Iterative Least-Squares Decoding of Analog Product Codes

机器学习题库一、 极大似然1、 ML estimation of exponential model (10)A Gaussian distribution is often used to model data on the real line, but is sometimesinappropriate when the data are often close to zero but constrained to be nonnegative. In such cases one can fit an exponential distribution, whose probability density function is given by()1xb p x e b-=Given N observations x i drawn from such a distribution:(a) Write down the likelihood as a function of the scale parameter b.(b) Write down the derivative of the log likelihood.(c) Give a simple expression for the ML estimate for b.2、换成Poisson 分布：()|,0,1,2,...!x e p x y x θθθ-==()()()()()1111log |log log !log log !N Ni i i i N N i i i i l p x x x x N x θθθθθθ======--⎡⎤=--⎢⎥⎣⎦∑∑∑∑3、二、 贝叶斯假设在考试的多项选择中，考生知道正确答案的概率为p ，猜测答案的概率为1-p ，并且假设考生知道正确答案答对题的概率为1，猜中正确答案的概率为1，其中m 为多选项的数目。

通信系统中的信道编码和解码技术在现代通信系统中，信道编码和解码技术起着至关重要的作用。

信道编码是一种将源数据进行编码的过程，以便在信道传输过程中提高信号的可靠性。

而在接收端，信道解码则是将接收到的编码数据进行解码，恢复为原始数据的过程。

本文将介绍通信系统中常用的信道编码和解码技术。

一、前向纠错编码（Forward Error Correction，FEC）前向纠错编码是一种能够在传输过程中主动纠正错误的编码技术。

其原理是通过在原始数据中添加冗余信息，使接收端能够在接收到有错误的数据包时，根据冗余信息进行纠错，从而恢复出正确的数据。

1. 常见的FEC编码方案（1）海明码（Hamming Code）海明码是一种最早被应用于通信领域的FEC编码方案。

它通过在原始数据中添加校验位，实现了单比特错误的纠正，并且能够检测多比特错误。

海明码的编解码算法相对简单，但纠错能力有限。

（2）LDPC码（Low-Density Parity Check Code）LDPC码是一种基于图论的FEC编码方案。

它通过在校验位的选择上使用低密度的校验矩阵，实现了较高的纠错能力。

LDPC码在现代通信系统中得到广泛应用，尤其是在卫星通信和无线通信领域。

（3）RS码（Reed-Solomon Code）RS码是一种广泛应用于磁盘存储和数字通信领域的FEC编码方案。

它通过在原始数据中添加冗余信息，实现了对一定数量的错误进行纠正。

RS码的编解码复杂度较高，但纠错能力强，适用于对信道质量较差的环境。

2. FEC编码的优势和应用FEC编码在通信系统中具有以下优势：（1）提高信号的可靠性：FEC编码能够在信道传输过程中纠正一定数量的错误，减少信号传输的错误率。

（2）节省带宽资源：通过添加冗余信息，FEC编码可以在一定程度上减少因错误重传导致的带宽浪费。

FEC编码在无线通信、卫星通信、光通信等领域广泛应用。

例如，在卫星通信系统中，由于信号传输距离较长，受到的干扰较多，采用FEC编码可以有效提高通信质量。

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The Method of Least SquaresHervéAbdi11IntroductionThe least square methods(LSM)is probably the most popular tech-nique in statistics.This is due to several factors.First,most com-mon estimators can be casted within this framework.For exam-ple,the mean of a distribution is the value that minimizes the sum of squared deviations of the scores.Second,using squares makes LSM mathematically very tractable because the Pythagorean theo-rem indicates that,when the error is independent of an estimated quantity,one can add the squared error and the squared estimated quantity.Third,the mathematical tools and algorithms involved in LSM(derivatives,eigendecomposition,singular value decomposi-tion)have been well studied for a relatively long time.LSM is one of the oldest techniques of modern statistics,and even though ancestors of LSM can be traced up to Greek mathe-matics,thefirst modern precursor is probably Galileo(see Harper, 1974,for a history and pre-history of LSM).The modern approach wasfirst exposed in1805by the French mathematician Legendre in a now classic memoir,but this method is somewhat older be-cause it turned out that,after the publication of Legendre’s mem-oir,Gauss(the famous German mathematician)contested Legen-1In:Neil Salkind(Ed.)(2007).Encyclopedia of Measurement and Statistics. Thousand Oaks(CA):Sage.Address correspondence to:HervéAbdiProgram in Cognition and Neurosciences,MS:Gr.4.1,The University of Texas at Dallas,Richardson,TX75083–0688,USAE-mail:herve@ /∼hervedre’s priority.Gauss often did not published ideas when he though that they could be controversial or not yet ripe,but would mention his discoveries when others would publish them(the way he did, for example for the discovery of Non-Euclidean geometry).And in1809,Gauss published another memoir in which he mentioned that he had previously discovered LSM and used it as early as1795 in estimating the orbit of an asteroid.A somewhat bitter anterior-ity dispute followed(a bit reminiscent of the Leibniz-Newton con-troversy about the invention of Calculus),which,however,did not diminish the popularity of this technique.The use of LSM in a modern statistical framework can be traced to Galton(1886)who used it in his work on the heritability of size which laid down the foundations of correlation and(also gave the name to)regression analysis.The two antagonistic giants of statis-tics Pearson and Fisher,who did so much in the early develop-ment of statistics,used and developed it in different contexts(fac-tor analysis for Pearson and experimental design for Fisher).Nowadays,the least square method is widely used tofind or es-timate the numerical values of the parameters tofit a function to a set of data and to characterize the statistical properties of esti-mates.It exists with several variations:Its simpler version is called ordinary least squares(OLS),a more sophisticated version is called weighted least squares(WLS),which often performs better than OLS because it can modulate the importance of each observation in thefinal solution.Recent variations of the least square method are alternating least squares(ALS)and partial least squares(PLS). 2Functionalfit example:regressionThe oldest(and still the most frequent)use of OLS was linear re-gression,which corresponds to the problem offinding a line(or curve)that bestfits a set of data points.In the standard formu-lation,a set of N pairs of observations{Y i,X i}is used tofind a function relating the value of the dependent variable(Y)to the values of an independent variable(X).With one variable and alinear function,the prediction is given by the following equation:ˆY=a+bX.(1) This equation involves two free parameters which specify the in-tercept(a)and the slope(b)of the regression line.The least square method defines the estimate of these parameters as the values wh-ich minimize the sum of the squares(hence the name least squares) between the measurements and the model(i.e.,the predicted val-ues).This amounts to minimizing the expression:E= i(Y i−ˆY i)2= i[Y i−(a+bX i)]2(2) (where E stands for“error"which is the quantity to be minimized). The estimation of the parameters is obtained using basic results from calculus and,specifically,uses the property that a quadratic expression reaches its minimum value when its derivatives van-ish.Taking the derivative of E with respect to a and b and setting them to zero gives the following set of equations(called the normal equations):∂E ∂a =2Na+2bXi−2Yi=0(3)and∂E ∂b =2bX2i+2aXi−2Yi X i=0.(4)Solving the normal equations gives the following least square esti-mates of a and b as:a=M Y−bM X(5) (with M Y and M X denoting the means of X and Y)andb= (Yi−M Y)(X i−M X)(Xi−M X)2.(6)OLS can be extended to more than one independent variable(us-ing matrix algebra)and to non-linear functions.2.1The geometry of least squaresOLS can be interpreted in a geometrical framework as an orthog-onal projection of the data vector onto the space defined by the independent variable.The projection is orthogonal because the predicted values and the actual values are uncorrelated.This is il-lustrated in Figure1,which depicts the case of two independent variables(vectors x1and x2)and the data vector(y),and shows that the error vector(y−ˆy)is orthogonal to the least square(ˆy)es-timate which lies in the subspace defined by the two independent variables.yFigure1:The least square estimate of the data is the orthogonal projection of the data vector onto the independent variable sub-space.2.2Optimality of least square estimatesOLS estimates have some strong statistical properties.Specifically when(1)the data obtained constitute a random sample from a well-defined population,(2)the population model is linear,(3)the error has a zero expected value,(4)the independent variables are linearly independent,and(5)the error is normally distributed and uncorrelated with the independent variables(the so-called homo-scedasticity assumption);then the OLS estimate is the b est l inear u nbiased e stimate often denoted with the acronym“BLUE"(the5 conditions and the proof are called the Gauss-Markov conditions and theorem).In addition,when the Gauss-Markov conditions hold,OLS estimates are also maximum likelihood estimates.2.3Weighted least squaresThe optimality of OLS relies heavily on the homoscedasticity as-sumption.When the data come from different sub-populations for which an independent estimate of the error variance is avail-able,a better estimate than OLS can be obtained using weighted least squares(WLS),also called generalized least squares(GLS). The idea is to assign to each observation a weight that reflects the uncertainty of the measurement.In general,the weight w i,as-signed to the i th observation,will be a function of the variance ofthis observation,denotedσ2i .A straightforward weighting schemais to define w i=σ−1i(but other more sophisticated weighted sch-emes can also be proposed).For the linear regression example, WLS willfind the values of a and b minimizing:E w=i w i(Y i−ˆY i)2=iw i[Y i−(a+bX i)]2.(7)2.4Iterative methods:Gradient descentWhen estimating the parameters of a nonlinear function with OLS or WLS,the standard approach using derivatives is not always pos-sible.In this case,iterative methods are very often used.These methods search in a stepwise fashion for the best values of the es-timate.Often they proceed by using at each step a linear approx-imation of the function and refine this approximation by succes-sive corrections.The techniques involved are known as gradient descent and Gauss-Newton approximations.They correspond to nonlinear least squares approximation in numerical analysis and nonlinear regression in statistics.Neural networks constitutes a popular recent application of these techniques3Problems with least squares,and alternativesDespite its popularity and versatility,LSM has its problems.Prob-ably,the most important drawback of LSM is its high sensitivity to outliers(i.e.,extreme observations).This is a consequence of us-ing squares because squaring exaggerates the magnitude of differ-ences(e.g.,the difference between20and10is equal to10but the difference between202and102is equal to300)and therefore gives a much stronger importance to extreme observations.This prob-lem is addressed by using robust techniques which are less sensi-tive to the effect of outliers.Thisfield is currently under develop-ment and is likely to become more important in the next future. References[1]Abdi,H.,Valentin D.,Edelman,B.E.(1999)Neural networks.Thousand Oaks:Sage.[2]Bates,D.M.&Watts D.G.(1988).Nonlinear regression analysisand its applications.New York:Wiley[3]Greene,W.H.(2002).Econometric analysis.New York:PrenticeHall.[4]Harper H.L.(1974–1976).The method of least squares andsome alternatives.Part I,II,II,IV,V,VI.International Satis-tical Review,42,147–174;42,235–264;43,1–44;43,125–190;43,269–272;44,113–159;[5]Nocedal J.&Wright,S.(1999).Numerical optimization.NewYork:Springer.[6]Plackett,R.L.(1972).The discovery of the method of leastsquares.Biometrika,59,239–251.[7]Seal,H.L.(1967).The historical development of the Gauss lin-ear model.Biometrika,54,1–23.。

关于Sethare 猜想的证明1 引言 本文考虑单位圆}1{<=∆z 上的拟共形影射，令∆是单位圆到自身的拟对称同胚，设Q (f)是所有设有边界值∆的拟共形影射f 的集合，拟共形影射0f :∆→ ∆是边界的极值影射，当对应的0h = ∆∂0f 是最大值的缩商Q(0f )的最小值时，记K[0f ]= inf{K[f]: f ∈ Q(0f )} ,这里K[f]是f 的最大值缩商，记A(∆) ={φ(z)是∆的全纯函数；φ=⎰⎰∆)(z φldxdy < ∞}.由文献[1]知f 是极值的当且仅当它的Beltrami 系数μ = z f /z f 有一个Bamilton 序列， {n φ(z) ∈A(∆) : n φ= 1},满足（1.1） 如果（1.1）式成立，亦称μ是极值Beltrami 系数。

如果拟共形影射f 具有下面形式的复特征K 是小于1的正常数，≠ϕ0在∆上全纯，则称它是Teichmiiller 影射。

如果)(z ϕ在∆中，则Teichmiiller 的影射的唯一极性值已经由Strebel[2]得到，当 ϕ不属于∆时，Teichmiiller 的影射极性值已经有许多作者考虑过了，如Sethares[3],Reich 和Strebel[1]，黄[5]，赖和吴[6]。

特别的，如果记ME(∆)等于ϕ(z)，ϕ在∆上全纯且在∆ 上全亚当，当ϕ∈ME(∆)时，在1868年，Sethares[3],猜想：f 是极值的当且仅当在∆∂上要么（i ）ϕ有一个二阶极点，要么（ii ）ϕ没有超过二阶的极点。

他自己证明了充分性部分，留下了必要性部分没解决，本文主要结果之一是证明了猜想的必要性问题。

定理1.假定若其对应的Teichmiiller 影射f 是极值的，则在∆∂上要么ϕ)(i 上有一个二阶极点，要么ϕ)(ii 没有超过二阶的极点。

本文将利用Fehlmann 的局部刻画，特别是有关本质边界点的知识来证明这个定理。

Iteratively reweighted least squaresModelsLinear regressionSimple regressionOrdinary least squaresPolynomial regressionGeneral linear modelGeneralized linear modelDiscrete choiceLogistic regressionMultinomial logitMixed logitProbitMultinomial probitOrdered logitOrdered probitPoissonMultilevel modelFixed effectsRandom effectsMixed modelNonlinear regressionNonparametricSemiparametricRobustQuantileIsotonicPrincipal componentsLeast angleLocalStatistics portalThe method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems. It solves objective functions of the form:by an iterative method in which each step involves solving a weighted least squares problem of the form:IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set. For example, by minimizing the least absolute error rather than the least square error.Although not a linear regression problem, Weiszfeld's algorithm for approximating the geometric median can also be viewed as a special case of iteratively reweighted least squares, in which the objective function is the sum of distances of the estimator from the samples.One of the advantages of IRLS over linear and convex programming is that it can be used with Gauss–Newton and Levenberg–Marquardt numerical algorithms.ExamplesL 1 minimization for sparse recoveryIRLS can be used for1 minimization and smoothed p minimization, p < 1, in the compressed sensing problems.It has been proved that the algorithm has a linear rate of convergence for1 norm and superlinear fort with t < 1,under the restricted isometry property, which is generally a sufficient condition for sparse solutions.[1][2]L p norm linear regressionTo find the parameters β = (β1, …,βk )T which minimize the L p norm for the linear regression problem,the IRLS algorithm at step t +1 involves solving the weighted linear least squares[3] problem:[4]where W (t ) is the diagonal matrix of weights with elements:In the case p = 1, this corresponds to least absolute deviation regression (in this case, the problem would be better approached by use of linear programming methods).[citation needed ]Notes[3]/%7Edispenser/cgi-bin/dab_solver.py?page=Iteratively_reweighted_least_squares&editintro=Template:Disambiguation_needed/editintro&client=Template:DnReferences•University of Colorado Applied Regression lecture slides (/courses/7400/2010Spr/lecture23.pdf)•Stanford Lecture Notes on the IRLS algorithm by Antoine Guitton (/public/docs/sep103/antoine2/paper_html/index.html)•Numerical Methods for Least Squares Problems by Åke Björck (http://www.mai.liu.se/~akbjo/LSPbook.html) (Chapter 4: Generalized Least Squares Problems.)•Practical Least-Squares for Computer Graphics. SIGGRAPH Course 11 (/~jplewis/lscourse/SLIDES.pdf)Article Sources and Contributors4Article Sources and ContributorsIteratively reweighted least squares Source: /w/index.php?oldid=536800201 Contributors: 3mta3, BD2412, BenFrantzDale, Benwing, Colbert Sesanker, DavidEppstein, Giggy, Grumpfel, Kiefer.Wolfowitz, Lambiam, Lesswire, LutzL, Melcombe, Michael Hardy, Oleg Alexandrov, RainerBlome, Salix alba, Serg3d2, Stpasha, Wesleyyin, 13 anonymous editsImage Sources, Licenses and Contributorsfile:linear regression.svg Source: /w/index.php?title=File:Linear_regression.svg License: Public Domain Contributors: SewaquFile:Fisher iris versicolor sepalwidth.svg Source: /w/index.php?title=File:Fisher_iris_versicolor_sepalwidth.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors: en:User:Qwfp (original); Pbroks13 (talk) (redraw)LicenseCreative Commons Attribution-Share Alike 3.0 Unported///licenses/by-sa/3.0/。

1.CT is a new technique in x-ray imaging that does not:(a)use a computer to reconstruct cross-sectional images of an object;(b)use mathematical methods to reconstruct cross-sectional images of an object;(c)use special detectors to detect x-rays that have passed through an object at multiple angles;(d)Use x-ray film as the image receptorCT is a new type of cross-sectional tomographic imaging in which all unwanted planes or layers of a body are completely eliminated using mathematical techniques;The goal of CT is to detect radiation that has passed through a body at multiple angles, and with the aid of a computer , to reconstruct a cross section of absorption values for that body section;2.who developed the first whole body CT scanner?(a)Hounsfield; (b)Ambrose;(c)Ledley; (d)general electric company3.which generation of CT scanners is based on the translate/rotate principle using a single x-ray beam?(a)First-generation; (b)second-generation(c)Third-generation; (d)fourth-generation4.The main difference of the fourth-generation scanner compared to the earlier scanners is:(a)the tube and detectors rotate together around the patient;(b)The use of slip ring;(c)the tube rotates with a stationery ring of detectors;(d)there is decrease in number of detectors;5.The first reconstruction from projections was used by:(a)Radon; (b)Hounsfield;(c)Roentgen; (d)Rowley6.why is a third-generation system sometimes referred to as a rotate-rotate scanner?(a)the tube first rotates in one direction,then stops to rotate in the opposite direction;(b)the tube continues to rotate in the same dirction;(c)The tube and detector rotate in unison,facing ipposite one another;(d)the tube and detector each rotate in the opposite direction;7.CT is superior to conventional film/screen radiography in its:(a)ability to provide radiographic images with a much lower radiation dose;(b)capacity for surveying many diverse anatomic areas quickly;(c)Ability to resolve small differences in tissue densities;(d)Spatial resolution;The goal of CT is to overcome the limitations of radiography and tomography by achieving the following: 1.minimizing the problem of superimpositionImproving the contrast of the image;Finding a way to record very small differences in tissue contrast.六1. the purpose of a rotating anode is to : (d)a)decrease the scan time;b)increase the photon quality;c)decrease th e patient radiation dose;d)increase the heat capacity;Review questions2. in a CT system, the x-ray that passes through the patient hits the detector and is converted into light. what type of detector has this system possess?a)solid-state;b)xenon gas;c)either solid-state or xenon gas; d)aluminum;Review questions3. which of the following are characteristics of solid-state detectors? (b,c,da)low efficiency;b)sensitive to temperature and moisture;c)may exhibit afterglow;d)also called scintillators.Review questions4. the purpose of filtration in CT is: (a,c)a)to absorb “soft” radiation; b)increase speed of detectors;c)to produce uniform x-ray transmissiond)decrease the scan time;Review questions5. what is the function of analog-to-digital converters? (c)a) they translate the raw data onto a matrix;b) they convert the light levels from a solid-state detector into an electric current;c) they convert electric signals into a digital format;d) they take digitized data and translate it into shades of gray;Review questions1. when compared to multiformat camera images,laser-printed images have: (b,d)a)lower resolution and greater distortion;b)higher resolution and less distortion;c)higher resolution and increased phosphor noise;d)the performance of using a narrow, non-divergent laser beam that exposes the film.七1.the first reconstruction from projections was used by:RadonHounsfield;Roentgen;Kuhl;2.which of the following is not an iterative method of image reconstruction from projections? Algebraic reconstruction technique;Simultaneous iterative reconstructive technique;Iterative least-squares techniqueTwo-dimensional Fourier reconstruction;3.in which of the following is a blurring pattern present?Back projection;Iterative method;Fourier method;Convolution method;4.which of the following involves applying suitable filtering formulas to the projection data before they are back projected to produce an image free of blurring;Back projection;Iterative method;Fourier method;Convolution method;5.with respect to accuracy,which of the following yields accurate results very quickly?Algebraic reconstruction technique;Back projection;Convolution method;Iterative least-squares technique;CT设备与原理第3章测验班级：姓名：分数：一、选择题1、旋转X射线管的主要优点是：（）A．缩短扫描时间；B.产生的线质硬；C.降低病人辐射剂量；D.增加热容量；2、滤过器的主要作用是：（）A、缩短扫描时间；B.缩短扫描时间；C、去除低能量的光子；D、提高探测器效率；3、采集和重建图像需要以下特别的硬件设备：（）A、X射线管和数字采集系统；B、滑环技术；C、CPU和光电放大管；4、在CT设备中应用最广泛的探测器类型有：（）A、电离室和盖式计数器；B、盖式探测器和荧光探测器；C、气体探测器和荧光探测器；5、CT设备中输入装置有：（）A、阵列处理器；B、中央处理单元（CPU）；C、微型处理器；D、键盘6、CT设备中探测器的作用：（）A、测量散射线；B、测量穿过病人的射线量；C、决定扫描层数D、以上全部7、中央处理单元的功能是：（）A、把模拟信号变成数字信号；B、探测转换后的光子；C、控制计算机的其它部件；D、把衰减系数取平均；8、在CT设备中决定层厚的有：（）A、准直器孔径大小；B、象素宽度；C、计算机程序；D、机架旋转速度。