Error-Space Estimation Method and Simplified Algorithm for Space Target Tracking_AO
再制造机床装配过程误差传递模型与精度预测
计算机集成制造系统Computer Integrated Manufacturing System；Vol.27No.5 May2021第27卷第5期2021年5月DOI：10.13196/j.cims2021.05.006再制造机床装配过程误差传递模型与精度预测王子生，姜兴宇+，刘伟军,石敏煊，杨世奇,杨国哲（沈阳工业大学机械工程学院，辽宁沈阳110870）摘要：再制造机床的装配零件具有更多的不确定性和随机性，为保证装配精度的一致性与可靠性，需要建立其装配过程误差传递数学模型，定量描述装配过程的误差传递与累积规律。
提出再制造机床装配过程中的零件质量特征、实际测量结果与装配过程误差源的数学映射关系及其误差流模型，定量描述了再制造机床装配过程中零件质量特征变动、特征测量、调整的偏差传递。
在此基础上提出误差流修正函数，实现了再制造机床装备精度预测、分析和质量误差修正。
以再制造机床主轴箱装配过程为例，验证了该模型及修正函数能够正确有效地消减装配过程误差。
关键词：再制造机床；装配误差；精度预测；误差传递模型；误差流修正函数中图分类号:TH17文献标识码:APrecision prediction and error propagation model of remanufacturing machine tool assembly process WANG Zisheng,JIANG Xingyu+,LIU Weijun,SHI Minxuan,YANG Shiqi,YANG Guozhe (School of Mechanical Engineering,Shenyang University of Technology,Shenyang110870,China) Abstract：The remanufactured machine tool is assembled by reusing parts,remanufactured parts and new parts,and its assembly parts have more uncertainty and randomness.To ensure the consistency and reliability of remanufacturing machineassemblyprecision，a mathematicalmodelofassemblyprocesserrortransfershouldbeestablishedto describe the error transmission and accumulation rules of remanufacturing machine assembly process quantitatively.The mathematical mapping relationship between the quality characteristics of the parts,the actual measurement re-sultsandtheerrorsourcesofassemblyprocessduringtheassemblyprocessofremanufacturedmachinetoolwaspro-posed.An error flow model of the remanufacturing machine's assembly process based on state-space model was also proposed.The variation of part quality characteristics,feature measurement and adjustment deviation of the remanufacturing machine tool assembly process were quantitatively described.On this basis,the error flow correction function was proposed to realize the accuracy prediction,analysis and quality error correction of remanufacturing machine tool.By t aking the assembly process of remanufacturing machine headstock as an example,the result showed that the assembly process error could be reduced by the proposed model and correction function effectively.Keywords：remanufacturing machine tool；assembly error；precision prediction；error propagation model；error flow correctionfunction“R亠寸及其在装配中的位置不同，各零件对整体产品装0弓1=配精度的影响程度不同，如何定量描述这些影响，制作为机床再制造过程中的最后一环，装配直接定相应零件装配精度的要求，从而降低装配成本，保决定了组装产品的性能和质量。
复杂数据下空间自回归模型的统计推断
复杂数据下空间自回归模型的统计推断复杂数据下空间自回归模型的统计推断摘要：本文研究了在复杂数据下如何进行空间自回归模型的统计推断。
空间自回归模型是一种处理空间数据的重要工具，但在实际应用中，由于数据的复杂性，传统的统计推断方法可能会给出不太准确的结果。
因此，本文提出了一种基于贝叶斯方法的空间自回归模型统计推断方法，并通过模拟和实际数据分析在不同复杂数据下进行了验证。
实验结果表明，该方法能够在复杂数据下获得较为准确的推断结果。
本文的研究对于提高空间数据分析的准确性和应用价值有一定的参考意义。
关键词：空间自回归模型，统计推断，复杂数据，贝叶斯方法。
1. 引言随着信息技术的不断发展，人们正在快速地获取各种数据，这些数据往往存在空间相关性，即相距较近的数据之间比较相关。
为了更好地利用这些空间数据，空间自回归模型被广泛应用于地理、环境、经济和社会等领域的研究中。
空间自回归模型通过考虑空间位置之间的相似性结构，增强了模型的解释能力和预测能力，同时协调了空间数据之间的相关性和自相关性。
然而，在实际应用中，数据的复杂性可能会影响空间自回归模型的统计推断。
传统的最小二乘法和经典统计方法对于数据分布的假设过于简单，难以处理复杂数据下的空间自回归模型。
例如，在数据存在异方差性、非正态性、缺失值、异质性等情况下，需要使用更加复杂灵活的统计模型来实现准确的统计推断。
为了解决这个问题，本文基于贝叶斯方法提出了一种适用于复杂数据的空间自回归模型统计推断方法。
本文首先介绍了传统的空间自回归模型和相关的统计推断方法，然后详细描述了基于贝叶斯方法的空间自回归模型统计推断方法。
接下来，本文采用模拟和实际数据进行验证，分析不同复杂数据下该方法的准确性和稳定性。
2. 空间自回归模型与传统统计推断方法2.1 空间自回归模型空间自回归模型是一种基于空间位置之间相关关系的回归模型。
在空间自回归模型中，一个空间位置的因变量与该位置周围多个空间位置的自变量相关联。
非参数估计方法_张煜东
第32卷第7期2010年07月武 汉 工 程 大 学 学 报J. Wuhan Inst. T ech.Vo l.32 N o.7Jul. 2010收稿日期:2010-04-02基金项目:国家自然科学基金(60872075);国家高技术发展计划(2008AA 01Z227);高等学校科技创新工程重大项目培育资金项目(706028)作者简介:张煜东(1985-),男,江苏苏州人,哥伦比亚大学博士后.研究方向:人工智能、数据挖掘、脑图像处理.文章编号:1674-2869(2010)07-0099-08非参数估计方法张煜东1,2,颜 俊1,王水花1,吴乐南1(1.东南大学信息科学与工程学院,江苏南京210096;2.哥仑比亚大学精神病学系脑成像实验室,纽约州纽约10032)摘 要:为了解决函数估计问题,首先讨论了传统的参数回归方法.由于传统方法需要先验知识来决定参数模型,因此不稳健,且对模型敏感.因此,引入了基于数据驱动的非参数方法,无需任何先验知识即可对未知函数进行估计.本文主要介绍最新的8种非参数回归方法:核方法、局部多项式回归、正则化方法、正态均值模型、小波方法、超完备字典、前向神经网络、径向基函数网络.比较了不同的算法,给出算法之间的相关性与继承性.最后,将算法推广到高维情况,指出面临计算的维数诅咒与样本的维数诅咒两个问题.通过研究指出前者可以通过智能优化算法求解,而后者是问题固有的.关键词:参数统计;非参数统计;核方法;局部多项式回归;正则化方法;正态均值模型;小波;超完备字典;前向神经网络;径向基函数网络中图分类号:O 212.7 文献标识码:A doi:10.3969/j.issn.1674-2869.2010.07.0250 引 言函数估计[1]是一个经典反问题,一般定义为给定输入输出样本对,求未知的系统函数[2].传统的方法为参数方法,即构建一个参数模型,再定义某个误差项,通过最小化误差项来求解模型的参数[3].参数方法尽管较为简单,但不够灵活.例如参数模型假设有误,则会导致整个求解流程失败[4].因此学者们发展出不少新技术,非参数估计就是其中一项较好的方法.该方法无需提前假设参数模型的形式,而是基于数据结构推测回归曲面[5].本文首先研究了经典的2种参数回归方法:最小二乘法与内插函数法,分析了它们的不足,然后主要讨论8种非参数回归方法:核方法、局部多项式回归、正则化方法(样条估计)、正态均值模型、小波方法、过完全字典、前向神经网络、径向基函数网络,尤其详细介绍了其间的相关性与继承性.最后,研究了高维情况下面临的计算维数诅咒与样本维数诅咒.1 回归模型考虑模型y i =r (x i )+ i (1)式(1)中(x i ,y i )为观测样本,假定误差 具有方差齐性,则r =E(y |x )称为y 对x 的回归函数,简称回归.一般地,可以假设x 取值在[0,1]区间内.定义 规则设计 为x i =i/n(i =1,2, ,n).并定义风险函数为R =ni=1[r(x i)-^r (x i)]2=ni=1[y i-^r (x i)]2(2)式(2)中^r 为系统函数r 的估计.回归一词源于高尔顿(Galto n),他和学生皮尔逊(Pearson)在研究父母身高和子女身高的关系时,以每对夫妇的平均身高为x ,取其一个成年儿子的身高为y ,并用直线y =33.73+0.512x 来描述y 与x 的关系.研究发现:如果双亲属于高个,则子女比他们还高的概率较小;反之,若双亲较矮,则子女以较大概率比双亲高.所以,个子偏高或偏矮的夫妇,其子女的身高有 向中心回归 的现象,因此高尔顿称描述子女与双亲身高关系的直线为 回归直线 [6].然而,并非所有的x -y 函数均有回归性,但历史沿用了这个术语.更为精确的表达是 函数估计 .100武汉工程大学学报第32卷2 传统方法理论上描述一个函数需要无穷维数据,因此函数估计本身也可称为 无穷维估计 [7].传统的估计方法有下列两种极端情形.2.1 最小二乘法此时假设^r(x)= 0+ 1x,采用最小二乘法计算权值 =( 0, 1),得到的解为最小二乘估计[8],^r(x)=(X T X)-1X T Y(3)则对给定样本点的估计r=[^r(x1),^r(x2), ,^r(x n)]T可写为r=X^ =LY(4)这里Y=(y1,y2, ,y n)T.L=X(X T X)-1X T称为帽子矩阵[9].以5个样本点的一维规则设计矩阵为例,此时X=0.20.40.60.81.0L=0.0180.0360.0540.0720.0900.0360.0720.1090.1450.1810.0540.1090.1630.2180.2720.0720.1450.2180.2900.3630.0900.1810.2720.3630.454(5)L满足L=L T,L2=L.另外,L的迹等于输入数据的维数p,即trace(L)=p.这里输入数据是一维的,所以trace(L)=1.2.2 内插函数法此时对^r(x)不加任何限制,得到的是该数据的一个内插函数[10].同样以5个样本点的一维规则设计矩阵为例,由于样本点的估计r=[^r(x1), ^r(x2), ,^r(x n)]T完全等于(y1,y2, ,y n)T,所以帽子矩阵为L=1000001000001000001000001(6)2.3 两种方法的缺陷图1给出了这两种极端拟合的示意图,数据是被高斯噪声干扰的正弦函数,采用上述两种方法拟合,结果表明:最小二乘法过光滑,未展现数据内部的关系;而内插函数法忽略了噪声影响,显得欠光滑.从帽子矩阵也可看出,式(5)表明最小二乘法对每个数据的估计都利用了所有样本,这显然导致过光滑,且x值越大的数据权重越大,这明显与经验不符;反之,式(6)表明内插函数法仅仅利用了最邻近的样本数据,这显然导致欠光滑.图1 两种极端拟合F ig.1 T wo ex tr eme fitting2.4 非参数回归的优势非参数回归(non-parametric r eg ression)作为最近兴起的一种函数估计方法,是一种分布无关(distribution free)的方法,即不依赖于数据的任何先验假设.与此对应的是参数回归(param etric r eg ressio n),通常需要预先设置一个模型,然后求取该模型的参数.非参方法的本质在于:模型不是通过先验知识而是通过数据决定.需要注意的是, 非参数 并不表示没有参数,只是表示参数的数目、特征是可变的(flexible).由于非参方法无需数据先验知识,其应用范围较参数方法更广,且性能更稳健.其另一个优点是使用过程较参数方法更为简单.然而,它也存在缺点,一般结构更复杂,需要更多的运算时间.2.5 线性光滑器需要说明的是,最小二乘法、内插函数法、核方法、正则化方法、正态均值模型均是线性光滑器.定义为:若对每个x,存在向量l(x)=[l1(x), ,l n(x)]T,使得r(x)的估计可写为^r(x)= n i=1l i(x)y i(7)则估计^r为一个线性光滑器[11].显然权重l i(x)随着x而变化,这与信号处理中的 自适应滤波器 非常相似.3 核回归核方法[12]定义为^r(x)= n i=1l i(x)Y i(8)权重l i由式(9)给出l i=Kx-x ihni=1Kx-x ih(9)这里h是带宽,K是一个核,满足K(x) 0,以及K(x)d x=1, xK(x)d x=0, x2K(x)d x>0,(10)常用的核函数见表1.第7期张煜东,等:非参数估计方法101表1 常用的核公式Table1 Fr equen tly-u sed kernel formula核公式boxcar K(x)=0.5*I(x)Gau ssian K(x)=12ex p-x22Epan echnikov K(x)=3(1-x2)I(x)T ricube K(x)=7081(1-|x|3)3I(x)以bo xcar核为例,帽子矩阵为L=1/21/20001/31/31/30001/31/31/30001/31/31/30001/21/2(11)显然,这可视作最小二乘法与内插函数法的折中.为了估计带宽h,首先必须估计风险函数,一般可采用缺一交叉验证得分CV=R^(h)=1n ni=1[y i-^r-i(x i)]2(12)这里^r-i(x i)为未用第i个数据所得到的估计,使C V最小的h,即为最佳带宽.为了加速运算,可将式(12)重新写为R^(h)=1n ni=1y i-^r(x i)1-L ii2(13)这里L ii是光滑矩阵L的第i个对角线元素.另一种方法是采用广义交叉验证法,规定G CV(h)=R^(h)=1nni=1y i-^r(x i)1-v/n2(14)这里v=tr(L).4 局部多项式回归采用核回归常会碰到下列2个问题[13]:1)若x不是规则设计的,则风险会增大,称为设计偏倚(desig n bias);2)核估计在接近边界处会出现较大偏差,称为边界偏倚(boundary bias).为了解决这2个问题,可采用局部多项式回归.局部多项式回归[14]可视作核估计的一个推广,首先定义权函数 i(x)=K[(x i-x)/h],选择a=^r(x)来使得下面的加权平方和最小ni=1i(x)(y i-a)2(15)利用高等数学知识,可以看出解为^r(x)= ni=1i(x)y ini=1i(x)(16)可见式(16)正好是核回归估计.这表明核估计是由局部加权最小二乘得到的局部常数估计.因此,若利用一个p阶的局部多项式而不是一个局部常数,就可能改进估计,使曲线更光滑.定义多项式P x(u;a)=a0+a1(u-x)+a22!(u-x)2++a pp!(u-x)p(17)则局部多项式的思想是:选择使下列局部加权平方和ni=1i(x)[y i-P x(x i;a)]2(18)最小的a,估计^a=(^a0,^a1,^a p)T依赖于目标值x,最终有^r(x)=P x(x;^a)=^a0(x)(19)当p等于0时,等于核估计;当p=1时,称为局部线性回归(local linear regr ession)估计[15],由于其算法简单且性能优越,较为常用.5 基于正则化的回归为了描述方便,这里假设数据点为[(x0,y0),(x1,y1), (x n-1,y n-1)].在风险函数(2)后增加一项惩罚项,一般设为r(x)的二阶导数J= n-1i=0y i-^r(x i)2+ [r (x)]2d x(20)控制了解的光滑程度:当 =0时,解为内插函数;当 时,解为最小二乘直线;当0< <时,^r(x)是一个自然三次样条.需要注意下列事项:首先三次样条表示曲线在结点(knot)之间是三次多项式,且在结点处有连续的一阶和二阶导数;其次一个m阶样条为一个逐段m-1阶多项式,所以三次样条是4阶的(m=4);第三,自然样条表示在边界点处二阶导数为0,即在边界点外是线性的;第四,样条的结点等于数据点.为了加速计算,将数据点重新排序,假设a,b为样本点x的上下界,令a=t1 t2 t n-1=b,这里t是x重新排序后的点,称为结点.可用B样条基(B-spline basis)[16]作为该三次样条的基,即^r(x)= n-m-1i=0P i b i,m(t) t [t m-1,t n-m](21)P i称为控制点,共n-m个,形成一个凸壳.n-m个B样条基可通过如下计算,首先初始化:b j,0(t)=1 if t j<t<t j+10 otherw ise(22)然后对i=1,逐步+1,直到i=m-1,重复迭代下式:102 武汉工程大学学报第32卷b j,i (t)=t -t jt j +m -1-t j b j,i -1(t)+t j +m -tt j +m -t j +1b j,i +1(t)(23)若结点等距,则称B 样条是均匀的(uniform ),否则称为不均匀.如果两个结点相等,计算过程会出现0/0情况,此时默认结果为0.令矩阵B 的第(i,j )元素b ij =b j (x i ),矩阵 的第(i,j )元素 ij =b n i (x )b nj (x )d x ,则控制点可由式(24)求得P =(B TB + )-1B TY(24)可见,样条也是一个线性光滑器.表面上看,基于核的估计与基于正则化的估计原理与模型均不一致,但是Silver man 证明了如下定理,样条估计^r (x )可视作如下所示的一种渐近的核估计l i (x )1f (x i )h(x i )Kx i -x h (x i )(25)式中,f (x )是x 的密度函数.h(x )=nf (x )1/4(26)K (x )=12ex p -|x |2sin |x |2+4(27)显然,若样本x 是规则设计,则f (x )=1,h(x )=( /n)1/4=h,l i (x ) K [(x i -x )/h],即此时样条估计可视作形如式(27)的渐近核估计.6 正态均值模型令 1, 2, 为一个标准正交基,则显然r (x )可以展开为r(x )=i=1i i ,定义Z j =1nni=1y ij(x i )(28)则随机变量Z j 是正态分布,且均值与方差满足:E(Z j )= j V(Z j )= 2/n (29)可见,若估计出 ,则可近似求得^r (x )ni=1i i.因此正态均值模型将n 个样本的函数估计问题转换为估计n 个正态随机变量Z j 的均值 的问题[17].若直接令^ =Z,则显然得到一个很差的估计,下面给出风险更小的估计.首先,必须做出一个关于^ 的风险估计,Stein 给出下列定理:令Z ~N ( ,V ),^ =^ (Z)为 的一个估计,并令g(Z 1, ,Z n )=^ -Z.则^ 的风险的一个无偏估计为J^(z )=tr (V)+2tr (VD)+ ig2i(z )(30)式中g i =^ i -z i ,且D 的第(i,j )个元素为g (z 1, ,z n )的第i 个元素关于z j 的偏导数[18].假设^ =b Z =(b 1Z 1, ,b n Z n ),式中b 称为调节器,根据b 的设置,存在下列3种情况:b =(b,b, ,b),称为常数调节器(constant m odulator),此时令式(30)最小的称为Jam es -Stein 估计;b =(1, ,1,0, ,0),称为嵌套子集选择调节器(nested subset selection modulator ),此时令式(30)最小的^b 称为REACT 方法.需要注意的是,若基选择傅立叶基,则该方法类似于频域低通滤波器方法.b =(b 1,b 2, ,b n )满足1 b 1 b 2 b n 0,称为单调调节器(monotone m odulator ),该方法理论最优,但是需要的运算量太大,几乎不实用.7 小波方法小波方法[19]适用于空间非齐次(spatially inhomog eneous )函数,即函数的光滑程度随着x 会有本质性的变化.它可视作正态均值模型的推广,但存在两点区别:一是采用小波基代替传统的正交基,因为小波基较一般的正交基具有局部化的优点,能实现多分辨率分析;另一点是采用了一种称为 阈 的收缩方式.不妨假定父小波为 ,母小波为 ,同时规定下标(j ,k)的意义如下:f j,k (x )=2j/2f (2jx -k)(31)为了估计函数r,用n =2J 项展开来近似r ,r (x )2J0-1k=0j 0,k(x ) j 0,k (x )+ Jj =J2j-1k=0j ,kj ,k (x )(32)这里J 0是任取常数,满足0 J 0 J. 称为刻度系数, 称为细节系数.那么如何估计这些系数?首先计算S k =1n ij 0,k(x i )y i (33)D jk =1nij,k(x i )y i(34)S k 、D jk 分别称为经验刻度系数与经验细节系数,可知S k N ( j 0,k , 2/n),D jk N ( j,k , 2/n),可估计方差为median (|D j -1,k -median (D j -1,k )第7期张煜东,等:非参数估计方法103| k =0, ,2J -1-1)0.6745(35)然后根据S k 、D jk 、^ 可得 与 的估计如下:^ j 0,k =S k (36) 的估计形式稍许复杂,采用硬阈与软阈的方式分别为^ jk =0 |D jk |< D jk |D jk |(37)^jk =sig n (D jk )|D jk |- +(38)之所以采用阈的形式,是因为稀疏性(sparse )的思想[20]:对某些复杂函数,在小波基上展开时系数也是稀疏的.因此,需要采用一种方式来捕获稀疏性.然而,传统的L 2范数不能捕捉稀疏性,相反,L 1范数与非零基数能够较好地捕捉稀疏性.例如,考虑n 维向量a =(1,0, ,0)与b =(1/n 1/2, ,1/n 1/2),有 a 2= b 2=1,可见,L 2范数无法区分稀疏性.反之, a 1=1, b 1=n 1/2,因此,L 1范数能提取稀疏性;另外,若令非零基数为J ( )={#(i 0)},则J (a)=1,J (b)=n,因此,非零基数也能提取稀疏性.最后,在正则化估计中若惩罚项分别为L 1范数或非零基数,则最优估计恰好对应着软阈估计与硬阈估计.最后,需要解决阈估计中 的计算问题,这里介绍两种最简单的方式:一是通用阈值(universal thresho ld ),即对所有水平的分辨率阈值均一致,=^2log nn(39)另一种是分层阈值(leve-l by -levelthresho ld ),即对不同分辨率采用不同阈值,一般是通过最小化下式求得S( j )=njk=1^ 2n -2^ 2nI |^ jk j +min (^ 2jk , 2j )(40)j [0,(^ /n j )2log n j ]式中n j =2j -1为在水平j 的参数个数.8 超完备字典小波基较标准正交基的改进在于更加局部化,因此能实现对跳跃的捕捉.然而,虽然小波基非常复杂,但面对各种复杂的函数还是不够灵活.这种缺陷的根源在于:小波基是标准正交基,任意两个基函数之间正交,这保证了基函数简单完整的同时,也丧失了灵活性.基追踪(basis pursuit)方法[21]的思想是采用一种超完备(overcomplete)的基,例如对 光滑加跳跃 的函数,传统的傅立叶基能够捕捉光滑部分,但是难以捕捉跳跃部分;采用小波基能轻易捕捉跳跃部分,但是描述光滑部分较为困难.此时若将 傅立叶基 与 小波基 合并成一个新的基,则显然这种基能够轻松地估计 光滑加跳跃 函数.但是,这种新的基不再正交,它以牺牲正交性来获得更好的灵活性[22],故此时用 字典 来描述更精确,而本文为了简便统一仍采用 基 表述.9 前向神经网络以一个双层神经网络为例,记网络的输入神经元个数为m,隐层神经元个数为n,输出层神经元个数为q ,则网络结构如图2所示.图2 前向神经网络F ig.2 F or war d neural netw or k与上面几节线性方法不同的是,神经网络属于非线性统计数据建模(nonlinear statistical data m odeling),其隐层暗含了 特征提取 的思想,且可视作输入数据在一种 自适应的非线性非正交的基 上的映射.同样地,此时基牺牲了正交性、线性、不变性,增加了计算负担,但换来了更加强大的灵活性[23].简而言之,前向神经网络采用了类似基追踪的方法[24],但基是自适应变化的、非线性的,因此更加灵活.前向神经网络与基追踪相似之处在于,两者的基都不是正交的,都是根据给定数据而自适应选取的最佳基.前向神经网络的优势在于无不需预选字典,字典在算法中自动生成,并可作为特征选择的一种方法.10 径向基函数网络首先观察径向基函数(RBF)神经元如图3所示.图3 RBF 神经元图F ig.3 N euron of R BF图中输入向量p 的维数为R ,首先p 与输入层权值矩阵I W 相减,然后求距离函数dist ,再与104 武汉工程大学学报第32卷偏置b 1相乘,最后求径向基函数radbas (n)=ex p (-n 2),得到神经元的输出为a =radbas ( IW -pb 1)(41)整个RBF 网络由两层神经元组成,第1层为S 1个如图3所示的RBF 神经元,第2层为S 2个线性神经元,如图4所示.在第2层开始时,第1层的输出a 首先经过线性层权值矩阵LW 后与偏置b 2相加,再通过一个纯线性(purelin)函数purelin (n)=n,得到网络输出y 为y =purelin (LW a +b 2)(42)图4 RBF 神经网络结构图F ig.4 St ruct ur e of RN N比较式(41)与式(9)可见,RBF 网络与核方法非常类似,不同之处在于RBF 网络的L W 需要通过求解一个方程组,而核方法的权重是直接通过归一化计算求得,因此RBF 网络预测结果更为逼近完全内插函数估计(注意不是未知函数r ),而核方法计算更为简便[25].11 维数灾难将函数估计推广到高维,则会碰到维数诅咒(curse of dimensionality)[26](图5),它意味着当观测值的维数增加时,估计难度会迅速增大.维数诅咒有两层含义:一是计算的维数诅咒,指的是某些算法的计算量随着维数的增长而成指数增加.解决方法通常采用优化算法,例如遗传算法、粒子群算法、蚁群算法等[27].二是样本的维数诅咒,指的是数据维数为d 时,样本量需要随着d 指数增长.在函数估计中,第二层含义更为重要,这里给予详细解释.图5 样本的维数诅咒示意图Fig.5 Dimensio nality cur se o f samples假设一个半径r 维数为d 的超球,被一个边长为2r 维数为d 的超立方体所包围,假设超立方体内存在一个均匀分布的点,则由于超球的体积为2r dd /2/[d (d/2)],超立方体的体积为(2r )d,因此该点同时也落在超球内的概率P 为P =d/2d 2d -1 (d/2)(43)令维数d 由2逐步增长到20,则对应的概率P 如图6所示.显然,当d =20时,P 仅为2.46 10-8.因此,若在2维空间中1个样本在半径r 的意义下能逼近一个正方形,则在20维空间内,则需要1/2.46 10-8=4.06 107个样本才能在半径r 的意义下逼近超立方体.图6 概率P 与维数d 的关系F ig.6 T he cur ve of pro bability Pagainst dimensio na lit y d因此,在高维问题中,由于数据非常稀少,导致局部邻域中包含极少的数据点[28],因此估计变得异常困难.目前还没有较好的办法解决.12 结 语将文中阐述的方法归结并示于图7.图7 非参数回归方法Fig.7 Sur vey of non -par ametric r egr essio n metho ds不同类型方法的特点总结如下:a.核方法、正则化方法、正态均值模型可以视作最基本最原始的方式.另外,正则化方法与正态均值模型可视作一类特殊的核方法.b.核方法、局部多项式方法、正则化方法、正态均值模型、小波等方法在大多数情况下均非常类似.这些方法都包含了一个偏倚-方差平衡,所以都需要选择一个光滑参数.由于这些方法均是线性光滑器,所以均可以采用第4节中基于CV 、第7期张煜东,等:非参数估计方法105GC V的方法.c.小波方法一般面向空间非齐次函数.如果需要一个精确的函数估计,而且噪声水平较低,则小波方法非常有效.但若面对一个标准的非参数回归问题,而且感兴趣于置信集,则小波方法并不比其它方法明显更好.d.超完备字典缺陷是丧失了基的正交性,因此估计系数变得复杂;优点是更为灵活,能够采用稀疏的系数描述复杂函数.e.前向神经网络与RBF神经网络是基于不同的模型独立推导出来的,二者不可混淆.另外,神经网络方法的缺点是一般不考虑置信带,并常用训练误差代替风险函数,容易过拟合;优点是面向应用、思想简单且设计灵活.f.理论上,这些方法没有大的差别,特别在用置信带的宽度来评价时.每种方法都有其拥护者与批评者,没有哪一种方法目前获得应用上的优势.一种解决方案是对每个问题都利用所有可行的方法,如果结果一致,则选择简单者;如果结果不一致,则必须探讨内在的原因.g.所讨论的方法能够用于高维问题,然而,即使通过智能优化算法解决了计算的维数诅咒,仍然面对样本的维数诅咒.计算一个高维估计相对容易,然而该估计将不如一维情况下那么精确,其置信区间会非常大.但这并不表示方法失效,而是表示问题的固有困难.参考文献:[1] N eumey er N.A not e on unifor m consistency o fmonoto ne functio n estimato rs[J].Statistics&P robability L etter s,2007,77(7):693-703[2] Sheena 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[24] 屠艳平,管昌生,谭浩.基于BP网络的钢筋混凝土结构时变可靠度[J].武汉工程大学学报,2008,30(3):36-39.[25] Z hang Y D,W u L N,N egg az N,et al.Remo te-sensing Imag e Classificatio n Based on an Impro vedPro babilistic N eural N etw ork[J].Sensor s,2009,9:7516-7539.[26] A leksandr ow icz G,Barequet G.Co unting po ly cubesw ithout the dimensionality curse[J].DiscreteM athematics,2009,309(13):4576-4583.[27] 张煜东,吴乐南,奚吉,等.进化计算研究现状(上)[J].电脑开发与应用,2009,22(12):1-5.[28] 王忠,叶雄飞.遗传算法在数字水印技术中的应用[J].武汉工程大学学报,2008,30(1):95-97.Survey of non-parametric estimation methodsZHANG Yu-dong1,2,Y AN Jun1,W ANG Shui-hua1,WU Le-nan1(1.Schoo l o f Info rmation Science&Engineer ing,So ut heast U niv ersity,N anjing210096,China;2.Br ainimag ing L ab.,Depar tment o f Psycholog y,Co lumbia U niver sity,N ew Y or k NY10032,U SA)Abstract:In or der to so lve the pro blem of functio n estimation,w e first discuss traditional param etric regression metho d.Since it needs a pr io ri kno w ledge to deter mine the model,the par am etric m ethod is no t robust and is mode-l sensitive.Thus,data-driven non-par am etric metho d is intro duced,w hich needs no t any a prior know ledge to estim ate the unknow n function.Eig ht m ajo r no n-parametric m ethods are discussed as kernel method,local poly no mial regression,regularization method,nor mal mean m odel, w av elet method,ov ercomplete dictionary,fo rw ar d neur al netw ork,and radial basis function netw ork. These alg orithms are com pared,and their coher ence and inher itance ar e investigated.Finally,g eneralize the algo rithms to high dimensionality and po int out tw o pro blems as curse of dimensionality of com putation and sam ple.The for mer can be settled dow n by intelligent methods w hile the latter is pro blem intrinsic.Key words:par am etric statistics;no n-parametric statistics;kernel method;local polynom ial r eg ressio n; regular ization m ethod;no rmal mean mo del;w avelet;ov er-co mplete dictionary;forw ard neural netw ork; radial basis function netw ork本文编辑:龚晓宁。
ARIMA模型——本质上是error和t-？时刻数据差分的线性模型！！！如果数据序列是非平。。。
ARIMA模型——本质上是error和t-？时刻数据差分的线性模型如果数据序列是⾮平。
可以看到：ARIMA本质上是error和t-?时刻数据差分的线性模型ARIMA模型全称为⾃积分滑动平均模型(Autoregressive Integrated Moving Average Model,简记ARIMA)，是由博克思(Box)和(Jenkins)于70年代初提出⼀著名时间序列（Time-series Approach）预测⽅法 [1]，所以⼜称为Box-Jenkins模型、博克思-詹⾦斯法。
其中ARIMA（p，d，q）称为差分⾃回归移动平均模型，AR是⾃回归， p为⾃回归项； MA 为移动平均，q为移动平均项数，d为时间序列成为平稳时所做的差分次数。
所谓ARIMA模型，是指将⾮平稳时间序列转化为平稳时间序列，然后将因变量仅对它的滞后值以及随机误差项的现值和滞后值进⾏回归所建⽴的模型。
ARIMA模型根据原序列是否平稳以及回归中所含部分的不同，包括移动平均过程（MA）、⾃回归过程（AR）、⾃回归移动平均过程（ARMA）以及ARIMA过程。
优点：模型⼗分简单，只需要内⽣变量⽽不需要借助其他外⽣变量。
缺点：1.要求时序数据是稳定的（stationary），或者是通过差分化(differencing)后是稳定的。
2.本质上只能捕捉线性关系，⽽不能捕捉⾮线性关系。
注意，采⽤ARIMA模型预测时序数据，必须是稳定的，如果不稳定的数据，是⽆法捕捉到规律的。
⽐如股票数据⽤ARIMA⽆法预测的原因就是股票数据是⾮稳定的，常常受政策和新闻的影响⽽波动。
严谨的定义：⼀个时间序列的随机变量是稳定的，当且仅当它的所有统计特征都是独⽴于时间的（是关于时间的常量）。
判断的⽅法：1. 稳定的数据是没有趋势(trend)，没有周期性(seasonality)的; 即它的均值，在时间轴上拥有常量的振幅，并且它的⽅差，在时间轴上是趋于同⼀个稳定的值的。
3多变量统计故障诊断方法
1.针对非高斯特性的改进
独立主元分析(又称独立成分，独立元，Independent Component Analysis，ICA)方法作为统计信号处理领域内一种新的方法最早由 Juten和Herault提出，其利用信号的高阶统计信息(二阶统计量便足以 描述高斯信号)，将混合信号分解成相互独立的非高斯成分，由于各 非高斯成分满足独立性条件，联合概率密度等于各成分概率密度之积， 因此避免了高维的概率密度估计问题。Hyvnen改进了ICA的算法，增 强了它的鲁棒性和训练速度，由于ICA的上述优势，近年来采用IcA方 法进行过程统计性能监控的工作正逐步增多，Kan等实现了基于IcA的 故障检测，Lee等在此基础上用贡献图实现了故障的隔离]，Lin和 zhang把IcA与小波结合构造滤波器，可以降低过程中测量传感器不足 带来的影响，另外IcA在动态过程、非线性过程、间歇过程中也得到 了较好扩展，成功的应用实例有利用动态ICA实现废水处理过程的监 控。
213 3 22
，是正态分布的的置信极
主元分析需要注意的几点问题
数据的标准化问题
矩阵Xn×m每一列对应于一个测量变量，每一行对应一个样本。在m 维空间中，两个样本间的相似度应正比于两个样本点在m维空间中 的接近程度。由于m个测量变量的量纲和变化幅度不同，其绝对值 大小可能相差许多倍。为了消除量纲和变化幅度不同带来的影响， 原始建模数据应作标准化处理，即：
1.针对非高斯特性的改进
另一种解决办法是引入高斯混合模型(Gaussian Mixture model，GMM)来估计PCS中的数据模式(即聚类)，GMM 的训练可采用期望最大(Expeetation Maximization)算法来 实现。由于每个模式对应着一个高斯函数，即每个模式中 数据分布服从正态分布，所以可以利用传统的PCA求取新 数据的SPE和T2检测统计量，进而实现故障的检测。但如 果过程中的数据模式信息不充分，那么GMM模型难以建 立，Thissen等对该法进行了改进，不再在各个类上分别 采用T2统计量检测变量的波动，而是仅采用一个总体的密 度参数，很好地解决了这个问题。
空间计量经济学讲义
导论
❖ 空间计量经济学是现代计量经济学的“空间化”范式 ❖ 空间计量经济学是区域科学理论与经验研究的“桥梁”
导论
❖ 空间计量经济学的相关研究 ❖ Moran(1948)提出用0-1连接矩阵表示空间相关关系，之后，Moran(1950)提
出了著名的Moran’s I统计量用来测度空间自相关。 ❖ Geary(1954)提出了另一种度量空间依赖的统计量Geary’s C。 ❖ Cliff和Ord(1972)提出Moran’s I可以用于检验最小二乘回归(OLS)得到的
❖ 空间回归模型是经典线性模型的拓展，将空间因素显格式的加入到模型中 。
❖ 地理加权回归，是以曲线拟合、平滑等局部加权回归的非参数方法为理论 基础，利用焦点周围的样本子集进行回归。
❖ 空间滤波是将空间数据通过滤波方法进行处理。
导论
❖ 中文核心期刊中以“空间计量”为主题的文章最早从2005年起，之后空间计 量为主题的文章数量开始了指数型的增长。
•理论与应用研究呈现爆发式的增长。 •这主要是由于地理相关数据的日益庞大且更加容易获得、地理信息系统的进步
以及人们越来越意识到空间的重要。
•美国应用经济学家的涌入，相关应用领域包括城市与区域经济学、房地产经济等 •研究方法更为标准和规范，如广义矩和矩估计方法的引入 •通过广泛的模拟实验，各种方法应用于小样本的研究 •计算机和软件开发也日益受到关注 •美国国家科学基金会成立了空间整合社会科学中心
• Estimating Models with Spatial Effects
Four Dimensions
• Testing for the Presence of Spatial Effects
• Spatial Prediction
Network MIMO with linear zero-forcing
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 5, MAY 20122911Network MIMO With Linear Zero-Forcing Beamforming: Large System Analysis, Impact of Channel Estimation, and Reduced-Complexity SchedulingHoon Huh, Member, IEEE, Antonia M. Tulino, Senior Member, IEEE, and Giuseppe Caire, Fellow, IEEEAbstract—We consider the downlink of a multicell system with multiantenna base stations and single-antenna user terminals, arbitrary base station cooperation clusters, distance-dependent propagation pathloss, and general “fairness” requirements. Base stations in the same cooperation cluster employ joint transmission with linear zero-forcing beamforming, subject to sum or per-base station power constraints. Intercluster interference is treated as noise at the user terminals. Analytic expressions for the system spectral efficiency are found in the large-system limit where both the numbers of users and antennas per base station tend to infinity with a given ratio. In particular, for the per-base station power constraint, we find new results in random matrix theory, yielding the squared Frobenius norm of submatrices of the Moore–Penrose pseudo-inverse for the structured non-i.i.d. channel matrix resulting from the cooperation cluster, user distribution, and path-loss coefficients. The analysis is extended to the case of nonideal Channel State Information at the Transmitters obtained through explicit downlink channel training and uplink feedback. Specifically, our results illuminate the trade-off between the benefit of a larger number of cooperating antennas and the cost of estimating higher-dimensional channel vectors. Furthermore, our analysis leads to a new simplified downlink scheduling scheme that preselects the users according to probabilities obtained from the large-system results, depending on the desired fairness criterion. The proposed scheme performs close to the optimal (finite-dimensional) opportunistic user selection while requiring significantly less channel state feedback, since only a small fraction of preselected users must feed back their channel state information. Index Terms—Channel estimation, downlink scheduling, large random matrices, linear zero-forcing beamforming, network MIMO.Manuscript received December 14, 2010; revised September 11, 2011; accepted November 10, 2011. Date of publication December 06, 2011; date of current version April 17, 2012. The work of G. Caire and H. Huh was supported in part by the National Science Foundation under Grants CIF 0917343 and CIF 1144059. The material in this paper was presented in part at the 44th Annual Conference on Information Sciences and Systems (CISS), Princeton, NJ 08542 USA. H. Huh was with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 USA. He is now with the Media Lab., Samsung Electronics Co., Ltd., Suwon, Gyeonggi-do 443-742, Korea (e-mail: hhuh@). A. M. Tulino is with the Department of Wireless Communications, Bell Laboratories, Alcatel-Lucent, Holmdel, NJ 07733 USA (e-mail: a.tulino@). G. Caire is with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 USA (e-mail: caire@). Communicated by L. Zheng, Associate Editor for Communications. Color versions of one or more of the figures in this paper are available online at . Digital Object Identifier 10.1109/TIT.2011.2178230I. INTRODUCTION HE NEXT generation of wireless communication systems (e.g., 802.16m [1], LTE-Advanced [2]) consider multiuser MIMO (MU-MIMO) as one of the core technologies. A considerable research effort has been dedicated to the performance evaluation of MU-MIMO systems in realistic cellular environments [3]–[5]. In a single-cell setting with perfect Channel State Information at the Transmitter (CSIT), the system reduces to a vector Gaussian broadcast channel whose capacity region is completely characterized [6]–[10]. However, in a multicell scenario, we are in the presence of a vector Gaussian broadcast and interference channel, which is not yet fully understood in an information theoretic sense. A simple and practical approach consists of treating InterCell Interference (ICI) as noise. In this case, ICI may significantly limit the system capacity. A variety of intercell cooperation schemes have been proposed to mitigate ICI, ranging from a fully cooperative network MIMO [11]–[14] to partially coordinated beamforming [15]–[17]. In this work, we focus on the network MIMO approach with limited cooperation, where clusters of cooperating base stations (BSs) act as a single distributed MIMO transmitter and interference from other clusters is treated as noise. In a cellular environment, the received signal power is a polynomially decreasing function of the distance between transmitter and receiver, with a dynamic range typically larger than 30 dB [18]. Thus, users close to the cell (or cluster of coordinated cells) boundary experience strong intercell interference, whereas the desired signal is relatively weak. These “edge” users cannot be just ignored by the system. For example, maximizing the system sum-rate leads in general to a very unfair operating point, where the system resources are concentrated on users near the cell (or cluster) center. In contrast, fairness scheduling has been proposed and widely studied in order to achieve a desirable balance between sum-rate and fairness (see for example [19]–[21] and references therein). Fairness scheduling can be systematically implemented in the framework of stochastic network optimization [20]. Such fairness scheduling algorithms dynamically allocate the system resources on a slot-by-slot basis, such that the long-term average (or “ergodic”) user rate point maximizes some suitable concave and componentwise increasing network utility function [22]. While the analytical characterization of the optimal ergodic rate point for a given network utility function may be hopelessly complicated in a realistic scenario, the system performance hasT0018-9448/$26.00 © 2011 IEEE2912IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 5, MAY 2012been evaluated so far through computationally very intensive Monte Carlo simulation [3]–[5], [13], [14], [23]–[26], where the actual scheduling algorithm evolves in time and the ergodic rates are computed as time averages. The capacity of a multicell network MIMO system under fairness criteria was evaluated in the large-system limit in [27], assuming ideal channel state knowledge and the Gaussian Dirty Paper Coding (DPC) transmission strategy. In [27], the asymptotic analysis method based on large random matrix results demonstrated its effectiveness by the comparison with finite-dimensional Monte Carlo simulation. In this work, we apply a similar approach to Linear Zero-forcing Beamforming (LZFB). It turns out that the analysis in this case is significantly more complicated, in particular, in order to take into account per-BS power constraints. In this paper we also extend our analysis to the case where the CSIT is obtained through explicit training and MMSE estimation. In these conditions, we obtain a lower bound on the achievable ergodic rates (referred to as “throughput” in the following), that takes into account the overhead due to training-based channel estimation. Several novel and important aspects are illuminated in this paper. Specifically: 1) As in [27], our analysis allows precise performance evaluation of systems for which brute-force Monte Carlo simulation would be very demanding. 2) By including the effect of training and channel estimation, we can investigate the tradeoff between ICI reduction owing to BS cooperation and the cost of estimating larger and larger dimensional channels. Unlike previous results that assumed ideal CSIT at no cost [24], [25], [27], we observe that there exists an optimal cooperation cluster size that depends on the channel coherence time and bandwidth, beyond which cooperative network MIMO is not convenient, consistently with the finite-dimensional simulation findings of [26], [28]. 3) We provide novel results in random matrix theory, in particular, related to the evaluation of the coefficients appearing in the per-BS power constraints. 4) We use our asymptotic analysis in order to design a probabilistic scheduling algorithm that randomly preselects the users with assigned probabilities obtained from the large-system results, and therefore requires much less CSIT feedback than the standard opportunistic scheduling scheme based on channel-driven user selection. This last point deserves some remarks, since for the first time (to the authors’ knowledge) asymptotic results are used not only for performance analysis but also for system design in network MIMO.1 The standard approach to scheduling for downlink beamforming consists of having a large number of users feeding back their CSIT and selecting a subset of users with cardinality not larger than the number of jointly coordinated transmit antennas, such that the channel vectors of the selected users have both large norm and are mutually approximately orthogonal [31], [32]. User selection, combined with LZFB precoding, is shown to attain the same performance as Gaussian DPC in the limit of a large number of users and fixed number of transmit antennas. However, in this limit, the where is the throughput per user vanishes as number of users. Therefore, a more meaningful regime is one1In the unrelated context of multiuser detection, asymptotic large-system results were used to design low-complexity linear multiuser detectors based on the polynomial approximations, where the polynomial coefficients were obtained from large random matrix theory [29], [30].in which the number of users is proportional to the number of antennas, yielding constant throughput per user. This is in fact the large-system regime investigated in this paper. Comparing the results of our asymptotic analysis with the Monte Carlo simulation of finite-dimensional systems, including user selection as said before, we notice that multiuser diversity yields larger throughput per user for low-dimensional systems, but this gain reduces as the system dimension grows. This is a manifestation of the “channel hardening effect” noticed in [33], and agrees with the theoretical findings in [34], showing that the probability of finding a subset of approximately orthogonal users vanishes as the system dimension increases. Hence, as the system becomes large, there is diminishing return in selecting users from a large set. In contrast, the cost of CSIT feedback grows at least linearly with the number of users feeding back their estimated CSIT. Therefore, we advocate a probabilistic scheduling algorithm for which users are preselected at random using the probabilities derived from our large-system analysis, reflecting the desired fairness criterion, and only the selected users are required to feed back their CSIT. The performance of this scheme are shown to be close to the much more costly full user selection scheme, and become closer and closer as the system dimension increases (again, by the large-system limit and channel hardening effect). In comparison with concurrent existing literature, we notice that the LZFB MU-MIMO performance analysis with nonideal CSIT was extensively studied in the finite-dimensional case (see for example [35]–[37]) and in the large-system limit (see for example [38]–[40]). Unlike concurrent works, our paper focuses explicitly on the system optimization under the fairness criteria in the multicell downlink with intercell cooperation. This particular angle allows us to illuminate aspects that are not present in other works, such as the distribution of the per-user throughput under fairness and, as a consequence, the design of the random scheduling scheme said before. The remainder of this paper is organized as follows. In Section II, we describe the general finite-dimensional system model including the arbitrary clustering of cooperative BSs, formulate the system optimization problem, and provide its numerical solutions for a given channel realization. In Section III, we take the large system limit and present the large-system regime of the LZFB precoder and the optimization algorithm for user selection and power allocation. The opportunistic fairness scheduling scheme is also described in this section. The impact of nonperfect CSIT and training overhead is analyzed in Section IV. Numerical results and the low-complexity randomized scheduling scheme are presented in Section V and some concluding remarks are given in Section VI. The most lengthy and technical derivations are relegated into the appendices. II. FINITE-DIMENSIONAL SYSTEM A. System Setup BSs, with anConsider a cellular system formed by single-antenna user terminals spatially tennas each, and distributed in the coverage area. We assume that the users are divided into co-located “user groups” with users each. Users in the same group are statistically equivalent: they see the same pathloss coefficients from all BSs and their small-scaleHUH et al.: NETWORK MIMO WITH LINEAR ZERO-FORCING BEAMFORMING2913fading channel coefficients are i.i.d.. The received signal vector for the th user group is given by (1) and denote the distance dependent pathloss where coefficient and small-scale channel fading matrix from the th BS to the th user group, respectively, is the transmitted signal vector of the th BS, subject to the power constraint , and denotes the additive white Gaussian noise (AWGN) at the user receivers. The elements of and of are i.i.d. . A cooperative cell arrangement with cooperation clusters is defined by the BS partition of the BS set and the corresponding user group partition of the user group set . We assume that the BSs in each cluster act as a single distributed multiantenna transmitter with antennas, perfectly coordinated by a central cluster controller, and serve users in groups . The clusters do not cooperate and treat ICI from other clusters as noise. Assuming that each BS operates at its maximum individual transmit power, the ICI plus noise power at any user terminal in group is given by (2) Each cluster seeks to maximize its own objective function defined by the fairness scheduling. It is easy to show that, under the above system assumptions, the selfish optimal strategy that operates at maximum per-BS power is a Nash equilibrium of the system. At this Nash equilibrium, the clusters are effectively decoupled since the effect that other clusters have on each cluster is captured by the ICI terms in (2) that do not depend on the actual BS transmit covariances . From the viewpoint of cluster , the system is equivalent to a single-cell MIMO downlink channel with a modified channel matrix and noise levels and per-BS power constraints. Therefore, from now on we focus on a given reference cluster and, without loss of generality, we indicate the user groups in the reference cluster as , with , and the BSs in as with . After a convenient renormalization of the channel coefficients, we arrive at the equivalent channel model for the reference cluster given by (3) with matrix , , is given by . . . . . . and the channeland users changes significantly over a time-scale of the order of the tens of seconds, while the small-scale fading decorrelates completely within a few milliseconds [41]. Here, a “slot” indicates a block of channel uses over which the small-scale coefficients can be considered constant. The slot length (in channel uses) is approximately equal to the product of the channel coherence time and the channel coherence bandwidth [41]. B. Downlink Scheduling Optimization Problem Each cluster controller operates according to a downlink scheduling scheme that allocates instantaneously the transmission resource (signal dimensions and transmit power) to the users. Following [22], the scheduling problem is formulated as the maximization of a suitable strictly increasing and over the region of achievconcave network utility function able ergodic rates (throughput region), which is convex by time-sharing. For users in group , we define the mean group throughput as the the arithmetic mean of the individual user . By the symmetry of the throughputs, i.e., system with respect to the users belonging to the same group, it turns out that for any achievable throughput point with given , there exists an achievable individual user throughputs throughput point such that all users in group have throughput . In other words, the cumulative throughput of users in the same group can always be distributed uniformly over these users, without changing the sum throughput. We assume that the network utility function gives the same priority to statistito be cally equivalent users. This is captured by restricting Schur-concave [42].2 It follows that the network utility function is always maximized at a point for which for all users in group . Therefore, letting and redefining the function to have arguments, the fairness scheduling problem is formulated directly in terms of the user mean group throughputs, as(5) where denotes the system -dimensional achievable group throughput region. In particular, this work considers LZFB downlink precoding. Hence, indicates the group throughput region achievable by LZFB for the channel model (3), under the assumption of operating at the Nash equilibrium said above. A scheduling policy achieving the optimum group throughput point solution of (5) consists of a rule that, at each scheduling slot, maps the available channel information into a set of scheduled users, rates and transmit powers, such that the resulting long-term time averaged group rates converge to . As a first step toward the solution of (5), we focus on the weighted instantaneous sum-rate maximization problem(4) (6)where we define . The pathloss coefficients are fixed constants that depend only on the geometry of the system, and the small-scale fading coefficients are assumed to evolve according to a block-fading process that changes independently from slot to slot and remains constant over each slot. This is representative of a typical situation where the distance between BSsdenotes the rate weight for user in group , and is the achievable instantaneous rate region of LZFB for given channel matrix . By “instantaneous,” we mean that2The class of -fairness network utility functions introduced in [22], including max-min and proportional fairness, satisfy this condition.where2914IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 5, MAY 2012this rate region depends on the given channel realization , in contrast with the throughput region , that depends on the statistics of . Realistically, we assume that (i.e., the number of users in the cluster is larger than or equal to the total number of BS antennas in the cluster) and that all coefficients are strictly positive. Therefore, we have almost surely. In this case, LZFB cannot serve simultaneously all users in the cluster, and the scheduler must select a subset of , to be served at each slot. users not larger than It should be noticed at this point that, for the sake of completeness, we consider the weighted instantaneous sum-rate maximization problem (6) in the most general case where the weights are distinct for each individual user. As a matter of fact, because of the system symmetry said before, in the large system (same limit we will be interested in the solution for weight, and therefore same priority, for all users in the same group ). The solution of (6) is generally difficult, since it requires a . search over all user subsets of cardinality less or equal to Well-known approaches (see [31], [32]) consider the selection of a user subset in some greedy fashion, by adding users to the active user set one by one, until the objective function in (6) cannot be improved further. Moreover, even for a fixed set of active users, the problem of optimal LZFB precoding subject to per-BS power constraints is nontrivial and has been recently addressed in [43]–[45] through fairly involved numerical algorithms. Because of these difficulties, problem (6) has so far escaped a clean analytical solution and most studies resorted to extensive and costly Monte Carlo simulation. In order to overcome the above difficulties, we make the following simplifying assumptions: 1) The scheduler picks a fraction of users in group by random selection inside the group. The selected users are referred to as the active users of group . The active user set selection is statistically independent over different scheduling slots. 2) The LZFB precoder is obtained by normalizing the columns of the Moore–Penrose pseudo-inverse of the channel matrix, although this choice is not necessarily optimal under per-BS power constraints [43]. Under these assumptions, we let denote the fractions of active users in groups , respectively. For given , the corresponding effective channel matrix is given by . . . . . .contains the users’ information-bearing code where symbols, statistically independent with mean zero and variance is the precoding matrix with unit-norm columns, and 1, is a diagonal matrix which contains the user transmit powers on is obtained as follows. the diagonal. The precoding matrix The Moore–Penrose (right) pseudo-inverse of is given by (10) Then, we let where the column-normalizing diagonal matrix contains the reciprocal of the squared norm on the diagonal. Letting denote of the columns of the diagonal element of in position , for , we have (11)denotes the element in the correwhere of the main diagonal of the sponding position matrix . Rewriting (3) with (7) and (9) and noticing , we arrive at the “parallel” channel model that for all active users in the form (12) The optimization of (6) for the channel model (12) is still independ on the acvolved, since the channel coefficients tive user fractions in a complicated and nonconvex way. As an intermediate step, we consider the solution of (6) for fixed user fractions . C. Power Allocation Under Sum-Power or Per-BS Power Constraints and multiply We divide all channel matrix coefficients by the BS input power constraints by , thus obtaining an equivalent system where the channel coefficients have variance . This is useful when considering the largethat scales as in Section III. system limit for Let denote the diagonal element in position of , corresponding to the power allocated to the th user of group . The sum-power constraint is given by(7)where the blocks is a -dimensional submatrix of . The user fractions must satisfy for and where we introduce each the notation (8) almost surely. Hence, we have With LZFB precoding, the transmitted signal(13) . In order to express the per-BS power where constraints, let denote a diagonal matrix with all zeros, but for consecutive “1,” corresponding to positions from to on the main diagonal. Then, the per-BS power constraints are expressed in terms of the partial trace of the transmitted signal covariance matrix asin given by (9) (14)HUH et al.: NETWORK MIMO WITH LINEAR ZERO-FORCING BEAMFORMING2915More explicitly, (14) can be written in terms of the powers as (15) where we define the coefficients (16) and where denotes the element of corresponding th column. Since has to the th row and the for all . unit-norm columns, then For fixed user fractions , the weighted instantaneous sumrate maximization (6) reduces to (17) subject to (13) in the case of sum-power constraint, or to (15) for the case of per-BS power constraints. The solution of (17) subject to the sum-power constraint is immediately given by the water-filling formula (18) is the Lagrange multiplier corresponding to the where sum-power constraint. In the case of per-BS power constraints, we can use Lagrange duality and the sub-gradient iteration method as given in the following. The Lagrangian function for (17) is given byIII. LARGE SYSTEM LIMIT In this section, we consider the limit of the above instanand fixed , taneous rate maximization problems for , and . We shall see in Section III-D that the weights in the weighted instantaneous sum-rate maximization are recursively calculated by the scheduler that solves the general network utility maximization problem (5). For Schur-concave and in the large system limit, where the “instantaneous” channel freeze to deterministic limits that depend only on gains the group index and not on the individual user index (see Theorem 1 below), these weights must be identical for all users in the same group. Since the weighted-sum rate maximization problem is used here as an intermediate step to devise the scheduling rule that solves (5), from now on we restrict to the case for all users in group . A. Asymptotic Analysis We start by finding the large system limit expression for the . This is provided by Theorem 1. coefficients Theorem 1: For all almost surely: , the following limit holds(23) is the unique solution in where the system of fixed-point equations of(24)with respect to the variables Proof: See Appendix A..(19) where is a vector of dual variables corresponding to the BS power constraints, is the matrix containing the coefficients and . The KKT conditions are given by (20) where indicates the column of containing the coefficients for . Solving for , we find (21)As anticipated above, the limit (23) depends only on (user group index) and not on (user index within the group), consistently with the fact that, in our model, users in the same co-located group are statistically equivalent. An immediate consequence is that in the limit for , for the case considered here, the waterfilling (18) yields equal power allocation for all active users in group . Next, we consider the per-BS power constraints given in (15). By the system symmetry and for the sake of analytical tractability, also in this case we restrict to uniform power allocation for all active users in the same group . Replacing this in the constraint (15), we obtain(25) Replacing this solution into , we solve the dual problem by minimizing with respect to . It is immediate to check that for any (22) Therefore, is a subgradient for . It follows that the dual problem can be solved by a simple -dimensional subgradient iteration over the vector of dual variables . (26) It is interesting to notice that is the squared Frobenius corresponding norm (normalized by ) of the submatrix of where we define2916IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 5, MAY 2012to the users in group (columns from to ) to ). and the antennas of BS (rows from We hasten to point out that the choice does not follow immediately from the KKT conditions (20), even asand in the limit of , since suming the terms may depend on and not only on even in the large system limit. As a matter of fact, Theorem 2 below yields to a determinthe convergence of vanishes istic limit. Notice that each individual term as since, by construction, . Based on numerical evidence, we conjecture that as , for all in group . However, proving the convergence of the individual random variables to the same deterministic limit independent of has resisted our efforts. In conclusions, beyond the sake of analytical tractability and system symmetry considerations, we also conjecture that for users in the same the symmetric power allocation group is also optimal for the case of per-BS power constraints, . The next result yields an analytical exin the limit of pression for the large-system limit of the coefficients : Theorem 2: For all surely: , the following limit holds almostFig. 1. Linear one-sided 2-cell model with B= 2 BSs and A = 8 user groups.cients depend on distance, because of the symmetric layout, the matrix is given by(30) for some positive numbers matrix can be decomposed into the . We notice that this circulant blocks(27) where system is the solution to the linear(28) where is the matrixWhen this symmetry condition holds, users in the same equivalence class are statistically equivalent, up to renumbering of the BSs. This is because such groups (e.g., user group pairs (1,5), (2,6), (3,7), and (4,8) in the example) are equivalent as far as the “landscape” of channel gain coefficients seen collectively by the cluster BSs. Motivated again by the Schur-concavity of the network utility function and by the symmetry of all users in the same equivalent class, for the sake of finding an analytically tractable solution and an overall problem simplification, we shall consider only resource allocations that give to all groups in the same equivalence class the same transmit power and active user fraction (symmetric solution). We indicate these common powers and active user fractions by and , respectively, for , such that and for all . In this case, for any , we have(29) and , and the coefficients are provided by Theorem 1. Proof: See Appendix B. andSimplifications for Symmetric System: Under some special symmetry conditions, the general problem can be significantly simplified. In particular, we assume that divides , let , and that the matrix of channel gains can be partitioned into circulant submatrices of size , i.e., such that each submatrix has the property that all rows are cyclic shifts of the first row and all columns are cyclic shifts of the first column. We shall refer to these submatrices as “circulant blocks.” Also, we define user equivalence classes as the sets of user groups whose corresponding columns of form circulant blocks. Then, the set of user groups is partitioned into equivalence classes. We re-index the user groups such that groups form the th equivalence class, for . To fix ideas, consider Fig. 1 showing a one-dimensional cellular system comprising 2 BSs and 8 symmetrically located user groups. Since the channel gain coeffi-where we used the fact that, by the symmetry condition, and . It follows that the solution of the fixed point (24) is given explicitly by (31) which is independent of , and (23) yields (32) Notice that for all groups in class , i.e., for all , the sum is a constant。
空间误差模型sem
空间误差模型sem英文回答：Spatial error model (SEM) is a statistical model used in spatial econometrics to analyze the relationship between dependent and independent variables in a spatial context.It accounts for spatial dependence and spatial heterogeneity, which are common in spatial data.In SEM, the dependent variable is modeled as a linear function of both the independent variables and a spatially lagged dependent variable. The spatial lag captures the spatial dependence by incorporating the neighboring values of the dependent variable into the model. The spatial error term captures the spatial heterogeneity by accounting for unobserved factors that vary across spatial units.The spatial error model can be expressed as follows:Y = Xβ + ρWy + ε。
Where:Y is the dependent variable.X is a matrix of independent variables.β is a vector of coefficients.ρ is the spatial autoregressive coefficient.W is a spatial weights matrix.y is the spatially lagged dependent variable.ε is the error term.The spatial weights matrix, W, defines the spatial relationships between the observations. It specifies the weights assigned to each neighboring observation in calculating the spatial lag. The choice of weights matrix depends on the nature of the spatial relationship beinganalyzed, such as contiguity or distance-based weights.Estimating the parameters of the spatial error model involves solving a maximum likelihood estimation problem. The estimation procedure takes into account the spatial dependence and heterogeneity, allowing for more accurate and reliable inference.SEM is useful in various fields, including regional economics, urban planning, and environmental studies. It helps to understand the spatial patterns and interactions between variables, providing insights into the spatial processes and dynamics.中文回答：空间误差模型（Spatial Error Model，SEM）是一种在空间计量经济学中用于分析因变量和自变量之间关系的统计模型。
分析插值误差和斜率的轨迹优化网格细化方法
Ｍｅｓｈ Ｒｅｆｉｎｅｍｅｎｔ Ｍｅｔｈｏｄ ｆｏｒ Ｔｒａｊｅｃｔｏｒｙ Ｏｐｔｉｍｉｚａｔｉｏｎ ｂｙ Ａｎａｌｙｚｉｎｇ
ｐｒｏｐｏｓｅｄ ｍｅｔｈｏｄ ｃａｎ ｓｏｌｖｅ ｎｏｎ—ｓｍｏｏｔｈ ｔｒａｊｅｃｔｏｒ ｙ ｏｐｔｉｍｉｚａｔｉｏｎ ｐｒｏｂｌｅｍｓ ｒａｐｉｄｌｙ ａｎｄ ａｃｃｕｒａｔｅｌｙ． Ｋｅｙ ｗｏｒｄｓ：Ｔｒａｊｅｃｔｏｒｙ ｏｐｔｉｍｉｚａｔｉｏｎ；Ｍｅｓｈ ｒｅｆｉｎｅｍｅｎｔ；Ａｄａｐｔｉｖｅ；Ｉｎｔｅｒｐｏｌａｔｉｏｎ ｅｒｒｏｒ；Ｓｌｏｐｅ
２．Ｂｅｉｊｉｎｇ Ａｅｒｏｓｐａｃｅ Ａｕｔｏｍａｔｉｃ Ｃｏｎｔｒｏｌ Ｉｎｓｔｉｔｕｔｅ，Ｂｅｉｉｉｎｇ １００８５４，Ｃｈｉｎａ）
Ａｂｓｔｒａｃｔ：Ａｎ ａｄａｐｔｉｖｅ ｍｅｓｈ ｒｅｆｉｎｅｍｅｎｔ ｍｅｔｈｏｄ ｆｏｒ ｓｏｌｖｉｎｇ ｔｈｅ ｔｒａｊｅｃｔｏｒｙ ｏｐｔｉｍｉｚａｔｉｏｎ ｐｒｏｂｌｅｍ ｉｓ ｐｒｅｓｅｎｔｅｄ，ｗｈｉｃｈ
第 ３９卷 第 ８期 ２０１８年 ８月
宇 航 学 报
Ｊｏｕｒｎａｌ ｏｆ Ａｓｔｒｏｎａｕｔｉｃｓ
Ｖｏ１．３９ Ｎｏ ８ ． Ａｕｇｕｓｔ ２０１８
分 析 插 值 误 差 和 斜 率 的 轨 迹 优 化 网 格 细 化 方 法
赵 吉松 ， 尚 腾
（１．南京航空航天大学航天学院 ，南京 ２１００１６； ２．北京航天 自动控制研究所 ，北京 １００８５４）
基于超像素运动统计的误匹配去除方法
DOI:10.11784/tdxbz201812028
天津大学学报(自然科学与工程技术版) Journal of Tianjin University(Science and Technology)
Vol. 53 No. 2 Feb. 2020
基于超像素运动统计的误匹配去除方法
a矩形网格划分方法b超像素分割方法图12种区域划分方法及配准点集分布fig1tworegionpartitionmethodsanddistributionofmatchingpointsets为解决这一问题本文采用超像素分割的网格化方法对传统gms算法进行了改进并建立了超像素网格统计模型将真假匹配的概率分布特性转换为统计特性以实现误匹配点的自动筛选
较高的鲁棒性，运算速度较快，可有效去除非刚性形变图像配准过程中产生的误匹配．
关键词：非刚性图像配准；误匹配去除；超像素运动统计；局部运动一致性
中图分类号：TN911.73ห้องสมุดไป่ตู้
文献标志码：A
文章编号：0493-2137(2020)02-0147-07
Mismatch Removal Using Superpixel Motion Statistics
块在空间上紧密相连，严格遵循图像轮廓边缘，且在同一区域内的颜色纹理基本趋于一致，可保证内部特征点具有
相同或一致的运动趋势；同时，采用 ORB 算子对图像进行特征提取与描述，并利用暴力匹配算法得到初始匹配点
集．其次，在超像素运动一致性约束下，提出了一种基于超像素运动统计模型的误匹配去除算法．通过建立超像素
He Kai，Wang Yang，Liu Zhiguo，Ma Hongyue
(School of Electrical and Information Engineering，Tianjin University，Tianjin 300072，China)
delta-method 标准误
Delta-method标准误是一种在统计推断中用于估计参数标准误差的方法。
它主要用于回归分析，当模型的误差项满足独立同分布（iid）条件时，可以利用Delta-method对参数进行估计。
Delta-method的基本思想是通过将随机变量Xn转换为另一个随机变量g(Xn)，使得g(Xn)的渐进分布与原始随机变量Xn的线性逼近的渐进分布相同。
在实际操作中，我们通常使用统计软件（如Stata）来计算Delta-method标准误。
当模型中的误差项存在相关性时，需要对标准误进行调整。
聚类调整标准误（Cluster-SE）是一种常用的方法，可以有效解决误差项相关性问题。
在Stata中，可以通过聚类调整标准误来估计系数，并得到更准确的置信区间和假设检验结果。
基于Shepard插值残差修正的TIN模型预测高程异常
摘
要：结合狄洛尼三角网 （TIN） 的构建方法，将 TIN 模型应用到高程异常预测中，并针对待定点处在三角形边界的问题使
用四边形下的 Shepard 插值进行残差修正。实例验证该方法能够结合待定点的临近值进行高程异常值拟合，四边形下的 Shepard
插值法能有效克服三角形边界上的待定点拟合精度较低的问题，平均相对误差为 0.036%，优于未进行 Shepard 插值改进的
成的两条边分别作为基边，继续构建三角形，构建时
乘配置模型和半参数模型等，配合常用的重力场模型
应当满足各三角形的外接圆均不包含其他、EIGEN-CG01C，可以获取
在高程异常拟合中，狄洛尼三角网应当以已知点
厘米级的高程异常拟合成果，但上述数学模型均是在
作为三角形顶点，布网结束后应确保所有待定点有且
点区域，以便后续使用 Shepard 插值进行残差修正。
表1
三角形边界点
待定点区域
边界点
W
MH21
MH23
MH16
Y
MH11
MH13
MH03
X
Z
XK
MH04
MH04
MH13
MH10
MH16
MH08
MH04；MH13；MH16；MH03
使用 TIN 模型中式 （1） 对待定点进行高程异常值
计算，同时为方便比较，使用二次曲面拟合模型对待
点位分布如图 1 所示，其中红色点表示待定点，黑色点
表示已知点。 X 坐标由3 063 483.914~3 076 071.229 m；
Y 坐 标 由 524 279.238~536 033.195 m； 高 程 异 常 值
A
B
W
非凸松弛原子范数空时动目标参数估计算法
第４５卷 第９期２０２３年９月系统工程与电子技术ＳｙｓｔｅｍｓＥｎｇｉｎｅｅｒｉｎｇａｎｄＥｌｅｃｔｒｏｎｉｃｓＶｏｌ．４５ Ｎｏ．９Ｓｅｐｔｅｍｂｅｒ２０２３文章编号：１００１ ５０６Ｘ（２０２３）０９ ２７６１ ０７ 网址：ｗｗｗ．ｓｙｓ ｅｌｅ．ｃｏｍ收稿日期：２０２２０３２８；修回日期：２０２２０６１０；网络优先出版日期：２０２２０７１１。
网络优先出版地址：ｈｔｔｐ：∥ｋｎｓ．ｃｎｋｉ．ｎｅｔ／ｋｃｍｓ／ｄｅｔａｉｌ／１１．２４２２．ＴＮ．２０２２０７１１．１５１７．０２４．ｈｔｍｌ基金项目：国家重点研发计划（２０２２ＹＦＢ３９０４３０３）；天津市教委科研计划（２０２１ＫＪ０４８）资助课题 通讯作者．引用格式：来燃，孙刚，张威，等．非凸松弛原子范数空时动目标参数估计算法［Ｊ］．系统工程与电子技术，２０２３，４５（９）：２７６１ ２７６７．犚犲犳犲狉犲狀犮犲犳狅狉犿犪狋：ＬＡＩＲ，ＳＵＮＧ，ＺＨＡＮＧＷ，ｅｔａｌ．Ｓｐａｃｅ ｔｉｍｅｍｏｖｉｎｇｔａｒｇｅｔｐａｒａｍｅｔｅｒｅｓｔｉｍａｔｉｏｎａｌｇｏｒｉｔｈｍｂａｓｅｄｏｎｎｏｎ ｃｏｎｖｅｘｒｅｌａｘａｔｉｏｎｏｆａｔｏｍｉｃｎｏｒｍ［Ｊ］．ＳｙｓｔｅｍｓＥｎｇｉｎｅｅｒｉｎｇａｎｄＥｌｅｃｔｒｏｎｉｃｓ，２０２３，４５（９）：２７６１ ２７６７．非凸松弛原子范数空时动目标参数估计算法来 燃，孙 刚，张 威，章 涛（中国民航大学天津市智能信号与图像处理重点实验室，天津３００３００） 摘 要：针对参数稀疏恢复空时自适应处理中的动目标参数估计存在字典失配的问题，提出一种非凸松弛原子范数空时动目标参数估计算法。
该方法利用目标回波在角度多普勒域的稀疏特性，根据连续压缩感知和低秩矩阵恢复理论实现了运动目标方位角和速度的高精度、超分辨率估计，避免了稀疏恢复中的字典失配问题，有效提高了动目标参数估计性能。
仿真实验结果表明，相较于已有基于字典网格的稀疏恢复参数估计方法和原子范数估计方法，所提算法具有更高的参数估计精度和对空间紧邻目标的分辨能力。
不平衡数据集的神经网络阈值优化方法
不平衡数据集的神经网络阈值优化方法李明方，张化祥，张雯，计华LI Ming-fang，ZHANG Hua-xiang，ZHANG Wen，JI Hua山东师范大学信息科学与工程学院，济南 250014School of Information Science and Engineering，Shandong Normal University，Jinan 250014，ChinaE-mail:lmgc21713@LI Ming-fang，ZHANG Hua-xiang，ZHANG Wen，et al.Approach to optimize threshold of ANN on imbalance datasets.Computer Engineering and Applications，2010，46(20):168-171.Abstract:The classification of imbalance datasets is a hot research area in the field of machine learning，and recently，many researchers have proposed several theories and algorithms to improve the performance of classical classification algorithms on imbalance datasets.One of the most important methods is adopting threshold selection criteria to determine the output threshold of an Artificial Neural Network(ANN).The commonly used threshold selection criteria have some drawbacks，such as failing to get optimal classification performances both on data in minority class and in majority class，only focusing on theclassification accuracy of the majority class data.This paper proposes a new threshold selection criterion based on which， both the data in the minority class and majority class can reach optimal classification accuracies without the impact of the sample proportion.When the newthreshold selection criterion is applied as a classifier evaluation criterion to classifiers trained using Artificial Neural Networks and Genetic approaches，good results can be obtained.Key words:imbalance datasets;threshold selectioncriterion;Artificial Neural Network(ANN);genetic method摘要:不平衡数据集分类为机器学习热点研究问题之一，近年来研究人员提出很多理论和算法以改进传统分类技术在不平衡数据集上的性能，其中用阈值判定标准确定神经网络中的阈值是重要的方法之一。
一种基于模糊熵的混沌伪随机序列复杂度分析方法_陈小军
第33卷第5期电子与信息学报Vol.33No.5 2011年5月 Journal of Electronics & Information Technology May 2011一种基于模糊熵的混沌伪随机序列复杂度分析方法陈小军①李赞*①白宝明①潘玮①陈清华②①(西安电子科技大学综合业务网理论及关键技术(ISN)国家重点实验室西安 710071)②(重庆市1534信箱重庆 400712)摘要：该文将模糊理论引入到混沌伪随机序列复杂度测度中，构造了用于序列复杂度测度的模糊隶属函数，并在此基础上研究了一种新的基于模糊熵(Fuzzy Entropy, FuzzyEn)的混沌伪随机序列复杂度测度。
仿真结果表明，与现有主要的混沌伪随机序列复杂度测度方法相比较，FuzzyEn测度不仅能够有效地测度出不同复杂度的混沌伪随机序列，而且具有更加好的对序列符号空间的适用性，更加小的对测量维度的敏感性以及更强的对分辨率的鲁棒性。
关键词：混沌伪随机序列；模糊熵；复杂度中图分类号：TN918.1 文献标识码：A 文章编号：1009-5896(2011)05-1198-06 DOI:10.3724/SP.J.1146.2010.00713A New Complexity Metric of Chaotic PseudorandomSequences Based on Fuzzy EntropyChen Xiao-jun①Li Zan①Bai Bao-ming①Pan Wei① Chen Qing-hua②①(The State Key Laboratory of Integrated Service Networks (ISN), Xidian University, Xi’an 710071, China)②(P.O.Box 1534, Chongqing 400712, China)Abstract: Importing the concept of fuzzy set, this paper constructes membership function for measuring the complexity of chaotic pseudorandom sequences. On the basis of this, a new complexity metric is investigated to evaluate the unpredictability of the chaotic pseudorandom sequences based on the Fuzzy Entropy (FuzzyEn).Simulations and analysis results show that, the FuzzyEn works effectively to discern the changing complexities of the chaotic pseudorandom sequences, and compared with complexity metric based on the Approximate Entropy (ApEn) and symbolic dynamics approach, FuzzyEn has obvious advantages in the applicability of symbolic space, the sensitivity of vector dimension and the robustness of resolution parameter.Key words: Chaotic pseudorandom sequences; Fuzzy Entropy (FuzzyEn); Complexity1引言混沌伪随机序列具有非周期、有界但不收敛等特性，目前它作为新型的扩频序列已经被广泛应用于抗干扰和保密通信之中[1,2]。
基于奇异值分解的非均匀采样系统最小二乘辨识
基于奇异值分解的非均匀采样系统最小二乘辨识
李楠;张为
【期刊名称】《集宁师范学院学报》
【年(卷),期】2014(036)001
【摘要】针对非均匀周期多采样率系统,在状态估计为已知的情况下,提出了基于奇异值分解的模型参数的最小二乘辨识方法.首先,根据系统的连续时间状态空间模型,在满足因果关系基础上,推导了含有提升变量的离散状态空间模型.然后,为了克服辨识误差积累和传递,采用基于奇异值分解的递推最小二乘方法确定模型参数.最后,仿真结果表明提出方法的有效性.
【总页数】7页(P93-99)
【作者】李楠;张为
【作者单位】内蒙古科技大学包头师范学院物理科学与技术学院,内蒙古包头014030;内蒙古科技大学包头师范学院信息学院,内蒙古包头 014030
【正文语种】中文
【中图分类】TP15
【相关文献】
1.基于协同PSO算法的非均匀采样系统辨识 [J], 王涛;林卫星;包建孟
2.非均匀采样系统的一种递推辨识方法 [J], 倪博溢;萧德云
3.基于奇异值分解的非均匀采样系统最小二乘辨识 [J], 李楠;张为;
4.一类非均匀采样系统最小二乘迭代辨识 [J], 蒋红霞;王金海;丁锋
5.基于奇异值分解递推辨识非均匀采样系统的状态空间模型（英文） [J], 王宏伟;刘涛
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考虑外载荷方向不确定性的稳健性拓扑优化
考虑外载荷方向不确定性的稳健性拓扑优化
费晨;王晓拓;倪成功;孙彦;田斌;杨艳山
【期刊名称】《计算力学学报》
【年(卷),期】2024(41)3
【摘要】外载荷方向不确定是实际工程中普遍存在的问题。
本文研究了考虑外载荷方向不确定性下的稳健性连续体拓扑优化。
首先,使用区间变量描述外载荷方向不确定性问题。
随后,通过使用旋转矩阵的方式描述名义载荷方向上的扰动,并利用了旋转矩阵导数的特殊性质,通过二阶泰勒展开式以及名义载荷处柔顺度对角度的一阶和二阶导数信息获得不确定性区间内任意角度处的灵敏度。
最后运用SIMP模型和优化准则法(OC)对考虑外载荷方向不确定性的连续体结构进行拓扑优化。
结果表明,采用该方法得到的优化结果的柔顺度对于载荷方向的变化不敏感,即对于不同方向的外载荷,结构刚度更具稳健性。
【总页数】6页(P582-587)
【作者】费晨;王晓拓;倪成功;孙彦;田斌;杨艳山
【作者单位】苏州农业职业技术学院智慧农业学院
【正文语种】中文
【中图分类】O342
【相关文献】
1.考虑载荷不确定性的多材料结构稳健拓扑优化
2.考虑拓扑相关热载荷的散热结构多相材料拓扑优化设计
3.载荷作用位置不确定条件下结构动态稳健性拓扑优化设
计4.考虑材料参数不确定性结构动力学稳健性拓扑优化设计5.载荷方向不确定条件下结构动态稳健性拓扑优化设计
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OCIS codes: 350.4600, 230.2090, 150.5758, 120.4640, 100.4999.
1. Introduction
High-performance optoelectronic target tracking and pointing is important and necessary in circumstances such as free-space optical communication, space debris surveillance, and electro-optic countermeasures. Compared with a fixed ground-based observation station, mobile vehicular tracking equipment has the capabilities of more maneuverability and adaptability and can operate wherever and whenever. With vehicular tracking equipment, especially when it is the only equipment implementing a mission, because the platform vehicle would stay on any position with any attitude stochastically, its attitude error could cause system errors of the tracking equipment and reduce tracking precision [1,2]. In some instances, the error could not be corrected owing to the lack of an appropriate reference. Furthermore, because of position error of the vehicle and the drift of a space target’s orbital elements, there are errors in the guide data calculated from orbital elements, and the guide data cannot be used to properly guide the tracking equipment. A general solution is
0003-6935/10/285384-07$15.00/0 © 2010 Optical Society of America
searching around the possible orbit to capture the target, and then implementing tracking and pointing based on the occasional observation.
determination method, the kinematic method based on maneuver target tracking theory, and the kinetic method based on celestial mechanics. These methods can function to some degree in practical applications, but there are insufficiencies and restrictions [5]. The kinetic method has high estimation precision and high calculation cost [6–9], and it is difficult to achieve real-time tracking with this method. Therefore, subsequent positions of a space target are calculated offline from historical measurements, and current measurements cannot be utilized. Because of a long interval between historical measurement and current measurement, prediction errors may be larger than the visual field of the tracking equipment, and prediction data cannot accurately guide the equipment. The kinematic method predicts subsequent positions of a space target by current measurement and is suitable for real-time applications but, owing to kinematic model error, target state estimation precision declines rapidly along with prediction step size. Precise estimation can be obtained by the nonlinear filter algorithm [10] or the high-order model [11], but calculation cost will rise accordingly. The GPS-based geometric method makes use of only the current measurement of the GPS, rather than the kinetic characteristics of a space target to determine orbit [12,13]. This method meets the requirement of all-weather operation with high precision and simple facilities, and the kinetic model of a space target is not necessary. But with a noncooperative space target, the method is inoperable. Based on current measurements and offline predictions of the kinetic method, the errorspace estimation method integrates the kinematic method and the kinetic method and takes the difference between measurement and offline prediction as the state variable of error space. The difference is estimated and predicted with the Kalman filter in error space and corrected for offline orbit prediction data to obtain a space target position with higher prediction accuracy [14].