the concept of distributive justice has been extensively studied in political philosophy and economics over the past few decades .one of the most important milestones in this field is rawls _ a theory of justice _ , which put forward equality as an outcome of the social contract that individuals behind a veil of ignorance should agree on .even though egalitarian doctrine that all human persons are equal in fundamental worth or moral status is a commonly shared idea , egalitarianism turns out to be a contested concept .there have been several divergent understandings of the meaning of equality , ways to achieve equality , or the metric to measure equality  .for instance , someone who puts more emphasis on equality of opportunity may have a very different opinion from those who put emphasis on equality of incomes in spite of the overall agreement on the concept of equality per se .  among numerous dimensions where egalitarianism varies, the question of what should be equalized is one that many different theories are competing on ; is it opportunity , capabilities , resource or welfare  ?dworkin , one of the most influential proponents of resource egalitarianism , admits the immediate appeal of the idea that it must ultimately be equality of welfare insofar as equality is important , and examines the logical consistency and practical applicability of this welfare egalitarianism  .according to dworkin , welfare egalitarianism , concerned with equality in every person s overall satisfaction , has an inconsistency in its logic .for example , if one accepts the idea that those who are handicapped need more resources to achieve equal welfare , the same argument should apply to those who have expensive tastes for the same reason .however , one should immediately recognize that the appeal of welfare egalitarianism becomes much less strong in the case of expensive tastes than in the case of the handicapped .the fact that the same idea can be accepted in some cases and seems disturbing in other cases reveals a logical inconsistency of welfare egalitarianism .dworkin also criticized welfare - based egalitarianism on that it inevitably relies on the possibility of interpersonal comparisons of utility , which places a large burden to a policy maker in practice .lastly , dworkin argues that it would probably prove impossible to reach a reasonable degree of equality in this conception in a community whose members held very different and very deeply felt political theories about justice in distribution .the last point is the one that we focus on in this paper .we will show that the existence of contradictory political theories does not immediately lead to the impossibility but can be formulated as dynamics which admits a unique solution under certain assumptions . by doing this, we will argue that the reasoning in can be regarded as a tool to analyze and advocate the idea of equality in welfare .to some extent , this is complementary to a previous work which argues that one can reach the idea of equality in welfare by starting from that of equality in resources  .following the logic that dworkin used when he showed the impossibility to reach an agreement on redistribution in terms of welfare , we also set aside the issues of logical inconsistency and inter - personal comparisons of individuals welfare . to focus on the relationship between individual preferences and resulting welfare, we also set aside the issue of impartiality ( see , e.g. , ) .we will further assume that individuals have preferences over the distribution of welfare among them  .many theoretical and experimental studies have shown that people are concerned with equality and fairness and often persist in fairness even when they lose monetary payoffs in doing so , e.g. , in the public games , ultimatum games or dictator games . this behavior can not be explained based on the assumption of self - regarding preferences but of others - regarding preferences or social preferences .our formulation in this paper can serve as a systematic description for such an approach , because it tells us a way to translate others payoffs into an individual s with respect to individual preference .let the concept of welfare be understood as the fulfillment of preferences  , including success in political preferences , i.e. , opinions of how welfare should be distributed .then the reasoning by dworkin  argues that such an egalitarian society , where everyone is concerned with the equality , will end up with supporting non - egalitarians by its own logic .suppose that a bigot enters an egalitarian society , with an opinion that some people deserve more than the others .this person will feel frustrated to see that her political preferences are not accepted by egalitarian neighbors , and her welfare becomes relatively lower than the others. if there is an official committed to compensating for inequality in welfare , by reallocating resources for example , the bigot should get extra resources from the official due to her political frustration , _ because _ she does not support the egalitarian idea of the society .this is called dworkin s paradox in this work .particularly we note that it can serve as an idealized model to represent our understanding of a modern democratic society. we will look into this hypothetical society a little closer .imagine a society of persons and an official .the official , representing a social institution , exists to mediate the global coordination .we assume that the total amount of welfare to be distributed among the persons is fixed as unity , and that the welfare is infinitely divisible , since we are interested only in relative fractions rather than absolute amounts that individuals have .the official herself does not take part in sharing the welfare , but only receives the persons opinions and find a way to distribute the welfare among them .let each person have a certain preference about how the welfare should be distributed , say with .we denote the actual welfare distribution as .the person s welfare is determined by the extent to which her preference is fulfilled . in other words, we consider an equation with a certain function , which is assumed to equally apply to all the persons .we suppose that the official wants to announce a stable welfare distribution such that each person s relative share remains unchanged after the announcement , which means that solves eq .( [ eq : f ] ) self - consistently .it is important to note that the preferences reported by each individual are assumed to be true and available to the official at every moment , which helps us to focus on basic ideas of the paradox .a few remarks are in order .first , we emphasize that the official plays only a passive role in this setup .as we will see below , the society reaches the same self - consistent solution as long as every person s welfare becomes public knowledge all the time .the official may guarantee such information to be accessible and accelerate the coordination but the official is basically assigned limited tasks compared to the original argument .second , related to the first point , we do not require the division of welfare to be impartial from a certain observer s point of view .our question is simply how much fulfillment one can get depending on her preference . in this sense , our approach differs from the impartial - division problem  and does not touch conceptual difficulties of impartiality ( see , e.g. , ) .finally , individuals are not behind the veil of ignorance .rather , each of them is supposed to construct a concrete opinion about every other individual using any kind of available information .although this can impose practical difficulties in a large society , it helps us avoid any theoretical ambiguity or conflict with the ethic of priority  found in the veil of ignorance  .  in order to give a more concrete form to eq .( [ eq : f ] ) , we first consider how to measure similarity or affinity between distributions and then plug it into eq .( [ eq : f ] ) .suppose two arbitrary distributions , and , with , , and .we define a suitable affinity function between them , whose specific functional form will be characterized by requiring the following four postulates  .first , we postulate separability , which means that one can refine affinity contribution from a certain bin by looking into the bin in a higher resolution without referring to the outside of the bin .second , we postulate invariance under permutation , because every bin is equivalent .third , the affinity should be non - negative , i.e. , , where if and only if is orthogonal to , whereas a maximum value is obtained if and only if . the distributions and are orthogonal when for every non - zero and vice versa .last , it should be symmetric in the sense that , which is intuitively justified .these four postulates characterize our affinity function as commonly known as the bhattacharyya measure  .while details of the derivation are shown in appendix , this functional form has a clear geometric interpretation : it can be viewed as a dot product of two vectors and , both of which are located on an -dimensional unit sphere by .it therefore becomes maximized when two vectors point in the same direction .it is plausible to assume that the function in eq .( [ eq : f ] ) will be a non - decreasing function of the affinity between and , so that  ] for any and such that is non - zero . otherwise , is reducible .such a reducible case is not our concern since a society is meaningful only when individuals interact with each other .henceforth , only irreducible cases are considered . then , unless everyone has zero self - interest , one can prove that there exists a unique stable distribution for every by using the perron - frobenius theorem  .in other words , is the only stable fixed point under the action of , so the official should distribute welfare as given by .let us consider a situation where an egalitarian with meets a selfish person with .the corresponding matrix formulation will be and its stable welfare distribution is obtained by analyzing eigenvectors as .we arrive at this even if we start from for the following reason : let be known to both the persons every time step .the egalitarian first feels happy to see the initial equality in , while the selfish person feels unsatisfied , which makes a difference at the next step .the drop in makes the selfish person even more upset , so her welfare continues to decrease until it reaches the stable value , .we now show that egalitarianism is the minimax solution of this two - person zero - sum game  .generalizing eq .( [ eq : wex ] ) as the eigenvalue analysis yields the converged share for the first person , , as shown in fig .[ fig : contour ] .it is a saddle - like shape and this person can minimize risk when she has demanded a moderate share of at the first place .the same is true for the other person as well .although we have assumed fixed preferences in developing the model , if the preferences can evolve in the long run to maximize individual welfare , therefore , this plot shows that this two - person case will lead to an equal welfare distribution .  in practice, a selfish person can be tempted to deceive the official by reporting a false preference to receive a larger share . providedthat person 2 has claimed her self - interest as a certain value , person 1 can always compute the best reply by looking up the maximal share at the given in fig .[ fig : contour ] .even if her true self - interest is higher than this false , she should still report to the official , knowing that she can not get better than in any way . when person 1 has chosen for this reason , the same consideration will lead person 2 to choose  ] .person s overall political preference can be described by another uniform random variable : for , this person is an egalitarian .if , she believes that the better deserve more , while means the opposite .in addition , is assumed to be uncorrelated with .the political preference is then assigned as , which satisfies . since is a relative quantity, one can always subtract an offset value to make , by which the normalization condition is satisfied .we can obtain the stable distribution by taking as a very large constant and assuming that is a function of only . by replacing the summation in by an integral , we get ^ 2 / ( 3\pi + 8),\ ] ] where the proportionality coefficient is determined by the normalization condition .this is an even function of with a maximum , implying the highest fulfillment for an egalitarian .one could point out that this describes just one possible distribution , not necessarily the stable one .however , we see that the integral is positive for all , and thus for all , and the perron - frobenius theorem tells us that this positivity is true only for the stable distribution  .this justifies our starting assumption that person s welfare is not determined by her innate part but by her political preference in this society .it is notable that critics have said that the idea of equality in welfare is insensitive to individual responsibility  .as explained in , one may consider a set of variables characterizing an individual and classify them into two categories : the first category consists of innate properties such as talents that an individual is hardly responsible for .the second category , on the other hand , includes choices and even some of preferences that we can connect to individual responsibility .if we regard as representing the first category while as representing the second category , this example shows that each individual does take responsibility for her political preference but not for her talents .it could be also argued that the limit squeezes into zero so that the whole problem reduces to the egalitarian society above , where everyone gets .that can be regarded as a first - order approximation of this problem .the calculation given here shows that an egalitarian indeed receives more than in the crude approximation .   in all the cases considered so far, preferences could be said to be neutral on the society level in the sense that there is no systematic bias over the whole society .let us now imagine that persons with identical preferences , , but not necessarily egalitarians .the other person indexed by has another type of preference , . by the similar reasoning as in the egalitarian society, the situation can be simplified to the following matrix : where means a summation over excluding .recall that the persons with an identical preference have the same amount of fulfillment so person should not distinguish them in order to maximize affinity between her preference and the welfare distribution among them .the above matrix means that we should only determine how to divide welfare between the person and the other persons .the eigenvalue analysis leads to with and .we can differentiate eq .( [ eq : rk2 ] ) with respect to to find the maximum .an easier alternative way is to observe from the separability that the maximum value is obtained when , because the question is how to match person s preference with the welfare distribution , where the second elements represent the whole persons . by solving with eq .( [ eq : rk2 ] ) , one can get maximizing as a solution of the following equation , w^4 - 2h w^3 + ( 1-h^2 ) w^2 + 2h w - 1 = 0 , \label{eq:4th}\ ] ] with .for the egalitarian with and , substituting satisfies eq .( [ eq:4th ] ) , consistently with the analysis of the egalitarians .we may also suppose that describes an unequal distribution so that the whole society is biased in a certain way . as a specific example , let us assume that , that is , almost everyone wants people with higher indices to have more welfare . with a normalization constant , it should mean that with , and we thus have inserting this into eq .( [ eq : rk2 ] ) , we plot in fig .[ fig : unequal ] , where the maximum is found at the crossing with .it is a little higher than for every .we can do the same calculation for a more severe situation of inequality by setting , which again yields the same conclusion with a bit larger .the difference between the optimal and does not vanish as whether or .this can be shown by inserting on the left - hand side of eq .( [ eq:4th ] ) and taking , which does not yield zero on the right - hand side .therefore , the person should demand a little more for herself than before and distribute the remainder of the preferences equally to the others , even though they are far from egalitarians , in order to get the maximum welfare .   in order to see whether egalitarians can eventually take over the society , we need to check whether the egalitarian preference remains as an attractive alternative when the society has both egalitarians and non - egalitarians with significant numberslet us imagine an inhomogeneous society where there are roughly two large groups : every person in one group of size occupies a high index and believes that the welfare should be proportional to . on the other hand , every person in the other group of size has a low index and an egalitarian preference .our question is what kind of preference is good for a person on the border , i.e. , with index . again , since people with identical preferences will get the same amount of welfare , the focal person on the border need not distinguish the members in each group : suppose that she wishes for each member in the egalitarian group and for each member in the non - egalitarian group .the normalization condition then determines her self - interest .depending on how she decides and , her final welfare will be calculated by analyzing the following matrix , where the second row describes this focal person .the conservation of the total welfare is imposed by setting .when is small , the maximum of is close to the egalitarian solution ( fig .[ fig : inhom]a ) .it agrees with the result of the homogeneously unequal preferences given above since the egalitarian preference is still an absolute minority .hence , if this person can choose her own preference , the society will possibly have one more egalitarian . as becomes larger , however , the situation gets different in that the maximum is located far from the egalitarian solution ( fig .[ fig : inhom]b ) .it implies that the transition process toward the egalitarian direction may exhibit transient behavior , instead of being smooth all the time .the theory of welfare is not an empty ideal as claimed in : dealing with a society where everyone has an identical non - egalitarian preference , we have found that the theory recommends something very similar to an egalitarian preference , instead of just rubber - stamping the dominant non - egalitarian opinion .in addition , this finding shows that the egalitarian society is in fact the _ only _ strict nash equilibrium .we therefore conclude that our analysis gives a strong support to equality of welfare by specifying which social and political conditions make it possible .on the other hand , our conclusion implies that a society can encourage egalitarianism by guaranteeing freedom of communication so that everyone can constantly express her fulfillment in public . in this respect ,we can perhaps mention one of the central messages in that `` liberty is essential to any process in which equality is defined and secured . '' in particular, we would like to put an extra emphasis on the communicative aspect of the liberty .  on a longer perspective ,our results suggest an explanation of how the concept of fairness could develop at a certain moment in the history of evolution when human beings became able to construct internal expectation for the future and understand others minds by communication .it is also worth stressing that our conclusion on egalitarianism as a strict nash equilibrium under certain well - defined conditions is strong enough to open further theoretical extensions and empirical tests .we are grateful to an anonymous reviewer for many helpful comments .is indebted to petter minnhagen and sang hoon lee for discussions .s.k.b . acknowledges the support from the swedish research council with grant no .621 - 2008 - 4449 .j.k.c . was supported by korea research foundation grant funded by the korean government ( nrf-2010 - 330-b00077 ) .b.j.k . was supported by the national research foundation of korea ( nrf ) grant funded by the korea government ( mest ) ( no .2010 - 0008758 ) .  10 [ 1]`#1 ` urlstyle [ 1]doi:#1  [ 1 ] [ 2 ]  _ _ _ _ _ _ _ _ _ _ _ _ _ _ key : # 1 + annotation : # 2 _ _ _ _ _ _ _ _ _ _ _ _ _ _  rawls j ( 1971 ) a theory of justice .cambridge , ma : harvard university press .sen a ( 1980 ) equality of what ?in tanner lectures on human values i , cambridge : cambridge university press , chapter 6 .195 - 220 .sen a ( 1997 ) on economic inequality .cambridge : cambridge university press , 1 - 23 pp .dworkin r ( 1981 ) what is equality ?part 1 : equality of welfare .philos public aff 10 : 185 - 246 .dworkin r ( 1981 ) what is equality ?part 2 : equality of resources .philos public aff 10 : 283 - 345 .hausman dm , mcpherson ms ( 2006 ) economic analysis , moral philosophy , and public policy .new york : cambridge university press , 185 - 186 pp .dworkin r ( 2000 ) sovereign virtue : the theory and practice of equality .cambridge , ma : harvard university press , 21 - 25 pp .roemer je ( 1986 ) equality of resources implies equality of welfare .q j econ 101 : 751 - 784 .de clippel g , moulin h , tideman n ( 2008 ) the impartial division of a dollar .j econ theory 139 : 176 - 191 .  mongin p( 2001 ) the impartial observer theorem of social ethics .econ philos 17 : 147 - 179 .fehr e , schmidt km ( 1999 ) a theory of fairness , competition , and cooperation .q j econ 114 : 817 - 868 .bolton ge , ockenfels a ( 2000 ) erc : a theory of equity , reciprocity , and competition .am econ rev 90 : 166 - 193 .singer p ( 1993 ) practical ethics .new york : cambridge university press , 83 - 109 pp .moreno - ternero jd , roemer je ( 2006 ) impartiality , priority , and solidarity in the theory of justice .econometrica 74 : 1419 - 1427 .moreno - ternero jd , roemerje ( 2008 ) the veil of ignorance violates priority .econ philos 24 : 233 - 257 .mathai am , rathie pn ( 1972 ) characterization of matusita s measure of affinity .ann inst statist math 24 : 473 - 483 .bhattacharyya a ( 1943 ) on a measure of divergence between two statistical populations defined by their probability distribution .bull calcutta math soc 35 : 99 - 110 .meyer cd ( 2000 ) matrix analysis and applied linear algebra .philadelphia : siam , 661 - 704 pp .davis md ( 1970 ) game theory : a nontechnical introduction .new york : basic books , 19 - 48 pp .nash jf , jr ( 1950 ) equilibrium points in n - person games .proc natl acad sci usa 36 : 48 - 49 .fleurbaey m , maniquet f ( 1996 ) fair allocation with unequal production skills : the no envy approach to compensation .math soc sci 32 : 71 - 93 .) * . the curves on the plane show contour lines.,scaledwidth=45.0% ]  ) as a function of when for .* the horizontal line shows .the maximum of is located at the crossing with the line ., scaledwidth=45.0% ]   obtained by solving eq .( [ eq : inhom ] ) within a region for . * ( a ) .( b ) .the crosses show . , title="fig:",scaledwidth=45.0% ]  obtained by solving eq . ([ eq : inhom ] ) within a region for . * ( a ) .( b ) .the crosses show . , title="fig:",scaledwidth=45.0% ]let us consider two arbitrary distributions , and , with , , and .we will define an affinity function between them , whose specific functional form will be characterized by requiring the following postulates .* p1 : separability ,\end{aligned}\ ] ] where and are positive with .in addition , is a non - negative differentiable function and converges to zero as or .the situation can be described as follows : we first observe and , instead of the real and .we can calculate affinity in this resolution .the question is how affinity will change when we come to know the real and .if the substructures inside and have an exact affinity by , nothing will change from the previous calculation , i.e. , .if they actually had no affinity by in this better resolution , on the other hand , the overall affinity should decrease by a certain amount , which will be a function of and .for example , for or , the decrement in will be also vanishingly small even if the subpopulations inside them look completely different .* p2 : invariance under permutation note that it is for , which is enough to prove the same property for general in combination with the other postulates . *p3 : non - negativity where if and only if is orthogonal to , whereas a maximum value is obtained if and only if .* p4 : symmetry   let us explain some direct consequences of p1 . for , it yields a trivial equality . for , we have .\ ] ] for , we will see two different ways to calculate affinity between and .the first way is the following : \nonumber\\ & = & \rho_2 \begin{pmatrix } p_3 , p_4\\ q_3 , q_4 \end{pmatrix } + \delta(p_3,q_3 ) \left\ { \rho_2 \begin{pmatrix } \frac{p_2}{p_3 } , \frac{p_3}{p_3}\\ \frac{q_2}{q_3 } , \frac{q_3}{q_3 } \end{pmatrix } \right .\nonumber\\ & & \left .+ \delta\left ( \frac{p_2}{p_3 } , \frac{q_2}{q_3 } \right ) \left [ \rho_2 \begin{pmatrix } \frac{p_1}{p_2 } , \frac{p_2}{p_2}\\ \frac{q_1}{q_2 } , \frac{q_2}{q_2 } \end{pmatrix } - 1 \right ] - 1 \right\}. \label{eq : rho4a}\end{aligned}\ ] ] we then consider another way to calculate the same quantity as follows : \nonumber\\ & = & \rho_2 \begin{pmatrix } p_3 , p_4\\ q_3 , q_4 \end{pmatrix } + \delta(p_3 , q_3 ) \left [ \rho_2 \begin{pmatrix } \frac{p_2}{p_3 } , \frac{p_3}{p_3}\\ \frac{q_2}{q_3 } , \frac{q_3}{q_3 } \end{pmatrix } -1 \right]\nonumber\\ & & + \delta(p_2 , q_2 ) \left [ \rho_2 \begin{pmatrix } \frac{p_1}{p_2 } , \frac{p_2}{p_2 } \\\frac{q_1}{q_2 } , \frac{q_2}{q_2 } \end{pmatrix }  - 1 \right ] .\label{eq : rho4b}\end{aligned}\ ] ] comparing eq .( [ eq : rho4a ] ) with eq .( [ eq : rho4b ] ) , we see that note that can take any value between zero and one , regardless of , and that is also independent of in the same way , as long as and are nonzero .now we define for  ] and .the above equality can be then rewritten as defining a function g(p , q ) \label{def : f}\ ] ] for , we apply eq .( [ eq : grs ] ) to to obtain note from eq .( [ eq : delta ] ) that hence , we see that \label{eq : f1}\\ & = & g(pr , qs ) + \delta(1-pr , 1-qs ) \left\ { g\left [ \frac{p(1-r)}{1-pr } , \frac{q(1-s)}{1-qs } \right ] \right.\nonumber\\ & & + \left .\delta \left ( \frac{1-p}{1-qr } , \frac{1-q}{1-qs } \right ) g(r , s ) \right\ } \label{eq : f2},\end{aligned}\ ] ] where the last equality comes from eq .( [ eq : grs ] ) .equation ( [ eq : g1x ] ) then gives us = g\left ( \frac{1-p}{1-pr } , \frac{1-q}{1-qs } \right),\ ] ] relating eqs .( [ eq : f1 ] ) and ( [ eq : f2 ] ) as follows : \\ & = & g(pr , qs ) + \delta(1-pr ,1-qs ) \left [ g\left ( \frac{1-p}{1-pr } , \frac{1-q}{1-qs } \right ) \right.\\ & & + \left .\delta \left ( \frac{1-p}{1-pr } , \frac{1-q}{1-qs } \right ) g(r , s ) \right].\end{aligned}\ ] ] in short , we have just confirmed that from the definition of in eq .( [ def : f ] ) , we see that for this to be true for every independent set of , it must be that \ ] ] with a certain constant .then p1 says that which is the lowest possible value of .however , according to the definition of in eq .( [ def : g ] ) and p3 , this lower bound should be . in other words , and we have restricted our variables as and , but it is not difficult to check that the above expression can be readily used in  ] when is characterized as will be discussed below. then p1 works as a recursive relation , yielding an expression for general : now we ask ourselves how should look .p1 and p4 imply that can be expanded as with coefficients and exponents .so let us write if we define , it means that should be of the following form , among every possible , it is only that satisfies the maximization condition in p3 .this can be shown by variational calculus using a lagrange multiplier , {q_i = p_i } = \alpha p_i^{2\alpha-1 } - \mu = 0,\ ] ] which is satisfied for any when and .hence we separate this term from the others in eq .( [ eq : over ] ) as where means that is excluded from the summation .there exists a maximum value in , which is obtained by noting that .since this value is the same for every , the last term should be kept constant for any , and we have to choose for every in the second term . to sum up , our affinity function is characterized as with a positive constant , the maximum value of .

Abstract: how to distribute welfare in a society is a key issue in the subject of distributional justice , which is deeply involved with notions of fairness . following a thought experiment by dworkin , this work considers a society of individuals with different preferences on the welfare distribution and an official to mediate the coordination among them . based on a simple assumption that an individual s welfare is proportional to how her preference is fulfilled by the actual distribution , we show that an egalitarian preference is a strict nash equilibrium and can be favorable even in certain inhomogeneous situations . these suggest how communication can encourage and secure a notion of fairness .


many unsupervised learning algorithms , such as spectral clustering , and low - dimensional embedding via laplacian eigenmaps and diffusion maps , , need as input a _ matrix of pairwise similarities _ between the different objects in our data . in some cases, obtaining the full matrix can be a costly matter .for example , may be based on some expensive - to - compute metric such as w2d ; based on some physical measurement ( such as in some computational biology applications ) ; or is given by a human annotator .in such cases , we would like to have a good approximation of the ( initially unknown ) matrix , while querying only a limited number of entries .an alternative but equivalent viewpoint is the problem of approximating an unknown weighted undirected graph , by querying a limited number of edges .this question has received previous attention in works such as and , which focus on the task of spectral clustering into two clusters , and assuming two such distinct clusters indeed exist ( i.e. that there is a big gap between the second and third eigenvalues of the laplacian matrix ) . in this workwe consider , both theoretically and algorithmically , the question of query - based graph approximation more generally , obtaining results relevant beyond two clusters and beyond spectral clustering .when considering graph approximations , the first question is what notion of approximation to consider .one important notion is _ cut approximation _ where we wish for every cut in the approximated graph to have weight close to the weight of the cut in the original graph up to a multiplicative factor . many machine learning algorithms ( and many more general algorithms ) such as cut based clustering , energy minimization , etc . are based on cuts , so this notion of approximation is natural for these uses .a stronger notion is _ spectral approximation _ , where we wish to uniformly approximate the quadratic form defined by the laplacian up to a multiplicative factor .this approximation is important for algorithms such as spectral clustering , laplacian eigenmaps , diffusion maps , etc .that use the connection between the spectral properties of the laplacian matrix and the graph .our theoretical analysis focuses on the number of queries needed for such approximations .we first consider the simple and intuitive strategy of sampling edges uniformly at random , and obtain results for both cut and spectral approximations , under various assumptions .we note that these results are considerably more general than the theoretical analysis in , which focuses on the behavior of the 2nd eigenvector of the laplacian matrix , and crucially rely on this large eigengap .we then consider how to extend these results to adaptive sampling strategies , and design a generic framework as well as a new adaptive sampling algorithm for clustering ( ` clus2k ` ) .compared to previous approaches , the algorithm is much simpler and avoids doing a costly full eigen - decomposition at each iteration , yet experimentally appears to obtain equal or even better performance on a range of datasets .our theoretical results build on techniques for graph sparsification ( , ) , where the task is to find a sparse approximation to a given graph .this is somewhat similar to our task , but with two important differences : first and foremost , we do not have access to the full graph , whereas in graph sparsification the graph is given , and this full knowledge is used by algorithms for this task ( e.g. using the sum of edge weights associated with each node ) .second , our goal is to minimize the number of edge sampled , not the number of edges in the resulting graph ( of course by sampling a smaller number of edges we will get a sparser graph ) .notice that if we wish to end with a sparse graph , one can always use any graph sparsification technique on our resulting graph and get a graph with guarantees on sparsity .for simplicity we will consider all graphs as full weighted graphs ( with zero weights at edges that do not exist ) and so any graph will be defined by a set of vertices and a weight matrix .we will start with a few basic definitions .let be a weighted graph and a subset of vertices , then the cut defined by , , is the sum of all the weights of edges that have exactly one endpoint in .let and be two graphs on the same set of vertices . is an -cut approximation of if for any we have let . the graph laplacian is defined as where is a diagonal matrix with values .the normalized graph laplacian is defined as .the laplacian holds much information about the graph .one of the main connections of the laplacian , and in particular the quadratic form it represents , to the graph is through the equation when this is easily seen to be the value of the cut defined by .many spectral graph techniques , such as spectral clustering , can be seen as a relaxation of such a discrete problem to .a graph is an approximation of if   we note that this is different than requiring ( using the matrix 2-norm ) as we can view it as a multiplicative error vs. an additive error term . in particular , it implies approximation of eigenvectors ( using the min - max theorem ) , which is relevant to many spectral algorithms , and includes the approximation of the 2nd eigenvector ( the focus of the analysis in ) as a special case moreover , it implies cut approximation ( via equation [ quadlap ] ) , and is in fact strictly stronger ( see for a simple example of a cut approximation which is not a spectral approximation ) .therefore , we will focus on spectral approximation in our theoretical results .our initial approximation strategy will be to uniformly at random sample a subset of edges , i.e. pick edges without replacement and construct a graph with weights for any and zero otherwise , where is the probability any edge is sampled .it is easy to see that the =w ] and all nondecreasing functions and , \leq \mathds{e}[f(x_i , i\in i)]\mathds{e}[g(x_j , j\in j)]\ ] ]  intuitively , a group of random variables are negatively dependent if when some of them have a high value , the others are more probable to have lower values . if we pick edges uniformly , each edge that has been pickedlowers the chances of the other edges to get picked , so intuitively the probabilities are negatively dependent .the edge picking probabilities have been shown to be indeed negatively dependent in .an important application of negative dependence is that the chernoff - hoeffding bounds , which hold for sums of independent random variables , also hold for negatively dependent variables .see supplementary material for details .we can now state the general spectral approximation theorem :  [ spectralbound ] let be a graph with weights  ] . from the markov s inequalitywe get that . if then the intersection is empty and we do not have an -approximation for any proving .this theorem proves that in order to get any reasonable approximation with a small budget ( at least with uniform sampling ) the original graphs minimal cut can not be too small and that samples are needed . comparing this to theorem [ spectralbound ] ( noticing ) we see that , for graphs with a lower bound on by sampling within a logarithmic factor of this lower bound is sufficient to ensure a good cut approximation .clustering algorithms assume a certain structure of the graph , generally they assume strongly connected components , i.e. the clusters , with weak connections between them ( the precise assumptions vary from algorithm to algorithm ) . while this is a bad scenario for approximation , as this normally means a small minimal cut ( and for spectral clustering a small ) , we will show how approximation can be used ( on the inner - cluster graphs ) to obtain useful results .we give two results , one geared towards spectral approximation , and the other towards cut approximation .assume a graph consists of clusters , define as the block diagonal matrix consisting of the similarity scores between same - cluster elements . and the off - diagonal elements .+  [ assumlambda ] define , the smallest of all the second normalized eigenvalues over all .assume for some constant .[ assumd ] where for some .[ outsmall ] assume that for some .assumption [ assumlambda ] implies well connected clusters , while assumption [ assumd ] excludes sparse , well - connected graphs , which we have already shown earlier to be hard to approximate .assumption [ outsmall ] essentially requires the between - cluster connections to be relatively weaker than the within - cluster connections .  under these assumptions ,the next theorem proves the clusters can be found .[ specclusterable ] let p be the zero eigenspace of corresponding to the clusters , the laplacian of the graph we get by sampling edges for , and the space spanned by the first eigenvectors of . under previous assumptions , .we simplified the statement in order not to get overwhelmed by notation . is a diagonal matrix whose diagonal values correspond to the canonical angles between the subspaces and , and is a common way to measure distance between subspaces .if , i.e. was spanned by eigenvectors of the full , then the theorem would be true by the sin - theta theorem using our assumptions .we need to show that this theorem can be used with .the sin - theta theorem states that where the `` noise '' factor , and the second eigenvalue of the `` signal '' factor . using theorem [ spectralbound ] and our first two assumptions we can approximate each and use to show that .we now only need to show .this can be done using the matrix chernoff inequality , by applying a result in that shows how it can be adapted to sampling without replacements .we note that the result in is limited to sampling without replacements as negative dependence has no obvious extension to random matrices . for further details see the supplementary material  this gives us a tradeoff between the number of edges sampled and the error .the theoretical guarantee from the sin - theta theorem for the complete graph is so for we have the same guarantee as if we used the full graph . for enough one can get as small as desired by using .cut based clustering , such as , have a different natural notion of `` clusterable '' .we will assume nothing on eigenvalues , making this more general than the previous section .[ asscut1 ] assume can be partitioned into clusters , within which the minimal cut is at least . furthermore , assume that any cut separating between the clusters of ( but does not split same cluster elements ) is smaller then , and that .these assumptions basically require the inner - cluster connections to be relatively stronger than between - cluster connections .[ clustering ] let be a graph with weights  ] are negatively dependent variables and define then |\geq\epsilon\mathds{e}[x]\right ) \leq 2\exp\left(-\frac{\epsilon^2\mathds{e}[x]}{3}\right ) \ ] ]  see 1.6 and 3.1 for details .+ first it is important to note a change in notation from in order to be consistent with notation used in .a -spectral approximation in our paper is weaker then a -spectral approximation in .+ we will now go over the main changes needed to prove theorem 6.1 in ( disregarding s.2 ) with negatively dependent sampling of edges and weights  ] instead of \leq\frac{1}{\gamma^{k+l-1}d_r} ] ) .+ the main change is to lemma 6.4 . using the modified lemma 6.6 and substituting negative dependents for independence one can prove \leq\frac{1}{\gamma^{k-|t|}}\sum_{\sigma\,valid\,for\,t,\tau}\prod_{s\in t}\frac{w_{v_{s-1},v_s}}{d_{v_{s-1}}}.\ ] ] instead of equation 10 in the paper .the last change is to pick proportional to instead of uniformly to prove that    instead of equation 11 .from there on all changes are straightforward .let . let be the zeros eigenspace of , which is the same as the zero eigenspace of , if all the are connected .let be the space spanned by the first eigenvectors of . according to the sin - theta theorem , where is the spectral norm of and is the second smallest unnormalized eigenvalue of . to prove the theorem we will show that and that .+ the first claim is through using the first two assumptions and the following lemma  [ nor_unnor ] let and be the second smallest normalized and unnormalized eigenvalues of , and then .  from the min - max theorem we have  and the lemma follows from the fact that .let be the number of edges needed in order to have an -spectral approximation of each inner - cluster matrix for with probability , then by theorem [ spectralbound ] we have that .using this fact , the first two assumptions and lemma [ nor_unnor ] , it is easy to see that .+ we now need to show that . the main tool would be the matrix chernoff inequlity for sampling matrixes without replacements .consider a finite sequence of hermitian matrices sampled uniformly without replacements from a finite set of matrices of dimension n. assume that define then }||)\leq n\cdot\left(\frac{e^\epsilon}{(1+\epsilon)^{1+\epsilon}}\right)^{||\mathds{e}[y]||/r}\ ] ]  this is an adaptation of theorem 5.1.1 from replacing the independence requirement to sampling without replacements . in order to adapt the proof we notice that the only place where independence is used in in lemma 3.5.1 ( subadditivity of the matrix cumulant generating functions ) where we need to prove that \leq tr\left(\exp\left(\sum\log\mathds{e}e^{\theta\bf x_i}\right)\right)\ ] ] using the result of , if are sampled uniformly at random without replacements for a finite set , and are sampled with the same probability with replacements then \leq \mathds{e}\left[tr\left(\exp\left(\sum\theta{\bf y_i}\right)\right)\right]\ ] ] so we can conclude that \leq\mathds{e}\left[tr\left(\exp\left(\sum\theta{\bf y_i}\right)\right)\right]\leq\ ] ] where the second inequality is from [ cgfs ] as are independent .we define for each edge connecting nodes in different clusters the matrix that is equal to zero with probability and is equal to with probability , where is the laplacian of a single edge graph with weight .then , =l^{out} ] , with minimal cut , and its approximation after sampling edges uniformly . if where , then the probability that is not an -cut approximation is smaller then .this is an adaptation of the proof in - consider a cut with weight .let the probability to sample a single edge .let where is an indicator whether edge on the cut was sampled and its weight .define the sum of on all the edges along the cut , then by the fact that edges are negatively dependent and theorem [ hoeffding ] , the probability that the cut is not an approximation is smaller then }{3}\right)=2\exp\left ( -\frac{\epsilon^2\alpha c p}{3}\right)\leq \\\leq 2\exp\left ( -\left(\ln\left(\frac{1}{\delta}\right)+k\right)\alpha\right)\cdot n^{-2\alpha } \end{split}\ ] ] define and let the number of cuts with value in the original graph .by the union bound the probability that some cut is not an approximation is less then ( notice that this sum is well defined since is non zero only in a finite number of values ) . defining then by the previous lemma .let be any measure on such that , then the integral is maximized when .this is due to the fact that is a monotonically decreasing function , so if the inequality is not tight at some point we could increase the value by picking for some appropriate and ( where is the dirac delta function ) . from thiswe can conclude that the probability of some cut not being an -approximation is bounded by    a drawback is that the theorem gives a bound that depends on the minimal cut , which we do not know , and unlike the situation in we can not approximate it using the full graph .we can prove a bound that uses only known data about the graph .the following theorem shows we can lower bound .[ cmin ] let be a graph with weights  ] and its approximation after sampling edges with minimal cut .define and as in previous theorems .if then the probability that is not an -cut approximation is smaller then .  this is just using lemma [ cmin ] with error probability and using that for theorem [ origkar ] with the same error probability and the union bound .this theorem gives a high probability bound that depends only on observable quantities . while the notation is a bit cumbersome , it is easy to see that if , i.e. the unscaled weight of the smallest cut is not too small , then and we have a bound that is almost as good as if we knew the real . + we will now prove theorem [ clustering ] in the paper .let be a graph with weights  ] . by definition of get that = \tilde{p}(e_i)\cdot\exp\left(\frac{t\rho w(e_i)}{\tilde{p}(e_i)}\right)+(1-\tilde{p}(e_i))\ ] ] one can easily verify that the right hand side of equation [ temp1 ] decreases monotonically with , so the fact that and implies that \leq \rho w(e_i)e^t+(1-\rho w(e_i))=\ ] ] where the last inequality is due to the fact that for .we can finish the proof since \leq\prod\limits_{i=1}\limits^l \mathds{e}[\exp(t\rho x_{ik})|\tilde{g}_{k-1 } ] \\ & \leq \exp(\rho c ( e^t-1 ) ) .\end{split}\ ] ] as .we can now prove the concentration of measure bound for a single cut  let be a graph such that and the output of a bounded adaptive sampling algorithm with replacements such that then the probability that a cut with weight in is not a -approximation is bounded by .we need to show that the proof is similar to the proof of the chernoff bound , replacing independence with lemma [ boundlemma ] .first look at . using the standard trick for all  by the markov inequality thisis bounded by }{\exp(t(1+\epsilon ) \rho m c)} ] .as is a deterministic function of this is equal to \exp\left(t\rho\sum\limits_{k=1}\limits^{m-1 } y_k \right)\right ] \\ & \leq \mathds{e}\left[\exp\left(t\rho\sum\limits_{k=1}\limits^{m-1 } y_k \right)\right]\exp(\rho c ( e^t-1 ) ) .\end{split}\ ] ] using lemma [ boundlemma ] . by inductionwe can conclude that the expectation is smaller then .we have shown that following the steps as in the standard chernoff bound proof one can show that this is smaller ( for the right ) then .the proof for this bound on is done in a similar fashion , and using the union bound we finish our proof .using one can now show similar theorems to what we shown in the previous section with this theorem replacing the ( negatively dependent ) chernoff bound .for a specific graph one can always design a bad biased sampling scheme .consider an adversarial scheme that always samples the largest weight edge between two constant clusters , it is easy to see that this can lead to bad cut clustering . to circumvent thiswe will consider graphs where the edge weights between the clusters , which we regard as noise , are picked randomly .let be the graph after sampling edges without replacements ( with probability of sampling uniformly ) with .let and be the minimal cut weight inside any cluster and the maximal cut weight between clusters , under previous assumptions the probability that is smaller then .  using theorem [ origkar ] on the edges sampled uniformly ( remembering that the biased sampling can only increase the cut weight ) we get that with probability greater then , .if we consider the weight of any cut between clusters , then the key observation is that because the edges are picked uniformly at random , then whatever the algorithm does is equivalent to running a uniform sampling of a constant edge set .we then get that the expected minimal cut weight is using lemma [ interlemma ] ( the upper bound is by looking as if all edges where picked from this cut ). we can now use the markov inequality to show .

Abstract: we consider the problem of learning from a similarity matrix ( such as spectral clustering and low - dimensional embedding ) , when computing pairwise similarities are costly , and only a limited number of entries can be observed . we provide a theoretical analysis using standard notions of graph approximation , significantly generalizing previous results ( which focused on spectral clustering with two clusters ) . we also propose a new algorithmic approach based on adaptive sampling , which experimentally matches or improves on previous methods , while being considerably more general and computationally cheaper .


it is known that student s _ t_-distributions fit the returns of stocks and equity indices better than normal distributions .prices for european options can be calculated for returns that follow a student s _ t_-distribution if the value of the underlying asset is capped or if the _ t_-distribution is truncated .capping the value of the asset or truncating the distribution keeps the integrals , which are required to price the option , finite .truncation is not an academic exercise .it is unphysical for the value of an asset to approach infinity .thus truncation is required for a model of values of assets and of returns on assets to reflect reality .capping has a similar effect as truncation and is simply a slightly different approach to creating a physical model for the price of an option .it is the `` fat tails '' of the student s _ t_-distribution that make truncation necessary .the normal pdf , on which the black - scholes formula for the price of an european option is based , decreases rapidly for values that are more than several standard deviations beyond the mean value .this rapid decrease keeps the integrals , which are required to price an option , finite and thus allows a price to be found without consideration of truncation or capping .however , this rapid decrease in the tails is not consistent with the observed distribution of returns , and thus calculations based on a normal pdf underestimate the probability of events that are several standard deviations beyond the mean .  the student s _ t_-distribution offers support from to + . however there is no requirement that the full domain be used when working with a distribution .if the data do not or can not exist over the full domain and yet the frequency of occurrence of the data do follow the distribution over a subset of the full domain , it is correct to truncate the domain to match the observations or physical constraints .one need only use a concept from conditional probability to maintain the normalization of the truncated pdf . in terms of the stock market , the magnitude of the returns and the value of an asset are truncated or capped . in practice , one does not observe a stock that is worth an infinite amount or daily returns on indices that are 30% .  in working with a student s _ t_-distribution to price european options ,it is necessary to decide upon the shape parameter and the scale parameter ( the scale parameter is the volatility ) for the _ t_-distribution , and to decide where to place the cap or the truncation .the decision on the value for the cap or truncation is based to a large extent on tolerance to risk or equivalently on the anticipated maximum value for the asset .one can choose to work with , e.g. , % confidence or 95% confidence , or choose the corresponding maximum value _x. the shape parameter and the volatility are a different story .these parameters should be chosen to match the underlying distribution .  in this paperwe discuss pricing options when the underlying distribution is a student s _ t_-distribution .we call a model to price european options when the underlying distribution is a _ t_-distribution a gosset model , in honour of w.s .gosset who published under the name student .we briefly discuss implementation , present the gosset equations in appendix b , discuss what is an appropriate shape parameter based on historical returns , and compare some of the greeks for the gosset and black - scholes prices for european call options .the gosset formulae for the prices of european options are similar to the black - scholes formula . in the limit as the shape parameter approaches infinity , the gosset formulae become the black - scholes formula . both formulae are based on tabulated functions .the black - scholes formula requires values for the error function .the gosset formulae require tabulated values for integrals of log student s _ t_-distributions .fortunately both the error function and the values for integrals can be calculated numerically in short time periods .appendix a is a copy of the output of a maple worksheet that was used to determine the time required to calculate and print to the screen the value of a call option for 100 different values for the asset _ s_. this script took 20 s to execute .one hundred evaluations of the maple intrinsic function blackscholes(_s _ , _ k _ , _ r _ , _ t _ , ) for 100 different values of _ s _ would take 5 ms to complete and to print to the screen .the execution time for pricing a call with a gosset formula could be reduced if optimized code is used to evaluate the integrals , similar to the optimized code that is used for evaluation of the error function to price an option using the black - scholes formula .one challenge with use of the gosset formula is that spreadsheets do not have built - in functions to evaluate the integrals , unlike the functions that can be used to price an option with the black - scholes formula .however , in time this shortcoming can be remedied .one advantage that the gosset formulae enjoy over the black - scholes formula is the inclusion of the `` fat tails '' that returns display .the gosset formulae can be viewed as extensions to the black - scholes formula .the gosset formulae price european options where the returns are normally distributed but the true ( parent ) volatility is not known _ a priori _( i.e. , the true volatility is estimated as the sample standard deviation ) or the true volatility is a random variable and is distributed as an inverse chi variate . the black - scholes formula is based on a volatility that is known ; there is no uncertainty in the volatility in the black - scholes approach .it is known that it is important to include stochastic volatility in pricing options e.g. _ , .  for the gosset formulae to hold strictly ,the volatility must be distributed as a chi random variable or as an inverse chi variate .the requirement of a chi or inverse chi pdf depends on the approach that is chosen to justify theoretically the fact that returns follow a student s _t_-distribution .one can postulate that the student s _ t_-distribution and the concomitant fat tails arise from the uncertainty introduced by estimating the true volatility from the sample standard deviation , or one can postulate that the volatility is a random variable .presumably the assumption that the volatility is a random variable takes into account the uncertainty in estimating the true volatility from the sample standard deviation .historical data show that the volatility follows , to a good approximation , an inverse chi distribution , which suggests that the true volatility is a random variable .the volatility is not constant in time .the markets show periods of calm and of wild swings in prices .these periods of calm and wild swings in price can not be predicted with certainty and thus the volatility is a random variable to an observer of the market .we find that both the inverse chi and the chi distributions fit the volatility data for the s&p 500 and the djia , with the inverse chi distribution providing a moderately better description of the volatility than the chi distribution .thus one would expect the returns to be described by a student s _ t_-distribution and not a normal pdf , as is observed .hence the predictions of the gosset formulae should be preferred over the predictions of the black - scholes formulae for the prices of european options .figure 1 gives the price of call and put options as a function of the strike price _k _ , asset value _s_ = 50.00 , confidence level _ p _= 0.999 , risk free rate , time to expiration , __ = 0.03 , volatility = 0.3 , and various values of the shape parameter for the student s _ t_-distribution .the prices for were calculated with the black - scholes formula .the solid curves for = 3 , 8 , and 21 were calculated using the gosset formulae , eqs .( b10 ) and ( b11 ) of appendix b , for a truncated pdf .the broken lines were calculated using the gosset formulae , eqs .( b6 ) and ( b7 ) of appendix b , for a capped distribution .clearly the difference between the gosset price and the black - scholes price increases as becomes smaller .as becomes smaller , the tails of a student s _ t_-distribution become `` fatter '' . it is also clear from fig .1 that the capped gosset price , i.e. , the price for an option that is calculated using a value for an asset that is capped and follows a log student s _ t_-distribution , is marginally greater than the truncated gosset price .the additional cost for a capped value allows for the possibility that the value of the asset may exceed the cap value .truncation assumes that the value of the asset will not exceed the cap .[ fig : fig1 ]   table 1 gives the critical values and increase in the value of the asset at the critical value for _ p _ = 0.999 for four different values of the shape parameter .the critical value and confidence level are defined such that the probability of drawing a return is .  to price an option with either the black - scholes approach or the gosset approach ,the value of the asset is assumed to vary as exp( ) , i.e. , the logarithm of the value of the asset is distributed as .table 1 shows that truncating the student s _ t_-distribution at _ p _ = 0.999 with = 0.3 allows for a maximum increase of the value of the asset of 21.421 times for = 3 . for the normal distribution = and at _ x _the relative increase in the value of the asset is 2.527 times .table 1 also shows that the truncation need not be restrictive , and thus pricing of options with the gosset formula is realistic .it is worth noting that one could truncate or cap at different values of _ p _ , say _= 0.99999 , which would allow for a maximum increase of the value of the asset of 106 times for = 3 , = 0.3 , or say _p _ = 0.95 , which would allow for a maximum increase of the value of the asset of 2.075 times for = 3 , = 0.3 the choice of the level for the cap or the truncation might depend on the underlying asset .penny stocks might increase dramatically in terms of percent increase , but little in terms of absolute value , and thus require a large critical value to capture the possible changes . = 21 & + critical value , _x _ & 10.215 & 4.501 & 3.527 & 3.090 + exp(0.3 _ ) & 21.421 & 3.858 & 2.881 & 2.527 +one question that arises naturally in pricing an option with a gosset formula is what value to use for the shape parameter . under the assumption that the returns are normally distributed , _ n _ where _ n _ is the number of independent samples that have been used to estimate the volatility .the argument for the maximum value is straightforward .let _ s _ be the sample standard deviation that has been calculated for _n _ independent samples from a normally distributed parent population . the sample standard deviation _ s _ follows a chi distribution with = _ n _ degrees of freedom , (_s _ ) , for data that are normally distributed .any measurement of _ s _ has an uncertainty and this uncertainty in _ s _ is reflected in the shape parameter of the sampling distribution . for normally distributed samples where the standard deviation of the parent distribution is estimated from the sample population , the sampling distribution is a student s _ t_-distribution with degrees of freedom , _f(_x , _ ) .  the uncertainty in the estimation of the true ( parent )standard deviation can be significant .the ( sample ) volatility is the sample standard deviation for the daily returns , and a symmetric 95% confidence interval for the true ( parent ) volatility is : 0.77 _ 1.43 _ for = 21 ; 0.83 _  1.27 _ for = 43 ; and , 0.87 _ 1.17 _ for = 87 , where _ s _ is the sample standard deviation ( volatility ) with degrees of freedom that has been calculated from the daily returns and it has been assumed that the returns are normally distributed .  the maximum value for , assuming that the volatility is estimated as the sample standard deviation of the returns , is obtained when the volatility is constant in time .praetz and gerig _ et al ._ found that the variance of returns is a random variable that changes slowly in time .our simple analysis of historical returns , which we present below , agrees with the findings of praetz and gerig _ et al .this randomness in the volatility adds weight to the tails of the distribution of the returns .additional uncertainty in the volatility leads to a lower value ( i.e. , fatter tails for the underlying distribution ) for the shape parameter in the gosset formula .the additional uncertainty arises when one wishes to price an option for exercise at some time _t _ in the future . since the volatility is a random variable , the volatility at time _ , is unknown and the additional uncertainty must be taken into account .we estimate values for the shape parameter from historical data from the djia ( oct 1928 to feb 2009 ) and s&p 500 ( jan 1950 to feb 2009 ) equity indexes .  in tables 2 to 5are mean and median values for 22 day volatilities .for the simulations , 22 samples were generated and the volatility ( i.e. , the sample standard deviation of the 22 samples ) was calculated .the process was repeated 1000 times , and the mean , median value , and the standard deviation _s _ of the 1000 22 day volatilities were calculated and are reported in the tables .  for each group of 22 values ,the minimum value and the maximum value were dropped to give 20 values , and the volatility was calculated for this truncated set of data .the process was repeated 1000 times and the mean , median value , and the standard deviation of the 1000 volatilities were calculated , as for the full set of 22 values .  for each group of 20 values ,the minimum value and maximum value were dropped to give 18 values .the calculations described above were repeated and the numbers entered into the tables .  for calculation of the expected volatility ,the pdf was assumed to be truncated at a critical value _x _ such that the probability of an observation _ x _ in the range _ _ x _ _ x _ was equal to _ n_/22 , i.e. , p\{ _ _ x _ _ x _ } = .there likely is a rigorous method to calculate the standard deviation for dropping the minimum and the maximum values from a finite number of random draws from a parent population , but the method that was adopted was simple and shows the trend .the expected volatility was calculated as    where _f_( ) is the appropriate probability density function and the denominator maintains the normalization of the truncated pdf .table 2 gives results of the simulation and calculation for the procedures described above .note that the normal distribution has been scaled to have the same variance as the _ t_-distribution .note also that the median 22 day volatility decreases as the maximum and minimum samples are dropped , and that the median value for the _ t_-distribution decreases faster than the median value for the normal distribution as the maximum and minimum samples are dropped .the standard deviation of the 1000 22-day volatilities decreases greatly for the _ t_-distribution as the maximum and minimum samples are dropped , but changes little for the normal distribution .this owes to the fat tails of the _ t_-distribution with  = 3 .   & & & _ t _ , = 3 & normal , =3 + & median & _ s _ & median & _ s _ & & + 22 & 1.48 & 0.65 & 1.70 & 0.28 & 1.73 & 1.73 + 20 ( drop 2 ) & 1.07 & 0.25 & 1.41 & 0.25 & 1.01 & 1.39 + 18 ( drop 4 ) & 0.87 & 0.19 & 1.21 & 0.24 & 0.816 & 1.18 +  table 3 shows the means , the medians , and the standard deviations for 1000 samples for the same _ t_-distribution and normal distribution in table 2 .the means are in agreement with the expected values in table 2 and show the same progression as the median when the minimum and maximum values are removed from each sample of 22 data points .   & mean & _ s _ & median & mean & _ s _ & median + 22 & 1.61 & 0.65 & 1.48 & 1.71 & 0.27 & 1.70 + 20 ( drop 2 ) & 1.09 & 0.25 & 1.07 & 1.43 & 0.24 & 1.41 + 18 ( drop 4 ) & 0.88 & 0.19 & 0.87 & 1.22 & 0.23 & 1.21 +  the data in the tables show that the fat tails of the _ t_-distribution result in large uncertainties of the volatility ( i.e. , the sample standard deviation _s _ of the 22 day volatility is large for the _ t_-distribution ) .the fat tails are also evident in the difference between the median and the mean values for _ n _ = 22 .note that the uncertainty in the volatility , as measured by the standard deviation _ s _, reduced dramatically for the data from a _ t_-distribution after truncation of the minimum and maximum values , but barely changed for a normal distribution .  table 4 compares values of the volatility for the s&p 500 daily returns to the simulations .the s&p data was broken into 676 non - overlapping segments of 22 values .the values were sorted and then the volatility ( i.e. , the sample standard deviation ) was calculated for the middle 22 , 20 , and 18 values of the 22 returns .the mean , median , and standard deviation _ s _ of the volatilities were calculated over the 676 volatilities .for ease of comparison , each column was normalized by the value for _ n _ = 22.  table 5 compares means and medians of the 22 day volatilities for the djia equity index to simulations .& mean & median & _ t _ , = 3 & normal + 22 & 1 & 1 & 1 & 1 + 20 ( drop 2 ) & 0.803 0.010 & 0.81 & 0.584 & 0.803 + 18 ( drop 4 ) & 0.677 0.011 & 0.68 & 0.471 & 0.683 +   & mean & median & _ t _ , = 3 & + 22 & 1 & 1 & mean median & 1 + 20 ( drop 2 ) & 0.799 0.008 & 0.802 & 0.67 0.72 & 0.83 + 18 ( drop 4 ) & 0.675 0.009 & 0.680 & 0.55 0.59 & 0.71 +  tables 4 and 5 show that the short - term djia and s&p 500 volatility behaves more like a normal pdf than a student s _ t_-distribution .the interpretation is that for a short term ( 22 returns ) , the returns are approximately normally distributed .however , the volatility changes with time , and for longer periods of time , the uncertainty in the volatility comes into play and the distribution becomes less normal like and more _this is evident from the data in table 6 , where the normalized means and medians are reported for calculations over longer time frames of 44 and 88 returns .& mean & median & mean & median + 44 & 1 & 1 & 1 & 1 + 40 & 0.773 0.012 & 0.792 & 0.776 0.016 & 0.793 + 36 & 0.646 0.012 & 0.658 & 0.649 0.016 & 0.648 + 88 & 1 & 1 & 1 & 1 + 80 & 0.748 0.017 & 0.748 & 0.753 0.023 & 0.786 + 72 & 0.622 0.016 & 0.625 & 0.626 0.021 & 0.663 +  table 6 shows that the means and medians of the truncated samples are smaller for longer time periods .this suggests that the data are departing from normally distributed .an error propagation equation was used to estimate the uncertainty in the ratio of the means . the uncertainty in _b_/_a _ was calculated as 2 , where ( _ s) = ( _ s)/_a_ + ( _ s)(_b_/_a_) ( _ b_/_a_)_s , to give a 95% confidence level assuming that _ a _ and _ b _ are normally distributed .the covariance _s _ was in the range of 0.97 to 0.91 for _ a _ equal to the mean for the full range and _ b _ equal to the mean for dropping the bottom and top 5% or 10% of the samples used to form _figure 2 is a plot of the normalized expected volatility as a function of the shape parameter for the student s _ t_-distribution .the long tic marks at 0.803 and 0.683 are the normalized expected volatilities for samples that are normally distributed .the upper curve is for = 20/22 = 0.9090 while the lower curve is for = 18/22 = 0.8182 the curves are normalized by the expected volatility for = 22/22 = 1.000  [ fig : fig2 ] .,title="fig : " ]  the normalized mean volatilities for the s&p 500 and djia equity indexes were placed on fig . 2 to match the normalized mean value of the volatility with the normalized expected volatility curves .this placement on the expected curve provides a means to estimate the shape parameter and the uncertainty in the shape parameter .the normalized mean volatilities are found in tables 4 , 5 , and 6 .the vertical bars give the 95% confidence intervals for the normalized mean values .the horizontal bars give the uncertainty in the shape parameters for the uncertainty in the mean value . from fig .2 it appears that a student s _ t_-distribution with a shape parameter = 13 4 would be appropriate for a 44 day volatility and that a student s _ t_-distribution with a shape parameter = 8 2 would be appropriate for an 88 day volatility , based on the historical data used in the analysis . for a 22 day volatility , the shape parameter should be  21 .  the data on fig .2 at = 2.65 0.11 and = 3.24 0.19 were obtained from fits to the full data sets for the djia and s&p 500 equity indexes .the shape parameters were obtained by fitting frequency of occurrence data to a student s _ t_-distribution . for fig .2 the returns were sorted from smallest to largest and the volatility _ s _ was calculated as the standard deviation over the _= 20186 returns in the djia data set and _ n _ = 14870 returns in the s&p 500 data set .the bottom 4.545% and top 4.545% of the data were truncated , and the standard deviations _s _ were calculated .normalized standard deviations of 0.583 0.006 and 0.634 0.007 were obtained by forming _s/s _ and are plotted on the figure . the bottom and top 4.545% returns were truncated from the previously truncated lists , and the normalized standard deviations were obtained .the uncertainties in the normalized standard deviations were estimated as 95% symmetric confidence intervals for a chi - squared variate with degrees of freedom equal to _n _ 1 . the normalized data obtained in this manner , where the degrees of freedom were obtained from independent fits to frequency of occurrence data , are consistent with the expected volatility curve .  given the above , it appears that the following model describes well the returns , in agreement with the results of praetz and gerig_et al .the returns are normally distributed with a defined volatility for a short term ( of order 22 consecutive returns ) .the volatility is a random variable that changes slowly in time .t_-distribution , which is a mixture of a normal pdf and a chi distribution for the reciprocal of the standard deviation ( i.e. , 1/volatility ) describes the pdf for the returns now and at some time in the future , where the volatility is uncertain owing to sampling and to the slow and random development of the volatility in time . in mathematical notation , the sampling distribution _f(_x , _ ) = (_x_ ) ( ) d is a students _ t_-distribution when the parent distribution _f_(_x_ ) is a normal distribution with standard deviation and follows an inverse chi distribution ( or , equivalently , the reciprocal of follows a chi distribution ) .figure 3 is a plot of a fit of a chi distribution to the reciprocal of the 22 day volatilities for the s&p 500 equity index from jan 1950 to feb 2009 .it can be observed that the reciprocal of the historical 22 day volatilities are , for the most part , distributed as a chi variate . since the returns over 22 days are found to be normal like( see the tables ) with a given volatility but the reciprocal of this volatility is distributed as a chi variate , the distribution of the returns is expected to follow a student s _ t_-distribution .a student s _t_-distribution for the returns is observed . in fig .3 the data deviate slightly from the best fit curve in the tail on the right hand side .the upward sloping curves are the cumulative density functions ( cdf ) for the data ( red ) and the best fit function ( black ) . for a perfect fit , the black and red cdf curves would overlap .  [fig : fig3a ]   figure 4 is a plot of a fit of a chi distribution to the 22 day volatilities ( the volatilities , not the reciprocals of the volatilities ) for the same s&p 500 data shown in fig .it is interesting to note that the chi distribution fits the 22 day volatilities reasonably well .the fit deviates in the tails on the right hand side , more so than the fit of fig 3 .fits of a chi distribution to synthetic data from an inverse chi distribution show that the inverse chi data has a fatter tail on the right hand side , similar to the fat tail on the right hand side that the s&p 500 data show in the figure .[ fig : fig3b ]   the shape parameter for the _ t_-distribution is , the number of degrees of freedom . from the analysis presented here ,the shape parameter is related to estimations of the distribution of the volatility .if the volatility does not change in time and is known ( i.e. , the volatility is not estimated as the sample standard deviations of the returns ) , then = and one recovers the black - scholes formula for the price of an european option . the black - scholes formula is derived on the assumption that the volatility is constant and known .if the true ( parent ) standard deviation ( volatility ) is not known and the returns are normally distributed , then a student s _t_-distribution should be used . if is small, then it is understood that the true volatility might take one of a broad range of values .the simple analysis presented also shows what might be reasonable values ( see fig .2 ) for the shape parameter of the student s _ t_-distribution .since the volatility is estimated as the sample standard deviation of the daily returns , it is expected that _n _ , where _n _ is the size of the sample that was used to compute the volatility .the gosset formulae for the price of an option by capping the value of the asset or by truncating the pdf are given in appendix b. these formulae are extended to include a floor or a truncation at the floor level .the notation is defined in appendix b.  the expression for delta for a truncated students _ t_-distribution , , is similar to the expression for for the black - scholes formula . in the limit of no truncation ( _ p _p _ = 1 , _ x _ = ) and = , the expression below reduces to for the black - scholes formula .  for the black - scholes formula , _ _= exp( ) , _ p _p _ = 1 , _ x _= , and the integral can be written in terms of the error function .the expression for for a capped asset price is    figure 5 plots for _ p _ = 0 , _ p _= 0.999 , _ k _ = 49.00 , _ r_ _ = 0.03 , and = 0.3 as a function of _ s_ and for _ p _ = 0.99 , 0.999 , and 0.9999 .a small value of was chosen to highlight the differences . the dashed lines plot delta for a capped distribution .[ fig : fig5 ]  for different levels of truncation .,title="fig : " ]   is also a function of the volatility . figure 7 plots for = 3 _ , p _ = 0 , _= 49.00 , _ r_ _= 0.03 as a function of _s_ for = 0.1 , 0.2 , and 0.3 the dashed lines gives , which is for a capped asset .the expression for is    which is similar to the expression for .[ fig : fig7 ] .,title="fig : " ]  [ fig : fig8 ]  for different volatilities.,title="fig : " ]  an expression for vega for a truncated _ t_-distribution , _ , is given below .  figure 10 is composed of plots of vega as a function of _s_ for the standard parameters used in this work .the expression for vega has a term owing to the _ z _ in the limits of integration .this term can not be neglected .vega increases as the shape parameter for the _ t_-distribution decreases .the curve for = 40 lies just above the curve for the black - scholes vega .the dashed lines are for vega for a capped distribution .an equation for vega for a capped asset , _ _ , is    [ fig : fig9 ]  for different values of the number of the degrees of freedom .,title="fig : " ]  the derivatives are messy .the number of terms and parameters makes it difficult to determine by looking at the expressions the importance of terms and parameters . as a practical matter , it is probably easier to compute the derivatives numerically as [ _ ( , , ... + , , ... ) ( , , ... _ , , ... ) ] / for a range of parameters and plot the values to build an understanding .figure 11 is plots of as calculated numerically as a function of _s_ for the standard parameters .theta increases as the shape parameter of the student s _ t_-distribution decreases .the dashed lines are for a capped distribution .theta was calculated as _ c_(_p _ , _ s _ , _ k _ , _r_(366/365 ) , (366/365) , ) _ c_(_p _ , _ s _ , _ k _ , _ r _ , _ , ) .[ fig : fig10 ] .,title="fig : " ]  figure 12 is plots of the numerical derivative of the price of a call with respect to the shape parameter as a function of _s_ for the standard parameters .this derivative is zero for an option priced with the black - scholes formula .not surprisingly , the price of a call is sensitive to the shape parameter for small .the dashed lines are for a capped distribution .[ fig : fig11 ]  as a function of the value of the asset for different values of the number of the degrees of freedom.,title="fig : " ]  figure 13 is plots of the logarithm of the derivative of the price of a call with respect to the truncation _p = p _ as a function of _s_ for the standard parameters .the dashed lines are for a capped distribution .note the large value of the magnitude of the derivative for _ p _ approaching unity .note also that the maximum value for is ( 1 _ p _ ) and that the change in the price of a call for a change in _ p _ of is the derivative .[ fig : fig12 ]  as a function of the value of the asset for different values for the level of truncation.,title="fig : " ]  the integrals that are required to price a call option contain terms with exp( _ x _ ) . as _ p approaches unity , _x _ approaches infinity , and the integrals required to price a call option assuming the returns follow a student s _ t_-distribution approach infinity .the case of _x _ approaching infinity is unphysical as this suggests that the value of the asset exp(  ) must also approach infinity . as _p _ approaches unity , the price of the option increases to account for the values of the asset that are being included .it is known that a student s _ t_-distribution fits the returns of stocks better than a normal pdf .a student s _ t_-distribution has `` fat tails '' , and the shape parameter and scale parameter of the _ t_-distribution can be selected to match the fat tails of the returns .prices for european options can be determined for assets that follow a log student s _t_-distribution if the value of the asset is capped or if the _ t_-distribution is truncated . capping or truncating the distribution keeps the integrals that are required to price an option finite .we call the formulae that use a student s _ t_-distribution to calculate the values of options gosset formulae , in honour of w.s . gosset .gosset used the pseudonym student to publish his work .  in the limit of large gosset formulae converge to the black - scholes formula for the values of european options .the gosset formula can be viewed as an extension of the black - scholes formula to allow for a volatility that has uncertainty or is a random variable .the black - scholes formula is based on the assumption that the volatility is known and constant .the returns must be normally distributed and the volatility must follow a chi or inverse chi distribution for the student s _ t_-distribution to hold strictly .the historical returns from the djia and s&p 500 appear to follow these distributions as the student s _ t_-distribution fits well the returns on these indexes .it is necessary to know the shape parameter to use the gosset formulae .the maximum value of  _ n _ where _ n _ is the number of independent returns that are used to calculate the volatility .the shape parameter , in general , will be less than the maximum value for large _n _ ( say _ 22 ) .the volatility is not constant in time , and for large _n _ , the uncertainty of the volatility broadens the pdf that describes the returns and hence reduces . also , for longer time to expiration _t _ , the volatility at time _ t _ is known with less accuracy , and a reduced is necessary to capture the uncertainty in the future value of the volatility .fits to the returns from the djia ( oct .1928 to feb . 2009 ) and the s&p 500 ( jan .1950 to feb . 2009 ) give of order 3 .thus it appears that typically should be in the range of to _ n_ . a method to estimate an appropriate value of by comparison of the standard deviations of the returns as the minimum and maximum values are droppedis given .some of the greeks for the gosset and black - scholes formulae for pricing european options are given and compared .the differences between the two approaches are the largest for small .this is not surprising , given that the difference between the two underlying distributions for the returns are the greatest for small .new greeks are required to characterize the price sensitivity to parameter change for the gosset approach .the gosset formula requires a cap or truncation and a shape parameter .changes in the value of a call option for changes in and the truncation or cap level are presented .[ [ section ] ]  a maple script to price a call option for 100 different values of the asset price is presented in this appendix .the script was used to find the execution time that is reported in sec .2 , implementation . the intrinsic function to price a call with the black - scholes formula is included as a comment in the loop .  [fig : script ]   [ [ section-1 ] ]  * * the equations to price an option by capping the value of an asset or truncating the underlying pdf are presented in a general form .the approach follows the work of cassidy , hamp , and ouyed .s _ be the price of a stock at time _ t _ , _t _ 0 .k _ be the strike price at time _t , _ where _ t _ is the time when the option expires .let _ s _ = _a( ) be the value of the stock where is a random variable and let the probability density function for the return be_ f( ) . if is a student s _ t _ variate , then    the value of a european call option , calculated at the time of expiration _= e\{(_s _  _ k) } , which is the expectation of the maximum value of \{_s _ _ k _ , 0}.the value of a european put option is _= e\{(_k _  _ s)}. these expressions for the values of the options follow from the arbitrage theorem .the values of the options at time _ t _= 0 are obtained from the expected time value of money .if _ r_(_t _ ) is the risk free rate , then _c_ = e\{_c _ exp((_t_)d_t _ ) } = _ c _ exp( ) when the risk free rate is assumed to be time independent , with a similar equation for _p.  the average value of a stock at time _ t _ = 0 when one wishes to price an option is _the average value of the stock at some time later is e\{_s _ } = e\{_a( ) } where the drift is contained in _a. for a martingale ( a fair wager ) , e\{_s _ } = _ s_(_r t _ ) .one may use the doob decomposition to offset the drift and require e\{_a( ) } = e\{_a(( ) ) } = _s_(_r t _ ) .this is equivalent to requiring that the pdf for _ s _ should be centred about , the drift owing to the risk premium . is not a random variable .the exp(_r t _ ) term takes into account the time value of money and allows for comparison of values at two different points in time .the difficulty with pricing an option for which the return follows a student s _ t_-distribution is that e\{_s _ } is infinite .the exp( ) term in the expectation dominates over the behaviour of the pdf for large .two methods for pricing an option are presented below .one method is to place a cap on the value of the asset .the other method is to truncate the pdf .truncation is physical because it is not possible for the price of an asset to approach infinity .the equations in this appendix are written in general terms .the approach to price an option by capping or truncating is not restricted to a student s _ t_-distribution .the approach can be used with other probability density functions . in this appendix ,provision is made for a floor _ x _ or for a truncation at the floor .let _ p _= p\{ _ x _ } and _ p = p\{ _ x _ } where _x _ and _ x _ are critical values . in terms of the pdf , _ p _ and _ p _ are equal to ( with _ i _ = _ p _ or _c _ )    for _ p _ = 0 , _p _ = 1 , and _f( ) a standard normal variate , the equations for the prices of european options presented reduce to the black - scholes formula .we call the formulae for prices of options that have been obtained with_ f( ) a student s _ t_-distribution gosset formulae , in honour of w. s. gosset .   assume a floor _ s _ = _ x _ for _ s _ _ x _ and a cap _ s _ = _ x _ for _ s _ _ x _ and let    a pre - subscript of _ c _ or _ t _ indicates that a quantity has been calculated with a cap or a truncation .one could write _ z _ as _ to indicate a floor as well as a cap . however , the extra symbol is not necessary here and is omitted .for a fair wager , the expected value of _s _ is found as    and the unknown _ _ can be found    the price of a european call option for a capped asset value is    assuming that ln(_k/_)/ _ x.  the price of a european put for a capped asset value is    in the limit as _p _ tends to zero , the floor disappears .note that a cap or truncation is necessary to price an option with a student s _ t_-distribution .the cap or truncation keeps the integrals , which are needed to price the options , finite .the floor is not necessary but may make for realistic pricing .the price of an option is quantized by the currency . for values less than the smallest unit of the currency, it may be appropriate to set the price equal to zero , i.e. , to use a floor .put - call parity holds for truncated or capped options . the equation for the price of a put is presented for completeness .assume that _s _ only takes values _ x _ _ s _ _x _ and thus _f( ) = 0 if  _x _ or  _ x. let    the denominator follows from the definition of conditional probability and is required to maintain the normalization of the pdf after the truncation : p\{_x _   _ x_+_dx _  _ x _    _ x _ } = p\{_x _    _ x_+_dx _  _ x _   _ x}/p\{_x _  _ x _ } = _ f()d_x_/(_p _ _ p _ ) if _ x _  _ x _ and zero otherwise .  the average value of the stock at _ t_ is e\{_s _ } = _ s_(_r t _ ) _= _  or    the price of a european call option for a truncated pdf is    assuming that ln(_k/)/ _ x.  the price of a european put for a truncated pdf is    note that _ is known in terms of _ s_ , _ r _ , and _ (_p(_x ) , _ p(_x _ ) ) and thus the option prices can be determined .note also that _ _ _ .this work was funded in part by the natural sciences and engineering research council of canada .bouchaud , j .-sornette d. , the black - scholes option pricing problem in mathematical finance : generalization and extensions for a large class of stochastic processes , _ j. phys .i france _ , 1994 , * 4 * , 863881 .de jong , c. and huisman , r. , from skews to a skewed _ : modelling option implied returns by a skewed student _ , computational intelligence for financial engineering ( cifer ) , proceedings of the ieee / iafe / informs 2000 conference ( 2000 ) .platen , e. and sidorowicz , r. , empirical evidence on student _ log of diversified world stock indices , quantitative finance research centre , research paper 194 , university of technology , sydney , broadway , n.s.w . ,

Abstract: european options can be priced when returns follow a student s _ t_-distribution , provided that the asset is capped in value or the distribution is truncated . we call pricing of options using a log student s _ t_-distribution a gosset approach , in honour of w.s . gosset . in this paper , we compare the greeks for gosset and black - scholes formulae and we discuss implementation . the _ t_-distribution requires a shape parameter to match the `` fat tails '' of the observed returns . for large , the gosset and black - scholes formulae are equivalent . the gosset formulae removes the requirement that the volatility be known , and in this sense can be viewed as an extension of the black - scholes formula .  : option pricing ; student s ; fat tails .


typical elm signals are shown in fig.[fig1 ] . here, we try to model type - i and type - iii elm signals of .jet typical elm signals ( [ perez2004 , thesis]).,width=321 ]  we just focus on the signal itself and ignore all the other information behind , i.e. , we just discuss how to use ( as simple as possible ) equations or other approaches to reproduce the shape of these signals .it is qualitative or at most semi - quantitative .this is a wide used approach for modeling . for example , the famous prey - predator model for fishbone and drift wave - zonal flow system .one may also extend it to pde to contain the spatial information .indeed , this way has been used to model elm and l - h transit in literatures , e.g. , [ diamond1994 ] , [ itoh1991 , 1993 , 1999 ] .  if we assume that the dynamic of signal is not explicit depend on time , then one equation is not enough because that is not single valued with .so , at least , we must use second order equation or use two first order equations . while , one second order equation is just a special case of two first order equations .in this example , we use the second option , i.e. , two first order equations , which is also used in [ diamond1994 ] .the equations are   a result is shown in fig . [fig : ode1 ] .ode for modeling elm , example 1.,width=321 ]  the physical explanations of the equations and parameters can be found in [ diamond1994 ] , however , where , .using one second order equation , e.g. , which is modified from stiff system equation , a result is shown in fig .[ fig : ode2 ] .the shape is very similar to the pde result in [ itoh1993 ] .but , an apparent drawback of ( [ eq : ode2 ] ) is that the signal is not always positive here .ode for modeling elm , example 2.,width=321 ]ddes contain derivatives which depend on previous time .we guess some equations here .the equations are from or modified from [ shampine2000 ] .there is also a type of dde has sawtooth solutions ( see e.g. , [ mallet - paret2011 ] ) , which may be also suitable for model the unsolved sawteeth phenomena in magnetic confined fusion study .the equation is   dde for modeling elm , example 1.,width=321 ]  results are shown in fig .[ fig : dde1 ] .equation is which is original to model four - year cycle of the population of lemmings .dde for modeling elm , example 2.,width=321 ]  the result is shown in fig .[ fig : dde2 ] .this is a more intuitive way to model elm , which combines different time scales using a more acceptable approach .the burst of elm signal is suggested related to peeling - ballooning mode in standard elm model ( [ connor1998 ] ) .so , we give two time scale : transport time scale and ballooning mode time scale .  to simplify the transport process , the transport time scale is modeled using   when pressure gradient exceeds marginal value ( using to represent it ) , the ballooning mode occurs .the signals will grow exponentially with the characteristic time .  when exceeds a marginal value , the plasma crash .and then , go to next cycle .we also add some randomicities to and to make the whole signal more like the actual experimental signal . a resultis shown in fig .[ fig : demc ] .demc modeling for elm.,width=321 ]  this simple demc model captures some fundamental properties of integrated simulations , such as [ chang2008 ] and [ lonnroth2009 ] .which one is more reasonable ? at present , i choose demc ode dde . while , maybe none of them can be final explanation of elm or h - mode .there is another math tool called fractional differential equation ( fde ) , which has been used for anomalous transport / diffusion problems in some areas . for plasma community, fde is not wide accepted yet , but it is possible in the future .fde is hard to give periodic solutions as elm at present .so , i havent given examples .chang , c. s. ; klasky , s. ; cummings , j. ; samtaney , r. ; shoshani , a. ; sugiyama , l. ; keyes , d. ; ku , s. ; park , g. ; parker , s. ; podhorszki , n. ; strauss , h. ; abbasi , h. ; adams , m. ; barreto , r. ; bateman , g. ; bennett , k. ; chen , y. ; azevedo , e. d. ; docan , c. ; ethier , s. ; feibush , e. ; greengard , l. ; hahm , t. ; hinton , f. ; jin , c. ; khan , a. ; kritz , a. ; krsti , p. ; lao , t. ; lee , w. ; lin , z. ; lofstead , j. ; mouallem , p. ; nagappan , m. ; pankin , a. ; parashar , m. ; pindzola , m. ; reinhold , c. ; schultz , d. ; schwan , k. ; silver , d. ; sim , a. ; stotler , d. ; vouk , m. ; wolf , m. ; weitzner , h. ; worley , p. ; xiao , y. ; yoon , e. and zorin , d. , toward a first - principles integrated simulation of tokamak edge plasmas , journal of physics : conference series , 2008 , 125 , 012042 .mallet - paret , j. and nussbaum , r. d. , superstability and rigorous asymptotics in singularly perturbed state - dependent delay - differential equations , journal of differential equations , 2011 , 250 , 4037 - 4084 .

Abstract: there is no wide accepted theory for elm ( edge localized mode ) yet . some fusion people feel that we may never get a final theory for elm and h - mode , since which are too complicated ( also related to the unsolved turbulence problem ) and with at least three time scales . the only way out is using models . ( this is analogous to that we believe quantum mechanics can explain chemistry and biology , but no one can calculate dna structure from schrodinger equation directly . ) this manuscript gives some possible mathematical approaches to it . i should declare that these are just math toys for me yet . they may inspire to good understandings of elm and h - mode , may not . useful or useless , i do nt know . one need not take too much care of it . just for fun and enjoying different interesting ideas .