the beowulf project began in 1994 at the nasa goddard space flight center .the history , current status and a list of about 100 clusters may be found at http://www.beowulf.org .this year at supercompting 2000 , gordon bell sponsored a prize for a computer constructed or purchased for under 50,000 in 1998 to build a 32-node linux cluster .the machine we built in nov .1998 is called candycane , which stands for cpus and network do your calculation and nothing else .candycane is an appropriate name because it was designed for the `` sweet spot , '' that is , components were picked to give the best price - performance ratio attainable .the cost per node was 2,000 , so the total cost was about 320 per node , or for approximately 275 .this node would have much better performance than the pii 350 ; however , the fastethernet would be a bottleneck on the milc code with kogut - susskind quarks .even so , a 32 node system with a minimum performance of 1280 and 1660 mflops , for and sites per node , respectively , could be built for under 6.3 and /mb ) .rambus was quite expensive at introduction , but has recently decreased quite a bit ( currently slightly more than 1 1,500 per node , except for quadrics which is over $ 3,000 per node . ) because of the big jump in price and performance , it is necessary to make sure the system maintains balance between cpu and network . regarding software choices , under fastethernet , mvia and gamma softwarehave reduced latency compared with standard tcp / ip .i have recently tried vmi under myrinet which is being developed at ncsa and found it to be superior to running under standard gm ( myrinet supplied ) driver .   turning next to single node performance, we note that it is easy to waste a lot of money on a poor system design . to illustrate this, we consider the various speed amd athlon processors available and their prices on two particular days .although we focus on athlon here , the same considerations apply to intel or other processors .figure [ fig : price_vs_speed ] shows that processor price is a rapidly increasing function of speed .it can also be a rapidly decreasing function of time , especially for the faster processors .dividing the price by the speed of the chip , we find that the relative expense rises rapidly for the faster chips .on april 7 , 2000 , there was an apparent sweet spot at 600 mhz .the faster chips have a higher price - performance ratio . depending upon the costs of the other components of the system, the entire system may have a higher or lower price - performance ratio .  for our qcd codes ,access to memory is quite important . in our first example, we compare a celeron chip with a 66 mhz fsb and a pentium ii chip with a 100 mhz fsb .the pentium s clock speed is only 6% faster than the celeron , but its performance is 38% faster for and 23% faster for .this is because of its larger cache and 52% faster fsb .when we overclock the celeron , it has a slower fsb but a faster cpu speed than the pii , and performance is about equal except for where the smaller cache probably is the source of the difference .this work was supported by the u.s .doe under grant de - fg02 - 91er 40661 .special thanks to the milc collaboration , the albuquerque high performance computer center , indiana university , llnl , national center for supercomputing applications , pittsburgh supercomputer center and san diego supercomputer center .

Abstract: small beowulf clusters can effectively serve as personal or group supercomputers . in such an environment , a cluster can be optimally designed for a specific problem ( or a small set of codes ) . we discuss how theoretical analysis of the code and benchmarking on similar hardware lead to optimal systems .   pacs / keywords : 12.38.gc , lattice qcd , linux , beowulf clusters , supercomputing


astrophysical opacities from the opacity project ( op ) have been updated by to include inner - shell contributions and an improved frequency mesh .the complete data set of monochromatic opacities and a suite of codes to compute mean opacities and radiative accelerations ( opcd_2.1 ) have also been publicly released by to make in - house calculations for arbitrary mixtures more versatile and expedient .regarding data accuracy , there is excellent overall agreement between the opal and op results as discussed by , and .rosseland mean opacities are sensitive to both the basic atomic data used and the assumed abundances of the chemical elements .what had been a good agreement between theory and the helioseismological data was found to be less good using revised solar abundances from .the revised op opacities have been instrumental in discussions of that problem .the modelling of stellar interiors , on the other hand , is being renewed with the solar experience .present ( _ wire _ , _ most _ , _ corot _ ) and future ( _ kepler _ ) space probes and the well established solar methods are giving the field of asteroseismology remarkable growth and the guarantee of invaluable data .in future work on stellar models it may be desirable to take account of revisions in abundances similar to those performed for the sun .  for some types of stars, models must take into account microscopic diffusion processes , e.g. radiative levitation , gravitational settling and thermal diffusion , as they can affect the internal and thermal structures , the depth of the convection zone , pulsations and give rise to surface abundance anomalies . as reviewed by ,such processes are relevant in the description of chemically peculiar stars , horizontal - branch stars , white dwarfs and neutron stars , and in globular cluster age determinations from population ii turnoff stars .furthermore , in order to solve the outstanding discrepancy of the atmospheric li abundance in old stars with that predicted in big - bang nucleosynthesis , have proposed li sinking deep into the star due to diffusion .this hypothesis has been recently confirmed in the observations by .the opcd_2.1 release includes data and codes to compute the radiative accelerations required for studies of diffusion processes .it should be noted that the radiative accelerations are summed over ionization stages and that data for the calculation of diffusion coefficients are calculated assuming that the distribution over ionization stages of the diffusing ions is the same as that in the ambient plasma .the validity of this approximation is discussed by .  in some cases ,particularly when element stratification depends on stellar depth , calculations of mean opacities and radiative accelerations must be repeated at each depth point of the model and at each time step of the evolution , and thus the use of codes more efficient than those in opcd_2.1 may be necessary .this becomes critical in the new distributed computing grid environments where the network transfer of large volumes of data is a key issue . in the present work we have looked into these problems , and , as a solution , report on the implementation of a general purpose , interactive server for astrophysical opacities referred to as opserver .it has been installed at the ohio supercomputer center , columbus , ohio , usa , from where it can be accessed through a web page or a linkable subroutine library .it can also be downloaded locally to be run on a stand - alone basis but it will demand greater computational facilities . in section 2we discuss the computational strategy of the codes in opcd_2.1 followed by a description of opserver in section 3 . in section 4we include some tests as an indication of its performance with a final summary in section 5 .we highlight here some of the key features of the codes in opcd_2.1 . for a chemical mixture specified by abundance fractions , they essentially compute two types of data : rosseland mean opacities ( rmo ) and radiative accelerations ( ra ) .   for the frequency variable where is the boltzmann constant , rmoare given by the harmonic mean of the opacity cross section of the mixture where is the mean atomic weight .the is a weighted sum of the monochromatic opacity cross sections for each of the chemical constituents and is conveniently tabulated on the -mesh where ^ 2}\ ] ] and .the rationale behind the -mesh is that it enhances frequency resolution where is large .similarly , the ra for a selected element can be expressed as where is its atomic weight and the speed of light . the function is given in terms of the effective temperature and fractional depth of the star by with the dimensionless parameter depends on the cross section for momentum transfer to the element - a_k(u)\ ] ] where is a correction to remove the contributions of electron scattering and momentum transfer to the electrons . both and , which are hereafter referred to as the mono data set (  gb ) ,are tabulated in equally spaced intervals to facilitate accurate interpolation schemes .the computational strategy adopted in the opcd_2.1 release is depicted in the flowcharts of figure 1 where it may be seen that calculations of rmo and ra are carried in two stages . in a time consuming stage 1 , rmo and ra are computed with the mixv and accv codes , respectively , on a representative tabulation of the complete temperature electron - density plane . in mixvthe chemical mixture is specified in the input file mixv.in as where and are the hydrogen and metal mass - fractions , the number of elements , and and are the metal nuclear charges and fractional abundances . in accv ,the input data ( accv.in ) are where now runs over the elements of the mixture , and and are respectively the nuclear charge and abundance multipliers of the test element .input data formats in either mixv.in or accv.in give the user flexible control over chemical mixture specifications .  as shown in figure 1 , the intermediate output files mixv.xx ( kb ) containing where is the mass - density , and acc.xx ( kb ) with are written to disk .they are then respectively read by the codes opfit and accfit in stage 2 for fast bicubic interpolations of rmo and ra on stellar depth profiles specified by the user in the opfit.in and accfit.in input files .the final output files are opfit.xx containing and accfit.xx with   in this computational approach , performance is mainly limited by the summation in equation ( [ sum ] ) which implies disk reading the mono data set ; for instance , in mixv it takes up to % of the total elapsed time .opcd_2.1 also includes other codes such as mx and ax which respectively compute rmo and ra for a star depth profile .the chemical mixture can be fully varied at each depth point using the specifications in equations ( [ mix1][mix2 ] ) , the rmo and ra being obtained in a one - step process using bicubic interpolations without splines .these methods are thus suitable for cases with multi - mixture depth profiles . further details of all the opcd codes are contained in the reference manual .in opserver the computational capabilities of the codes in opcd_2.1 are greatly enhanced by the following innovative adaptations .  1 .the codes are restructured within a client server network architecture whereby the time consuming stage 1 is performed on a powerful processor while the fast stage 2 is moved to the client , e.g. a web server or a user workstation . in this arrangement performancecould be affected by the client server transfer of the mixv.xx and acc.xx intermediate files , but since they are never larger than 0.5 mb , it is not expected to be a deterrent with present - day bandwidths . in a local installation where both the client and server reside on the same machine, communication is managed through shared buffers in main memory ;i.e. the corresponding data in mixv.xx and acc.xx are not written to disk .the codes are transcribed as a subroutine library to be referred to hereafter as the oplibrary which can be linked by the user stellar modelling code for recurrent subroutine calls that avoid data writing on disk .that is , the input data in the mixv.in , accv.in , opfit.in and accfit.in files and the output tables in the opfit.xx and accfit.xx files ( see figure 1 ) are now handled as subroutine parameters while the intermediate mixv.xx and acc.xx files are passed via shared main - memory buffers .chemical mixtures are again specified with the formats of equations ( [ mix1][mix2 ] ) which allow full variation at each depth point in a single subroutine call .rmo / ra are computed with the complete mono data set always loaded in main memory thus avoiding lengthy and repeated disk readings .this is achieved by implementing opserver on a dedicated server where mono is permanently resident in ram , or in the case of a local installation , by disk - reading it once at the outset of a modelling calculation .when accessing the remote server , client data requests are addressed through the http protocol , i.e. in terms of a uniform resource locator ( url ) .this allows data fetching from the central facility through an interactive web page or a network access subroutine , the latter being particularly suitable for a stellar model code that is to be run in a distributed grid environment .the do - loop that computes the summation of equation ( [ sum ] ) has been parallelized in openmp which provides a simple , scalable and portable scheme for shared - memory platforms .as shown in figure 2 , the current opserver enterprise is implemented as a client server model at the ohio supercomputer center ( osc ) .the web server communicates with the supercomputer via a socket interface .earlier versions were developed on an sgi origin2000 server with the powerfortran parallelizing compiler .the current version runs on a linux system with fortran openmp directives .opserver offers three user modes with full functionality except when otherwise indicated in the following description .* mode a * : :  in this mode opserver is set up locally on a stand - alone basis ( see  figure 2 ) .the facilities of the osc are not used .a new opcd release  ( opcd_3.3 ) is  downloaded , followed by ( i ) installation of both the oplibrary and the  mono data set and ( ii ) linking of the oplibrary to the user modelling  code .computations of rmo / ra are preceded by the reading of the  complete mono data set from disk and therefore requires at least 1 gb  of ram .* mode b * : :  in this mode , the oplibrary is downloaded , installed and linked to the  user code , but stage 1 is performed remotely at the osc ( see  figure 2 ) .this option has been customized for stellar modelling in a  distributed grid environment that would otherwise imply ( i.e. mode a )  the network transfer , installation and disk - reading of the mono data  set at runtime .it is also practical when local computer capabilities  ( ram and/or disk space ) are limited .the functions provided by the mx  and ax codes have not been implemented .* mode c * : :  in this mode rmo / ra computations at the osc are requested through an  interactive web page which allows  both stage 1 and stage 2 to be carried out remotely or , alternatively ,  stage 2 locally by downloading the mixv.xx/acc.xx intermediate files  ( see figure .1 ) with the browser for further processing with local  opfit / accfit executables .opserver benchmarks were initially carried out on an sgi origin2000 multiprocessor at the osc with an earlier release of opcd .for the standard s92 mixture , the mixv code took up to 140 s to compute the mixv.xx file , of which 126 s were dedicated to disk - reading and 14 s to the actual computing of the mean opacities .opserver took on average  s to compute mixv.xx which was not written to disk unless requested . in figure3 we show the acceleration obtained on the origin2000 through parallelization where the calculation of mean opacities is reduced to 2 s with 8 processors .further significant acceleration is prevented by data transfer overheads .on more recent workstations , the local performances of the codes in opcd_2.1 and opserver depend on processor speed and ram and cache sizes .for instance , on a powermac g5 ( powerpc 970fx processor at 2.0 ghz , 1 gb of ram and l2 cache of 512 kb ) the first time mixv is run it takes for a single s92 mixture 103.8 s to compute the rmo , but on subsequent runs the elapsed time is reduced to an average of  s. similarly , opserver takes 103.3 s which is then reduced to  s on subsequent runs .once the mono data set is loaded in ram by opserver ( mode a ) , calculations of rmo for a single s92 mixture only take  s and  s for ra for the test element ar . in modeb , where stage 1 is carried out remotely at the osc and the mixv.xx and accv.xx files are transferred at the relatively low rate of 1.88 kb / s , computations of rmo and ra take  s and  s , respectively .the noticeable longer time taken for the latter is due to the transfer time taken for the larger accv.xx file .rosseland mean opacities and radiative accelerations can be computed from op data in any one of the following ways .( i ) download the original opcd_2.1 package as described by and perform all calculations locally .\(ii ) mode a , download the upgraded opcd_3.3 package , install opserver and perform all calculations locally by linking the subroutines in the oplibrary. calculations with opserver are more efficient but require large local computer memory .\(iii ) mode b , as mode a but with stage 1 performed remotely at the osc .mode b is convenient if fast calculations are required but local computer memory is limited or when stellar modelling is to be carried out in a grid environment .\(iv ) mode c , perform all calculations remotely at the osc through an interactive web page whereby files are downloaded locally with the browser .we acknowledge the invaluable assistance of juan luis chaves and gilberto daz of cecalcula during the initial stages of opserver .we are also much indebted to the ohio supercomputer center , columbus , ohio , usa , for hosting opserver and for technical assistance ; to the centre de donnes astronomiques de strasbourg , france , for hosting the opcd releases ; and to drs josslen aray , manuel bautista , juan murgich and fernando ruette of ivic for allowing us to test the opserver installation on different platforms .fd would like to thank s. rouchy for technical support .akp and fd have been partly supported by a grant from the us national science foundation .asplund m. , grevesse n. , sauval a. j. , 2005 , in barnes t. g. , bash f. n. , eds , asp conference series vol .336 , cosmic abundances as records of stellar evolution and nucleosynthesis . astronomical society of the pacific , san francisco , p.25 badnell n. r. , bautista , m. a. , butler , k. , delahaye f. , mendoza c. , palmeri p. , zeippen c. j. , seaton , m. j. , 2005 , mnras , 360 , 458

Abstract: codes to compute mean opacities and radiative accelerations for arbitrary chemical mixtures using the opacity project recently revised data have been restructured in a client server architecture and transcribed as a subroutine library . this implementation increases efficiency in stellar modelling where element stratification due to diffusion processes is depth dependent , and thus requires repeated fast opacity reestimates . three user modes are provided to fit different computing environments , namely a web browser , a local workstation and a distributed grid .   [ firstpage ]  atomic processes radiative transfer stars : interior .


many statistical procedures have been developed to classify data into two or more given classes .generally , if the data arise from a known class of distributions , properties of the classifiers are established through either theoretical considerations or simulation studies . by this , alternative classifiers are compared and procedures identified that are optimal under properly chosen assumptions . however , real - world data do often not fit into standard parametric distribution models . classifying themrequires nonparametric procedures , while , due to the lack of established general properties , selecting a good classifier has to be mainly based on empirical evidence .usually such evidence is sought from simulation studies mimicking certain features of the data that arise in practical applications . at best , in a given field of applicationso called ` stylized facts ' are identified and translated into a simulation setting .but often such ` stylized facts ' do not exist .then , we can not learn much from simulation studies how a statistical procedure really works in practice .the adequacy and fitness of the procedure can only be demonstrated when it is applied to real - world data , and its _ general _ fitness can be only established by successful application to a _ large variety _ of such data .  in the sequelthis is done for a newly developed nonparametric classifier , the -procedure .it is applied to fifty binary classification problems regarding real - world benchmark data .the -procedure first transforms the data from their original property space into a _ depth space _ , which is a low - dimensional unit cube , and then separates them by a projective invariant procedure , called _-procedure_. to each data point the transformation assigns its depth values with respect to the given classes .the depth coordinates of the data reflect their degree of centrality w.r.t .each of the classes .this central ordering is carried out using a properly chosen depth function .the subsequent separation in the depth space accounts only for differences in the depth values : if a binary separator is determined by the -procedure .the -procedure stepwise selects pairs of _ extended depth properties _ ( that is , depth coordinates and powers and products of them ) and separates them by a linear rule .the separator is a hyperplane in the extended depth space , which corresponds to a polynomial line in the basic depth plane , both containing the origin as an element . with classes , -separations can be performed and a majority rule applied ; alternatively one - against - all separators can be used .we restrict the present study to the case , see  for .  in the zonoid depth is applied , which is efficiently computed also in higher dimensions . herewe employ four alternative depths : the mahalanobis depth , the spatial depth , the projection depth and the tukey depth .the first three depths are positive everywhere , while the tukey depth ( like the zonoid depth ) vanishes outside the convex hull of the data .however , the tukey depth reflects the shape of the data much better than the previous three do and it is more robust against outliers than these ( and than the zonoid depth as well ) . for computational reasons we use the _ random tukey depth _ , which approximates the tukey (= location ) depth by minimizing univariate tukey depths over a finite number of directions .when using the random tukey depth ( or another depth that vanishes outside the convex hull of the data ) a first practical question is how outsiders , that is data points having zero depth in all classes , should be treated .these points are , by construction , represented by the origin of the depth space and , hence , arbitrarily assigned . with real data , often a large portion of the data turn out to be such outsiders . as our task is to classify all points , we need either a depth that does not produce outsiders , or a supplementary treatment of outsiders . by definition ,the -procedure includes a treatment of outsiders if necessary . for the -classifier with the random tukey depth, several possible treatments are introduced in the sequel .the paper considers the respective variants of the -classifier and compares them with the -classifiers based on mahalanobis , spatial and projection depths .recall that the latter depths , as they are positive on the entire , do not yield outsiders .a second question is how many directions should be chosen to approximate the tukey depth and how they should be generated ; it is addressed in by means of a simulation study .( for the random projection depth this question is less important as it has no outsiders .) broad numerical experience is provided about the relative usefulness of the classifiers .further we investigate how many features in the extended depth space are needed on an average to satisfactorily separate the data .finally we demonstrate the robustness of our procedure when applied to real data containing substantial amounts of outliers .  for comparisonseveral indicators are introduced , two of which refer to a combination of classical procedures as benchmark . to evaluate the performance of the -procedure under different depths and outsider treatments ( when using the random tukey depth ) we have set up experiments with a large number of binary classification tasks with real data .these data sets have been selected from open internet sources and different fields of application . some of them have already served as benchmark sets in other classification studies . by thisthey are well suited for evaluating and comparing our new approach .the data can be downloaded in standardized form from www.wisostat.uni-koeln.de/28969.html .the complete -procedure is available as an r - package named ddalpha .the -classifiers are also compared with three traditional procedures : _ linear ( lda ) _ and _ quadratic ( qda ) discriminant analysis _ and the _ -nearest - neighbors ( knn ) _ classifier , as in many cases ( including the data sets considered here ) at least one of them performs satisfactorily .we exclude neural network methods because they offer too many possible architectures , among which it is difficult to select in an automatic and computationally feasible way . while we expect that , given the specific data set , a properly adapted neural network performs rather well , such an approach affords a by hand tuning for each data set .therefore we do not regard neural networks as fair competitors to the -classifiers .we exclude as well the usual support vector machine ( svm ) as a classifier because for each data set it has to be specially tuned .  overview : sect .[ sec : ddaclassifier ] describes the training phase of the -classifier , which consists of the depth transformation and the -separation in the extended depth space .the problem of generating directions for the random tukey depth is discussed .[ sec : treatments ] regards the classification phase , where the problem of outsiders arises .several classical approaches to classify the outsiders ( lda , maximum mahalanobis depth , knn ) are introduced , as well as a simplified svm approach , which liaises with the -separation in two ways .[ sec : realdata ] first describes the 50 classification tasks , which vary by absolute and relative sizes of training classes and include different portions of outliers and ties .then the settings and results of the empirical study are presented . in sect .[ sec : evidence ] further evidence on outsider treatments and the number of ( extended ) depth properties needed is discussed .[ sec : conclusions ] concludes .consider a -class classification problem , .the -classifier has been recently proposed by .its classification phase consists of two parts : a transformation of the data from the original space into the _ depth space _ ( depth transformation ) and their subsequent separation using a modified version of the -procedure ( -separation ) , see , , , .this procedure is a projective invariant method to separate the depth transformed data .   the _ depth transformation _ maps into ^q ] is called the _depth plot_.  first we briefly regard three depths whose empirical versions take positive values beyond the convex hull of the data . for a point and a random vector in ( especially one having an empirical distribution on a set of -variate observations ) the _ mahalanobis depth _ of w.r.t . is defined as where measures the location ( e.g. the mean ) of , and the scatter ( e.g. the covariance matrix ) of .  the _ affine invariant spatial depth _ of regarding is defined as \|\,.\ ] ] here for and , and is the covariance matrix of . as the mahalanobis and spatial depths lack robustness when using standard moment estimates for and , we consider also robustified versions of them , where and are estimated by mcd ( minimum covariance determinant ) .  the _ projection depth _ of regarding is given by with where denotes the ( univariate ) median of and the ( univariate ) median of the absolute deviation of from its median .the _ tukey depth _ or _ location depth _ of w.r.t . is defined as the minimal probability of lying in a halfspace bounded by a hyperplane through , where is the probability distribution of .( a closed halfspace is the set of all points that lie on one side of a hyperplane including that hyperplane . )if is an empirical distribution , this means that is the minimal portion of the data that can be cut off by a hyperplane through . obviously , in general , the tukey depth vanishes outside the convex hull of the distributions support .the mahalanobis , spatial , projection depths are everywhere positive ; thus outsiders can not occur .however they are not very sensitive to the shape of the underlying distribution , which is illustrated in figure [ fig : depths ] .it exhibits the data of the `` non - donating '' class in the `` blood - transfusion '' task .the figure shows the level sets of mahalanobis , spatial , projection and tukey depths .observe that the mahalanobis depth yield ellipses , while with the spatial and projection depth rather symmetric , ellipsis - like level sets are obtained .the asymmetric shape of the data is much more reflected by the tukey depth ( bottom right ) ; therefore it appears to be better suited to extracting the relevant information from the training classes .note that , by construction , the mahalanobis depth has exact elliptical regions and the projection depth contains a symmetric factor , namely mad , which accounts for the quasi - symmetric shape of the level sets .moreover , the projection depth comes with an enormous computational cost . for these reasonswe include the tukey depth in our study , in spite of its need for extra outsider treatments .  ; spatial depth using the same mcd estimates ; tukey depth .the level sets are pictured for the depth values for the tukey depth , and for for the rest of the depths.,title="fig : " ] ; spatial depth using the same mcd estimates ; tukey depth .the level sets are pictured for the depth values for the tukey depth , and for for the rest of the depths.,title="fig : " ] ; spatial depth using the same mcd estimates ; tukey depth .the level sets are pictured for the depth values for the tukey depth , and for for the rest of the depths.,title="fig : " ] + ; spatial depth using the same mcd estimates ; tukey depth .the level sets are pictured for the depth values for the tukey depth , and for for the rest of the depths.,title="fig : " ] ; spatial depth using the same mcd estimates ; tukey depth .the level sets are pictured for the depth values for the tukey depth , and for for the rest of the depths.,title="fig : " ] ; spatial depth using the same mcd estimates ; tukey depth .the level sets are pictured for the depth values for the tukey depth , and for for the rest of the depths.,title="fig : " ]   and are easily computed . to estimate ( for ) and we use empirical moments and minimum covariance determinant ( mcd ) estimates that have outlyingness parameter , see , .the tukey depth as well as the projection depth satisfy the weak projection property , i.e. the depth of a point can be represented as the minimum of the depths on all unidimensional projections .based on this we approximate the tukey depth by the random tukey depth , which is the minimum univariate tukey depth over a set of unidimensional projections in randomly selected directions . as the exact calculation of is rather elaborate , we approximate it in the same way .   for each binary separation-procedure _ constructs a decision hyperplane in the extended property space ^r ] .let and be two training classes in having and elements respectively .each data point is transformed to ^ 2 ] , the depth space was extended up to degree , which yields the extended depth space ^ 5 ] and find a separating straight line minimizing _ emr _ in each of them .again , among these planes we choose the one that yields the smallest _ emr _ and obtain the second projection axis .it separates the data at its zero point , as the first axis did in the initial step ; see figure [ figurealpha ] ( right ) for the pima example . similarly , after step 2 the extended space is reduced to an -dimensional unit hypercube .the steps are iterated , and properties are selected from the extended space , as long as the minimum _ emr _ decreases and the remaining extended space is non - void .note that with the pima data , the procedure stops after step 2 .-separation ; step 1 ( left ) and step 2 ( right).,title="fig:"]-separation ; step 1 ( left ) and step 2 ( right).,title="fig : " ]   as mentioned above , we approximate the -variate tukey depth by the minimum univariate depth of the ( on lines in several directions ) projected data .the random tukey depth inherits the robustness from the tukey depth . when implementing the random tukey depth we have to answer the following questions in a way that makes our classifier reasonable and computationally feasible .  1 .how should the random directions be generated ? 2 .  how many directions should we consider ? 3 .shall we generate a new set of directions for each point ?ad ( 1 ) : as in we generate directions that are uniformly and independently distributed on the unit sphere and independent of the data .alternatively we could proceed as in , i.e. search through the normals of randomly formed -simplices .however we find it more efficient to spend computational time on generating random directions and evaluating univariate depths rather than computing exact directions from the data ; moreover , a moderate number of random directions proves to be enough .ad ( 2 ) : we mention two qualitative arguments concerning this number .firstly , a larger dimension of the space needs a larger number of directions , which is obvious from the geometry of .however , we are not able to indicate the precise dependency of this number on the dimension .secondly , we point out that there exists a trade - off between the number of directions used for the random tukey depth and the number of outsiders .a simulation study that illuminates this trade - off has been conducted in . if the data stems from a centrally symmetric distribution ( e.g. gaussian or another elliptical distribution ) the depth can be rather precisely approximated with a small number of directions .but if the data exhibit asymmetries and possible outliers , as real - world data mostly do , we need a larger number of directions to adequately represent them in the depth space . the degree of asymmetry and fat - tailedness found in the data may guide us in choosing this number ; see . to be able to compare the procedure on different data sets we have chosen a fixed number of directionsthis number has been set to 10 000 as a practical compromise between accuracy of the depth calculation and computational load .it is obvious ( see e.g. ) that the random tukey depth of a point can widely deviate from the tukey depth . on the other hand ,the -separation is rather robust , since the only invariant it uses is whether a point belongs to a class or not .therefore an upward bias at a few points does not much influence the final separation .further , a trade - off exists between the number of directions used for the random tukey depth and the number of effective outsiders , which favors a moderate number of directions .ad ( 3 ) : we use the same set of directions for each data point .though the generation of a new set of directions for each point produces different depth values , there is no reason to expect them to be more precise .by keeping the set of directions constant we increase the speed of calculations in the training phase .the computations are boosted by avoiding the generation of the directions and projection of the points onto them .let be the number of random directions .when instantly generating the direction set for each point the complexity of the depth calculation amounts to .if the direction set is not constantly changed , first , the time for their generation is saved .second , recall that in the training phase the depth of all points of the training sample w.r.t .each class has to be computed .hence , univariate projections are ordered , and all the univariate depths on each projection can be determined in a single pass .this yields complexity .note that holds .therefore , for all , and that are large enough to suppress eventual constants , the constant direction approach is substantially faster , see also .it also enhances the classifier s stability in the classification phase , as the same directions are used to approximate the depth of a new point to be classified .exact calculation of the projection depth appears to be a heavy computational task .therefore we approximate the projection depth in the same way as the tukey depth by minimizing the univariate depth of projections in randomly drawn directions .compared with the tukey depth , many more directions are needed to produce a reasonable approximation of the projection depth : when traversing the unit sphere the projection depth ( which is a piecewise linear function ) changes direction much more frequently because of the median and mad estimates . to be able to compare the procedure on different data sets we have chosen two fixed numbers of directions .these have been set to 10 000 and 100 000 as a practical compromise between accuracy of the depth calculation and computational load .the _ classification phase _ proceeds as follows .consider a point to be classified .first the depth transform of is calculated as . if is zero but not , the point is assigned to class , and viceversa .if both and are non - zero the point is classified according to the separation rule determined by the -procedure .this is always the case when using the mahalanobis , spatial or projection depths .when employing the random tukey depth some may have , then is regarded as an _outsider_. an outsider , being represented by the origin of the depth plot , can not be readily classified but needs some special treatment .specifically , a point that , in the original data space , lies outside the convex hulls of the two training sets has zero tukey depth in both classes and , thus , is an outsider . if a point is no outsider it is mentioned as an _insider_. insiders are instantaneously classified by the -procedure .  as in , outsiders may be classified by determining their nearest neighbors . in doing so ,euclidean and mahalanobis distances can be employed , the latter to account for scatter within the classes . alternatively outsiders can be classified according to their maximum mahalanobis depth , which is always positive . a depth function , which is the maximum of the zonoid depth and a properly scaled mahalanobis depth , and thus circumvent the outsider problem . propose an approach that avoids the outsider problem as well . in the classification phase , for each point to be classified , ( 1 ) the sample is extended by reflecting the training classes symmetrically at center , ( 2 ) the depth of points in the extended sample is considered , and ( 3 ) a -nearest - neighbor rule that uses depth in place of distance is applied for classifying . here not only the classification phase is computationally hard ( by instantaneous calculation of the depths of all data points ) , but also the training phase , where the classifier has to be validated in order to determine .this requires onerous computations .  in the sequelwe compare several alternative outsider treatments , which are classifiers applied to data in the original space .the treatments include three well known classifiers : linear discriminant analysis , maximum mahalanobis depth , and -nearest neighbors as well as a new one , which we call _ svm - simplified_. the performance of the -classifier with mahalanobis , spatial and projection depth is contrasted as well .note that all depths ( tukey , mahalanobis , spatial and projection ) are affine invariant as well as all treatments used ( lda , knn with an affine invariant distance , mahalanobis depth and the support vector machine ) .therefore all considered -classifiers are affine invariant ( under appropriate moment assumptions ) , if the exact versions of the depths are calculated .since the random tukey depth and the random projection depth converge to the exact versions , using them makes the -classifiers approximately affine invariant .  the random tukey depth ( rtd )used for the depth transform is very efficiently calculated , but yields outsiders . as it approximates the tukey depth ( td ) from above , some td - outsiders will have non - zero rtd and , by this , be assigned to one of the classes. a smaller number of directions yields a worse approximation of the td , but reduces the number of outsiders .the remaining rtd - outsiders still need a special treatment , though .thus , when using the rtd , we face a trade - off between the quality of depth approximation and the extent of outsider treatment needed . as we will see below with real data , choosing a moderate number of random directions gives best results.   _ linear discriminant analysis _ ( lda ) , introduced in , separates the classes by a hyperplane in the original data space ; see also .the lda classifier is particularly simple .it is optimal if the data follow a gaussian or , more general , a unimodal elliptical distribution and the classes differ by location shifts only .however , in classifying real data it is often outperformed by other approaches . in many applications , the real data can not be assumed to be gaussian and ask for procedures different from lda .however , after having classified the rtd - insiders , the remaining task of classifying the outsiders appears to be a much less exigent task and may be successfully done by a simple procedure like lda .  the _ maximum - mahalanobis - depth _classifier is given by where is the prior probability for class .the priors are estimated by the training class portions .again , this classifier has optimality properties under ellipticity .applied to outsiders it is expected to perform satisfactorily .the _ -nearest - neighbors _ classifier is still another option for treating outsiders .its parameter , the number of the nearest neighbors , has to be chosen by cross - validation .often a relatively small is enough ; see , e.g. , , where already produces satisfying results . to make the procedure affine invariant we use mahalanobis distances ( based on the pooled data set ) for finding nearest neighbors .   as another way to handle the outsider problem we propose to supplement the -classifier by an additional svm - rule , which is restricted to classifying the outsiders .it has a particularly simple structure .recall that the -procedure delivers a separator which is a hyperplane in the extended depth space .this hyperplane induces a decision rule in the original data space .next , we remove all training points which are not correctly classified by this rule ( so that _ _emr__ ) and subject the remaining points to an additional svm classification step that involves determining a single kernel parameter but no box - constraint .this new approach is named _ svm - simplified _ ( _ svm - s _ ) . as the _svm - s _ rule is defined on the whole , it is able to assign points which are outsiders in the -classification .figure [ figsvm13 ] , left panel , shows two classes , each containing 250 points , which are simulated from n,\bigl[\begin{smallmatrix } 1 & 1\\ 1 & 4 \end{smallmatrix}\bigr]) ] ( blue ) , together with the separating lines of the optimal bayes ( dashed ) and ( solid ) classifiers .the left panel regards the original data , while the right panel exhibits the data after the removal step .the _ svm - s _ step consists in solving the following quadratic programming problem : subject to the constraints  here we notate , , and . stands for the -dimensional vector of responses , and is a symmetric -matrix with elements where is a gaussian kernel . note that no box - constraint condition is needed here as the points are separable without error .but still the kernel parameter has to be chosen . for given a solution of ( [ eqn : svmproblem ] ) is obtained , provided the two classes are linearly separable in the reproducing kernel hilbert space that corresponds to .every such solution determines a margin between the classes and a number of support vectors , .  in figure [ figsvm2 ] , depending on , the values of ( dashed line ) and the corresponding numbers of support vectors ( solid line ) are plotted for the above example .a zero value of the margin or of the number of support vectors indicates that with the given no errorless discrimination is possible .( logarithm to the base 10 ) . ]  loosely speaking , controls the complexity of the _ svm - s _ rule ; for small values of the decision rule is not able to separate the classes at all .therefore it suggests itself to use the simplest separating rule , i.e. selecting the smallest for which the classes are separated without error ( as indicated by the small circle in figure [ figsvm2 ] ) .this also comes out as a most stable decision rule .figure [ figsvm13 ] exhibits the corresponding decision rule in its right panel as a solid line , while the dashed line indicates the same optimal bayes decision rule as on the left panel of the figure ; the rules appear to be very similar .most important , calculating the _ svm - s _ rule needs no parameter tuning besides selecting the parameter , which is a straightforward task .needless to say , with the usual support vector machine ( svm ) a solution can be obtained that is at least as good as the one achieved here . however , obtaining this solution needs a tuning of parameters that is computationally much more intensive .  to summarize the above procedure : the training phase consists of two steps , first determining the -classifier and then determining an _ svm - s _ rule based on the correctly -classified points .note that the classification performance of this procedure is determined by the -classifier , andthe _ svm - s _ step just extrapolates this classifier to treat the outsiders . in our experiments the whole training phase took between a few seconds and several minutes of computation time ( 64-bit , 1 kernel of the icore 7 - 2600 having enough operative memory ) .the time reached a maximum of 10 minutes with four very large data sets only .  in the classification phasewe have two choices : either using the obtained _ svm - s _rule for all points to be classified , or first check for each whether it is an insider or an outsider and then classify it with the -rule if it is an insider and with the _ svm - s _ rule otherwise .the first choice yields a particularly fast procedure , as the _ svm - s _ rule does not involve any depth calculations in the classification phase .we choose the second one as it is in line with the application of the outsider treatments mentioned above .its results are presented in column ` svm - s ' of tables [ tab : performance1 ] and [ tab : performance2 ] .to evaluate the -procedures with different outsider treatments and to judge their usefulness in practical applications we have set up experiments based on a large variety of real data sets .the methodology is applied to 50 binary classification tasks , which have been obtained from partitioning 33 freely accessible data sets , see tables [ tab : datasets1 ] and [ tab : datasets2 ] .the subset of the `` pima indian diabetes '' described above is included in the table as `` pima '' ( no .the authors and introducers of the accessed data sets are ( `` biomed '' ) , ( `` cloud '' ) , ( `` irish - ed '' ) , ( `` kidney '' ) , ( `` plasma - retinol '' ) , ( `` socmob '' ) and ( `` veteran - lung - cancer '' ) ; these data sets have been downloaded from + lib.stat.cmu.edu/datasets . + data sets `` chemdiab '' and `` hemophilia '' have been taken from the r - packages ` locfit ' and ` rrcov ' respectively .the `` pima '' data set constitutes a training subsample of the `` diabetes '' ( see below ) and can be downloaded from + www.stats.ox.ac.uk/pub/prnn . +datasets `` baby '' , `` banknoten '' , `` crab '' , `` gemsen '' , `` groessen '' , `` tennis '' , `` tips '' and `` uscrime '' have been downloaded from the teaching data base + stat.ethz.ch/teaching/datasets .+ the rest of the data sets is taken from + archive.ics.uci.edu/ml ; + it in particular originates from ( `` blood - transfusion '' ) , ( `` breast - cancer - wisconsin '' ) and ( `` vowel '' ) .multiclass problems were reasonably split into binary classification problems , and some of the data sets were slightly processed by removing correlated attributes , by dropping objects with missing values , and by selecting prevailing classes . for detailed descriptions of the data considered we refer to the corresponding literature and public repositories ; the fifty tasks together with short descriptions of the data can be found on the web page + www.wisostat.uni-koeln.de/28969.html .as we see from tables [ tab : datasets1 ] and [ tab : datasets2 ] , the classification tasks are much different .the tables show their basic parameters : dimension of the original space , log ratio of the cardinalities and of the training classes ( so that the sign reflects which is larger ) , total sample length , percentages of outliers and outsiders .as we see , up to 13 attribute dimensions are considered .the total sample sizes range from 47 to 1349 , while the relative size of the two classes varies between 1 and 6.5 .  [ cols=">,<,>,>,>,>,>,>,>",options="header " , ]a fast classification procedure , the -procedure , has been introduced that is essentially nonparametric , robust , and computationally feasible for any dimension of attributes .the -procedure is available in the r - package ddalpha .the -classifier is particularly robust for two reasons : first , as the classification is done by the -procedure , which is _ per se _ robust ; second , as the data are transformed into a low - dimensional compact space . generally ,two cases are to be distinguished with the depth transform : non - vanishing depth or depth vanishing beyond the convex hulls of the training classes .non - vanishing depths , e.g. mahalanobis or spatial , often induce a spurious symmetry and are intrinsically non - robust ( though , can be robustified ) .the projection depth , which is non - vanishing , also produces ellipsis - like regions .it is robust , but computationally inefficient when .the last problem is faced by the tukey depth , which best reflects the shape of the data . in place of the exact versions of projection and tukey depth we employ their random versions by minimizing univariate depths in directions that are uniformly distributed on the sphere .a very large number of these directions is needed for the calculation of the projection depth , while for the random tukey depth the number of directions can be kept low , as there is a tradeoff between this number and the number of points being classified by their depth values .  on the other hand the random tukey depth yields outsiders when classifying , i.e. points lying outside the convex hulls of all classes , which can not be readily classified and need additional treatment . in real data applicationsthe percentage of outsiders can be large ( see the introduced measure of outsider proneness ) and thus substantially influence the classification performance .therefore , the choice of the treatment is important when applying the random tukey depth .the treatments considered subject the outsiders either to linear discriminant analysis ( lda ) , classification according to -nearest neighbors ( knn ) , maximum mahalanobis depth classification based on moment or mcd estimates , or the newly introduced svm - simplified procedure ( _ svm - s _ ) . the latter is very fast as it needs no tuning of a box - constraint ; only the smallest separating has to be computed . additional calculations ( not included here , see also ) show , that regarding the other possible outsider treatments , the choices of number in knn as well as of the covering parameter in mcd do not much influence their performance .thus the -procedure needs practically no parameter tuning .the degree of the separating polynomial is chosen by cross - validation within the depth representation only , where the modified -procedure , on each of the planar subspaces considered , has a quick sort complexity , o , and by that is very fast .the above introduced variants of the -procedure are challenged by 50 binary classification problems that arise from a broad range of real data .the tasks are complicated by the presence of outliers and ties . as competitors of the -procedure three traditional classification methods (lda , qda , and knn ) are evaluated with the same data .naturally , none of the classifiers is best at all tasks in terms of the average error rate , but each classifier is best at some of them .our results also show that no single depth or outsider treatment dominates the others . just for almost all datasets the classification of outsiders according to their maximum moment - mahalanobis depth is outperformed by the same with the mcd - mahalanobis depth .this can be explained by the outliers present .five goodness measures are introduced aggregating performance of the classifiers over the 50 classification tasks w.r.t .different aspects and allowing for direct comparison of the classifiers .clearly , classification performance greatly depends on the choice of the depth and , if needed , the outsider treatment .the measures point out two -classifiers ( starting with the best ) : based on spatial and mahalanobis depth ( both using moment estimates ) .the rest of the classifiers show varying performance for different goodness measures , which is demonstrated by the goodness visualization .the experience with real - data problems tells us further that , in most practical cases , the separation procedure in the depth space stops after a few steps . in most casesthe subspace spanned by the depth features needed has dimension two , which points out the high stability of the separating rule .the problems and solutions investigated in this study are also of interest in more general settings : the problem of coping with outsiders is common to any statistical procedure that is based on depth plots and involves a notion of depth vanishing outside the convex hull . using the random tukey depth as an efficient approximation of the tukey depth and selecting the random directionshas many applications .available algorithms for exactly calculating the tukey depth are computationally expensive , but can serve as a benchmark .finally , the svm - simplified method is introduced and appears as a simple and efficient way to avoid the computational burden of tuning the svm .we thank the maintainers and contributors of the repositories + archive.ics.uci.edu/ml , + lib.stat.cmu.edu/datasets , + stat.ethz.ch/teaching/datasets .we are also grateful to oleksii pokotylo , master student at the national technical university of ukraine , for his help in preparation and maintaining the r - package ddalpha .valuable comments of two anonymous referees and an editor are greatly acknowledged .

Abstract: the -classifier , a nonparametric fast and very robust procedure , is described and applied to fifty classification problems regarding a broad spectrum of real - world data . the procedure first transforms the data from their original property space into a depth space , which is a low - dimensional unit cube , and then separates them by a projective invariant procedure , called -procedure . to each data point the transformation assigns its depth values with respect to the given classes . several alternative depth notions ( spatial depth , mahalanobis depth , projection depth , and tukey depth , the latter two being approximated by univariate projections ) are used in the procedure , and compared regarding their average error rates . with the tukey depth , which fits the distributions shape best and is most robust , ` outsiders ' , that is data points having zero depth in all classes , appear . they need an additional treatment for classification . evidence is also given about the dimension of the extended feature space needed for linear separation . the -procedure is available as an r - package .   * keywords : * classification , supervised learning , alpha - procedure , data depth , spatial depth , projection depth , random tukey depth , outsiders , features