over the optical wavelength range 365 711 nm , with 2 nm spectral bins ( 174 bins ) .since the shapes of the nebular emission lines are not predicted by pgase - ii we simulate them using gaussians with a fwhm 5 nm .this treatment is not required for archetypal analysis , but allows convenient visual presentation .  to generate a model composite spectrum ,the five fiducial spectra are first normalised to the same integrated flux . a composite spectrumis then generated via the superposition of fiducial spectra using a set of five quasi - random weighting factors . to satisfy the constraint that the sum of the five weighting factors be unity whist having quasi - uniform distribution, we generate the weighting factors according to the relationship ( 1-\sum_{n=1}^{i-1}w_n) ] denotes an uniformly distributed random number in [ 0,1 ] .up to 5 random numbers are generated this way , or they are set to zero when .these pseudo - random values are then randomly assigned as the five weighting factors .since one of the five random values is never scaled , it retains an uniform distribution within [ 0,1 ] while the sum is constrained to be unity .the sample of model galaxies is generated by repeating the process with different random numbers for each galaxy .in addition to determining the spectrum for each model galaxy , we also calculate the standard photometric colours . the distribution of the model galaxies in the two - colour plane is shown in fig .the fiducial spectra themselves are displayed in the first 5 panels across the top row of fig .the sixth panel exhibits an example of a composite spectrum , composed from ratios of 0.14 , 0.66 , 0.00 , 0.13 , and 0.08 of the fiducial spectra .having generated sets of composite spectra we then apply archetypal analysis .sample sizes of 50 , 100 and 900 composite spectra have been explored .they show comparable results , and we present only the analysis of 900 composite spectra .we first explore the use of different numbers of archetypes . to illustrate the key results , fig .2 shows a cloud of data points corresponding to 900 composite spectra projected onto the two - colour plane .the corresponding projections of the five fiducial spectra are also shown .these are identical , to machine precision , to the projections of the archetypes when _ five _ archetypes are used . the projection of the archetype convex hull on the two - colour plane can be determined by noting that each edge of this polytope is a mixture of the archetypes lying at either end , with no contribution from the other three archetypes . the dashed line traces this projection of the 5-archetype convex hull .its curvature arises from the non - linear relationship between spectral energy distribution and colour .  when _ four _ archetypes are used to represent the data , two of the archetypes remained essentially unchanged ( 7 myr and 45 myr ) while the other two are mixtures of the three remaining fiducial spectra in the forms 81% 300 myr + 19% 2 gyr and 14% 2 gyr + 85% 14 gyr .three _ archetypes are used , the 7 myr fiducial spectrum is recovered , while the other two are the intermediate representations 80% 45 myr + 20% 300 myr and 44% 2 gyr + 56% 14 gyr . the four archetype and three archetype representations produce the archetype convex hulls shown as projections in fig .2 .  in summary , the use of 5 archetypes recovers exactly the 5 fiducial spectra . when fewer archetypes are used , one or more of them represents the `` younger '' fiducial spectra with high purity , while the other archetypes appear as intermediate representations of the older fiducial spectra .this result is due primarily to the presence of prominent spectral features ( especially emission lines ) in the younger spectra , and the absence of marked differences between the older fiducial spectra .archetypal analysis is designed to focus attention on the outliers of the data set .this emphasis on outliers raises the question of sensitivity to noise , which potentially makes a larger contribution in the outliers of the data set .we examine this question in this section , showing that archetypal analysis is robust in the presence of the types of noise characterising astronomical spectroscopy .we explore the effects of noise by adding four distinct kinds of contamination to the sample of 900 synthetic spectra described above .they are ( i ) random noise with gaussian statistics , ( ii ) random noise with poisson statistics , ( iii ) narrow , strong contamination dubbed `` cosmic rays '' and ( iv ) a pervasive fixed pattern dubbed `` sky subtraction residual '' .the amplitude of the random noise models was selected to provide data with a characteristic signal - to - noise ( s / n ) ratio of 10 . in particular , for poisson random noise , the mean noise amplitude added to a any wavelength bin is determined from the flux of that wavelength bin .here we have ignored sky spectrum contributions .we simulated a 5% cosmic ray contamination rate , assuming poisson statistics for the number of contaminating events per synthetic spectra . in this way ,44/900 spectra suffered single - pixel events , represented as a spike of three times the amplitude of the peak flux density in the corresponding spectrum . in one spectrum ( 1/900 )two spikes were added .the wavelength of each spike was chosen at random .sky subtraction residuals were simulated using the sky spectral profile published by . in the analysis of real spectra , sky lines would be quasi - randomly redshifted relative to the galaxy lines , an effect we model by randomly blueshifting the sky - line spectrum with the range .the shifted sky spectra were rebinned in wavelength and scaled to represent random sky subtraction errors with relative amplitudes of ( uniform distribution ) , and added to the synthetic spectra .the combination of random redshifts and wavelength rebinning reduces the wavelength span from 174 to 139 spectral bins .3 and fig .4 illustrate respectively the effects of noise on ( a ) the archetypes themselves , and ( b ) the coefficients that represent the data in terms of the archetypes . in fig .3 , the first five panels of the top row display the fiducial spectra , which are essentially identical to the five archetypes found in the absence of noise . beneath this row ,each column shows the corresponding archetype derived from the contaminated data set , as well as a difference spectrum with respect to fiducial spectrum .  with the exception of the `` 2 gyr , cosmic rays '' case which is discussed below, there is an evident correlation between the `` input '' and `` derived '' weights , but there is also substantial scatter .some of this scatter reflects the propagation of error into the archetypes themselves , although the recovered archetypes are generally quite similar to those derived in the absence of contamination .most of the scatter arise from the propagation of error in fitting each spectrum as a mixture of archetypes .the appearance of large scatter for the `` older '' spectral components reflects the fact that the three `` older '' archetypes are similar to one another .the rightmost panel in the top row of fig .3 illustrates one of the 900 synthetic spectra . in the column below , the corresponding contaminated synthetic spectra are exhibited , as well as the difference from the uncontaminated spectrum .the chosen spectrum is one that suffers a single cosmic ray event , in that form of contamination . to the right of these panelsare the corresponding spectra obtained as the weighted sum of the archetypes . in all cases ,the archetypal representation preserves spectral detail while reducing the amplitude of the contamination in the original spectrum .this result confirms the potential efficacy of archetypal analysis for compacting the size of a data set .the results of archetypal analysis on the inferred statistical properties of the entire sample of 900 spectra is illustrated in fig .4 . in part 4(a ) , each small panel is a scatter plot of points whose abscissa are the weights applied to the fiducial spectra in forming synthetic spectra , and whose ordinates are the derived weights of the corresponding archetypal representation .deviations from perfect correlation reflect the propagation of errors due to contamination of the spectra into the inferred star formation history of each model galaxy . in fig .4(b ) , the corresponding distribution functions of the difference between the `` input '' and `` derived '' weight are exhibited .  the results illustrated in fig .4 reveal both the strengths and weaknesses of archetypal analysis .the tight correlations revealed in the top row reflect the essentially perfect correspondence between the fiducial spectra and the archetypes when uncontaminated data are analysed using five archetypes .the relatively tight correlations in the first two columns show that even in the presence of significant contamination , the spectral trace of `` younger '' populations can be measured with high precision .the distribution of points in the 300 myr to 14 gyr columns reflects the difficulty of extracting accurately the spectral trace of single , old star formation events in data having relatively poor s / n ratio .this problem is well understood , and the results are unlikely to be inferior to those that might derived from other methods .it is illuminating to interpret these results in terms of the topology of the data and archetype convex hulls .the data can be regarded as 900 points ( the number of spectra ) in 174- ( or 139- ) dimensional ( the number of wavelengths ) space . when weak or moderate levels of gaussian or poisson noise are present , or a weak sky spectrum suffering random wavelength shifts are present , the data points including the vertices of the data convex hull are displaced by distances that are relatively small compared with their median distance from the origin .consequently , the archetype convex hull differs only mildly from the corresponding archetype convex hull in the absence of contamination .the archetypes resemble those of the uncontaminated data , and because the number of archetypes is far smaller than the number of wavelengths , data representations are significantly smoothed .  the `` 2 gyr , cosmic rays '' panels in fig . 3 and fig .4 are different from all other panels . in this case, archetypal analysis has derived as one of the archetypes a spectrum with two cosmic ray events in it .because no other spectrum in the sample contains this pattern , the archetype does not contribute significantly to any of the data representations .the analysis thus yields results that closely approximate those that would be found were _ four _ archetypes used .this case illustrates the sensitivity of archetypal analysis to outliers with strong and unusual noise characteristics .however , it also shows that the method is robust even in the presence of such noise , and of course reveals a way to identify and excise noisy data in a systematic manner .in view of its emphasis on representing data sets as mixtures of pure types , archetypal analysis offers considerable promise in analysing galaxy spectra that are , presumably , superpositions of the spectra from different emission processes ( agn , nebulae , stars ) and stellar populations of different ages .it provides near - perfect accuracy and precision in extracting star formation histories from noise - free synthetic data .the presence of noise , particularly with strongly non - gaussian statistics , compromises this high accuracy but our studies suggest that the technique can be applied with success to galaxy spectral data with typical noise properties .there are extensions to archetypal analysis that we have explored in a limited way .for example , the results of archetypal analysis depend to some extent on the pre - conditioning or standardisation of the input data . we found that with cosmic ray contamination , the best results are obtained when the data are standardised , while this seems unnecessary with gaussian or poisson random noise and for the sky subtraction residual set .we have shown that cosmic ray events may be effectively identified through their obvious presence in some archetypes .this invites exploration of an iterative approach to data conditioning in which non - physical components of the archetypes are progressively edited from the data set , providing a potentially powerful new methodology for peeling away contaminating components in the data .archetypal analysis was designed to seek a representation using members of the data set . however , in some cases there are preferred _ models _ for the data .we have confirmed that archetype analysis is still applicable in this case , with the models replacing the archetypes .used in this way , archetypal analysis shares many of the features of moped .we are currently using archetypal analysis to analyse real galaxy spectra from the sample and the 2dfgrs survey .we thank matthew colless , nicole bordes and bernard pailthorpe for advice and comments .colless m. , et al .( the 2dfgrs team ) , 2001 , , 328 , 1039 .cutler a. , breiman l. , 1994 , , 36 , 338 .fioc m. , rocca - volmerange b. , 1999 , preprint ( astro - ph/9912179 ) .heavens a. , jimenez r. , lahav o. , 2000 , , 317 , 965 .kennicutt c.j . , 1992 , , 79 , 255 .lahav o. , 2001 , in banday a.j . , zaroubi s. , bartelmann m. , eds , proc .mpa / eso / mpe conference at garching , germany , 2000 . mining the sky .heidelberg : springer - verlag , p.33 .preprint ( astro - ph/0012407 ) .lejeune t. , cuisinier f. , buser r. , 1997 , , 125 , 229 .lejeune t. , cuisinier f. , buser r. , 1998 , , 130 , 65 .madgwick d.s . , lahav o. , 2001 , in treyer , tresse , eds , where s the matter ?tracing dark and bright matter with the new generation of large scale surveys . frontier group .preprint ( astro - ph/0110220 ) .murtagh f. , heck a. , 1987 , in boyd r.l.f .et al . , eds , astrophysics and space science library , multivariate data analysis .d. reidel pub . ,dordrecht , holland , p.19 .reichardt c. , jimenez r. , heavens a , 2001 , , 327 , 849 .ronen s. , aragn - salamanca a. , lahav o. , 1999 , , 303 , 284 .salpeter e. , 1955 , , 121 , 161 .slonim n. , somerville r. , tishby n. , lahav o. , 2001 , , 323 , 270 .

Abstract: archetypal analysis represents each individual member of a set of data vectors as a mixture ( a constrained linear combination ) of the _ pure types _ or _ archetypes _ of the data set . the archetypes are themselves required to be mixtures of the data vectors . archetypal analysis may be particularly useful in analysing data sets comprising galaxy spectra , since each spectrum is , presumably , a superposition of the emission from the various stellar populations , nebular emissions and nuclear activity making up that galaxy , and each of these emission sources corresponds to a potential archetype of the entire data set . we demonstrate archetypal analysis using sets of composite synthetic galaxy spectra , showing that the method promises to be an effective and efficient way to classify spectra . we show that archetypal analysis is robust in the presence of various types of noise .   methods : data analysis methods : statistical galaxies : evolution galaxies : fundamental parameters galaxies : stellar content


modelling and estimating the underlying spatial dependence structure of the observed data is a key element to spatially interpolate data and perform simulations .its description is commonly carried out using statistical tools such as the covariance or variogram calculated on the whole domain of interest .simplifying assumptions are often made on the spatial dependence structure .they include stationarity assumption where the second order association between pairs of locations is assumed to depend only on the vector between these locations .however , it has become increasingly clear that this assumption is driven more by mathematical convenience than by reality . in pratice, it often happens that the stationarity assumption may be doubtful .non - stationarity can occur due to many factors , including specific landscape and topographic features of the region of interest or other localized effects .these local influences can be observed by computing local variograms , whose characteristics may vary across the domain of observations .more practionners choose to replace this convenient assumption with more realistic assumption of non - stationarity .they subdivide the modelling area into stationary domains , thereby increasing the required professional time for modelling and potentially producing disjointed domains that are globally inconsistent .various approaches have been developped over the years to deal with non - stationarity through second order moments ( see , for a review ) .one of the most popular methods of introducing non - stationarity is the convolution approach .it consists of taking a spatial white noise , then averaging it using weights that vary spatially to thereby obtain a non - stationary random function . in this way , the resulting spatial dependence structure is non - stationary . the covariance or variogram as a whole is allowed to vary in some way between different locations . use spatially varying gaussian kernel function to induce a non - stationary covariance structure .the resulting covariance function has a closed - form , but is infinitely differentiable , which may not be desirable for modelling real phenomena . use a family of spatially varying modified bessel kernel functions to produce a non - stationary covariance function with local smoothness characteristics that are similar to the stationary matrn covariance functions class .one limitation of this approach is that the non - stationary covariance function does not have a closed - form and in general can only be evaluated by numerical integration . moreover, this approach does not take into account a spatially varying anisotropy .furthermore , explicit expressions of some non - stationary covariance functions class that include a non - stationary version of the stationary matrn covariance function have been introduced by .this was further developed by , and .however , these classes of analytical non - stationary covariance functions do not directly derived from a random function model like the convolution representation or the spectral representation . thus , this does not facilitate their understanding and interpretability .it is helpful to have a constructive approach for random functions admitting such closed - form non - stationary covariance functions .moreover , the inference of the parameters of these latter remains a critical problem . enumerate some difficulties and suggest some possible methods including the moving windows approach based on the local variogram or local likelihood . mention two typical problems arising with moving windows method .first , the range of validity of a stationary approximation can be too small to contain enough local data to estimate the spatial dependence structure reliably .second , it can produce non - smooth parameter estimates , leading to discontinuities on the kriging map which is undesirable in many cases . develop a weighted local likelihood approach for estimating parameters of the non - stationary version of the stationary matrn covariance function .this approach downweight the influence of distant observations and allows irregular sampling observations .a few drawbacks of their approach are the computational burden of inverting covariance matrices at every location for parameter estimation and the gaussian distributional assumption for analytical tractability .  in this work, we are interested primarily in the modelling of second order non - stationary random functions , using a constructive approach in the spirit of convolution model .we present a new model for second order non - stationary random functions as a convolution of an orthogonal random measure , with a spatially varying stochastic weighting function .this is an extension of the classical convolution model where a deterministic weighting function is used .by this modelling approach , the resulting class of non - stationary covariance functions is very general , flexible and allows to retrieve classes of closed - form non - stationary spatial covariance functions known from the literature , for a suitable choices of the stochastic weighting functions family .these latter classes show locally a stationary behaviour and their parameters are allowed to vary with location , yielding local variance , range , geometric anisotropy and smoothness . in this way, it naturally allows stationary local covariance parameters to be knitted together into a valid non - stationary global covariance .thus , we establish a direct link between the existing explicit classes of non - stationary covariance functions and the convolution approach .this construction bears some resemblance to the moving average model with stochastic weighting function introduced by to generate stationary covariance functions like cauchy and matrn families .secondly , we develop a procedure of estimating parameters that govern these classes of closed - form non - stationary covariance functions under a single realization and local stationarity framework , through a step by step approach .first , we compute local variograms by a non - parametric kernel estimator .then , it is used in a weighted local least squares procedure for estimating the parameters at a reduced set of representative points referred to as anchor points .finally , a kernel smoothing method is used to interpolate the parameters at any location of interest .the estimation method proposed is free distribution and no matrix inversions or determinants calculation are required .it is applied to two real datasets : soil and rainfall data .the outline of the paper is as follows : a generalized convolution model is described through its basic ingredients and main properties in section [ sec2 ] . in section [ sec3 ] ,we show how the model allows us to construct explicit classes of non - stationary covariance functions . in section [ sec4 ] ,parameter inference of these explicit classes of non - stationary covariance is detailed .we tackle the spatial predictions and conditional simulations in section [ sec5 ] . in section [ sec6 ] ,two real datasets are used to illustrate the performances of the proposed method and its potential .finally , section [ sec7 ] outlines concluding remarks .let be a random function defined on a fixed continuous domain of interest of the euclidean space . the random function is defined as follows : where is an orthogonal random measure on with ; is the dirac measure at and . is a family of independent and identically distributed random variables , and independent of , taking their values in the subset of according to a probability measure . is a family of square - integrable functions on , for all . for fixed , is a random variable whose first two moments are assumed finite and integrable on .  under these assumptions ,the stochastic convolution defined in ( [ eq1 ] ) is well - defined in , that is . to fix ideas , here are two examples of orthogonal random measure which can be used to simulate the random function  :  1 .consider a homogeneous poisson point process on with intensity .let the number of points that falls in a measurable set .the function defined by is an orthogonal random measure , where is the -dimensional lebesgue measure . 2 .gaussian white noise on is an orthogonal random measure such that : is normally distributed with mean and variance ; when and are disjoint and are independent and .[ prop1 ] under the model specified in ( [ eq1 ] ) , we have the following moment properties :   the proposition [ prop1 ] gives a very general and flexible class of non - stationary covariance functions .this latter is defined at any two locations and is valid on , that is to say positive definite .therefore , the random function is non - stationary through its second order moment .the proof of the above properties is deferred to the [ appendix - sec1 ] .note that the class of non - stationary covariance functions obtained in ( [ eq2b ] ) is a mixture of convolutions .it allows to generate different forms of regularity at the origin including low degree of regularity .indeed , the operation of randomization by the positive measure produces models that can be less regular than the base model induced by , but never more regular .consider a mapping from to the set of real - valued positive definite -dimensional square matrices .let be a continuous isotropic stationary correlation function , positive definite on , for all .let and .consider the families of functions and for all such that and is the -variate gaussian density function centered at , with covariance matrix .the following propositions and corollaries are proven in the [ appendix - sec1 ] .[ prop2 ] if , then the class of non - stationary covariance functions generated through is :   proposition [ prop2 ] provides a very general class of non - stationary covariance functions similar to that proposed by . the corollaries which will followshow that appropriate choices for the function and the positive measure give explicit classes of non - stationary covariance functions known from the literature .we thus establish a direct link between these latter and the convolution model .[ cor1 ] if and , for a finite positive measure on such that + , then the class of non - stationary correlation functions generated through is :   we thus find the class of closed - form non - stationary correlation functions introduced by .the intuition behind this class is that to each input location is assigned a local gaussian kernel matrix and the correlation between two targets and is calculated by averaging between the two local kernels at and . in this way, the local characteristics at both locations influence the correlation of the corresponding target values .thus , it is possible to account for non - stationarity .it is done by specifying the mapping which models the anisotropy of the correlation function .the resulting kernel matrix at each point is interpreted as a locally varying geometric anisotropy matrix .it controls the anisotropic behavior of the random function in a small neighborhood around .the positive measure defined in the corollary [ cor1 ] is the measure associated with the schoenberg representation of the correlation function which is a continuous , positive definite and radial function on , for all  .an approach based on a class of non - stationary correlation models defined in does not allow the use of isotropic stationary correlation models that are not valid in all dimensions .thus , we lost the isotropic stationary correlation functions with compact support such that the spherical model ( valid in ) .[ ex1 ] by a specific choice of the positive measure in corollary [ cor1 ] , we obtain some subclasses of closed - form non - stationary correlation functions :  1 . ; 2 . ; 3 . , with the density of the gamma distribution , where is the modified bessel function of second kind with order  ; 4 . , with the density of the inverse gamma distribution   the example [ ex1 ] shows that a suitable choice of the positive measure produces non - stationary versions of some well known stationary correlation functions : gaussian , exponential , matrn and cauchy .these examples can also be deduced from the following corollaries [ cor2 ] and [ cor3 ] .[ cor2 ] if and , with the density of gamma distribution , then the class of non - stationary correlation functions generated through is : where ; is the modified bessel function of second kind with order .  from corollary [ cor2 ], we retrieve the non - stationary version of the stationary matrn correlation function introduced by .this one allows both local geometric anisotropy and degree of differentiability to vary spatially .when the function is constant , we obtain the non - stationary matrn correlation function with constant regularity parameter shown in example [ ex1 ] . however , the associated function and positive measure are different .[ cor3 ] if et , with the density of inverse gamma distribution , then the class of non - stationary correlation functions generated through is : where .the corollary [ cor3 ] gives a non - stationary version of the cauchy correlation function introduced by .both the local geometric anisotropy and long - range dependence are allowed to vary spatially .for the constant long - range dependence parameter , we obtain the cauchy non - stationary correlation function shown in example [ ex1 ] .  from above formulas , it is straightforward to construct a closed - form non - stationary covariance function including a standard deviation function . doing so , the non - stationary covariance is defined as follows : where is a standard deviation function and a closed - form non - stationary correlation function .  notethat , the closed - form non - stationary covariance functions defined in have the desirable property of including stationary covariance functions as a special case .in this section , we propose a procedure for estimation of spatially variable parameters that govern the classes of closed - form non - stationary second order structure model presented in section [ sec3 ] .although it could be used for all of these classes , we focus on the one defined in which is flexible enough for many applications . the proposed estimation methodology allows us to deal simultaneously with other types of non - stationarity : mean and variance .thus , we consider an extended model built from the model specified in which includes a non - stationarity both in mean and variance . the estimation procedure is based on the local stationarity assumption and is achieved using a three - step estimation scheme .first , a non - parametric kernel estimator of the local variogram is built .then , it is used in a weighted local least squares procedure to estimate parameters at a representative set of points referred to as anchor points .next , a kernel smoothing approach is used to interpolate the parameters at any location of interest .   to include a non - stationarity both in mean and variance, we consider a random function described as follows :    where is an unknown fixed function . is an unknown positive fixed function . is a zero - expectation , unit variance random function with correlation function defined in :   thus , the random function carries the spatial dependence structure of the random function .the model formulation defined in leads to a first and second order moments written as follows :   from the expressions ( [ eq10a ] ) and ( [ eq10b ] ) , we see that the non - stationarity of the random function is characterized by the parameters , and defined at any location of the domain of interest .let be a vector of observations from a unique realization of the random function , associated to known locations .the goal is to use the data to infer the standard deviation function , the correlation function determined by and the mean function .next , we use them to predict the value of the random function at unsampled locations . from now on , without loss of generality , we assume that .the estimation of the parameters , and relies on the quasi - stationarity or local stationarity assumption which allows certain simplifications .specifically , we work with the following definition of local stationarity introduced by .   a random function will be locally stationary if it has an expectation and covariance such that :  1 . is a very regular function varying slowly in space at the scale of the available information ; more precisely , can be considered as constant in a neighborhood of ; 2 .  there is a function of three arguments such that and such that , for a given , is a very regular and slowly varying ( in the same sense as in 1 . ) function of the two arguments and .in other words , for locations and not too far from each other , only depends on , as is if the covariance was stationary .the intuitive idea behind this definition is that if a random function is locally stationary , then at any location there exists a neighborhood where the random function can be approximated by a stationary random function .thus , ; where is a stationary covariance function and the limit represents the radius of the quasi - stationarity neighborhood . in this way , the parameters are assumed to be very smooth functions which vary slowly over the domain .the expectation of the random function being approximately equal to a constant inside the quasi - stationarity neighborhood , the resulting local covariance structure at any location is written as follows :   in more practical terms , we can define moving neighborhoods within which the expectation and the covariance can be considered stationary , and where the available information is sufficient to make the inference .the quasi - stationarity assumption is a compromise between the distances of homogeneity of the studied phenomenon and the density of the available information .indeed , it is always possible to reach the stationarity by reducin the size of neighborhoods .but most of these neighborhoods will contain almost no data ; therefore inference of parameters will not be possible in these neighborhoods .   locally , the non - stationary spatial dependence structure defined in is thus reduced to an anisotropic stationary one .the anisotropy function is parameterized using the spectral decomposition . by this way, the positive definiteness is guaranteed and the locally varying geometric anisotropy can be captured . precisely , at any location , , where is the diagonal matrix of eigenvalues and is the eigenvector matrix .then , we have   and   doing so , the scale parameter of the isotropic stationary correlation function in the expression is set to one to avoid any overparameterization .at each point , the square roots of the eigenvalues control the local ranges and the eigenvector matrix specify the local orientations .thus , the anisotropy function is characterized by the functions , and .it is convenient to estimate the second order spatial structure through the variogram . under the local stationarity framework, we define a non - parametric kernel moment estimator of the stationary local variogram at a fixed location and lag , as follows : }^2}{2\sum_{v(\mathbf{h})}k^{\star}_{\epsilon}(\mathbf{x}_0,\mathbf{s}_i)k^{\star}_{\epsilon}(\mathbf{x}_0,\mathbf{s}_j ) } , \\|\mathbf{h}\| \leq b,\ ] ] where the average is taken over , the set of all pairs of locations separated by vector ; are standardized weights ; is a non - negative , symmetric kernel on with bandwidth parameter . for irregularly spaced data there ,are generally not enough observations separated by exactly .then , is usually modified to , where is a tolerance region of surrounding .this moment estimator of the local variogram at any location is a kernel weighted local average of squared differences of the regionalized variable .the kernel function is used to smoothly downweight the squared differences ( for each lag interval ) according to the distance of these paired values from a target location .we assign to each data pair a weight proportional to the product of the individual weights .observation pairs near to the target location have more influence on the local variogram estimator than those which are distant .  note that when and ; where is the diameter of the domain of interest , we obtain the classical matheron moment estimator for a global stationary structure . for and ,we get the moving window estimator .  to calculate the non - parametric kernel estimator , we opt for an isotropic stationary gaussian kernel : .the latter has a non - compact support and therefore considers all observations .thus , the local variogram estimator is not limited only to the local information , distant points are also considered .this avoids artifacts caused by the only use of observations close to the target location .it also reduces instability of the obtained local variogram at regions with low sampling density .furthermore , it provides a smooth parameter estimate and then is compatible with the quasi - stationarity assumption . concerning the size of the quasi - stationarity neighborhood , it is set with respect to the bandwidth .we take such that the standard deviation of the isotropic stationary gaussian kernel match the isotropic stationary uniform kernel ( with compact support ) .another possible choice for is to take a quantile of the isotropic stationary gaussian kernel ( e.g. ) or full width at half maximum ( ) .we want to estimate the vector of structural parameters and the mean parameter at any location of the domain of interest .the estimation of the parameters vector which characterizes the stationary local variogram at a fixed location are found via the following minimization problem : where is the product term by term ; }^t ] ; }^t ] ; are given lag vectors ; is the vector of unknown parameters and is an open parameter space .  the estimation of the structural parameters and depends on the bandwidth parameter through the kernel function .the bandwidth parameter controls the range of validity of the stationary approximation and its selection is adresssed in section [ ssec3 ] .note that the estimation of the structural parameters does not require the prior estimation of the mean function .moreover , no model is specified for this latter .concerning the estimation of the parameter , the mean at a fixed location is approximatively equal to a constant inside the quasi - stationarity neighborhood .thus , using the estimate of the vector of structural parameters obtained in , is estimated explicitly by a local stationary kriging of the mean . specifically we have : where  ] .   for the spatial prediction purpose , one needs to compute the parameters , and at prediction and observation locations . in practice, it is unnecessary to solve the minization problem at each target location .indeed , doing so is computationally intensive and redundant for close locations , since these estimates are highly correlated . to reduce the computational burden ,the proposed idea consists in obtainning the parameter estimates only at some reduced set of representative points referred to as anchor points defined over the domain .then , using the estimates obtained at anchor points , a kernel smoothing method is used to make available estimates at any location of interest . since the parameters and are supposed to vary slowly in space ( quasi - stationarity ), the nadaraya - watson kernel estimator seems appropriate , in addition to being relatively simple .however , other smoothers can be used as well ( local polynomials , splines , ) .the nadaraya - watson kernel estimator of at any location is given by : where is a kernel on ; is a smoothing parameter ; are the raw estimates of parameter at anchor points .we similarly define kernel smoothing estimator , and at any location . for the specific case of the orientation parameter , the kernel estimator at any location given through the minimization of the following criteria : where are the raw estimates of parameter at anchor points ; is the -dimensional sphere of unit radius its centre at the origin ; is a distance between two orientations . for , we can take the .  since, the parameters , and are assumed to vary slowly and regularly across the domain of interest ( local stationarity ) , we choose as an isotropic stationary gaussian kernel .the selection of its smoothing parameter is discussed in section [ ssec3 ] .   in the proposed method, a crucial point is the determination of the bandwidth parameter used in the computation of the local variogram non - parametric kernel estimator defined in .indeed , the size of the local stationarity neighborhood is expressed in terms of the bandwidth parameter .we are also interest in the choice of the bandwidth used to smooth the parameter estimates .concerning the bandwidth parameter , in some applications it may be appropriate to choose it subjectively , but in general , it is desirable to have available methods for choosing it automatically from the data . the data - driven method used to select the bandwidth consists of leaving out one data location and using a form of cross - validation . because the estimation of the spatial dependence structure is rarely a goal per se but an intermediate step before kriging , we want to choose the bandwidth that gives the best cross - validation mean square error ( mse ) .more explicitly , we minimize with respect to the leave - one - out cross - validation score : where denotes the spatial predictor computed at location using all observations except . the prediction method is described in section [ ssec4 ] .the choice of the smoothing bandwidth associated to the parameter is done using the following cross - validation criteria : where are respectively the raw and smoothed estimates of the parameter at anchor points .the cross - validation score function is minimized over a grid of smoothing bandwidths to choose the optimal value .theoretically , smoothing bandwidths associated to each local parameter can be different .our numerical examples indicate that choosing the same smoothing bandwidth for all local parameters in order to reduce the computational burden , makes little difference in terms of prediction performance .this remark was already highlighted by .the main purposes of modelling and estimating the spatial dependence structure is to spatially interpolate data and perform conditional simulations .the expected benefit using the closed - form non - stationary covariance model is to obtain spatial predictions and variance estimation errors more realistic than those based on a inadequate stationary covariance . in this section ,a description of kriging and conditional simulations based on the non - stationary model is presented .let the non - stationary covariance function of the random function and its mean .given the vector of observtaions at fixed locations , the point predictor for the unknown value of at unsampled location is given by the optimal linear predictor :   the kriging weight vector  ] ; }$ ] .   despite its optimality , the estimator obtained by kriging smoothes reality .thus , the spatial texture of the reality ( captured by the covariance or variogram ) can not be reproduced from data points only . however , in some situations , it is more important to reproduce the spatial variability than obtain the best accuracy .we want to have a realization which has the same degree of variability than the actual phenomenon and coincides with observations at data points . to do this, we use conditional simulations .  herewe assume that is a gaussian random function , with mean and non - stationary covariance structure .we want to simulate at a large number of locations a gaussian random function with same mean and covariance , and ensure that the realization honors the observed values .this can be accomplished from an unconditional simulation of the random function as follows :  1 .  realize a unconditional simulation of the random function ; 2.  carried out a simple kriging of from its values taken at the data points , using and ; 3 .add the unconditional simulation and the result of kriging .we have , where is a gaussian random function with zero expectation , unit variance and non - stationary correlation function .thus , to simulate the gaussian random function ( step 1 of the previous algorithm ) , we need to known how we can simulate .simulation of the gaussian random function can be carried out using a propagative version of the gibbs sampler proposed by .this algorithm allows to simulate a gaussian vector at a large number of locations ( comparatively to the existing classical algorithms such as cholesky method or gibbs sampler ) without relying on a markov assumption ( it does not need to have a sparse precision matrix ) .the algorithm proposed in requires neither the inversion nor the factorization of a covariance matrix .note that simulation methods such as spectral method or turning bands method are not adapted to the non - stationary case . the representation that underlies these methods relies on the stationarity assumption .formally , the algorithm proposed by is outlined below :  let be a standardized gaussian vector , with covariance matrix .the underlying idea is that instead of simulating the vector , we can simulate the vector .this latter is a gaussian vector with covariance matrix . noting that the inverse of is precisely , the gibbs sampler is applied to and is updated accordingly .this gives the following algorithm :  1 .set ; 2 .generate ; 3 .  generate and take for each ; 4 .take and go to 2 .thus , at each step of the algorithm , a gaussian value is assigned randomly to a pivot , and then spread to the other components .in this section , we illustrate the advantages of our proposed approach and its potential on two datasets : a soil data and a rainfall data .a comparison scheme of kriging under stationary and non - stationary models is carried out through a validation sample .next , we compare prediction and prediction standard deviation maps on a fine grid . the aim is to compare the behaviour of the models in terms of spatial interpolation and associated uncertainty .we now study a dataset coming from a soil gamma radiometric potassium concentration ( in shots per second , cps ) survey in the region of the hunter valley , nsw , australia .this data set was used by to illustrate their approach .we have a training data ( 537 observations ) which serves to calibrate the model and a validation data ( 1000 observations ) which serves only to assess the prediction performances .raw estimates of parameters , and at anchor points are shown respectively on figures [ fig1b ] , [ fig1c ] and [ fig1d ] .they are based on the non - stationary exponential covariance function ( example [ ex1 ] ) . concerning the estimated anisotropy function at anchor points ,it is represented by ellipses as shown in figure [ fig1d ] .indeed , covariance ( positive definite ) matrices have an appealing geometrical interpretation : they can be uniquely identified with an ellipsoid . based on these estimates , non - stationarity in the datais quite visible .especially , from figure [ fig1d ] where we can clearly see the spatially varying azimuth .such directional effects are also quite apparent on data ( figure [ fig1a ] ) .note that the stationary approach has not detected a global geometric anisotropy .0.35  at anchor points ; ( c ) estimated variance function at anchor points ; ( d ) estimated anisotropy function at anchor points where the ellipses were scaled to ease vizualisation .( potassium concentration data)[fig1],title="fig:",scaledwidth=100.0% ]  0.35  at anchor points ; ( c ) estimated variance function at anchor points ; ( d ) estimated anisotropy function at anchor points where the ellipses were scaled to ease vizualisation .( potassium concentration data)[fig1],title="fig:",scaledwidth=100.0% ]  0.35  at anchor points ; ( c ) estimated variance function at anchor points ; ( d ) estimated anisotropy function at anchor points where the ellipses were scaled to ease vizualisation .( potassium concentration data)[fig1],title="fig:",scaledwidth=100.0% ]  0.35  at anchor points ; ( c ) estimated variance function at anchor points ; ( d ) estimated anisotropy function at anchor points where the ellipses were scaled to ease vizualisation .( potassium concentration data)[fig1],title="fig:",scaledwidth=100.0% ]  following the hyper - parameters selection procedure presented in section [ ssec3 ] , the bandwidth associated with non - parametric gaussian kernel estimator of the local variogram is m. figure [ fig2 ] shows the maps of smoothed parameters over the whole domain of observations : mean , variance , anisotropy ratio and azimuth .the optimal smoothing bandwidth associated to the gaussian kernel smoothing corresponds to m , following the selection procedure described in section [ ssec3 ] .0.35   0.35   0.35   0.35   a visualization of the covariance at certain points for estimated stationary and non - stationary models is presented in figure [ fig3 ] .we can see how the non - stationary spatial dependence structure changes the shape from one place to another as compared to the stationary one .the stationary model is a nested isotropic model ( nugget effect , exponential and spherical ) while the non - stationary model corresponds to the non - stationary exponential covariance function ( example [ ex1 ] ) .the kriged values and the kriging standard deviations for the stationary and non - stationary models are shown in figure [ fig4 ] .the overall look of the predicted values and prediction standard deviations associated with each model differ notably . in particular, the proposed method takes into account certain local characteristics of the regionalization that the stationary approach is unable to retrieve .this example shows the ability of our non - stationary approach to manage data with locally varying anisotropy .figure [ fig5 ] shows some conditional simulations in the gaussian framework , based on the estimated non - stationary model .  to assess the predictive performances of our approach ,the regionalized variable is predicted at a hold - out sample ( 1000 points ) .table [ tab1 ] presents the summary statistics for the external validation results using the classical stationary approach and the new proposed one .some well - known discrepancy measures are used , namely the mean absolute error ( mae ) , the root mean square error ( rmse ) , the normalized mean square error ( nmse ) , the logarithmic score ( logs ) and the continued rank probability score ( crps ) . for rmse , logs and crps , the smaller the better; for mae , the nearer to zero the better ; for nmse the nearer to one the better . table [ tab1 ] shows that the proposed approach outperforms the stationary one with respect to all the measures. the cost of non - using the non - stationary approach in this case can be substantial : in average the prediction at validation locations is about 22% better for the non - stationary model than for the stationary model , in terms of rmse ..external validation on a set of 1000 locations .( potassium concentration data ) [ cols="<,^,^",options="header " , ]the proposed convolution approach provides in terms of modelling a new model for second order non - stationary random functions .this new model generalizes the classical convolution model and provides explicit and flexible classes of non - stationary covariance functions known from the literature .moreover , a statistical methodology for estimating these latter is developed .the estimation method offers an integrated treatment of all aspects of non - stationarity ( mean , variance , spatial continuity ) in the modelling process .the estimation procedure relies on the mild hypothesis of quasi - stationarity and does not impose any distributional assumptions except the existence of the first and second moment . furthermore , no matrix inversions or determinants are required , and hence the inference is practicable even for large datasets .the proposed method provides an exploratory analysis tool for the non - stationarity .indeed , mapping of the parameter values ( mean , variance , azimuth ) by location allows to exhibit or describe the non - stationarity .a plot of variance versus mean allows to identify the common relationships between the mean and variance which is known as proportional effect .the performance of our proposed method has been demonstrated on two real datasets : soil and rainfall data .the two applications have revealed an increased prediction accuracy when compared to the standard stationary method , and demonstrated the ability to extract the underlying non - stationarity from a single realization .a comparison of predictions and prediction standard deviations maps indicates that our non - stationary method captures some varying spatial features ( such as locally varying anisotropy ) in the data that can be not present detected with the stationary method , the resulting outcome appears much more realistic . beyond the spatial predictions ,we also show how conditional simulations can be carried out in this non - stationary framework .  in the proposed convolution method ,the estimation relies on the local variogram non - parametric kernel estimator .to better adapt to the variable sampling density in the domain of observations , it would be interesting to work with a non - parametric locally adaptive kernel estimator .the idea is to increase the bandwidth in low sample density areas and to narrow it in highly sampled areas .the local stationarity assumption is the basis of the proposed methodology , then it works well for smoothly varying non - stationarity. however it can be difficult to apply on sparse data or data that present rapid or abrupt spatial structure variations . in these cases, it may be advisable to proceed under a stationary framework. it would be interesting to extend the proposed method for inclusion of covariates .this can be achieved by setting the variance and anisotropy functions for covariates .this can be done by following the work of and . to date, the non - stationary modelling convolution approach does not provide closed - form non - stationary covariance functions with compact support , this remains an open problem .indeed , the use of such covariances considerably reduce the computational burden of kriging when dealing with large data sets .the authors would like to thank dr budiman minasny at the faculty of agriculture & environment at the university of sydney in australia , for providing the first data used in this paper .the integral for all exists and is finite , because the product of two square integrable functions is integrable .furthermore , it is easy to show that for all , is positive definite on . since , is integrable on with respect to the positive measure for every pair . then , is a valid covariance function on , according to .

Abstract: standard geostatistical models assume second order stationarity of the underlying random function . in some instances , there is little reason to expect the spatial dependence structure to be stationary over the whole region of interest . in this paper , we introduce a new model for second order non - stationary random functions as a convolution of an orthogonal random measure with a spatially varying random weighting function . this new model is a generalization of the common convolution model where a non - random weighting function is used . the resulting class of non - stationary covariance functions is very general , flexible and allows to retrieve classes of closed - form non - stationary covariance functions known from the literature , for a suitable choices of the random weighting functions family . under the framework of a single realization and local stationarity , we develop parameter inference procedure of these explicit classes of non - stationary covariance functions . from a local variogram non - parametric kernel estimator , a weighted local least - squares approach in combination with kernel smoothing method is developed to estimate the parameters . performances are assessed on two real datasets : soil and rainfall data . it is shown in particular that the proposed approach outperforms the stationary one , according to several criteria . beyond the spatial predictions , we also show how conditional simulations can be carried out in this non - stationary framework .   non - stationarity , convolution , covariance , kernel , kriging , simulation .