青少年风湿性关节炎新药上市疗效达85%

合集下载

r语言quarter函数

r语言quarter函数在R语言中，quarter函数用于将日期向下舍入到指定季度。

它是一种非常常用的时间处理函数，可用于将日期数据进行分组和统计分析。

quarter函数的使用方法非常简单，只需将日期作为参数传入即可。

下面是quarter函数的语法：quarter(x, ...)参数x表示需要处理的日期数据，可以是日期向量、日期字符串或日期对象。

被处理的日期数据可以是单个日期，也可以是一个日期向量。

另外，quarter函数还可以接受其他参数，例如：- "start"参数用于指定每个季度的起始月份，取值范围是1-12，默认值为1。

例如，指定start=1表示每年的第一个月是1月，指定start=3表示每年的第一个月是3月。

- "fiscalyear"参数用于指定是否按照财务年度计算季度，取值为TRUE或FALSE，默认为FALSE。

如果设置为TRUE，则季度将以财务年度为基准计算，财务年度通常不同于日历年度。

下面是一些使用quarter函数的例子，以帮助理解其功能：# 使用日期向量作为参数dates <- c("2021-01-01", "2021-03-15", "2021-06-30", "2021-10-20")quarters <- quarter(dates)print(quarters)# 输出：[1] 1 1 2 4# 将日期向下舍入到对应的季度。

2021-01-01属于第一季度，2021-03-15也属于第一季度，2021-06-30属于第二季度，2021-10-20属于第四季度。

# 使用日期字符串作为参数date <- "2022-07-01"quarter <- quarter(date)print(quarter)# 输出：[1] 3# 将日期向下舍入到对应的季度。

Supertraces on the Algebras of Observables of the Rational Calogero Model with Harmonic Pot

a r X i v :h e p -t h /9512038v 1 6 D e c 1995Supertraces on the Algebras of Observables of the Rational Calogero Model with Harmonic Potential S.E.Konstein and M.A.Vasiliev I.E.Tamm Department of Theoretical Physics,P.N.Lebedev Physical Institute,117924Leninsky Prospect 53,Moscow,Russia.Abstract We deﬁne a complete set of supertraces on the algebra SH N (ν),the algebra of observables of the N -body rational Calogero model with harmonic interaction.This result extends the previously known results for the simplest cases of N =1and N =2to arbitrary N .It is shown that SH N (ν)admits q (N )independent supertraces where q (N )is a number of partitions of N into a sum of odd positive integers,so that q (N )>1for N ≥3.Some consequences of the existence of several independent supertraces of SH N (ν)are discussed such as the existence of ideals in associated W ∞-type Lie superalgebras.I IntroductionIn this paper we investigate some properties of the associative algebras which were shown in [1,2,3]to underly the rational Calogero model [4]and were denoted as SH N (ν)in [5].Algebra SH N (ν)is the associative algebra of polynomials constructed from arbitrary ele-ments σof the symmetric group S N and the generating elements a αi obeying the following relations σa αi =a ασ(i )σ,(1) a αi ,a βj =ǫαβA ij ,(2)where i,j =1,...,N ,α,β=0,1,ǫαβ=−ǫβα,ǫ01=1andA ij =δij +ν˜A ij ,˜Aij =δij N l =1K il −K ij .(3)1Here K ij∈S N with i,j=1,...,N,i=j,are the elementary permutations i↔j satisfying the relationsK ij=K ji,K ij K ij=1,K ij K jl=K jl K li=K li K ijfor i=j=l=i andK ij K kl=K kl K ijif i,j,k,l are pairwise diﬀerent.Note that in this paper repeated Latin indices i,j,k,... do not imply summation.The deﬁning relations(1)-(3)are consistent.In particular,the Jacobi identities[aαi,[aβj,aγk]]+[aβj[aγk,aαi]]+[aγk,[aαi,aβj]]=0(4) are satisﬁed.An important property of SH N(ν)which allows one to solve the Calogero model[4] is that this algebra possesses inner sl2automorphisms with the generators1Tαβ=the notation SH1).Properties of this algebra are very well studied(see e.g.[13]).Note that since the center of mass coordinates1/N N i=1aαi decouple from everything else in the deﬁning relations(1)-(3),the associative algebra SH N(ν)has the structure SH N(ν)= SH1⊗SH′N(ν)where,by deﬁnition,SH′N(ν)is the algebra of elements depending only on the relative coordinates aαi−aαj.The properties of SH′2(ν)are well studied too[14].The algebra SH′2(ν)is deﬁned by the relations[aα,aβ]=ǫαβ(1+2νK),(8) where K is the only nontrivial element of S2while aαare the relative motion oscillators. For the particular case ofν=0one recovers the algebra SH1in the sector of the K independent elements.In[14]it was shown that SH′2(ν)admits a unique supertrace operation deﬁned by the simple formulastr(1)=1,str(K)=−2ν,str(W)=str(W K)=0(9) for any polynomial W∈SH′2of the formW=∞n=1Wα1...αn aα1...aαn(10)with arbitrary totally symmetric multispinors Wα1...αn .For the particular case ofν=0one recovers the supertrace on SH1.Furthermore it was shown in[14]by explicit evaluation of the invariant bilinear form B(x,y)def=str(xy)that forν=l+116(4ν2−1)is an arbitrary constant.In its turn this observation clariﬁed the origin of the ideals of SH′2(ν)atν=l+1Let us note that an attempt to deﬁne diﬀerently graded traces like,e.g.,an ordinary trace (π≡0)unlikely leads to interesting results.Knowledge of the supertrace operations on SH N(ν)is useful in various respects.One of the most important applications of the supertrace is that it gives rise to n-linear invariant formsstr(a1a2...a n)(14) that allows one to work with the algebra essentially in the same way as with the ordinary ﬁnite-dimensional matrix algebras and,for example,construct Lagrangians when work-ing with dynamical theories based on SH N(ν).Another useful property is that since null vectors of any invariant bilinear form span a both-side ideal of the algebra,this gives a powerful device for investigating ideals which decouple from everything under the super-trace operation as it happens in SH2(ν)for half-integerν.It is also worth mentioning that having an explicit form of the trilinear form in one or another basis is practically equivalent to deﬁning a star-product law in the algebra.An important motivation for the analysis of the supertraces of SH N(ν)is due to its deep relationship with the analysis of the representations of this algebra,which in its turn gets applications to the analysis of the wave functions of the Calogero model.For example,given representation of SH N(ν),one can speculate that it induces some super-trace on this algebra as(appropriately regularized)supertrace of(inﬁnite)representation matrices.When the corresponding bilinear form degenerates this would imply that the representation becomes reducible.As we show,the situation for SH N(ν)is very interesting since starting from N=3 it admits more than one independent supertrace in contrast to the cases of N=1and N=2.This fact is in agreement with the results of[5]where it was shown that there exist many inequivalent lowest-weight type representations of SH N(ν)for higher N(these representations are classiﬁed according to the representations of S N.)Another important consequence of this phenomenon is that the Lie superalgebras W N,∞(ν)are not simple while appropriate their simple subalgebras possess non-trivial outer automorphisms.The paper is organized as follows.In Section II we analyze consequences of S N and sl2 automorphisms of SH N(ν).In Section III we discuss general properties of the supertraces and consequences of the existence of several independent supertraces.In Section IV we study the restrictions on supertraces of the group algebra of S N considered as a subalgebra of SH N(ν),which follow from the deﬁning relations of SH N(ν).These restrictions are called ground level conditions(GLC).They play a fundamental role in the problem since as we show in Section V every solution of GLC admits a unique extension to some supertrace on SH N(ν).In Appendix A it is shown that the number of independent supertraces on SH N(ν)equals to the number of partitions of N into a sum of odd positive integers.Some technical details of the proof of Section V are collected in Appendices B and C.II Finite-Dimensional Groups of Automorphisms The group algebra of S N is theﬁnite-dimensional subalgebra of SH N(ν).The elements σ∈S N induce inner automorphisms of SH N(ν).It is well known,that anyσ∈S N can4be expanded into a product of pairwise commuting cyclesσ=c1c2c3...c t,(15) where c w,w=1,...,t,are cyclic permutations acting on distinct subsets of values of indices i.For example,a cycle which acts on theﬁrst s indices as1→2→...→s→1 has the formc=K12K23...K(s−1)s.(16) We use the notation|c|for the length of the cycle c.For the cycle(16),|c|=s.We take a convention that the cycles of unit length are associated with all values of i such that σ(i)=i,so that the relation w|c w|=N is true.Given permutationσ∈S N,we introduce a new set of basis elements Bσ={b I}instead of{aαi}in the following way.For every cycle c w in the decomposition(15)(w=1,...,t), let usﬁx some index l w,which belongs to the subset associated with the cycle c w.The basis elements bαw j,j=1,...,|c w|,which realize1-dimensional representations of the commutative cyclic group generated by c w,have the formbαw j=1|c w||c w|k=1(λw)jk aαl(w,k),(17)where l(w,k)=c−k w(l w)andλw=exp(2πi/|c w|).(18) From the deﬁnition(17)it follows thatc w bαw j=(λw)j bαw j c w,(19)c w bαn j=bαn j c w,for n=w(20) and thereforeσbαw j=(λw)j bαw jσ.(21) In what follows,instead of writing bαw j we use the notation b I with the label I ac-counting for the full information about the indexα,the index w enumerating cycles in(15),and the index j which enumerates various elements bαw j related to the cycle c w,i.e.I(I=1,...,2N)enumerates all possible triples{α,w,j}.We denote the indexα, the cycle and the eigenvalue in(19)corresponding to someﬁxed index I asα(I),c(I),andλI=(λw)j,respectively.The notationσ(I)=σ0implies that b I∈Bσ0.B1is the original basis of the generating elements aαi(here1is the unit permutation).Let M(σ)be the matrix which maps B1−→Bσin accordance with(17),b I= i,αM I iα(σ)aαi.(22) Obviously this mapping is ing the matrix notations one can rewrite(21)asσb Iσ−1=2NJ=1ΛI J(σ)b J,∀b I∈Bσ,(23) 5whereΛJ I(σ)=δJ IλI.Every polynomial in SH N(ν)can be expanded into a sum of monomials of the formb I1b I2...b I sσ,(24) where allσ(I k)=σ.Every monomial of this form realizes some one-dimensional repre-sentation of the Abelian group generated by all cyclesc w in the decomposition(15).The commutation relations for the generating elements b I follow from(2)and(3)b I,b J =F IJ=C IJ+νf IJ,(25)whereC IJ=ǫα(I)α(J)δc(I)c(J)δλIλ−1J(26) andf IJ= i,j,α,βM I iα(σ)M J jβ(σ)ǫαβ˜A ij.(27)The indices I,J are raised and lowered with the aid of the symplectic form C IJ µI= J C IJµJ,µI= JµJ C JI; M C IM C MJ=−δJ I.(28)Note that the elements b I are normalized in(17)in such a way that theν-independent part in(25)has the form(26).Another importantﬁnite-dimensional algebra of inner automorphisms of SH N(ν)is the sl2algebra which acts on the indicesα.It is spanned by the S N-invariant second-order polynomials(5).Evidently,SH N(ν)decomposes into the inﬁnite direct sum of only ﬁnite-dimensional irreducible representations of this sl2spanned by various homogeneous polynomials(24).From the deﬁning relations(1)-(3)it follows that SH N(ν)is Z2-graded with respect to the automorphismf(aαj)=−aαj,f(K ij)=K ij(29) which gives rise to the parityπ(13).In applications to higher-spin models,this automor-phism distinguishes between bosons and fermions.The algebra SH N(ν)admits the antiautomorphismρ,ρ(aαk)=iaαk,ρ(K ij)=K ij,(30) which leaves invariant the basic relations(1)-(3)provided that an order of operators is reversed according to the deﬁning property of antiautomorphisms:ρ(AB)=ρ(B)ρ(A). From(15),(16)and(21)it follows thatρ(σ)=σ−1,ρ(b I)=ib J,(31) where J is related to I in such a way thatα(J)=α(I),σ(J)=(σ(I))−1,c(J)=(c(I))−1 andλJ=λ−1I.Note that in higher-spin theories the counterpart ofρdistinguishes between odd and even spins[16].6III General Properties of SupertraceIn this section we summarize some general properties to be respected by any supertrace in SH N(ν).Let A be an arbitrary associative Z2graded algebra with the parity functionπ(x)=0 or1.Suppose that A admits some supertrace operations str p where the label p enumeratesdiﬀerent nontrivial supertraces.We call a supertrace str even(odd)if str(x)=0∀x∈Asuch thatπ(x)=1(0).Let T A be a linear space of supertraces on A.We say that dim T A is the number of supertraces on A.Given parity-preserving(anti)automorphismτand supertrace operation str on A, str(τ(x))is some supertrace as well.For inner automorphismsτ(τ(x)=pxp−1,π(p)=0)it follows from the deﬁning property of the supertrace that str(τ(x))=str(x).Thus,T A forms a representation of the factor-group of the parity preserving automorphisms andantiautomorphisms of A over the normal subgroup of the inner automorphisms of A.Applying this fact to the original parity automorphism(−1)πone concludes that T A can always be decomposed into a direct sum of subspaces of even and odd supertraces,T A=T0A⊕T1A and that T1A=0if the parity automorphism is inner.In the sequel we only consider the case where dim T A<∞and there are no nontrivialodd supertraces.Let A=A1⊗A2with the associative algebras A1and A2endowed with some even supertrace operations t1and t2,respectively.The supertrace on A can bedeﬁned by setting str(a1⊗a2)=t1(a1)t2(a2),∀a1∈A1,∀a2∈A2.As a result,one concludes that T A=T A1⊗T A2.In the case of SH N(ν)one thus can always separate out a contribution of the center of mass coordinates as an overall factor(SH1admits theunique supertrace).If A isﬁnite-dimensional then the existence of two diﬀerent supertraces indicates thatA admits non-trivial both-side ideals.Actually,consider the bilinear form B(f,g)=α1str1(fg)+α2str2(fg)with arbitrary parametersα1,α2∈C and elements f,g∈A. The determinant of this bilinear form is some polynomial ofα1andα2.Therefore it vanishes for certain ratiosα1/α2orα2/α1according to the central theorem of algebra. Thus,for these values of the parameters the bilinear formB degenerates and admits non-trivial null vectors x,B(x,g)=0,∀g∈A.It is easy to see that the linear space I of all null vectors x is some both-side ideal of A.For inﬁnite-dimensional algebras the existence of several supertraces does not necessarily imply the existence of ideals.As mentioned in introduction the existence of several supertrace operations may be related to the existence of inequivalent representations.Also it is worth mentioning that for the case of inﬁnite-dimensional algebras and representations under investigation it can be diﬃcult to use the standard(i.e.matrixwise)deﬁnition of the supertrace.In this situation the formal deﬁnition of the supertraces on the algebra we implement in this paper is the only rigorous one.Let l A be the Lie superalgebra which is isomorphic to A as a linear space and is endowedwith the product law(12).It contains the subalgebra sl A∈l A spanned by elements g such that str p(g)=0for all p.Evidently sl A forms the ideal of l A.The factor algebra t A=l A/sl A is a commutative Lie algebra isomorphic to T∗A as a linear space.Elements of t A diﬀerent from the unit element of A(which exist if dim T A>1)can induce outer automorphisms of sl A.Let us note that it is this sl A Lie superalgebra which usually has7physical applications.For the case of SH N(ν)under consideration the algebra l SHN(ν)isidentiﬁed with the algebra W N,∞(ν)introduced in[8].We therefore conclude that these algebras are not simple for N>2because it is shown below that SH N(ν)admits several supertraces for N>2.Instead one can consider the algebras sW N,∞(ν).Let l A contain some subalgebra L such that A decomposes into a direct sum of irre-ducible representations of L with respect to the adjoint action of L on A via supercom-mutators.Then,only trivial representations of L can contribute to any supertrace on A. Actually,consider some non-trivial irreducible representation R of L.Any r∈R can be represented asr= j[l j,r j},l j∈L,r j∈R(32)since elements of the form(32)span the invariant subspace in R.From(11)it follows then that str(r)=0,∀r∈R.From the deﬁnition of the supertrace it follows thatstr(a1a2)+str(a2a1)=0(33) for arbitrary odd elements a1and a2of A.A simple consequence of this relation is that str(a1a2...a n+a2...a n a1+...+a n a1...a n−1)=0(34) is true for an arbitrary even n if all a i are some odd elements of A.Since we assume that the supertrace is even(34)is true for any n.This simple property turns out to be practically useful because,when odd generating elements are subject to some commutation relations with the right hand sides expressed via even generating elements like in(2),it often allows one to reduce evaluation of the supertrace of a degree-n polynomial of a i to supertraces of lower degree polynomials.Another useful property is that in order to show that the characteristic property of the supertrace(11)is true for any x,g∈A,it suﬃces to show this for a particular case where x is arbitrary while g is an arbitrary generating element of someﬁxed system of generating elements.Then(11)for general x and g will follow from the properties that A is associative and str is linear.For the particular case of SH N(ν)this means that it is enough to set either g=aαi or g=K ij.Let us now turn to some speciﬁc properties of SH N(ν)as a particular realization of A.By identifying L with sl2(5)and taking into account that SH N(ν)decomposes into a direct sum of irreducibleﬁnite-dimensional representations of sl2,one arrives at the followingLemma1:str(x)can be diﬀerent from zero only when x is sl2-singlet,i.e.[Tαβ,x]=0. Corollary:Any supertrace on SH N(ν)is even.Analogously one deduces consequences of the S N symmetry.In particular,one proves Lemma2:Given c∈S N such that cF=µF c for some element F and any constant µ=1,str(F)=0.Given monomial F=b I1b I2...b I sσwith b I k∈Bσand a cycle c0in the decomposition(15)ofσone concludes that str(F)=0if k:c(I k)=c0λI k=1where λIkare the eigenvalues(21)of b I k.8IV Ground Level ConditionsLet us analyze restrictions on a form of str(a),a∈S N,which follow from the deﬁning relations of SH N(ν).Firstly,we describe supertraces on the group algebra of S N.Let some permutationσdecomposes into n1cycles of length1,n2cycles of length2,...and n N cycles of length N.The non-negative integers n k satisfy the relationNk=1kn k=N(35) andﬁxσup to some conjugationσ→τστ−1,τ∈S N.Thusstr(σ)=ϕ(n1,n2,...,n N),(36) whereϕ(n1,n2,...,n N)is an arbitrary function.Obviously the linear space of invariant functions on S N(i.e.such that f(τστ−1)=f(σ))coincides with the linear space of supertraces on the group algebra of S N.Therefore,the dimension of the linear space of supertraces is equal to the number p(N)of independent solutions of(35),the number ofconjugacy classes of S N.One can introduce the generating function for p(N)as P(q)= ∞n=0p(n)q n= ∞k=11Lemma 3:Let c 1and c 2be two distinct cycles in the decomposition (15).Let indices i 1and i 2belong to the subsets of indices associated with the cycles c 1and c 2,respectively.Then the permutation c =c 1c 2K i 1i 2is a cycle of length |c |=|c 1|+|c 2|.Lemma 4:Given cyclic permutation c ∈S N ,let i =j be two indices such that c k (i )=j ,where k is some positive integer,k <|c |.Then cK ij =c 1c 2where c 1,2are some non-coinciding mutually commuting cycles such that |c 1|=k and |c 2|=|c |−k .Using the deﬁnition (17),the commutation relations (1)-(3)and Lemmas 3and 4one reduces GLC to the following system of equations:n 2k ϕ(n 1,...,n 2k ,...,n N )=−νn 2k 22k −1 s =k,s =1O s ϕ(n 1,...,n s +1,...,n 2k −s +1,...,n 2k −1,...,n N )+2O k ϕ(n 1,...,n k +2,...,n 2k −1,...,n N )+N s =2k ;s =1sn s ϕ(n 1,...,n s −1,...,n 2k −1,...,n 2k +s +1,...,n N )+2k (n 2k −1)ϕ(n 1,...,n 2k −2,...,n 4k +1,...,n N )(39)where O k =0for k even and O k =1for k odd.Let us note that by virtue of the substitution ϕ(n 1,...,n N )=νE (σ)˜ϕ(n 1,...,n N ),(40)where E (σ)is the number of cycles of even length in the decomposition of σ(15),i.e.E (σ)=n 2+n 4+ (41)one can get rid of the explicit dependence of νfrom GLC (39).As a result,there are two distinguishing cases,ν=0and ν=0.For lower N the conditions (39)take the formϕ(0,1)+2νϕ(2,0)=0(42)for N =2(cf.(9)),ϕ(1,1,0)+2νϕ(3,0,0)+νϕ(0,0,1)=0(43)for N =3andϕ(2,1,0,0)+2νϕ(4,0,0,0)+2νϕ(1,0,1,0)=0ϕ(0,2,0,0)+2νϕ(2,1,0,0)+2νϕ(0,0,0,1)=0ϕ(0,0,0,1)+4νϕ(1,0,1,0)=0for N =4.As a result one ﬁnds 1-parametric families of solutions for N =1and N =2and 2-parametric families of solutions for N =3and N =4.Let G N be the number of independent solutions of (39).As we show in the next section G N =dimT SH N (ν)for all ν.In other words all other conditions on the supertrace do not10impose any restrictions on the functionsϕ(n1,...,n N)but merely express supertraces of higher order polynomials of aαi in terms ofϕ(n1,...,n N).In the Appendix A we prove the followingTheorem1:G N=q(N)where q(N)is a number of partitions of N into a sum of odd positive integers,i.e.the number of the solutions of the equation ∞k=0(2k+1)n k=N for non-negative integers n i.One can guess this result from the particular case ofν=0where GLC tell us that ϕ(n1,...,n N)can be nonvanishing(and arbitrary)only when all n2k=0.Interestingly enough,G N remains the same forν=0.V Supertrace for General ElementsIn this section we proveTheorem2:dimT SHN(ν)=G N where G N is the number of independent solutions of theground level conditions(39).The proof of the Theorem2will be given in a constructive way by virtue of the following double induction procedure:(i).Assuming that GLC are true and str{b I,P p(a)σ}=0∀P p(a),σand I provided that b I∈Bσandλ(I)=−1;p≤k orλ(I)=−1,E(σ)≤l,p≤k orλ(I)=−1;p≤k−2,where P p(a)is an arbitrary degree p polynomial of aαi(p is odd)and E(σ)is the number of cycles of even length in the decomposition(15)ofσ,one proves that there exists such a unique extension of the supertrace that the same is true for l→l+1.(ii).Assuming that str{b I,P p(a)σ}=0∀P p(a),σand b I such thatσ(I)=σ,p≤k one proves that there exists such a unique extension of the supertrace that the assumption (i)is true for k→k+2and l=0.As a result this inductive procedure extends uniquely any solution of GLC to some supertrace on the whole SH N(ν).(Let us remind ourselves that the supertrace of any odd element of SH N(ν)is trivially zero by sl2invariance).The inductive proof of the Theorem2is based on the S N covariance of the whole setting and the following importantLemma5:Given permutationσwhich has E(σ)cycles of even length in the decompo-sition(15),the quantity f IJσforσ(I)=σ(J)=σandλI=λJ=−1can be uniquely expanded as f IJσ= qαqσq whereαq are some coeﬃcients and E(σq)=E(σ)−1∀q.Lemma5is a simple consequence of the particular form of the structure coeﬃcients f IJ(27)and Lemmas3and4.The proof is straightforward.Let us stress that it is Lemma5which accounts for the speciﬁc properties of the algebra SH N(ν)in the analysis of this section.In practice it is convenient to work with the exponential generating functionsΨσ(µ)=str e Sσ ,S=2N L=1(µL b L),(44)11where σis some ﬁxed element of S N ,b L ∈B σand µL ∈C are independent parameters.By diﬀerentiating over µL one can obtain an arbitrary polynomial of b L in front of σ.The exponential form of the generating functions implies that these polynomials are Weyl ordered.In these terms the induction on a degree of polynomials is equivalent to the induction on a degree of homogeneity in µof the power series expansions of Ψσ(µ).As a consequence of the general properties discussed in the preceding sections the generating function Ψσ(µ)must be invariant under the S N similarity transformationsΨτστ−1(µ)=Ψσ(˜µ),(45)where the S N transformed parameters are of the form˜µI = J M (τστ−1)M −1(τ)Λ−1(τ)M (τ)M −1(σ) J I µJ (46)and matrices M (σ)and Λ(σ)are deﬁned in (22)and (23).In accordance with thegeneralargumentof SectionIII the necessary and suﬃcient conditions for the existence of even supertrace are the S N -covariance conditions (45)and the condition thatstr b L,(expS )σ =0for any σand L.(47)To transform (47)to an appropriate form,let us use the following two general relations which are true for arbitrary operators X and Y and the parameter µ∈C :Xexp (Y +µX )=∂∂µexp (Y +µX )− t 1exp (t 1(Y +µX ))[X,Y ]exp (t 2(Y +µX ))D 1t (49)with the convention that D n −1t =δ(t 1+...+t n −1)θ(t 1)...θ(t n )dt 1...dt n .(50)The relations (48)and (49)can be derived with the aid of the partial integration (e.g.over t 1)and the following formula∂∂µL Ψσ(µ)= (λL t 1−t 2)str exp (t 1S )[b L ,S ]exp (t 2S )σ D 1t.(53)12This condition should be true for anyσand L and plays the central role in the analysis of this section.There are two essentially distinguishing cases,λL=−1andλL=−1.In the latter case,the equation(53)takes the form0= str exp(t1S)[b L,S]exp(t2S)σ D1t,λL=−1.(54) In Appendix B we show by induction that the equations(53)and(54)are consistent in the following sense∂(1+λK)∂µK str exp(t1S)[b L,S]exp(t2S)σ D1t=0,λL=−1.(56) Note that this part of the proof is quite general and does not depend on a concrete form of the commutation relations of aαi in(2).By expanding the exponential e S in(44)into power series inµK(equivalently b K) one concludes that the equation(53)uniquely reconstructs the supertrace of monomials containing b K withλK=−1(from now on called regular polynomials)via supertraces of some lower order polynomials.The consistency conditions(55)and(56)then guarantee that(53)does not impose any additional conditions on the supertraces of lower degree polynomials and allow one to represent the generating function in the formΨσ=Φσ(µ)(57) + L:λL=−1 10µL dτConsider the part of str b I,(expS′)σ which is of order k inµand suppose that E(σ)= l+1.According to(54)the conditions(60)give0= str exp(t1S′)[b I,S′]exp(t2S′)σ D1t.(61) Substituting[b I,S′]=µI+ν M f IMµM,where the quantities f IJ andµI are deﬁned in(25)-(28),one can rewrite the equation(61)in the formµIΦσ(µ)=−ν str exp(t1S′) M f IMµM exp(t2S′)σ D1t.(62)Now we use the inductive hypothesis(i).The right hand side of(62)is a supertrace of at most a degree k−1polynomial of aαi in the sector of degree k polynomials inµ. Therefore one can use the inductive hypothesis(i)to obtainstr exp(t1S′) M f IMµM exp(t2S′)σ D1t= str exp(t2S′)exp(t1S′) M f IMµMσ D1t,where we made use of the simple fact that str(S′Fσ)=−str(FσS′)=str(F S′σ)due to the deﬁnition of S′.As a result,the inductive hypothesis allows one to transform(60)to the following formX I≡µIΦσ(µ)+νstr exp(S′) M f IMµMσ =0.(63) By diﬀerentiating this equation with respect toµJ one obtains after symmetrization ∂∂µJX I(µ)+∂∂µJ µIΦσ(µ)+(I↔J)=−ν2 L,M(t1−t2)str exp(t1S′)F JLµL exp(t2S′)f IMµMσ D1t+(I↔J).(65) The last term on the right hand side of this expression can be shown to vanish under the supertrace operation due to the factor of(t1−t2),so that one is left with the equationL IJΦσ(µ)=−νwhereR IJ(µ)= M str exp(S′){b J,f IM}µMσ +(I↔J)(67) andL IJ=∂∂µIµJ.(68)The diﬀerential operators L IJ satisfy the standard sp(2E(σ))commutation relations [L IJ,L KL]=− C IK L JL+C IL L JK+C JK L IL+C JL L IK .(69) We show by induction in Appendix C that this algebra is consistent with the right-hand side of the basic relation(66)i.e.that[L IJ,R KL]−[L KL,R IJ]=− C IK R JL+C JL R IK+C JK R IL+C IL R JK .(70)Generally,these consistency conditions guarantee that the equations(66)express Φσ(µ)in terms of R IJ in the following wayΦσ(µ)=Φσ(0)+νt(1−t2E(σ))(L IJ R IJ)(tµ),(71)provided thatR IJ(0)=0.(72) The latter condition must hold for the consistency of(66)since its left hand side vanishes atµI=0.In the formula(71)it guarantees that the integral on t converges.In the case under consideration the property(72)is indeed true as a consequence of the deﬁnition (67).Taking into account Lemma5and the explicit form of R IJ(67)one concludes that the equation(71)expresses uniquely the supertrace of special polynomials via the supertraces of polynomials of lower degrees or via the supertraces of special polynomials of the same degree with a lower number of cycles of even length provided that theµindependent term Φσ(0)is an arbitrary solution of GLC.This completes the proof of Theorem2. Comment1:The formulae(57)and(71)can be eﬀectively used in practical calculations of supertraces of particular elements of SH N(ν).Comment2:Any supertrace on SH N(ν)is determined unambiguously in terms of its values on the group algebra of S N.Corollary:Any supertrace on SH N(ν)isρ-invariant,str(ρ(x))=str(x)∀x∈SH N(ν), for the antiautomorphismρ(30).This is true due to the Comment2becauseσandσ−1=ρ(σ)belong to the same conjugacy class of S N so that str(ρ(σ))=str(σ).15。

赫斯特指数

8
Hurst 指数的计算方法
由原始数据计算 R (T) /S (T) ，T= 2，3，⋯，
R T mX a t,x T mX itn ,T
1 t T
1 t T
STT1tT1ξtξT21/2
然后在 ln(R /S) - lnT 坐标系中用直线拟合观测点。
该直线斜率即为H 指数的值。
在完全有效的资本市场上，证券价格完全能够反映信息蕴涵的价值。收益率的波动不能用过去的收益率来预测。股票的收益率此时是随机游走，服从布朗运动模型、正态分布。
21
目前世界上大多数国家股票市场的实践都证明股票收益率分布具有尖峰肥尾以及存在长期记忆效应等特征，传统的有效市场理论显然已经不合时宜。
此时，时间序列有混沌性。过去的增量与未来的增量是正相关的，序列在下一时刻极有可能仍将保持原方向不变。因此，一定范围的记录会持续相当长的时期，从而形成一个个大的循环。但是这些循环没有固定的周期，难以依靠过去的数据预测未来的变化。
H = 1：完全预测。此时, 时间序列为一条直
线。未来完全可以用现在进行预测。
学者们在非线性分析思维的启示下，提出了与有效市场理论相对应的分形市场理论，其代表人物有Mandelbrot 和Edgar E. Peters 。
22
Mandelbrot(1964) 对资本市场的统计特性进行了开创性的探索，创立了分形几何学，提出了分形理论；
Peters( 1994) 在Mandelbrot 的基础上进一步对资本市场统计特性进行了研究，提出了分形市场假说( FMH) 。
而东方明珠(600832) 的收益率是Antipersistent 的。这表明它们的收益率趋向于返回过去的记录，收益率变化的增量发散较慢。

R语言时间序列中文教程

R语言时间序列中文教程R语言是一种广泛应用于统计分析和数据可视化的编程语言。

它提供了丰富的函数和包，使得处理时间序列数据变得非常方便。

本文将为大家介绍R语言中时间序列分析的基础知识和常用方法。

R语言中最常用的时间序列对象是`ts`对象。

通过将数据转换为`ts`对象，可以使用R语言提供的各种函数和方法来分析时间序列数据。

我们可以使用`ts`函数将数据转换为`ts`对象，并指定数据的时间间隔、起始时间等参数。

例如，对于按月份记录的时间序列数据，可以使用以下代码将数据转换为`ts`对象：```Rts_data <- ts(data, start = c(2000, 1), frequency = 12)```在时间序列分析中，常用的一个概念是平稳性。

平稳性表示时间序列的均值和方差在时间上不发生显著变化。

平稳时间序列的特点是，它的自相关函数(ACF)和偏自相关函数(PACF)衰减得很快。

判断时间序列是否平稳可以通过绘制序列的线图和计算序列的自相关函数来进行。

我们可以使用R语言中的`plot`函数和`acf`函数来实现。

例如，对于一个名为`ts_data`的时间序列数据，可以使用以下代码绘制序列的线图和自相关函数图：```Rplot(ts_data)acf(ts_data)```在进行时间序列分析时，经常需要进行模型拟合和预测。

R语言提供了一些常用的函数和包，用于时间序列的模型拟合和预测。

其中，最常用的方法是自回归移动平均模型(ARIMA)。

ARIMA模型是一种广泛应用于时间序列分析的统计模型，它可以描述时间序列数据中的长期趋势、季节性变动和随机波动等特征。

我们可以使用R语言中的`arima`函数来拟合ARIMA模型，并使用`forecast`函数来进行预测。

以下是一个使用ARIMA模型进行时间序列预测的示例代码：```Rmodel <- arima(ts_data, order = c(p, d, q))forecast_result <- forecast(model, h = 12)```以上代码中，`p`、`d`和`q`分别表示ARIMA模型的自回归阶数、差分阶数和移动平均阶数。

分形布朗运动和hurst指数

分形布朗运动和hurst指数
分形布朗运动是一种随机过程，其特性与布朗运动相似，但具有更复杂的分形结构。

布朗运动是指微观粒子在液体或气体中由于受到分子的不断碰撞而进行的无规则、连续且随机的运动。

而分形布朗运动则是在这种运动过程中引入了分形结构，使得其具有更为复杂的运动模式。

Hurst指数是用来描述分形布朗运动的一个重要参数。

它表示分形布朗运动在时间序列上的长期依赖性或持久性。

Hurst指数的值介于0和1之间，其中0.5表示随机游走，小于0.5表示负持久性，即过去的变化趋势对未来的影响逐渐减弱，而大于0.5则表示正持久性，即过去的变化趋势对未来的影响逐渐增强。

在金融领域中，分形布朗运动和Hurst指数被广泛应用于模拟股票价格等金融时间序列。

由于股票价格具有分形结构和持久性，因此分形布朗运动可以很好地描述股票价格的波动特征。

通过估计Hurst指数，我们可以了解股票价格的波动趋势和未来价格的变化情况。

除了金融领域，分形布朗运动和Hurst指数还在其他领域得到广泛应用。

例如，在地球物理学中，它们被用于模拟地震和海浪等自然现象；在生物学中，它们被用于描述生物种群的增长和变化趋势等。

此外，分形布朗运动和Hurst 指数还被应用于图像处理、信号处理等领域。

总之，分形布朗运动是一种具有复杂分形结构的随机过程，其特性与布朗运动相似但更为复杂。

Hurst指数是描述分形布朗运动的一个重要参数，可以用来估计时间序列的持久性和变化趋势。

在金融、地球物理学、生物学等领域中，分形布朗运动和Hurst指数得到了广泛应用，为我们提供了更准确、更有效的分析方法和工具。

r语言简介

R的缺点
• 需要编程，不傻瓜 • 不如s-plus输出的画图效果好
第一步，交互式使用R
• 一个R程序需要你输入命令时，默认的提示符是 > • 退出R程序的命令是>q() • 此时R会话会问你是否需要保存数据：
Help
• R也有类似man的内嵌帮助工具 • 比如得到solve函数的帮助： • >help(solve) 此时会发出请求，得到html格式的帮助或者？solve 作用一样的
低级图形函数
• 向已经存在的图形中添加自己定义的信息
• • • • •
points(x,y) 在图形上增加点 lines(x,y) 在图形上连接线 abline(a,b) #画y=a+bx 的直线 abline(h=y) #画过所有点的水平直线 text(x,y,labels) 在图形上给定的x,y位置添加文字，labels经常是整数或者字符向量。这个功能经常用于下面的命令： >plot(x,y,type=“n”);text(x,y,names) 图形参数type=“n”不让点显示，text()使x,y 点位置上的符标由names决定
• a[,,]表示整个数组，和忽略下标直接使用a效果一样
数据框
• • • • 数据框通常是矩阵形式的数据，但矩阵各列可以是不同类型的每列是一个变量，每行是一个观测数据框生成：数据框可以用data.frame()函数生成，其用法与list()函数相同，各自变量变成数据框的成分，自变量可以命名，成为变量名。例如： • > d <- data.frame(name=c("李明", "张聪", "王建"), age=c(30, 35, 28), height=c(180, 162, 175)) • >d

nr中的tbsize计算公式

nr中的tbsize计算公式TBSIZE是一种用于计算网络传输速度的公式，它可以帮助我们评估网络的带宽和传输效率。

在计算TBSIZE时，我们需要考虑到传输速度、网络延迟和数据包大小等因素。

下面将详细介绍TBSIZE的计算公式及其在网络传输中的应用。

我们需要了解TBSIZE的定义。

TBSIZE是指传输缓冲区大小（Transmission Buffer Size），它表示在一段时间内可以发送的最大数据量。

传输缓冲区是指在发送数据时，存储待发送数据的内存区域。

通过调整传输缓冲区的大小，可以对网络传输速度进行优化。

TBSIZE的计算公式如下：TBSIZE = (RTT * BDP) / 8其中，RTT表示网络往返时间（Round Trip Time），BDP表示带宽延迟乘积（Bandwidth-Delay Product）。

公式中的8是将单位从比特转换为字节。

带宽延迟乘积（BDP）是指网络带宽和网络延迟的乘积，它反映了网络传输的效率。

网络带宽是指在单位时间内传输的数据量，通常以比特为单位。

网络延迟是指数据从发送端到接收端所需的时间，通常以毫秒为单位。

通过计算TBSIZE，我们可以确定在一段时间内可以发送的最大数据量。

这对于优化网络传输速度和提高用户体验非常重要。

在实际应用中，我们可以根据需求和网络情况来调整传输缓冲区的大小，以达到最佳的传输效果。

在网络传输中，TBSIZE的优化对于提高数据传输速度和减少数据丢失非常重要。

通过合理调整传输缓冲区的大小，可以在保证传输稳定性的同时，提高传输效率。

例如，在视频流传输中，通过增大传输缓冲区的大小，可以减少视频卡顿和数据丢失的情况，提供更流畅的观看体验。

TBSIZE的计算还可以帮助我们评估网络的带宽和延迟状况。

通过监测网络往返时间（RTT）和带宽延迟乘积（BDP），我们可以了解网络的传输效率和瓶颈所在，进而采取相应的优化措施。

总结起来，TBSIZE是一种用于计算网络传输速度的公式，它可以帮助我们评估网络的带宽和传输效率。

hurst指数2篇

hurst指数第一篇：Hurst指数简介及应用领域Hurst指数是一种用于衡量时间序列数据的长期记忆性的统计量，其应用广泛于金融分析、水文学、信号处理等领域。

本文将对Hurst指数进行详细介绍，并探讨其应用领域。

Hurst指数最初是由数学家H.E. Hurst于1951年提出的，其用于衡量时间序列数据的波动性和相关性。

时间序列数据是指一组按时间顺序排列的观测值，例如股票价格、气温记录等。

Hurst指数的取值范围在0到1之间，其中0表示完全反序列相关，1表示完全正序列相关，0.5表示完全随机。

Hurst 指数越接近于0.5，说明时间序列数据的波动性越接近于随机，没有长期记忆性；而越接近于0或1，说明时间序列数据存在较强的趋势性，即具有长期记忆性。

Hurst指数的计算需要借助于重叠子序列的均值计算，具体步骤如下：首先，将时间序列数据分解成不同长度的子序列；然后，计算每个子序列的均值；最后，计算不同子序列长度下的均值之比。

根据计算得到的比值，可得到Hurst指数。

在金融分析中，Hurst指数常被用于衡量股票价格的长期记忆性和预测性。

通过计算Hurst指数，可以评估股票价格的波动性，进而辅助投资者进行风险管理和决策制定。

例如，当股票价格的Hurst指数较高时，说明价格具有较强的趋势性，投资者可以选择更长期的持有策略，以获得更大的收益。

此外，Hurst指数在水文学领域也得到了广泛的应用。

水文学研究常关注各种水文变量的波动性，例如降水量、水位等。

通过计算Hurst指数，可以评估水文变量的长期趋势，进而为水资源管理、洪水预测等提供科学依据。

除金融分析和水文学外，Hurst指数在信号处理、网络分析等领域也有着重要的应用价值。

例如，对于信号处理，Hurst指数可以用于评估信号的分形特性和自相似性，从而指导滤波、数据压缩等算法的设计与优化。

综上所述，Hurst指数是一种用于衡量时间序列数据长期记忆性的统计量，在金融分析、水文学、信号处理等领域有广泛的应用。

hurst 指数python

hurst 指数python摘要：1.Hurst 指数简介2.Python 在Hurst 指数计算中的应用3.Hurst 指数的计算方法4.Python 代码示例5.总结正文：1.Hurst 指数简介Hurst 指数是一种用来描述时间序列数据的长期记忆特性的指标。

它是由英国统计学家Hurst 在1951 年提出的，被广泛应用于金融、气象、水文等领域。

Hurst 指数的取值范围为0 到1 之间，当指数大于0.5 时，表示时间序列具有正长时相关性，即具有趋势；当指数等于0.5 时，表示时间序列无关；当指数小于0.5 时，表示时间序列具有负长时相关性，即具有反趋势。

2.Python 在Hurst 指数计算中的应用Python 作为一门广泛应用于数据分析和科学计算的语言，拥有丰富的库和工具，可以方便地实现Hurst 指数的计算。

在使用Python 计算Hurst 指数时，常用的库有NumPy、Pandas 和Statsmodels 等。

3.Hurst 指数的计算方法Hurst 指数的计算方法有多种，其中较为常见的有以下几种：- R/S分析法：R/S分析法是Hurst指数计算中最常用的方法，其基本思想是将时间序列数据进行分段，计算各分段的平均值，然后计算各分段平均值之间的相关性。

- 波动率法：波动率法是通过计算时间序列数据的波动率来估计Hurst 指数的方法。

波动率的计算可以采用简单的方差计算，也可以采用更为复杂的GARCH 模型等。

- 功率谱法：功率谱法是通过计算时间序列数据的功率谱来估计Hurst 指数的方法。

功率谱可以反映时间序列在不同时间尺度上的能量分布，从而为Hurst 指数的估计提供依据。

4.Python 代码示例以下是一个使用Python 和Pandas 库计算Hurst 指数的简单示例：```pythonimport pandas as pdfrom scipy import stats# 创建一个简单的时间序列数据data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]# 将数据转换为Pandas 的Series 对象series = pd.Series(data)# 计算R/S分析法的Hurst指数rs_result = stats.rs_one_step(series)hurst_rs = rs_result[1]print("Hurst 指数（R/S 分析法）：", hurst_rs)```5.总结本文介绍了Hurst 指数的计算方法和Python 在Hurst 指数计算中的应用，并通过一个简单的Python 代码示例展示了如何使用Pandas 和Scipy 库计算Hurst 指数。

urwtest参数说明

urwtest参数说明English answer:urwtest is a program that tests the functionality of the Unicode Regular Expressions (URE) library. The library is used by many programs to perform complex text processing tasks, such as finding and replacing text, searching for patterns, and validating input.The urwtest program can be used to test the following aspects of the URE library:Character classes: Character classes are used to match characters that have certain properties, such as being a letter, a digit, or a whitespace character.Anchors: Anchors are used to match characters at the beginning or end of a string, or at the beginning or end of a line.Quantifiers: Quantifiers are used to match characters that occur a certain number of times.Grouping: Grouping is used to group characters together so that they can be treated as a single unit.Backreferences: Backreferences are used to match characters that have been previously matched.The urwtest program can be used to test the URE library by providing a regular expression and a string to match. The program will then output whether the regular expression matches the string.The urwtest program has a number of options that can be used to control its behavior. These options include:-v: Verbose output. This option causes the urwtest program to output more information about the regular expression and the string being matched.-i: Case-insensitive matching. This option causes theurwtest program to ignore the case of the characters in the regular expression and the string being matched.-m: Multiline matching. This option causes the urwtest program to treat the string being matched as a multiline string.-s: Dotall matching. This option causes the urwtest program to treat the dot (.) character in the regular expression as matching any character, including newline characters.-x: Extended syntax. This option causes the urwtest program to allow the use of whitespace characters and comments in the regular expression.The urwtest program can be a useful tool for testing the functionality of the URE library. It can be used to verify that regular expressions are working as expected, and to troubleshoot problems with regular expressions.Here are some examples of how to use the urwtestprogram:$ urwtest 'abc' 'abc'。

(2013,pr)EDCircles_A real-time circle detector with a false detection control

EDCircles:A real-time circle detector with a false detection controlCuneyt Akinlar n,Cihan TopalDepartment of Computer Engineering,Anadolu University,Eskisehir26470,Turkeya r t i c l e i n f oArticle history:Received9April2012Received in revised form21September2012Accepted26September2012Available online3October2012Keywords:Circle detectionEllipse detectionReal-time image processingHelmholtz PrincipleNFAa b s t r a c tWe propose a real-time,parameter-free circle detection algorithm that has high detection rates,producesaccurate results and controls the number of false circle detections.The algorithm makes use of thecontiguous(connected)set of edge segments produced by our parameter-free edge segment detector,theEdge Drawing Parameter Free(EDPF)algorithm;hence the name EDCircles.The proposed algorithmﬁrstcomputes the edge segments in a given image using EDPF,which are then converted into line segments.The detected line segments are converted into circular arcs,which are joined together using two heuristicalgorithms to detect candidate circles and near-circular ellipses.The candidates areﬁnally validated by an acontrario validation step due to the Helmholtz principle,which eliminates false detections leaving onlyvalid circles and near-circular ellipses.We show through experimentation that EDCircles works real-time(10–20ms for640Â480images),has high detection rates,produces accurate results,and is very suitablefor the next generation real-time vision applications including automatic inspection of manufacturedproducts,eye pupil detection,circular trafﬁc sign detection,etc.&2012Elsevier Ltd.All rights reserved.1.IntroductionDetection of circular objects in digital images is an importantand recurring problem in image processing[1]and computervision[2],and has many applications especially in such automa-tion problems as automatic inspection of manufactured products[3],aided vectorization of line drawing images[4,5],pupil and irisdetection[6–8],circular trafﬁc sign detection[9–11],and manyothers.An ideal circle detection algorithm should run with aﬁxed set ofinternal parameters for all images,i.e.,require no parameter tuningfor different images,be very fast(real-time if possible),detectmultiple small and large circles,work with synthetic,natural andnoisy images,have high detection rate and good accuracy,andproduce a few or no false detections.The circle detection algorithmpresented in this paper satisﬁes all of these properties.Traditionally,the most popular circle detection techniques arebased on the famous circle Hough transform(CHT)[12–16].Thesetechniquesﬁrst compute an edge map of the image using atraditional edge detector such as Canny[17],map the edge pixelsinto the three dimensional Hough circle space(x,y,r)and extractcircles that contain a certain number of edge pixels.Not onlyCHT-based techniques are very slow and memory-demanding,butthey also produce many false detections especially in the presenceof noise.Additionally,these methods have many parameters thatmust be preset by the user,which greatly limits their use.To overcome the limitations of the classical CHT-based meth-ods,many variants have been proposed including probabilistic HT[18,19],randomized HT[20,21],fuzzy HT[22],etc.There are alsoapproaches based on HT and hypothesisﬁltering[23–25].Allthese methods try to correct different shortcomings of CHT,butare still memory-demanding and slow to be of any use in real-time applications.Apart from the CHT-based methods,there are several rando-mized algorithms for circle detection.Chen et al.[26]propose arandomized circle detection(RCD)algorithm that randomly selectsfour pixels from the edge map of an image,uses a distance criterionto determine whether there is a possible circle in the image.Theythen use an evidence-collecting step to test if the candidate circle isa real-circle.RCD produces good results,but is slow.Recently,Chunget al.[27,28]have proposed efﬁcient sampling and reﬁnementstrategies to speed up RCD and increase the accuracy of RCD’sresults.Although the new RCD variants named GRCD-R,GLRCD-R[28]have good detection rates and produce accurate results,theystill are far from being real-time.Furthermore,all RCD-variantswork on the edge map of an image computed by a traditional edgedetector such as the Sobelﬁlter or the Canny edge detector,whichhave many parameters that must be set by the user.Recently,many efforts have concentrated on using geneticalgorithms and evolutionary computation techniques in circledetection[29–36].Ayala-Ramirez et al.[30]proposed a geneticalgorithm(GA)for circle detection,which is capable of detectingmultiple circles but fails frequently to detect small or imperfectContents lists available at SciVerse ScienceDirectjournal homepage:/locate/prPattern Recognition0031-3203/$-see front matter&2012Elsevier Ltd.All rights reserved./10.1016/j.patcog.2012.09.020n Corresponding author.Tel.:þ902223213550x6553.E-mail addresses:cakinlar@.tr(C.Akinlar),cihant@.tr(C.Topal).Pattern Recognition46(2013)725–740circles.Dasgupta et al.[31–33]developed a swarm intelligence technique named adaptive bacterial foraging optimization (ABFO)for circle detection.Their algorithm produces good results but is sensitive to noise.Cuevas et e discrete differential evolution (DDE)optimization [34],harmony search optimization (HSA)[35]and an artiﬁcial immune system optimization technique named Clonal Selection Algorithm (CSA)[36]for circle detection.Although these evolutionary computation techniques have good detection rates and accurate results,they usually require multiple runs to detect multiple circles,and are quite slow to be suitable for real-time applications.Just like RCD,these algorithms work on an edge map pre-computed by a traditional edge detection algorithm with many parameters.Frosio et al.[37]propose a real-time circle detection algorithm based on maximum likelihood.Their method is fast andcan detect partially occluded circular objects,but requires that the radius of the circles to be detected be predeﬁned,which greatly limits its applications.Wu et al.[41]present a circle detection algorithm that runs 7frames/s on 640Â480images.The authors claim to achieve high success rate,but there is not much experi-mental validation to back their claims.Zhang et al.[38]propose an ellipse detection algorithm that can be used for real-time face detection.Liu et al.[39]present an ellipse detector for noisy images and Prasad et al.[40]present an ellipse detector using the edge curvature and convexity information.While both algorithms produce good results,they are slow and not suitable for real-time applications.Vizireanu et al.[42–44]make use of mathematical morphol-ogy for shape decomposition of an image and use the morpholo-gical shape decomposition representation of the image for recognition of different shapes and patterns in the image.While their algorithms are good for the detection of general shapes in an image,they are not suitable for real-time applications.Desolneux et al.[60]is the ﬁrst to talk about the a contrario circular arc detection.Recently,Patraucean et al.[45,46]propose a parameter-free ellipse detection algorithm based on the a contrario framework of Desolneux et al.[58].The authors extend the line segment detector (LSD)by Grompone von Gioi et al.[63]to detect circular and elliptic arcs in a given image without requiring any parameters,while controlling the number of false detections by the Helmholtz principle [58].They then use the proposed algorithm (named ELSD [46])for the detection identiﬁcation of Bubble Tags [47].In this paper,we present a real-time (10–20ms on 640Âimages),parameter-free circle detection algorithm that has high detection rates,produces accurate results,and has an a contrario validation step due to the Helmholtz principle that lets it control the number of false detections.The proposed algorithm makes use of the contiguous (connected)set of edge segments produced by our parameter-free edge segment detector,the edge drawing parameter free (EDPF)[48–53];hence the name EDCircles [54,55].Given an input image,EDCircles ﬁrst computes the edge segments of the image using EDPF.Next,the resulting edge segments are turned into line segments using our line segment detector,EDLines [56,57].Computed lines are then converted into arcs,which are combined together using two heuristic algorithms to generate many candidate circles and near-circular ellipses.Finally,the candidates are vali-dated by the Helmholtz principle [58–63],which eliminates false detections leaving only valid circles and near-circular ellipses.2.The proposed algorithm:EDCirclesEDCircles follows several steps to compute the circles in a given image.The general idea is to extract line segments in an image,convert them into circular arcs and then combine these arcs to detect circles and near-circular ellipses.General outline ofEDCircles algorithm is presented in Algorithm 1and we will describe each step of EDCircles in detail in the following sections.Algorithm 1.Steps of EDCircles algorithm.1.Detect edge segments by EDPF and extract complete circles and ellipses.2.Convert the remaining edge segments into line segments.3.Detect arcs by combining line segments.4.Join arcs to detect circle candidates.5.Join the remaining arcs to detect near-circular ellipse candidates.6.Validate the candidate circles/ellipses using the Helmholtz principle.7.Output the remaining valid circles/ellipses.2.1.Edge segment detection by edge drawing parameter free (EDPF)Given an image,the ﬁrst step of EDCircles is the detection of the edge segments in the image.To achieve this,we employ our recently proposed,real-time edge/edge segment detector,edge drawing (ED)[48–51].Unlike traditional edge detectors,e.g.,Canny [17],which work by identifying a set of potential edge pixels in an image and eliminating non-edge pixels through operations such as non-maximal suppression,hysteresis thresh-olding,erosion,etc.,ED follows a proactive approach and works by ﬁrst identifying a set of points in the image,called the anchors,and then joins these anchors using a smart routing procedure;that is,ED literally draws edges in an image.ED outputs not only a binary edge map similar to those output by traditional edge detectors,but it also outputs the result as a set of edge segments each of which is a contiguous (connected)pixel chain [49].ED has many parameters that must be set by the user,which requires the tuning of ED’s parameters for different types of images.Ideally,one would want to have a real-time edge/edge segment detector which runs with a ﬁxed set of internal parameters for all types of images and requires no parameter tuning.To achieve this goal,we have recently incorporated ED with the a contrario edge validation mechanism due to the Helmholtz principle [58–60],and obtained a real-time parameter-free edge segment detector,which name edge drawing parameter free (EDPF)[52,53].EDPF works running ED with all ED’s parameters at their extremes,which all possible edge segments in a given image with many false positives.We then validate the extracted edge segments by the Helmholtz principle,which eliminates false detections leaving only perceptually meaningful edge segments with respect to the a contra-rio approach.Fig.1(a)shows a 424Â436grayscale synthetic image contain-ing a big circle obstructed by four rectangular blocks,a small ellipse obstructed by three rectangular blocks,a small circle,an ellipse and an arbitrary polygon-like object.When this image is fed into EDPF,the edge segments shown in Fig.1(b)are produced.Each color in the edge map represents a different edge segment,each of which is a contiguous chain of pixels.For this image,EDPF outputs 15edge segments in just 3.7ms in a PC with 2.2GHz Intel 2670QM CPU.Notice the high quality nature of the edge map with all details clearly visible.Each edge segment traces the boundary of one or more objects in the ﬁgure.While the boundary of an object may be traced by a single edge segment,as the small circle,the ellipse and the polygonal object are in Fig.1(b),it is also possible that an object’s boundary be traced by many different edge segments.This is the case for the big circle as the circle’s boundary is traced by four different edge segments,and the small obstructed ellipse,which is traced by three different edge segments.The result totally depends on the structure of the objects,the amount of obstruction and noise in the image.That is,there is noC.Akinlar,C.Topal /Pattern Recognition 46(2013)725–740726ellipse and the polygon,the entire boundary of anobject in the image is returned as a closed curve;that is,the edge segment starts at a pixel on the boundary of an object,traces its entire boundary and ends at where it starts.In other words,the ﬁrst and last pixels of the edge segment are neighbors of each other.It is highly likely that such a closed edge segment traces the boundary of a circle,an ellipse or a polygonal shape as is the case in Fig.1.So as the ﬁrst step after the detection of the edge segments,we go over all edge segments,take the closed ones and see if the closed edge segment traces the entire boundary of a circle or an ellipse.Processing of a closed edge segment follows a very simple idea:We ﬁrst ﬁt a circle to the entire list of pixels in the edge segment using the least squares circle ﬁt algorithm [64]and compute the root mean square error.If the circle ﬁt error,i.e.,the root mean square error,is smaller than some threshold (ﬁxed at 1.5pixels for the proposed algorithm),then we add the circle to the list of circle candidates.Just because the circle ﬁt error is small does not mean that the edge segment is an actual circle;it is just a candidate yet and needs to go through circle validation by the Helmholtz principle to be returned as a real circle.Section 2.6describes the details of circle validation.If the circle ﬁt fails,then we try ﬁtting an ellipse to the pixels of the edge segment.We use the ellipse ﬁt algorithm described in [65],which returns an ellipse equation of the form Ax 2þBxy þCy 2þDx þEy þF ¼0.If the ellipse ﬁt error,i.e.,the root mean square error,is smaller than a certain threshold (ﬁxed at 1.5pixels for the proposed algorithm),then we add the ellipse to the list of ellipse candidates,which also needs to go through validation by the Helmholtz principle before being returned as a real ellipse.If the edge segment is accepted either as a circle or an ellipse candidate,it is removed from the list of edge segments and is not processed any further.Otherwise,the edge segment is used in further processing along with other non-closed edge segments.2.2.Conversion of edge segments into line segmentsAfter the removal of the closed edge segments,which are taken as circle or ellipse candidates,the remaining edge segments are converted into line segments (lines for short in the rest of the paper).The motivation for this step comes from the observation that any circular shape is approximated by a consecutive set of lines (as seen in Fig.1(c)),and these lines can easily be turned into circular arcs by a simple post-processing step as described in the next section.Conversion of an edge segment into a set of lines follows the algorithm given in our line detector,EDLines [56,57].The idea is to start with a short line that satisﬁes a certain straightness criterion,and extend the line for as long as the root mean square error is smaller than a certain threshold,i.e.,1pixel error.Refer to EDLines [56,57]for the details of line segment extraction,where we validate the lines after detection using the Helmholtz principle to eliminate invalid detections.In EDCircles,though,we do not validate the lines after detection.The reason for this decision comes from our observation that the line segment validation algorithm due to the Helmholtz principle usually eliminates many short lines,which may be valuable for the detection of small circles in an image.So,unlike EDLines,we do not eliminate any detected lines and use all detected lines for further processing and detection of arcs.Fig.1(c)shows the lines extracted from the image shown in Fig.1(a).Clearly,circular objects are approximated by a set of consecutive lines.In the next section,we describe how these lines can be converted into circular arcs by processing of consecutive lines.2.3.Circular arc detectionWe at least three consecutive lines that Using this deﬁnition,we detect a list of lines making up an edge segment,simply walk over the lines and compute the angle between consecutive lines and the direction of turn from one line to the next.If at least three lines turn in the same direction and the angle between the lines is in-between certain thresholds,then these lines may form a circular arc.Fig.2illustrates a hypothetical edge segment being approxi-mated by 18consecutive line segments,labeled l 1through l 18.To compute the angle between two consecutive lines,we simply threat each line as a vector and compute the vector dot product.Similarly,to compute the turn of direction from one line to the next,we simply compute the vector cross product and use the sign of the result as the turn direction.Fig.3(a)illustrates the approximation of the blue right vertical edge segment in Fig.1(b)by 11consecutive line segments,labeled v1through v11.Fig.3(b)shows the details of the 11lines:their lengths,the angle between consecutive lines and the direction of the turn going from one line to the next,where a ‘þ’denotes a left turn,and ‘À’denotes a right turn.Our arc detection algorithm is based on the following idea:For a set of lines to be a potential arc candidate,they all must have the same turn direction (to the left or to the right)and the angle between consecutive lines must be in-between certain thresh-olds.If the angle is too small,we assume that the lines are collinear so they cannot be part of an arc;if the angle is too big,we assume that the lines are part of a strictly turning object such as a square,a rectangle,etc.For the purposes of our current implementation,we ﬁx the low angle threshold to 61,and the high angle threshold to 601.These values have been obtained byFig.1.(a)A sample image (424Â436).(b)Edge segments (a contiguous chain of pixels)extracted by EDPF.Each color represents a different edge segment.EDPF outputs 15edge segments in 3.7milliseconds (ms).(c)Lines approximating the edge segments.A total of 98lines are extracted.(For interpretation of the references to color in this ﬁgure caption,the reader is referred to the web version of this article.)C.Akinlar,C.Topal /Pattern Recognition 46(2013)725–740727experimentation on a variety ofimages containing various circular objects.The bottom part of Fig.2depicts the angles between consecutive lines of the edge segment shown at the top of Fig.2,and the turn of direction from one line to the next.The angles smaller than the low angle threshold or bigger than the high angle threshold,e.g.,y 1,y 2,y 9,and y 16,have been colored red;all other angles have been colored either blue or green depending on the turn of direction.Speciﬁcally,if the next line turns to the left,the angle has been colored blue,and if the next line turns to the right,then the angle has been colored green.Having computed the angles and the turn of direction infor-mation,we simply walk over the lines of an edge segment lookingfor a set of at least three consecutive lines which all turn in the same direction and the turn angle from one line to the next is in-between the low and high angle thresholds.In Fig.2,lines v 3through v 7satisfy our criterion and is a potential arc candidate.Similarly,lines v 10through v 16make up for another arc candidate.Given a set of at least three lines that satisfy our arc candidate constraints,we ﬁrst try ﬁtting a circle to all pixels making up the lines using the circle ﬁt algorithm in [64].If the circle ﬁt succeeds,i.e.,if the root mean square error is less than 1.5pixels,then the extracted arc is simply added to the list of arcs,and we are done.Otherwise,we start with a short arc consisting of only of three lines and extend it line-by-line by ﬁtting a new circle [64]until the root mean square error exceeds 1.5pixels.At this point,the to the list of arcs,and we continue processing detect more circular ing this algorithm,in Fig.2:Lines v 3through v 7form arc A1with center ðx c A 1,y c A 1Þand radius r A 1.Similarly,lines v 10through v 16form arc A 2with center ðx c A 2,y c A 2Þand radius r A 2.In a complex image consisting of many edge segments,we will have hundreds of arcs.Fig.4shows the arcs computed from the lines of Fig.1(c),and Table 1gives the details of these arcs.An arc spans a part between (StartAngle,EndAngle)of the great circle speciﬁed by (Center X,Center Y,Radius).The arc is assumed to move counter-clockwise from StartAngle to EndAngle over the great circle.As an example,A 2covers a total of 611from 911to 1521of the great circle with center coordinates (210.6,211.3)and radius ¼182.6.2.4.Candidate circle detection by arc joinAfter the computation of the arcs,the next step is to join the arcs into circle candidates.To do this,we ﬁrst sort all arcs with respect to their length in descending order,and start extending the longest arc ﬁrst.The motivation for this decision comes from the observation that the longest arc is the closest to a full circle,so it must be extended and completed into a full circle before theFig.3.(a)An illustration of the blue right vertical segment in Fig.1(b)being approximated by 11consecutive line segments labeled v 1through v 11.The angle between line segments v 1and v 2(y 1),v 3and v 4ðy 3Þ,v 7and v 8ðy 7Þ,and v 10and v 11(y 10)are also illustrated.(b)Lines making up the blue right vertical segment in Fig.1(b).Fig.2.(a)A hypothetical edge segment being approximated by 18consecutive line segments labeled l 1through l 18.(b)The angle y i between v i and v i þ1are illustrated and colored with red,green or blue.If the angle is bigger than a high threshold,e.g.,y 1,y 2and y 9(colored red),or if the angle is smaller than a low threshold,e.g.,y 16(also colored red),then these lines cannot be part of an arc.Otherwise,if three or more consecutive lines turn to the left,e.g.,lines v 3through v 7(angles colored blue),then these lines may form an arc.Similarly,if three or more consecutive lines turn to the right,e.g.,lines v 10through v 16(angles colored green),then these lines may form an arc.(For interpretation of the references to color in this ﬁgure caption,the reader is referred to the web version of this article.)C.Akinlar,C.Topal /Pattern Recognition 46(2013)725–740728other arcswould.During the extension of an arc,the idea is to look for arcs having similar radii and close centers,and collect a list of candidate arcs that may be combined with the current arc.Given an arc A 1to extend into a full circle,we go over all detected arcs and generate a set of candidate arcs that may be joined with A 1.We have two criterions for arc join:(1)Radius difference constraint:The radius difference between A 1and the candidate arc A 2must be within some threshold.Speciﬁcally,if A 2’s radius is within 25%of A 1’s radius,then A 2is taken as a candidate for join;otherwise A 2cannot be joined with A 1.As an example,if A 1’s radius is 100,then all arcs whose radii are between 75and 125would be taken as candidates for arc join.(2)Center distance constraint:The distance between the center of A 1and the center of the candidate arc A 2must be within some threshold.Speciﬁcally,we require that the distance between the centers of A 1and A 2must not exceed 25%of A1’s radius.As an example,if A 1’s radius is 100,then all arcs whose centers are within 25pixels of A 1’s center would be taken as candidates for arc join assuming they also satisfy the radius difference constraint.Fig.5illustrates possible scenarios during arc join for circle detection.In Fig.5(a),we illustrate a case where all potential arc candidates satisfy the center distance constraint,but one fails the radius difference constraint.Here,A 1is the arc to be extended with A 2,A 3and A 4as potential candidates for arc join.As illustrated,the centers of all arcs are very close to each other;that is,the distance of the centers of A 2,A 3and A 4from the center of A 1are all within the center distance threshold r T .As for the radius difference constraint,only A 3and A 4satisfy it,while A 2’s radius falls out of the radius difference range.So in Fig.5(a),only arcs A 3and A 4would be selected as candidates for joining with A 1.In Fig.5(b),we illustrate a case where all potential arc candidates satisfy the radius difference constraint,but one fails the center distance constraint.Here,A 1is the arc to be extended with A 2,A 3and A 4as potential candidates for arc join.As illustrated,the radii ofall arcs are very close to each other,so they all satisfy the radius difference constraint.As for the center distance constraint,only A 2and A 4satisfy it,while A 3’s center falls out of the center distance threshold r T .So in Fig.5(b),only arcs A 2and A 4would be selected as candidates for joining with A 1.After the computation of the candidate arcs,the next step is to combine them one-by-one with the extended arc A 1by ﬁtting a new circle to the pixels making up both of the arcs.Instead of trying the join in random order,we start with the arc whose either end-point is the closest to either end-point of A 1.The motivation for this decision comes from the observation that if there is more than one arc that is part of the same great circle,it is better to start the join with the arc closest to the extended arc A 1.In Fig.5(a)for example,we would ﬁrst join A 1with A 4and then with A 3.Similarly,in Fig.5(b)we would ﬁrst join A 1with A 2and then A 4.After an arc A 1is extended with other arcs on the same great circle,we decide at the last step whether to make the extended arc a circle candidate.Here,we take the view that if an arc spans at least 50%of the circumference of its great circle,then we make the arc a circle candidate.Otherwise,the arc is left for circular ellipse detection.In Fig.5(a)for example,when A 1joined with A 4and A 3,the extended arc would span more 50%of the circumference of its great circle.So the extended arc would be made a circle candidate.In Fig.5(c)however,when A 1,A 2and A 3are joined together,we observe that the extended arc does not span at least 50%of the circumference of its great circle,i.e.,y 1þy 2þy 3o p ;so the extended arc is not taken as a circle putation of the total arc span is performed by simply looking at the ratio of the total number of pixels making up the joined arcs to the circumference of the newly ﬁtted circle.If this ratio is greater than 50%,then the extended arc is taken as a circle candidate.To exemplify the ideas presented above,here is how the seven arcs depicted in Fig.4(a)and detailed in Table 1would be processed:we ﬁrst take A 1,the longest arc,as the arc to be extended,with A 2,A 3,A 4,A 5,A 6and A 7as the remaining arcs.Since the radii of A 2,A 3and A 4are within 25%of A 1’s radius and their center distances are within the center distance threshold,only these three arcs would be taken as candidates for join.We next join A 1and A 2since A 2’s end-point is closest to A 1(refer to Fig.4(a)).After A 1and A 2are joined,the extended arc would now be joined with A 3since A 3’s end-point would now be closest to the extended arc.Finally,A 4would be joined.Since the ﬁnal extended arc covers more than 50%of its great circle,it is taken as a circle candidate.Continuing similarly,the next longest remaining arc is A 5,so we try extending A 5with A 6and A 7being the only remaining arcs in our list of arcs.The only candidateFig.4.(a)Arcs computed from the lines of Fig.1(c).(b)Candidate circles and ellipses before validation (overlayed on top of the image with red color).(For interpretation of the references to color in this ﬁgure caption,the reader is referred to the web version of this article.)Table 1Details of the arcs shown in Fig.4(a).An arc spans a part between (StartAngle,EndAngle)of a great circle speciﬁed by (center X,center Y,Radius).The arc moves counter-clockwise from StartAngle to EndAngle over the circle.ArcCenter XCenter YRadiusStart angle (deg.)End angle (deg.)A 1210.3211.8182.232586A 2210.6211.3182.691152A 3212.2215.9178.5275312A 4210.7211.6183.0173264A 5111.1267.552.3275312A 6120.1291.434.9141219A 7139.4288.649.294143C.Akinlar,C.Topal /Pattern Recognition 46(2013)725–740729。

R语言时间序列中文教程共34页

R 语言时间序列中文教程2019特别声明：R 语言是免费语言，其代码不带任何质量保证，使用 R 语言所产生的后果由使用者负全责。

前言R 语言是一种数据分析语言，它是科学的免费的数据分析语言，是凝聚了众多研究人员心血的成熟的使用范围广泛全面的语言，也是学习者能较快受益的语言。

在 R 语言出现之前，数据分析的编程语言是 SAS。

当时 SAS 的功能比较有限。

在贝尔实验室里，有一群科学家讨论提到，他们研究过程中需要用到数据分析软件。

SAS 的局限也限制了他们的研究。

于是他们想，我们贝尔实验室的研究历史要比 SAS 长好几倍，技术力量也比 SAS 强好几倍，且贝尔实验室里并不缺乏训练有素的专业编程人员，那么，我们贝尔实验室为什么不自己编写数据分析语言，来满足我们应用中所需要的特殊要求呢？于是，贝尔实验室研究出了 S-PLUS 语言。

后来，新西兰奥克兰大学的两位教授非常青睐 S-PLUS 的广泛性能。

他们决定重新编写与 S-PLUS 相似的语言，并且使之免费，提供给全世界所有相关研究人员使用。

于是，在这两位教授努力下，一种叫做 R 的语言在奥克兰大学诞生了。

R 基本上是 S-PLUS 的翻版，但 R 是免费的语言，所有编程研究人员都可以对 R 语言做出贡献，且他们已经将大量研究成果写成了 R 命令或脚本，因而 R 语言的功能比较强大，比较全面。

研究人员可免费使用 R 语言，可通过阅读 R 语言脚本源代码，学习其他人的研究成果。

笔者曾有幸在奥克兰大学受过几年熏陶，曾经向一位统计系的老师提请教过一个数据模拟方面的问题。

那位老师只用一行 R 语句就解答了。

R 语言的强大功能非常令人惊讶。

为了进一步推广 R 语言，为了方便更多研究人员学习使用 R 语言，我们收集了 R 语言时间序列分析实例，以供大家了解和学习使用。

当然，这是非常简单的模仿练习，具体操作是，用复制粘贴把本材料中 R 代码放入 R 的编程环境；材料中蓝色背景的内容是相关代码和相应输出结果。

Image Super-Resolution via Sparse Representation

1Image Super-Resolution via Sparse Representation Jianchao Yang,Student Member,IEEE,John Wright,Student Member,IEEE Thomas Huang,Life Fellow,IEEEand Yi Ma,Senior Member,IEEEAbstract—This paper presents a new approach to single-image superresolution,based on sparse signal representation.Research on image statistics suggests that image patches can be well-represented as a sparse linear combination of elements from an appropriately chosen over-complete dictionary.Inspired by this observation,we seek a sparse representation for each patch of the low-resolution input,and then use the coefﬁcients of this representation to generate the high-resolution output.Theoretical results from compressed sensing suggest that under mild condi-tions,the sparse representation can be correctly recovered from the downsampled signals.By jointly training two dictionaries for the low-and high-resolution image patches,we can enforce the similarity of sparse representations between the low resolution and high resolution image patch pair with respect to their own dictionaries.Therefore,the sparse representation of a low resolution image patch can be applied with the high resolution image patch dictionary to generate a high resolution image patch. The learned dictionary pair is a more compact representation of the patch pairs,compared to previous approaches,which simply sample a large amount of image patch pairs[1],reducing the computational cost substantially.The effectiveness of such a sparsity prior is demonstrated for both general image super-resolution and the special case of face hallucination.In both cases,our algorithm generates high-resolution images that are competitive or even superior in quality to images produced by other similar SR methods.In addition,the local sparse modeling of our approach is naturally robust to noise,and therefore the proposed algorithm can handle super-resolution with noisy inputs in a more uniﬁed framework.I.I NTRODUCTIONSuper-resolution(SR)image reconstruction is currently a very active area of research,as it offers the promise of overcoming some of the inherent resolution limitations of low-cost imaging sensors(e.g.cell phone or surveillance cameras)allowing better utilization of the growing capability of high-resolution displays(e.g.high-deﬁnition LCDs).Such resolution-enhancing technology may also prove to be essen-tial in medical imaging and satellite imaging where diagnosis or analysis from low-quality images can be extremely difﬁcult. Conventional approaches to generating a super-resolution im-age normally require as input multiple low-resolution images of the same scene,which are aligned with sub-pixel accuracy. The SR task is cast as the inverse problem of recovering the original high-resolution image by fusing the low-resolution images,based on reasonable assumptions or prior knowledge about the observation model that maps the high-resolution im-age to the low-resolution ones.The fundamental reconstruction Jianchao Yang and Thomas Huang are with Beckman Institute,Uni-versity of Illinois Urbana-Champaign,Urbana,IL61801USA(email: jyang29@;huang@).John Wright and Yi Ma are with CSL,University of Illinois Urbana-Champaign,Urbana,IL61801USA(email:jnwright@; yima@).constraint for SR is that the recovered image,after applying the same generation model,should reproduce the observed low-resolution images.However,SR image reconstruction is gen-erally a severely ill-posed problem because of the insufﬁcient number of low resolution images,ill-conditioned registration and unknown blurring operators,and the solution from the reconstruction constraint is not unique.Various regularization methods have been proposed to further stabilize the inversion of this ill-posed problem,such as[2],[3],[4].However,the performance of these reconstruction-based super-resolution algorithms degrades rapidly when the desired magniﬁcation factor is large or the number of available input images is small.In these cases,the result may be overly smooth,lacking important high-frequency details[5].Another class of SR approach is based on interpolation[6],[7], [8].While simple interpolation methods such as Bilinear or Bicubic interpolation tend to generate overly smooth images with ringing and jagged artifacts,interpolation by exploiting the natural image priors will generally produce more favorable results.Dai et al.[7]represented the local image patches using the background/foreground descriptors and reconstructed the sharp discontinuity between the two.Sun et.al.[8]explored the gradient proﬁle prior for local image structures and ap-plied it to super-resolution.Such approaches are effective in preserving the edges in the zoomed image.However,they are limited in modeling the visual complexity of the real images. For natural images withﬁne textures or smooth shading,these approaches tend to produce watercolor-like artifacts.A third category of SR approach is based on ma-chine learning techniques,which attempt to capture the co-occurrence prior between low-resolution and high-resolution image patches.[9]proposed an example-based learning strat-egy that applies to generic images where the low-resolution to high-resolution prediction is learned via a Markov Random Field(MRF)solved by belief propagation.[10]extends this approach by using the Primal Sketch priors to enhance blurred edges,ridges and corners.Nevertheless,the above methods typically require enormous databases of millions of high-resolution and low-resolution patch pairs,and are therefore computationally intensive.[11]adopts the philosophy of Lo-cally Linear Embedding(LLE)[12]from manifold learning, assuming similarity between the two manifolds in the high-resolution and the low-resolution patch spaces.Their algorithm maps the local geometry of the low-resolution patch space to the high-resolution one,generating high-resolution patch as a linear combination of ing this strategy,more patch patterns can be represented using a smaller training database.However,using aﬁxed number K neighbors for reconstruction often results in blurring effects,due to over-or under-ﬁtting.In our previous work[1],we proposed a method2for adaptively choosing the most relevant reconstruction neigh-bors based on sparse coding,avoiding over-or under-ﬁtting of [11]and producing superior results.However,sparse codingover a large sampled image patch database directly is too time-consuming.While the mentioned approaches above were proposed forgeneric image super-resolution,speciﬁc image priors can be incorporated when tailored to SR applications for speciﬁcdomains such as human faces.This face hallucination prob-lem was addressed in the pioneering work of Baker and Kanade[13].However,the gradient pyramid-based predictionintroduced in[13]does not directly model the face prior,and the pixels are predicted individually,causing discontinuities and artifacts.Liu et al.[14]proposed a two-step statisticalapproach integrating the global PCA model and a local patch model.Although the algorithm yields good results,the holistic PCA model tends to yield results like the mean face and theprobabilistic local patch model is complicated and compu-tationally demanding.Wei Liu et al.[15]proposed a newapproach based on TensorPatches and residue compensation. While this algorithm adds more details to the face,it also introduces more artifacts.This paper focuses on the problem of recovering the super-resolution version of a given low-resolution image.Similar to the aforementioned learning-based methods,we will relyon patches from the input image.However,instead of work-ing directly with the image patch pairs sampled from high-and low-resolution images[1],we learn a compact repre-sentation for these patch pairs to capture the co-occurrence prior,signiﬁcantly improving the speed of the algorithm.Our approach is motivated by recent results in sparse signal representation,which suggest that the linear relationships among high-resolution signals can be accurately recoveredfrom their low-dimensional projections[16],[17].Although the super-resolution problem is very ill-posed,making preciserecovery impossible,the image patch sparse representation demonstrates both effectiveness and robustness in regularizing the inverse problem.a)Basic Ideas:To be more precise,let D∈R n×K be an overcomplete dictionary of K atoms(K>n),and suppose a signal x∈R n can be represented as a sparse linearcombination with respect to D.That is,the signal x can be written as x=Dα0where whereα0∈R K is a vector with very few( n)nonzero entries.In practice,we might onlyobserve a small set of measurements y of x:y.=L x=L Dα0,(1) where L∈R k×n with k<n is a projection matrix.In our super-resolution context,x is a high-resolution image(patch), while y is its low-resolution counter part(or features extracted from it).If the dictionary D is overcomplete,the equation x=Dαis underdetermined for the unknown coefﬁcientsα. The equation y=L Dαis even more dramatically under-determined.Nevertheless,under mild conditions,the sparsest solutionα0to this equation will be unique.Furthermore,if D satisﬁes an appropriate near-isometry condition,then for a wide variety of matrices L,any sufﬁciently sparse linear representation of a high-resolution image patch x interms Fig.1.Reconstruction of a raccoon face with magniﬁcation factor2.Left: result by our method.Right:the original image.There is little noticeable difference visually even for such a complicated texture.The RMSE for the reconstructed image is5.92(only the local patch model is employed).of the D can be recovered(almost)perfectly from the low-resolution image patch[17],[18].1Fig.1shows an example that demonstrates the capabilities of our method derived fromthis principle.The image of the raccoon face is blurred anddownsampled to half of its original size in both dimensions. Then we zoom the low-resolution image to the original sizeusing the proposed method.Even for such a complicated texture,sparse representation recovers a visually appealingreconstruction of the original signal.Recently sparse representation has been successfully appliedto many other related inverse problems in image processing, such as denoising[19]and restoration[20],often improving onthe state-of-the-art.For example in[19],the authors use theK-SVD algorithm[21]to learn an overcomplete dictionary from natural image patches and successfully apply it to theimage denoising problem.In our setting,we do not directlycompute the sparse representation of the high-resolution patch. Instead,we will work with two coupled dictionaries,D h forhigh-resolution patches,and D l for low-resolution ones.The sparse representation of a low-resolution patch in terms of D l will be directly used to recover the corresponding high-resolution patch from D h.We obtain a locally consistent solution by allowing patches to overlap and demanding that thereconstructed high-resolution patches agree on the overlappedareas.In this paper,we try to learn the two overcomplete dictionaries in a probabilistic model similar to[22].To enforce that the image patch pairs have the same sparse representations with respect to D h and D l,we learn the two dictionaries simultaneously by concatenating them with proper normal-ization.The learned compact dictionaries will be applied to both generic image super-resolution and face hallucination to demonstrate their effectiveness.Compared with the aforementioned learning-based methods,our algorithm requires only two compact learned dictionaries, instead of a large training patch database.The computation, mainly based on linear programming or convex optimization, is much more efﬁcient and scalable,compared with[9],[10], [11].The online recovery of the sparse representation uses the low-resolution dictionary only–the high-resolution dictionary 1Even though the structured projection matrix deﬁned by blurring and downsampling in our SR context does not guarantee exact recovery ofα0, empirical experiments indeed demonstrate the effectiveness of such a sparse prior for our SR tasks.3is used to calculate theﬁnal high-resolution image.The computed sparse representation adaptively selects the most relevant patch bases in the dictionary to best represent each patch of the given low-resolution image.This leads to superior performance,both qualitatively and quantitatively,compared to the method described in[11],which uses aﬁxed number of nearest neighbors,generating sharper edges and clearer textures.In addition,the sparse representation is robust to noise as suggested in[19],and thus our algorithm is more robust to noise in the test image,while most other methods cannot perform denoising and super-resolution simultaneously.b)Organization of the Paper:The remainder of this paper is organized as follows.Section II details our formula-tion and solution to the image super-resolution problem based on sparse representation.Speciﬁcally,we study how to apply sparse representation for both generic image super-resolution and face hallucination.In Section III,we discuss how to learn the two dictionaries for the high-and low-resolution image patches respectively.Various experimental results in Section IV demonstrate the efﬁcacy of sparsity as a prior for regularizing image super-resolution.c)Notations:X and Y denote the high-and low-resolution images respectively,and x and y denote the high-and low-resolution image patches respectively.We use bold uppercase D to denote the dictionary for sparse coding, speciﬁcally we use D h and D l to denote the dictionaries for high-and low-resolution image patches respectively.Bold lowercase letters denote vectors.Plain uppercase letters denote regular matrices,i.e.,S is used as a downsampling operation in matrix form.Plain lowercase letters are used as scalars.II.I MAGE S UPER-R ESOLUTION FROM S PARSITYThe single-image super-resolution problem asks:given a low-resolution image Y,recover a higher-resolution image X of the same scene.Two constraints are modeled in this work to solve this ill-posed problem:1)reconstruction constraint, which requires that the recovered X should be consistent with the input Y with respect to the image observation model; and2)sparsity prior,which assumes that the high resolution patches can be sparsely represented in an appropriately chosen overcomplete dictionary,and that their sparse representations can be recovered from the low resolution observation.1)Reconstruction constraint:The observed low-resolution image Y is a blurred and downsampled version of the high resolution image X:Y=SH X(2) Here,H represents a blurringﬁlter,and S the downsampling operator.Super-resolution remains extremely ill-posed,since for a given low-resolution input Y,inﬁnitely many high-resolution images X satisfy the above reconstruction constraint.We further regularize the problem via the following prior on small patches x of X:2)Sparsity prior:The patches x of the high-resolution image X can be represented as a sparse linear combination in a dictionary D h trained from high-resolution patches sampled from training images:x≈D hαfor someα∈R K with α 0 K.(3) The sparse representationαwill be recovered by representing patches y of the input image Y,with respect to a low resolution dictionary D l co-trained with D h.The dictionary training process will be discussed in Section III.We apply our approach to both generic images and face images.For generic image super-resolution,we divide the problem into two steps.First,as suggested by the sparsity prior(3),weﬁnd the sparse representation for each local patch,respecting spatial compatibility between neighbors. Next,using the result from this local sparse representation, we further regularize and reﬁne the entire image using the reconstruction constraint(2).In this strategy,a local model from the sparsity prior is used to recover lost high-frequency for local details.The global model from the reconstruction constraint is then applied to remove possible artifacts from theﬁrst step and make the image more consistent and natural. The face images differ from the generic images in that the face images have more regular structure and thus reconstruction constraints in the face subspace can be more effective.For face image super-resolution,we reverse the above two steps to make better use of the global face structure as a regularizer. Weﬁrstﬁnd a suitable subspace for human faces,and apply the reconstruction constraints to recover a medium resolution image.We then recover the local details using the sparsity prior for image patches.The remainder of this section is organized as follows:in Section II-A,we discuss super-resolution for generic images. We will introduce the local model based on sparse represen-tation and global model based on reconstruction constraints. In Section II-B we discuss how to introduce the global face structure into this framework to achieve more accurate and visually appealing super-resolution for face images.A.Generic Image Super-Resolution from Sparsity1)Local model from sparse representation:Similar to the patch-based methods mentioned previously,our algorithm tries to infer the high-resolution image patch for each low-resolution image patch from the input.For this local model, we have two dictionaries D h and D l,which are trained to have the same sparse representations for each high-resolution and low-resolution image patch pair.We subtract the mean pixel value for each patch,so that the dictionary represents image textures rather than absolute intensities.In the recovery process,the mean value for each high-resolution image patch is then predicted by its low-resolution version.For each input low-resolution patch y,weﬁnd a sparse representation with respect to D l.The corresponding high-resolution patch bases D h will be combined according to these coefﬁcients to generate the output high-resolution patch x. The problem ofﬁnding the sparsest representation of y can be formulated as:min α 0s.t. F D lα−F y 22≤ ,(4)4 where F is a(linear)feature extraction operator.The main roleof F in(4)is to provide a perceptually meaningful constraint2on how closely the coefﬁcientsαmust approximate y.We willdiscuss the choice of F in Section III.Although the optimization problem(4)is NP-hard in gen-eral,recent results[23],[24]suggest that as long as the desiredcoefﬁcientsαare sufﬁciently sparse,they can be efﬁcientlyrecovered by instead minimizing the 1-norm3,as follows:min α 1s.t. F D lα−F y 22≤ .(5)Lagrange multipliers offer an equivalent formulationminαF D lα−F y 22+λ α 1,(6)where the parameterλbalances sparsity of the solutionandﬁdelity of the approximation to y.Notice that this isessentially a linear regression regularized with 1-norm on thecoefﬁcients,known in statistical literature as the Lasso[27].Solving(6)individually for each local patch does notguarantee the compatibility between adjacent patches.Weenforce compatibility between adjacent patches using a one-pass algorithm similar to that of[28].4The patches areprocessed in raster-scan order in the image,from left to rightand top to bottom.We modify(5)so that the super-resolutionreconstruction D hαof patch y is constrained to closelyagree with the previously computed adjacent high-resolutionpatches.The resulting optimization problem ismin α 1s.t. F D lα−F y 22≤ 1,P D hα−w 22≤ 2,(7)where the matrix P extracts the region of overlap betweenthe current target patch and previously reconstructed high-resolution image,and w contains the values of the previouslyreconstructed high-resolution image on the overlap.The con-strained optimization(7)can be similarly reformulated as:minα˜Dα−˜y 22+λ α 1,(8)where˜D=F D lβP D hand˜y=F yβw.The parameterβcontrols the tradeoff between matching the low-resolution input andﬁnding a high-resolution patch that is compatible with its neighbors.In all our experiments,we simply set β=1.Given the optimal solutionα∗to(8),the high-resolution patch can be reconstructed as x=D hα∗.2Traditionally,one would seek the sparsestαs.t. D lα−y 2≤ .For super-resolution,it is more appropriate to replace this2-norm with a quadratic norm · F T F that penalizes visually salient high-frequency errors.3There are also some recent works showing certain non-convex optimization problems can produce superior sparse solutions to the 1convex problem,e.g., [25]and[26].4There are different ways to enforce compatibility.In[11],the values in the overlapped regions are simply averaged,which will result in blurring effects. The greedy one-pass algorithm[28]is shown to work almost as well as the use of a full MRF model[9].Our algorithm,not based on the MRF model, is essentially the same by trusting partially the previously recovered high resolution image patches in the overlapped regions.Algorithm1(Super-Resolution via Sparse Representation). 1:Input:training dictionaries D h and D l,a low-resolutionimage Y.2:For each3×3patch y of Y,taken starting from the upper-left corner with1pixel overlap in each direction,•Compute the mean pixel value m of patch y.•Solve the optimization problem with˜D and˜y deﬁnedin(8):minα ˜Dα−˜y 22+λ α 1.•Generate the high-resolution patch x=D hα∗.Putthe patch x+m into a high-resolution image X0. 3:End4:Using gradient descent,ﬁnd the closest image to X0 which satisﬁes the reconstruction constraint:X∗=arg minXSH X−Y 22+c X−X0 22.5:Output:super-resolution image X∗.2)Enforcing global reconstruction constraint:Notice that (5)and(7)do not demand exact equality between the low-resolution patch y and its reconstruction D lα.Because of this,and also because of noise,the high-resolution image X0produced by the sparse representation approach of the previous section may not satisfy the reconstruction constraint (2)exactly.We eliminate this discrepancy by projecting X0 onto the solution space of SH X=Y,computingX∗=arg minXSH X−Y 22+c X−X0 22.(9) The solution to this optimization problem can be efﬁciently computed using gradient descent.The update equation for this iterative method isX t+1=X t+ν[H T S T(Y−SH X t)+c(X−X0)],(10) where X t is the estimate of the high-resolution image after the t-th iteration,νis the step size of the gradient descent. We take result X∗from the above optimization as our ﬁnal estimate of the high-resolution image.This image is as close as possible to the initial super-resolution X0given by sparsity,while respecting the reconstruction constraint.The entire super-resolution process is summarized as Algorithm1.3)Global optimization interpretation:The simple SR algo-rithm outlined in the previous two subsections can be viewed as a special case of a more general sparse representation framework for inverse problems in image processing.Related ideas have been proﬁtably applied in image compression, denoising[19],and restoration[20].In addition to placing our work in a larger context,these connections suggest means of further improving the performance,at the cost of increased computational complexity.Given sufﬁcient computational resources,one could in prin-ciple solve for the coefﬁcients associated with all patches simultaneously.Moreover,the entire high-resolution image X itself can be treated as a variable.Rather than demanding that X be perfectly reproduced by the sparse coefﬁcientsα,we can penalize the difference between X and the high-resolution image given by these coefﬁcients,allowing solutions that5are not perfectly sparse,but better satisfy the reconstruction constraints.This leads to a large optimization problem:X ∗=arg min X ,{αij }SH X −Y 22+λi,jαij 0+γi,jD h αij −P ij X 22+τρ(X ) .(11)Here,αij denotes the representation coefﬁcients for the (i,j )th patch of X ,and P ij is a projection matrix that selects the (i,j )th patch from X .ρ(X )is a penalty function that encodes additional prior knowledge about the high-resolution image.This function may depend on the image category,or may take the form of a generic regularization term (e.g.,Huber MRF,Total Variation,Bilateral Total Variation).Algorithm 1can be interpreted as a computationally efﬁcient approximation to (11).The sparse representation step recovers the coefﬁcients αby approximately minimizing the sum of the second and third terms of (11).The sparsity term αij 0is relaxed to αij 1,while the high-resolution ﬁdelity term D h αij −P ij X 2is approximated by its low-resolution version F D l αij −F y ij 2.Notice,that if the sparse coefﬁcients αare ﬁxed,the third term of (11)essentially penalizes the difference between the super-resolution image X and the reconstruction given by the coefﬁcients: i,j D h αij −P ij X 22≈ X 0−X 22.Hence,for small γ,the back-projection step of Algorithm 1approximately minimizes the sum of the ﬁrst and third terms of (11).Algorithm 1does not,however,incorporate any prior be-sides sparsity of the representation coefﬁcients –the term ρ(X )is absent in our approximation.In Section IV we will see that sparsity in a relevant dictionary is a strong enough prior that we can already achieve good super-resolution per-formance.Nevertheless,in settings where further assumptions on the high-resolution signal are available,these priors can be incorperated into the global reconstruction step of our algorithm.B.Face super-resolution from SparsityFace image resolution enhancement is usually desirable in many surveillance scenarios,where there is always a large distance between the camera and the objects (people)of interest.Unlike the generic image super-resolution discussed earlier,face images are more regular in structure and thus should be easier to handle.Indeed,for face super-resolution,we can deal with lower resolution input images.The basic idea is ﬁrst to use the face prior to zoom the input to a reasonable medium resolution,and then to employ the local sparsity prior model to recover details.To be precise,the solution is also approached in two steps:1)global model:use reconstruction constraint to recover a medium high-resolution face image,but the solution is searched only in the face subspace;and 2)local model:use the local sparse model to recover the image details.a)Non-negative matrix factorization:In face super-resolution,the most frequently used subspace method for mod-eling the human face is Principal Component Analysis (PCA),which chooses a low-dimensional subspace that captures as much of the variance as possible.However,the PCA bases are holistic,and tend to generate smooth faces similar to the mean.Moreover,because principal component representations allow negative coefﬁcients,the PCA reconstruction is often hard to interpret.Even though faces are objects with lots of variance,they are made up of several relatively independent parts such as eyes,eyebrows,noses,mouths,checks and chins.Nonnegative Matrix Factorization (NMF)[29]seeks a representation of the given signals as an additive combination of local features.To ﬁnd such a part-based subspace,NMF is formulated as the following optimization problem:arg min U,VX −UV 22s.t.U ≥0,V ≥0,(12)where X ∈R n ×m is the data matrix,U ∈R n ×r is the basismatrix and V ∈R r ×m is the coefﬁcient matrix.In our context here,X simply consists of a set of pre-aligned high-resolution training face images as its column vectors.The number of the bases r can be chosen as n ∗m/(n +m )which is smaller than n and m ,meaning a more compact representation.It can be shown that a locally optimum of (12)can be obtained via the following update rules:V ij ←−V ij (U T X )ij(U TUV )ij U ki ←−U ki (XV T )ki(UV V T )ki,(13)where 1≤i ≤r ,1≤j ≤m and 1≤k ≤n .The obtained basis matrix U is often sparse and localized.b)Two step face super-resolution:Let X and Y denote the high resolution and low resolution faces respectively.Y is obtained from X by smoothing and downsampling as in Eq.2.We want to recover X from the observation Y .In this paper,we assume Y has been pre-aligned to the training database by either manually labeling the feature points or with some automatic face alignment algorithm such as the method used in [14].We can achieve the optimal solution for X based on the Maximum a Posteriori (MAP)criteria,X ∗=arg max Xp (Y |X )p (X ).(14)p (Y |X )models the image observation process,usually with Gaussian noise assumption on the observation Y ,p (Y |X )=1/Z exp(− SHU c −Y 22/(2∗σ2))with Z being a nor-malization factor.p (X )is a prior on the underlying high resolution image X ,typically in the exponential form p (X )=exp(−cρ(X )).Using the rules in (13),we can obtain the basis matrix U ,which is composed of sparse bases.Let Ωdenote the face subspace spanned by U .Then in the subspace Ω,the super-resolution problem in (14)can be formulated using the reconstruction constraints as:c ∗=arg min cSHU c −Y 22+ηρ(U c )s.t.c ≥0,(15)where ρ(U c )is a prior term regularizing the high resolution solution,c ∈R r ×1is the coefﬁcient vector in the subspace Ω。

Statistics of natural image categories

I NSTITUTE OF P HYSICS P UBLISHING N ETWORK:C OMPUTA TION IN N EURAL S YSTEMS Network:Comput.Neural Syst.14(2003)391–412PII:S0954-898X(03)53778-2Statistics of natural image categoriesAntonio Torralba1and Aude Oliva21Artiﬁcial Intelligence Laboratory,MIT,Cambridge,MA02139,USA2Department of Psychology and Cognitive Science Program,Michigan State University,East Lansing,MI48824,USAE-mail:torralba@ and aoliva@Received16September2002,inﬁnal form30January2003Published12May2003Online at /Network/14/391AbstractIn this paper we study the statistical properties of natural images belonging todifferent categories and their relevance for scene and object categorization tasks.We discuss how second-order statistics are correlated with image categories,scene scale and objects.We propose how scene categorization could becomputed in a feedforward manner in order to provide top-down and contextualinformation very early in the visual processing chain.Results show howvisual categorization based directly on low-level features,without grouping orsegmentation stages,can beneﬁt object localization and identiﬁcation.We showhow simple image statistics can be used to predict the presence and absence ofobjects in the scene before exploring the image.(Someﬁgures in this article are in colour only in the electronic version)1.IntroductionFigure1shows a collection of mean images created by averaging pictures from the same semantic category.According to the seminal work of Rosch and collaborators(1976),people recognize most of the objects at the same level of abstraction:the basic level(e.g.car,chair). It has been shown that objects of the same basic-level category share similar common features and usually they have a similar shape.This is illustrated in the averaged images or‘prototypes’shown inﬁgure1.In each prototype image,information about the level of spatial similarity existing between local features(e.g.distribution of coloured regions and intensity pattern)is demonstrated by the degree of sharpness.Object categories like faces and pedestrians are much more regular in terms of distribution of pixel intensities than other object groups(e.g.chairs).Similarly to objects,natural images depicting environmental scenes can also be classiﬁed in basic-level categories(Tversky and Hemenway1983),and as such,are expected to share common features(Jepson et al1996).Despite the fact that the organization of parts and regions in environmental scenes is much less constrained than in objects,the resulting prototype images are not spatially stationary.A strong relationship exists between the category of a scene picture 0954-898X/03/030391+22$30.00©2003IOP Publishing Ltd Printed in the UK391392ATorralbaandAOlivaScenesObjectsObjects in scenesMountain Beach Forest Street IndoorHighway Animalin natural scene Car inurban scene Far pedestrian in urban scene Close-up person in urban scene T reein urban scene Lamp inindoor scene Face PedestrianCarCowHand ChairFigure 1.Averaged pictures of categories of objects,scenes and objects in scenes,computed with 100exemplars or more per category.Exemplars were chosen tohave the same basic level and viewpoint in regard to an observer.The group objects in scenes (third row)represent examples of the averaged peripheral information around an object centred in the image.Statistics of natural image categories393 and the distribution of coloured regions in the image.In a similar vein,the distribution of structural and colour patterns of the background scene around an object is constrained.The third row of averaged prototypes shown inﬁgure1(objects in scenes)have been created by averaging hundreds of images constrained to have a particular object at one scale present in the image(before averaging,the images were translated in order to have the object of interest in the centre).Because of the correlation existing between an object and its context the background does not average to a uniformﬁeld(Torralba2003).On the contrary,the background exhibits the texture and colour pattern that is common to all environments where a speciﬁc object is usually found.Figure1illustrates the degree of regularities found in natural image categories when ecological conditions of viewing are considered.The statistics of natural image categories depend not only on how the visual world is built to serve a speciﬁc function,but also the viewpoint that the observer adopts.Because different environmental categories are built with different objects and materials,and the point of view of the observer imposes its own constraints (such as its size and motion),we expect toﬁnd strong biases in the statistics distribution of the image information.Those statistics might have different biases for different animal species.In this paper,we illustrate how simple statistics of natural images vary as a function of the interaction between the observer and the world.The paper is organized as follows:section2 describes the statistical properties of natural images per scene category.Sections3and4 introduce,respectively,the spectral principal components of natural images and scene-tuned ﬁlters,and describe how these methods could be used to perform simple scene categorization tasks.Section5summarizes computational approaches of scene categorization,and section6 shows the robustness of simple statistics in performing object detection tasks.2.Statistical properties of natural categories2.1.1/f spectra of natural imagesStatistics of natural images have been found to follow particular regularities.Seminal studies (Burton and Moorhead1987,Field1987,1994,Tolhurst et al1992)have observed that the average power spectrum of natural images falls with a form1/fαwithα∼2(orα∼1 considering the amplitude spectrum,seeﬁgure2(a)).Related studies found a bias in the distribution of orientations,illustrated in the power spectra ofﬁgure2.In real-world images,including both natural landscapes and man-made environments,vertical and horizontal orientations are more frequent than obliques(Baddeley 1997,Switkes et al1978,van der Schaaf and van Hateren1996,Oliva and Torralba2001).A more complete model of the mean power spectra(using polar coordinates)can be written as E[|I(f,θ)|2] A s(θ)/fαs(θ)(1) in which the shape of the spectra is a function of orientation.The function A s(θ)is an amplitude scaling factor for each orientation andαs(θ)is the frequency exponent as a functionof orientation.Both factors contribute to the shape of the power spectra.The model of equation(1)is needed when considering the power spectra of man-made and natural scene images separately(cfﬁgure2and table1,also Baddeley1996,1997).Table1shows that the values of the slopeαand the amplitude A vary with orientation and also with the type of environment3.The anisotropic distribution of orientations is also compatible with 3The database used in this study contains about12000pictures of scenes and objects.Images were256×256pixels in size.They come from the Corel stock photo library,pictures taken from a digital camera and images downloaded from the web.394A Torralba and AOliva-b--d-Figure2.(a)Mean power spectrum averaged from12000images(vertical axis is in logarithmicunits).Mean power spectra computed with6000pictures of man-made scenes(b)and6000picturesof natural scenes(d);(c)and(e)are their respective spectral signatures.The contour plots represent50and80%of the energy of the spectral signature.The contour is selected so that the sum of thecomponents inside the section represents50%(and80%)of the total.Units are in cycles per pixel(cf also Baddeley1996).Table1.Averageαand A values for images representing man-made and natural environments.The valuesαand A were obtained byﬁtting the model A/fαto the power spectrum of each singleimage at three orientations(horizontal,f x,oblique and vertical,f y).Theﬁt was performed inthe frequency interval[0.02,0.35]cycles/pixel.The amplitude factor A is normalized so thatthe maximum averaged value is unity.Averages were computed with more than3500images percategory(cf alsoﬁgures2(b),(c)).Similar values are obtained when theﬁt is performed on theaveraged power spectrum.The numbers in parenthesis give the standard deviation.H O VNaturalα 1.98(0.58) 2.02(0.53) 2.22(0.55)A0.96(0.40)0.86(0.38)1(0.35)Man-madeα 1.83(0.58) 2.37(0.45) 2.07(0.52)A1(0.32)0.49(0.24)0.88(0.29)neurophysiological data showing that the number of cells in early cortical stages varies in regard to the spatial scale tuning and the orientation(e.g.more vertical and horizontal tuned cells than oblique in the fovea,DeValois and DeValois1988).2.2.Spectral signatures of image categoriesDifferent categories of environments also exhibit different orientations and spatial frequency distributions,captured in the averaged power spectra(Baddeley1997,Oliva et al1999,Oliva and Torralba2001).Figure3shows that the differentiation among various man-made categories resides mainly in the relationship between horizontal and vertical contours at different scales, while the spectral signatures of natural environments have a broader variation in spectral shapes.Statistics of natural image categories395Natural object River andwaterfallForest FieldMountain Beach CoastMan-made object Portrait IndoorsceneHighbuildingStreet City-view HighwayFigure3.Spectral signatures of14different image categories.Each spectral signature is obtained by averaging the power spectra of a few hundred images per category.The contour plots represent 60,80and90%of the energy of the spectral signatures(energy is obtained by adding the square of the Fourier components).The size of the spectral signature is correlated with the slope(α).A large value ofαproduces a fast decay of the energy at high spatial frequencies,which produces a smaller contour.The overall shape is a function of bothα(θ)and A(θ).The particular spectral signature per scene category is even more striking when considering basic-level classes of environmental scenes such as streets,highways,buildings,forests etc. From the contour plots ofﬁgure3,we can see that the dominant spatial scales and dominant orientations are typical of classes of scenes representing different volumes or depth ranges.The spectral signatures of pictures of large-scale scenes(e.g.,beach,coast,ﬁeld)are dominated by the horizon.When the scene background becomes closer to the observer(from mountains to enclosed landscapes and natural objects),the spectral signatures become isotropic and denser in high spatial frequencies.The shape of the spectral signatures is correlated with the scale (e.g.size)at which the main components of the image should be found(e.g.ﬁner texture in forest,coarser texture in waterfalls).2.3.Scene scale and image scaleImage statistics also vary when considering scenes at different scales.Figure4shows the spectral signatures of scenes sharing similar depth range.These signatures have been obtained from a database of images for which four subjects were asked to provide the mean depth or volume of the environment represented in the image(Torralba and Oliva2002).Scene scale is measured in metres.Each spectral signature was computed by averaging the power spectra of scene pictures within a similar distance range.When considering large changes in scale(a factor larger than10),signiﬁcant differences exist between the spatial and the spectral statistics of pictures depicting scenes and objects at different scales.There are at least two factors that can explain the dependence between structure and depth range.First,the point of view that any given observer adopts on a speciﬁc scene is constrained by the volume of the scene.Many real world objects can be observed from an inﬁnite number of viewpoints as long as the observer is directly capable of manipulating the object.However,as distance and scale increase,the possible viewpoints of a human observer become increasingly limited and predictable.For instance,tall buildings are usually observed from the ground,or a window of another building.Second,the parts or objects that compose one scene differ strongly from one scale to another scale,due to functional constraints and to the physical processes that shape the space at each scale.396ATorralbaandAOliva 5-50 m 50-500 m > 500 m > 500 m1-5 m 5-50 m 50-500 m 1-5 m Figure 4.Averaged spatial images and spectral signatures as a function of scene scale.Scene scale refers to the mean distance between the observer and the principalelements that compose the scene.Each image average and spectral signature was calculated with 300–400images.Statistics of natural image categories397 Figure4emphasizes the differences between man-made and natural environments at different scale ranges.Close-up views on man-made objects tend to produce images that are composed ofﬂat and smooth surfaces.Consequently,the energy of the power spectra for close-up views is concentrated mainly in low spatial frequencies.As distance between the observer and the scene background increases,the visualﬁeld comprehends a larger space,that is likely to encompass more objects.The perceived images of man-made scenes appear as a collection of surfaces that break down into smaller pieces(objects,walls,windows etc).Thus, the spectral energy corresponding to high spatial frequencies increases as the scene becomes more cluttered due to the increase,with distance,in the area covered by the visualﬁeld.In contrast,spectral signatures of natural environments behave differently while increasing depth. Figure4shows that when the distance between the observer and the background grows,natural structures become larger and smoother(small grain disappears due to the spatial sampling of the image).Therefore,on average,with an increment of distance,the level of clutter decreases, as does the energy in the high spatial frequencies.In addition,the pattern of orientation varies with the scale.Close-up views on natural structures have a tendency to be isotropic in orientations(and the point of view of the observer is unconstrained).As distance grows, there is an increased bias towards vertical and horizontal orientations,together with the point of view of the observer becoming more constrained.As distance continues to increase,energy is concentrated mainly in vertical spatial frequencies,as very large environmental scenes are organized along horizontal layers.In order to recognize the scene or to navigate such panoramic environments,faced with point of view limitations,an observer might consider looking towards the horizon to visually embrace the whole scene.Several studies have examined the scale invariance properties of natural image statistics (e.g.Field1987,Ruderman1997,among others).These studies have focused on the similarity between the statistics of wavelet outputs at different image scales.Results indicate that some image statistics are scale invariant.Here,we differentiate between image scale,that refers to scales in terms of spatial frequencies,and scene scale,that refers to the mean distance between the observer and the elements that compose the scene.Note that for the range of distances we consider(from1m to several kilometres)the problem of distance cannot be modelled as a scaling factor.With each change on an order of magnitude in distance,images perceived also belong to different scene semantic categories(single objects,rooms,places,large outdoors and panoramic scenes).Therefore,we can expect that the statistics of images might evolve when changing scene scale and provide useful categorical information about the probable depth range of the scene(see sections5and6,and also Torralba and Oliva2002).Figure5shows how the output energy of oriented wavelets may vary when considering both image scale and scene scale.The wavelets used are oriented Gaborﬁlters tuned at a radial frequency of1/4cycles/pixel at12different orientations.Changes in image scale are obtained by subsampling the images from2562to322pixels by factors of two.Changes in scene scale are obtained by averaging the outputs obtained from scene pictures with different depth ranges (from close-up views of objects and textures to panoramic views and natural landscapes).The most important modiﬁcation of the spectral signatures is observed across scene scales.When modifying scene scale,the shape of the polar plots evolves by changing the amount of energy at each orientation.However,across image scales,there is little variation.This observation is more striking for natural environments than man-made scenes.2.4.Non-stationary statisticsAnother important characteristic of natural images is how the image statistics change with spatial location.When considering all the possible directions of the eye-camera,statistics of398A Torralba and AOlivaI m a g e s c a l e (c /i )Scene scale (m)Scene scale (m)Figure 5.Polar plots of responses of multiscale oriented Gabor ﬁlters.The magnitude of eachorientation corresponds to the total output energy averaged across the entire image.The energiesare normalized across image scale by multiplying by a constant so that noise with 1/f amplitudespectrum has the same polar plots at all imagescales.1 m 10 m 100 m 1000 m 10000 mFigure 6.Illustration of the non-stationarity of image statistics in groups of man-made (top)and natural (bottom)environments at different depth scales (from left to right,close-up viewsto panoramic views).The spectral signatures were obtained by averaging the windowed powerspectra at 4×4locations in the images.As scene scale increases,the image statistics becomenon-stationary.natural images are expected to be scale invariant (Field 1994,1999,Ruderman 1994,1997)and stationary (the features are equally distributed in regard to locations,Field (1994),(1999)).This is the case indeed with the statistics of images of close-up views of objects that are,on average,stationary,as there is no preferred point of view for the camera (cf ﬁgure 6,scene scale of 1metre).However,for images of scenes that embrace a large volume,the probable points of view that a human observer will adopt become much more restrained,because of its height and its probable location (on the ﬂoor).If the task of the observer is to recognize the identity of a large space scene,most of the useful information will be given while looking towards the horizon.Therefore,different image statistics will characterize the top and bottom half of the image (e.g.smooth texture of the sky,long vertical contours of skylines at the top,cluttered forms at the bottom).Figure 7shows an example of how image inversion affects theStatistics of natural image categories399Figure7.Top–down effect on depth judgments.The image on the left is generally recognizedas close-up view on bushes and maybe a spider’s web on top.On the right-hand side,the imageis categorized as the inside of a forest,corresponding to a larger distance than the image on theleft.The image on the left corresponds to the image on the right after inversion upside down andleft–right.The upside-down inversion affects the perception of concavities and convexities due tothe assumption of light from above,and,therefore,modiﬁes the perceived relative3D structure ofthe scene.But,moreover,the wrong recognition affects the absolute scale of the perceived space. perception of the absolute depth of a scene.Note that it is not just the relative shape of the scene that is misperceived but also the absolute scale.The image on the left appears as a closer structure than the image on the right for most observers.As image statistics are a function of the observer,images of large-scale scenes taken from a bird’s-eye view should elicit almost stationary distribution of features as the point of view becomes totally unconstrained in regard to the possible orientation of the perceived images. In the case of a human observer,standing,the viewpoint is strongly constrained,producing images with non-stationary statistics,as shown inﬁgure6.The property of non-stationarity is very relevant to sensory and cognitive systems,as it may provide an invariant signature ofa speciﬁc kind of environment.3.Principal components of real-world images and power spectraPrincipal component analysis(PCA)has been extensively used in vision problems for coding and recognition.In the face recognition domain,when faces are correctly aligned and scaled, PCA gives a reduced set of orthogonal functions able to reconstruct faces(Craw and Cameron 1991,Turk and Pentland1991).This operation facilitates the recognition procedure that is performed in a low-dimensional space(Sirovich and Kirby1987,Swets and Weng1996).PCA has also been used for obtaining efﬁcient codes of the visual input,adapted to the statistics of natural stimuli(e.g.Hancock et al1992,Liu and Shouval1994).Image principal components (IPCs)decompose the image asi(x,y)=Pn=1v n IPC n(x,y)(2)where i(x,y)is the intensity distribution of the image along spatial variables x and y.P N2 is the number of total IPCs and N2=2562is the number of pixels of the image.The IPC n(x,y)400A Torralba and AOliva (a)Figure 8.(a)The ﬁrst eight principal components of images (IPCs)and the (b)energy spectraof images (SPCs).The frequency f x =f y =0is located at the centre of each image (SPC).Frequencies are deﬁned in the interval [−1/2,1/2].The amplitude at each frequency is normalizedwith respect to its standard deviation before applying the PCA.are the eigenvectors of the covariance matrix:T =E [(i −m )(i −m )T ],where i are the pixels of the image rearranged in a column vector.E is the expectation operator.m =E [i ]is the mean of the images.v n are the coefﬁcients for describing the image i (x ,y ).Figure 8(a)shows the IPCs computed from 5000pictures of real-world scenes.As discussed by Field (1994),stationarity of natural images is responsible for IPC shape (ﬁgure 8(a))which corresponds to the Fourier basis.Here,we compute the power spectrum of an image by taking the squared magnitude of its discrete Fourier transform (DFT):(k x ,k y )=1N2|I (k x ,k y )|2(3)where I (k x ,k y )=1N 2N −1 x =0N −1 y =0i (x ,y )exp −j2πN(xk x +yk y ) (4)f x =k x /N and f y =k y /N are the discrete spatial frequencies.The power spectrum 4, (k x ,k y ),encodes the energy density for each spatial frequencies and orientations over the whole image.PCA applied to power spectra gives the main components that take into account the structural variability between images.First,the power spectrum is normalized with respect to its variance for each spatial frequency: (k x ,k y )= (k x ,k y )/std[ (k x ,k y )],with std[ (k x ,k y )]= E [( (k x ,k y )−E [ (k x ,k y )])2].This normalization compensates for the 1/f αshape of the power spectrum.The spectral principal components (SPCs)decompose the normalized power spectrum of an image ass (k x ,k y )=Pn =1u n SPC n (k x ,k y ).(5)P is the number of SPCs.4Although not reﬂected in equation (4),for computing the spectral signatures of the precedent sections we have applied a circular Hamming window to the images in order to avoid boundary artifacts.Statistics of natural image categories401Figure9.Projection of images into the space represented by the second and the third principalcomponents of the power spectra.Images are organized according to spectral properties:SPC2putsimages with dominant energy in the f y axis on top of theﬁgure opposed to images with dominantenergy in the f x axis which are at the bottom.SPC3opposes images with energy in the f x and f yaxis(cross shape)with respect to images with energy at oblique orientations.A coarse organizationof scenes emerges:man-made versus natural scenes and open versus closed environments.Figure8(b)shows the resulting SPCs.In accordance with results of section2,the threeﬁrst principal components exhibit vertical and horizontal spectral component dominance.Figure9 shows a set of scene images organized according to the projections of their power spectra along the second and the third spectral principal components.Along the SPC2axis,scenes with a dominant horizon,such as open landscapes and suburban open scenes,stand at the top of theﬁgure,in opposition to scenes deﬁning closed and enclosed environments(with isotropic power spectra).Along the SPC3axis,images having a cross shape power spectrum(mostly urban areas)stand at one extreme whereas natural landscapes stand at the other extreme.An organization of broad environmental categories emerges,suggesting that the variability in the second-order statistics of natural images may be relevant for natural scenes classiﬁcation tasks.A linear combination of theﬁrst three SPCs is able to separate man-made scenes from natural scenes with an accuracy of80%.402A Torralba and A Oliva 4.Receptiveﬁelds for scene recognitionThe level of organization achieved by the SPCs suggests that the variability in the second-order statistics of natural images may be relevant for categorization purposes.As illustrated inﬁgures2–6,we have observed that second-order image statistics vary along the naturalness dimension(man-made versus natural landscapes)and openness dimension(related to the depth,Baddeley1997,Oliva and Torralba2001).This suggests that the categorical status of an environmental view,along those two perceptual dimensions,could be computed in a feedforward manner,from a set of low-level detectors encoding information similar to the one provided by the power spectrum(see also section6).In this section,we look for the best spectral statistics providing discrimination between man-made,natural,open and closed scene categories.Linear discrimination between two scene categories,using the normalized image power spectra (k x,k y),can be achieved asw=N−1k x=0N−1k y=0(k x,k y)DST(k x,k y)=N−1k x=0N−1k y=0(k x,k y)DST (k x,k y)(6)with DST (k x,k y)=DST(k x,k y)/std[ (k x,k y)].DST (k x,k y)is the weighing of the spectral components needed to discriminate between the two classes(DST standing for discriminant spectral template).w is the most discriminant feature and is obtained as a weighted sum of the power spectrum of the image.As SPCs deﬁne a complete orthogonal basis for describing the normalized power spectra,we can write the DST asDST(k x,k y)Pn=1a n SPC n(k x,k y).(7)The coefﬁcients a n indicate how to weight of each SPC in order to build a speciﬁc templateDST.Here,we used theﬁrst P=16SPCs.The coefﬁcients a n are determined by a supervisedlearning stage.In the learning stage,each image is represented by a column vector of features u={u n},u n being the projection of the power spectrum of the image into the n th SPC.Then, we deﬁne two groups of images of different scene categories(e.g.,pictures of man-made andnatural environments,the same image database as described in section3).The parameters of the DST,a n,can be learnt by applying Fisher linear discriminant analysis(e.g.Ripley1996, Swets and Weng1996)that looks for the coefﬁcients a n giving the best classiﬁcation rate. After training,the classiﬁcation rate for man-made scenes versus natural landscapes goes to 93%(versus80%when using the SPCs only).Training and testing are respectively done on different set of images,of thousands of exemplars each.When applying the discrimination analysis to the differentiation between open and closed and enclosed environments,the correct classiﬁcation rate reaches94%.Although more complicated classiﬁers could be used,the linear classiﬁer allows for a simple analysis.Due to the linearity of equation(6),the discrimination performed in the spectral domain(DST )can be written in the spatial domain.It is then possible to compute receptiveﬁelds tuned to global scene statistics that discriminate between the categories of natural scenes.The output energy of a discreteﬁlter with transfer function H(k x,k y)can be computed asE=N−1k x=0N−1k y=0(k x,k y)|H(k x,k y)|2.(8)This expression is similar to equation(6)used to compute the structural feature w.However,as the squared magnitude of the transfer function of aﬁlter cannot have negative values,the DST。

Hurst指数概念及计算方法

公式六注：
n=每个片段的规格
8.计算Hurst指数 a. 将每种分段方法的片段大小（size）和ARS对10取对数 b. 这样我们就有了6组对数序列。将lgARS作为被解释变量Y，lgSize作为解释变量X，线性回归估计斜率H，H就是Hurst指数。
一、什么是Hurst指数
Hurst指数概念及计算方法
Hurst指数，简单来讲就是有偏的随机游走。Hurst指数最初是由英国水利学家[Harold Edwin Hurst](https:///wiki/Harold_Edwin_Hurst)提出，并以他的名字洪水过程是时间系列曲线，具有与时间相关的长记忆性。即干旱愈久，就可能出现持续的干旱；大洪水年过后仍然会有较大洪水这一指数的发现是基于Hurst对尼罗河进行长期的水文观测，并在此基础上提出了用重标极差（R/S）分析法来建立Hurst指数，作为判断是随机游走还是有偏的随机游走。
三、Hurst指数在中国股市的应用
在中国股市上，通常是把Hurst指数作为一个短期指标来运用。同时它可以更好地配合其他技术指标来使用，如开盘价、最高价、最低价、收盘价、成交量和成交金额等。通过大量的统计发现，在中国市场上，股市具有延续性。
四、Hurst指数的计算方法
1. 将时间序列分割成不同的片段。比如取沪深三百日回报率(daily return)一百天内的数据作为时间序列，将其按照以下六种规格分割： a. 单个片段大小是整个序列，分成1组； b. 单个片段大小是序列的1/2，分成2组； c. 单个片段大小是序列的1/4，分成4组； d. 单个片段大小是序列的1/8，分成8组； e.单个片段大小是序列的1/16，分成16组； f. 单个片段大小是序列的1/32，分成32组；
注：

Choosing the legal retirement age in presence of unemployment

2
1
Introduction
Both the political debate and the scientiﬁc literature aim at ﬁnding sustainable reforms to solve the ﬁnancial unbalance of public Pay-As-You-Go (PAYG) pensions. Three policies are usually pointed out: raising contributions, cutting beneﬁts and raising the retirement age. Among those, a rise in the retirement age has received a special attention. A general conclusion of the literature (Cremer and Pestieau (2003)) is to advocate policies that encourage continued activity as a way to improve ﬁnancial problems of pension systems. This literature rests on the assumption that labour markets are competitive and can completely absorb increases in the labour stock. However, if we depart from the assumption of perfectly competitive labour markets and consider unemployment, the issue of the retirement age cannot set aside labour market features. A popular argument stresses the fact that early retirement schemes were introduced to make the old leave the labor force, hence providing more jobs for the young. Symmetrically, the reluctancy to increase the retirement age is often explained by its detrimental eﬀect on the employment opportunities of the young. This mechanism is studied in details in Gruber, Milligan, and Wise (2009). There is however no clear empirical evidence on the link between the rate of unemployment and the retirement age. Keuschnigg and Keuschnigg (2004) ﬁnd a negative general equilibrium eﬀect on unemployment of an increase in R. But Diamond (2005), using data from Gruber and Wise (1998), ﬁnds no relationship between the unemployment rate for males and the log of the tax force. Taking this absence of clear evidence into account, we assume that the unemployment rate does not depend on the age of retirement. Another crucial issue pointed out in the political debate is that raising the legal retirement age is not feasible if there is no demand for older labour force. This paper sheds light on this argument since the central question we are interested in is weather it is still desirable to increase the retirement age when workers face the risk of being unemployed. Our framework also allows us to study the impact of the unemployment rate on the chosen retirement age. In this purpose we study the design of the PAYG system, in particular concerning the legal retirement age,1 when there is unemployment. More speciﬁcally, we are interested in the choice of the retirement age (mandatory and common to all) when the level of

p85值计算方法

p85值计算方法P85值是统计学中常用的一种统计指标，用于描述数据的分布。

它是指当一个特定百分比的观察值高于一组数据时，所对应的数值。

在实际应用中，P85值通常用于分析经济指标、财务数据以及其他领域的数据集。

P85值计算方法基本上可以分为以下几个步骤：步骤1：确定数据集首先，需要确定要计算的数据集。

这可以是一个样本数据集或整体数据集，具体取决于研究的目的和数据的可获得性。

步骤2：排序数据将数据按照数值的大小进行排序，从小到大或从大到小都可以，选择哪一种排序方式应根据具体情况而定。

步骤3：确定P值根据需要计算的P值，确定计算P85值。

P85值表示85%的观测值高于它。

因此，P85值可以看作是将数据切割成两个部分的点，其中85%的观测值在P85值之下，15%的观测值在P85值之上。

步骤4：计算P85值根据数据集的大小，计算P85值。

P85值可以通过以下两种方法进行计算：方法一：使用百分位数计算P85当数据量较大时，可以使用百分位数的方法来计算P85值。

百分位数是一种用于描述数据分布的统计量，它表示某一特定百分比的观测值在该数值下面。

对于P85值的计算，可以将百分位数设定为85%，然后找到该百分位数对应的数值。

方法二：使用插值计算P85当数据量较小时，可以使用插值的方法来计算P85值。

插值是一种通过已知数值之间的关系来估计未知数值的方法。

对于P85值的计算，可以先找到离85%最近的两个数值，然后通过插值来估计P85值所对应的数值。

步骤5：解释计算结果最后，根据计算得到的P85值，对数据的分布进行解释。

P85值越高，说明数据集的分布越向右偏移，即在较高数值的一侧有更多的数据点。

总结：P85值是一种用于描述数据分布的统计指标，在统计学和数据分析领域中具有广泛的应用。

计算P85值的方法可以通过使用百分位数或插值来实现，具体方法选择应根据数据集的大小和特点来决定。

通过计算P85值，可以更准确地分析数据的分布特征，从而提供有价值的信息和洞察力。

nr中的tbsize计算公式

nr中的tbsize计算公式
在NR系统中，TBS（Transport Block Size）是指每个时隙内可传输的数据块大小，是一种衡量系统效率的重要指标。

TBS的计算公式如下：
TBS = {NPRB x NR x 12 x log2(MOD)} - {OH + G + RI} 其中，NPRB是RB数量，NR是每RB的符号数，12是每符号的比特数，MOD是调制方式（如QPSK、16QAM、64QAM等），OH是协议开销，G是保护间隔（Guard Interval），RI是循环前缀（Cyclic Prefix）长度。

需要注意的是，不同的调制方式和保护间隔会对TBS产生影响，因此在具体计算时需要根据系统参数进行调整。

此外，TBS还受到其他因素的影响，如信道质量、信道带宽等，因此TBS的计算公式只是一种理论计算，实际传输效果需要进行实测验证。

- 1 -。

r subset函数提取位置元素

r subset函数提取位置元素（原创版）目录1.介绍 R 语言和 r subset 函数2.解释位置元素的概念3.演示如何使用 r subset 函数提取位置元素4.讨论 r subset 函数的优点和局限性正文R 语言是一种广泛应用于数据处理和统计分析的编程语言。

在 R 语言中，有许多功能强大的函数可以帮助我们处理和分析数据，其中 r subset 函数就是其中一个常用的函数。

它可以帮助我们从一个数据集中提取出满足特定条件的子集，使得我们可以更加高效地分析和利用数据。

在介绍 r subset 函数之前，我们需要先了解一下位置元素的概念。

在 R 语言中，数据集通常是以矩阵或者数据框的形式存在的。

而位置元素，就是指这些矩阵或者数据框中的行和列。

例如，对于一个二维矩阵，它的位置元素包括行号和列号。

而对于一个数据框，它的位置元素则包括行号和列名。

接下来，我们将演示如何使用 r subset 函数提取位置元素。

假设我们有一个二维矩阵，它的行号和列名分别是 1,2,3 和 A,B,C。

我们可以使用 r subset 函数来提取出第二行和第三列的元素，代码如下：```Rmatrix(c(1,2,3,4,5), nrow=3, ncol=2)```这个函数的作用是创建一个新的数据框，其中包含了原始数据集中满足条件的元素。

在这个例子中，满足条件的元素就是第二行和第三列的元素。

r subset 函数的优点在于，它可以帮助我们从一个复杂的数据集中快速地提取出我们需要的子集，从而提高我们的工作效率。

同时，它也非常灵活，可以满足我们各种不同的需求。

然而，r subset 函数也存在一些局限性。

首先，它只能处理数据框和矩阵这两种数据类型。

其次，它提取的元素是基于位置的，而不是基于内容的。

因此，如果我们需要根据数据的内容来提取元素，就需要使用其他的函数。

总的来说，r subset 函数是一个非常实用的函数，可以帮助我们快速地提取位置元素。

青少年风湿性关节炎新药上市 疗效达85%

r语言quarter函数

Supertraces on the Algebras of Observables of the Rational Calogero Model with Harmonic Pot

赫斯特指数

R语言时间序列中文教程

分形布朗运动和hurst指数

r语言简介

nr中的tbsize计算公式

hurst指数2篇

hurst 指数python

urwtest参数说明

(2013,pr)EDCircles_A real-time circle detector with a false detection control

R语言时间序列中文教程共34页

Image Super-Resolution via Sparse Representation

Statistics of natural image categories

Hurst指数概念及计算方法

Choosing the legal retirement age in presence of unemployment

p85值计算方法

nr中的tbsize计算公式

r subset函数提取位置元素

青少年风湿性关节炎新药上市疗效达85%