Matrix Identities on Weighted Partial Motzkin Paths
Published
Peter Sollich
Abstract
We present a new method for obtaining the response function G and its average G from which most of the properties of learning and generalization in linear perceptrons can be derived. We rst rederive the known results for the `thermodynamic limit' of in nite perceptron size N and show explicitly that G is self-averaging in this limit. We then discuss extensions of our method to more general learning scenarios with anisotropic teacher space priors, input distributions, and weight decay terms. Finally, we use our method to calculate the nite N corrections of order 1=N to G and discuss the corresponding nite size e ects on generalization and learning dynamics. An important spin-o is the observation that results obtained in the thermodynamic limit are often directly relevant to systems of fairly modest, `real-world' sizes.
摘录不饱和聚酯文献中的经典句子
1.To access the description of a composite material, it will be necessary to specify the nature of components and their properties, the geometry of the reinforcement, its distribution, and the nature of the reinforcement–matrix interface.2. However, most of them are not chemically compatible with polymers3. That’s why for many years, studies have been conducted on particles functionalization to modulate the physical and/or chemical properties and to improve the compatibility between the filler and the matrix [7].4. Silica is used in a wide range of products including tires, scratch-resistant coatings, toothpaste,medicine, microelectronics components or in the building5. Fracture surface of test specimens were observed by scanning electron microscopy6.Test specimens were prepared by the following method from a mixture composed with 40 wt% UPE, 60 wt% silica Millisil C6 and components of ‘‘Giral.’7.Grafted or adsorbed component amounts on modified silica samples were assessed by thermogravimetric analysis (TGA) using a TGA METTLER-TOLEDO 851e thermal system. For the analysis, about 10–20 mg of samples were taken and heated at a constant rate of 10 C/min under air (purge rate 50 mL/min) from 30 to 1,100 C.8.Nanocomposites with different concentrations of nanofibers wereproduced and tested, and their properties were compared with those of the neat resin.9.Basically, six different percentages were chosen, namely 0.1, 0.3, 0.5, 1, 2, 3 wt %.10.TEM images of cured blends were obtained with a Philips CM120 microscope applying an acceleration voltage of 80 kV.Percolation threshold of carbon nanotubes filled unsaturated polyesters 11.For further verification, the same experiment was carried out for the unmodified UP resin, and the results showed that there were no endothermic peaks12.The MUP resin was checked with d.s.c, scanning runs at a heating rate of 10°C min 1. Figure 4a shows that an endothermic peak appeared from 88 to 133°C, which indicates bond breaking in that temperature range.13.On the basis of these results, it is concluded that a thermally breakable bond has been introduced into the MUP resin and that the decomposition temperature is around I lO°C.14.The structures of the UP before and after modification were also checked with FTi.r. Figure 5 shows a comparison of the i.r. spectra of the unmodified and modified UP resins.15This is probably a result of the covalent bonding ofthe urethane linkage being stronger than the ionic bondingof MgO.16.These examples show that different viscosity profiles can be designed with different combinations of the resins and thickeners according to the needs of the applications.17. A small secondary reaction peak occurred at higher temperatures, probably owing to thermally induced polymerization. 18.Fiber-reinforced composite materials consist of fibers of high strength and modulus embedded in or bonded to a matrix with a distinct interfaces between them.19.In this form, both fibers and ma-trix retain their physical and chemical identities,yet they provide a combination of properties that cannot be achieved with either of the constituents acting alone.20.In general, fibers are the principal load-bearing materials, while the surrounding matrix keep them in the desired location, and orientation acts as a load transfer medium between them and protects them from environmental damage.21.Moreover, both the properties, that is,strength and stiffness can be altered according to our requirement by altering the composition of a single fiber–resin combination.22.Again, fiber-filled composites find uses in innumerable applied ar- eas by judicious selection of both fiber and resin.23.In recent years, greater emphasis has been rendered in the development of fiber-filled composites based on natural fibers with a view to replace glass fibers either solely or in part for various applications. 24.The main reasons of the failure are poor wettability and adhesion characteristics of the jute fiber towards many commercial synthetic resins, resulting in poor strength and stiffness of the composite as well as poor environmental resistance.25.Therefore, an attempt has been made to overcome the limitations of the jute fiber through its chemical modification.26.Dynamic mechanical tests, in general, give more information about a composite material than other tests. Dynamic tests, over a wide range of temperature and frequency, are especially sensitive to all kinds of transitions and relaxation process of matrix resin and also to the morphology of the composites.27.Dynamic mechanical analysis (DMA) is a sensitive and versatile thermal analysis technique, which measures the modulus (stiffness) and damping properties (energy dissipation) of materials as the materials are deformed under periodic stress.28.he object of the present article is to study the effect of chemical modification (cyanoethylation)of the jute fiber for improving its suitability as a reinforcing material in the unsaturated polyesterres in based composite by using a dynamic mechanical thermal analyzer.30.General purpose unsaturated polyester resin(USP) was obtained from M/S Ruia Chemicals Pvt. Ltd., which was based on orthophthalic anhydride, maleic anhydride, 1,2-propylene glycol,and styrene.The styrene content was about 35%.Laboratory reagentgrade acrylonitrile of S.D.Fine Chemicals was used in this study without further purification. 31.Tensile and flexural strength of the fibers an d the cured resin were measured by Instron Universal Testing Machine (Model No. 4303).32.Test samples (60 3 11 3 3.2 mm) were cut from jute–polyester laminated sheets and were postcured at 110°C for 1 h and conditionedat 65% relative humidity (RH) at 25°C for 15 days.33.In DMA, the test specimen was clamped between the ends of two parallel arms, which are mounted on low-force flexure pivots allowing motion only in the horizontal plane. The samples in a nitrogen atmosphere were measured in the fixed frequency mode, at an operating frequency 1.0 HZ (oscillation amplitude of 0.2 mm) and a heating rate of 4°C per min. The samples were evaluated in the temperature range from 40 to 200°C.34.In the creep mode of DMA, the samples were stressed for 30 min at an initial temperature of 40°C and allowed to relax for 30 min. The tem- perature was then increased in the increments of 40°C, followed by an equilibrium period of 10min before the initiation of the next stress relax cycle. This program was continued until it reached the temperature of160°C. All the creep experiments were performed at stress level of20 KPa (approximate).35.The tensile fracture surfaces of the composite samples were studied with a scanning electron microscope (Hitachi Scanning electron Microscope, Model S-415 A) operated at 25 keV.36.The much im proved moduli of the five chemically modified jute–polyester composites might be due to the greater interfacial bond strength between the ma trix resin and the fiber.37.The hydrophilic nature of jute induces poor wettability and adhesion characteristics with USP resin, and the presence of moisture at the jute–resin interface promotes the formation of voids at the interface. 38.On the other hand, owing to cyanoethylation, the moisture regain capacity of the jute fiber is much reduced; also, the compatibility with unsaturated polyester resin has been improved and produces a strong interfacial bond with matrix resin and produces a much stiffer composite.39.Graphite nanosheets(GN), nanoscale conductive filler has attracted significant attention, due to its abundance in resource and advantage in forming conducting network in polymer matrix40.The percolation threshold is greatly affected by the properties of the fillers and the polymer matrices,processing met hods, temperature, and other related factors41.Preweighted unsaturated polyester resin and GN were mixed togetherand sonicated for 20 min to randomly disperse the inclusions.42.Their processing involves a radical polymerisation between a prepolymer that contains unsaturated groups and styrene that acts both asa diluent for the prepolymer and as a cross-linking agent.43.They are used, alone or in fibre-reinforced composites, in naval constructions, offshore applications,water pipes, chemical containers, buildings construction, automotive, etc.44.Owing to the high aspect ratio of the fillers, the mechanical, thermal, flame retardant and barrier properties of polymers may be enhanced without a significant loss of clarity, toughness or impact strength.45.The peak at 1724 cm-1was used as an internal reference, while the degree of conversion for C=C double bonds in the UP chain was determined from the peak at 1642 cm-1and the degree of conversion for styrene was calculated through the variation of the 992 cm-1peak46. Paramount to this scientific analysis is an understanding of the chemorheology of thermosets.47．Although UPR are used as organic coatings, they suffer from rigidity, low acid and alkali resistances and low adhesion with steel when cured with c onventional ‘‘small molecule’’ reagents.48．Improvements of resin flexibility can be obtained by incorporating long chain aliphatic com-pounds into the chemical structure of UPR. 47．In this study, both UPR and hardeners were based on aliphatic andcycloaliphatic systems to produce cured UPR, which have good durability with excellent mechan-ical properties.50．UPR is one of the widely used thermoset polymers in polymeric composites, due to their good mechanical properties and relatively inexpensive prices.51．[文档可能无法思考全面，请浏览后下载，另外祝您生活愉快，工作顺利，万事如意!]。
矩阵逆学习资料woodbury公式
矩阵逆学习资料woodbury公式关于矩阵逆的补充学习资料：Woodbury matrix identity本⽂来⾃维基百科。
the Woodbury matrix identity, named after Max A. Woodbury[1][2] says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report.[3]The Woodbury matrix identity is[4]where A, U, C and V all denote matrices of the correct size. Specifically, A is n-by-n, U is n-by-k, C is k-by-k and V is k-by-n. This can be derived using blockwise matrix inversion.In the special case where C is the 1-by-1 unit matrix, this identity reduces to the Sherman–Morrison formula. In the special case when C is the identity matrix I, the matrix is known innumerical linear algebra and numerical partial differential equations as the capacitance matrix.[3]Direct proofJust check that times the RHS of the Woodbury identity gives the identity matrix:Derivation via blockwise eliminationDeriving the Woodbury matrix identity is easily done by solving the following block matrix inversion problemExpanding, we can see that the above reduces to and, which is equivalent to .Eliminating the first equation, we find that , which can be substituted into the second to find. Expanding and rearranging, we have, or . Finally, we substitute into our , and we have. Thus,We have derived the Woodbury matrix identity.Derivation from LDU decompositionWe start by the matrixBy eliminating the entry under the A (given that A is invertible) we getLikewise, eliminating the entry above C givesNow combining the above two, we getMoving to the right side giveswhich is the LDU decomposition of the block matrix into an upper triangular, diagonal, and lower triangular matrices.Now inverting both sides givesWe could equally well have done it the other way (provided that C is invertible) i.e.Now again inverting both sides,Now comparing elements (1,1) of the RHS of (1) and (2) above gives the Woodbury formulaApplicationsThis identity is useful in certain numerical computations where A?1has already been computed and it is desired to compute (A+ UCV)?1. With the inverse of A available, it is only necessary tofind the inverse of C?1+ VA?1U in order to obtain the result using the right-hand side of the identity. If C has a much smaller dimension than A, this is more efficient than inverting A+ UCV directly. A common case is finding the inverse of a low-rank update A+ UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A+ B where the matrix can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.This is applied, e.g., in the Kalman filter and recursive least squares methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new observation is processed at a time. This significantly speeds up the often real time calculations of the filter.See also：Sherman–Morrison formulaInvertible matrixSchur complementMatrix determinant lemma, formula for a rank-k update to a determinantBinomial inverse theorem; slightly more general identity. Notes：1.Jump up ^ Max A. Woodbury, Inverting modified matrices, MemorandumRept. 42, Statistical Research Group, Princeton University,Princeton, NJ, 1950, 4pp MR381362.Jump up ^ Max A. Woodbury, The Stability of Out-Input Matrices.Chicago, Ill., 1949. 5 pp. MR325643.^ Jump up to: a b Hager, William W. (1989). "Updating the inverse of amatrix". SIAM Review31 (2): 221–239. doi:10.1137/1031049.JSTOR2030425. MR997457.4.Jump up ^Higham, Nicholas (2002). Accuracy and Stability ofNumerical Algorithms (2nd ed.). SIAM. p. 258. ISBN978-0-89871-521-7. MR1927606.Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 2.7.3. Woodbury Formula", Numerical Recipes: The Artof Scientific Computing (3rd ed.), New York: CambridgeUniversity Press, ISBN978-0-521-88068-8External links：Some matrix identitiesWeisstein, Eric W., "Woodbury formula", MathWorld.src="///doc/2742834010661ed9ad51f3e6.html /wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;" />Retrieved from"/doc/2742834010661ed9ad51f3e6.html /w/index.php?title=Woodbury_matrix_identity&oldid=627695139"Categories:Linear algebraLemmasSherman–Morrison formula（来⾃维基百科。
[转载]图的拉普拉斯矩阵学习-Laplacian
![[转载]图的拉普拉斯矩阵学习-Laplacian](https://img.taocdn.com/s3/m/528874d72dc58bd63186bceb19e8b8f67c1cefa5.png)
[转载]图的拉普拉斯矩阵学习-Laplacian Matrices of Graphs We all learn one way of solving linear equations when we first encounter linearalgebra: Gaussian Elimination. In this survey, I will tell the story of some remarkable connections between algorithms, spectral graph theory, functional analysisand numerical linear algebra that arise in the search for asymptotically faster algorithms.I will only consider the problem of solving systems of linear equationsin the Laplacian matrices of graphs. This is a very special case, but it is also avery interesting case. I begin by introducing the main characters in the story.1. Laplacian Matrices and Graphs. We will consider weighted, undirected,simple graphs G given by a triple (V,E,w), where V is a set of vertices, Eis a set of edges, and w is a weight function that assigns a positive weight toevery edge. The Laplacian matrix L of a graph is most naturally defined bythe quadratic form it induces. For a vector x ∈ IRV , the Laplacian quadraticform of G is：Thus, L provides a measure of the smoothness of x over the edges in G. Themore x jumps over an edge, the larger the quadratic form becomes.The Laplacian L also has a simple description as a matrix. Define theweighted degree of a vertex u by：Define D to be the diagonal matrix whose diagonal contains d, and definethe weighted adjacency matrix of G by:We haveL = D − A.It is often convenient to consider the normalized Laplacian of a graph insteadof the Laplacian. It is given by D−1/2LD−1/2, and is more closely related tothe behavior of random walks.Regression on Graphs. Imagine that you have been told the value of afunction f on a subset W of the vertices of G, and wish to estimate thevalues of f at the remaining vertices. Of course, this is not possible unlessf respects the graph structure in some way. One reasonable assumption isthat the quadratic form in the Laplacian is small, in which case one mayestimate f by solving for the function f : V → IR minimizing f TLf subjectto f taking the given values on W (see [ZGL03]). Alternatively, one couldassume that the value of f at every vertex v is the weighted average of f atthe neighbors of v, with the weights being proportional to the edge weights.In this case, one should minimize:|D-1Lf|subject to f taking the given values on W. These problems inspire manyuses of graph Laplacians in Machine Learning.。
Introduction to Linear Algebra
»a = 5 a= 5
A vector is a mathematical quantity that is completely described by its magnitude and direction. An example of a three dimensional column vector might be 4 b= 3 5 uld easily assign bT to another variable c, as follows:
»c = b' c= 4 3 5
A matrix is a rectangular array of scalars, or in some instances, algebraic expressions which evaluate to scalars. Matrices are said to be m by n, where m is the number of rows in the matrix and n is the number of columns. A 3 by 4 matrix is shown here 2 A= 7 5 5 3 2 3 2 0 6 1 3 (3)
»a = 5;
Here we have used the semicolon operator to suppress the echo of the result. Without this semicolon MATLAB would display the result of the assignment:
»A(2,4) ans = 1
The transpose operator “flips” a matrix along its diagonal elements, creating a new matrix with the ith row being equal to the jth column of the original matrix, e.g. T A = 2 5 3 6 7 3 2 1 5 2 0 3
Matrix Derivative_wiki
Matrix calculusIn mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. This notation was to describe systems of differential equations, and taking derivatives of matrix-valued functions with respect to matrix variables. This notation is commonly used in statistics and engineering, while the tensor index notation is preferred in physics.NoticeThis article uses another definition for vector and matrix calculus than the form often encountered within the field of estimation theory and pattern recognition. The resulting equations will therefore appear to be transposed when compared to the equations used in textbooks within these fields.NotationLet M(n,m) denote the space of real n×m matrices with n rows and m columns, such matrices will be denoted using bold capital letters: A, X, Y, etc. An element of M(n,1), that is, a column vector, is denoted with a boldface lowercase letter: a, x, y, etc. An element of M(1,1) is a scalar, denoted with lowercase italic typeface: a, t, x, etc. X T denotes matrix transpose, tr(X) is trace, and det(X) is the determinant. All functions are assumed to be of differentiability class C1 unless otherwise noted. Generally letters from first half of the alphabet (a, b, c, …) will be used to denote constants, and from the second half (t, x, y, …) to denote variables.Vector calculusBecause the space M(n,1) is identified with the Euclidean space R n and M(1,1) is identified with R, the notations developed here can accommodate the usual operations of vector calculus.•The tangent vector to a curve x : R→ R n is•The gradient of a scalar function f : R n→ RThe directional derivative of f in the direction of v is then•The pushforward or differential of a function f : R m→ R n is described by the Jacobian matrix The pushforward along f of a vector v in R m isMatrix calculusFor the purposes of defining derivatives of simple functions, not much changes with matrix spaces; the space of n×m matrices is isomorphic to the vector space R nm. The three derivatives familiar from vector calculus have close analogues here, though beware the complications that arise in the identities below.•The tangent vector of a curve F : R→ M(n,m)•The gradient of a scalar function f : M(n,m) → RNotice that the indexing of the gradient with respect to X is transposed as compared with the indexing of X. The directional derivative of f in the direction of matrix Y is given by•The differential or the matrix derivative of a function F : M(n,m) → M(p,q) is an element of M(p,q) ⊗ M(m,n), a fourth-rank tensor (the reversal of m and n here indicates the dual space of M(n,m)). In short it is an m×n matrix each of whose entries is a p×q matrix.is a p×q matrix defined as above. Note also that this matrix has its indexing and note that each ∂F/∂Xi,jtransposed; m rows and n columns. The pushforward along F of an n×m matrix Y in M(n,m) is thenas formal block matrices.Note that this definition encompasses all of the preceding definitions as special cases.According to Jan R. Magnus and Heinz Neudecker, the following notations are both unsuitable, as the determinants of the resulting matrices would have "no interpretation" and "a useful chain rule does not exist" if these notations are being used:[1]1.2.The Jacobian matrix, according to Magnus and Neudecker,[1] isIdentitiesNote that matrix multiplication is not commutative, so in these identities, the order must not be changed.•Chain rule: If Z is a function of Y which in turn is a function of X, and these are all column vectors, then•Product rule:In all cases where the derivatives do not involve tensor products (for example, Y has more than one row and X has more than one column),ExamplesDerivative of linear functionsThis section lists some commonly used vector derivative formulas for linear equations evaluating to a vector.Derivative of quadratic functionsThis section lists some commonly used vector derivative formulas for quadratic matrix equations evaluating to a scalar.Related to this is the derivative of the Euclidean norm:Derivative of matrix tracesThis section shows examples of matrix differentiation of common trace equations.Derivative of matrix determinantRelation to other derivativesThe matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The Fréchet derivative is the standard way in the setting of functional analysis to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree up to translation of notations. As is the case in general for partial derivatives, some formulae may extend under weaker analytic conditions than the existence of the derivative as approximating linear mapping.UsagesMatrix calculus is used for deriving optimal stochastic estimators, often involving the use of Lagrange multipliers. This includes the derivation of:•Kalman filter•Wiener filter•Expectation-maximization algorithm for Gaussian mixtureAlternativesThe tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except one writes only a single component at a time. It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation. Note that a matrix can be considered simply a tensor of rank two.Notes[1]Magnus, Jan R.; Neudecker, Heinz (1999 (1988)). Matrix Differential Calculus. Wiley Series in Probability and Statistics (revised ed.).Wiley. pp. 171–173.External links•Matrix Calculus (/engineering/cas/courses.d/IFEM.d/IFEM.AppD.d/IFEM.AppD.pdf) appendix from Introduction to Finite Element Methods book on University of Colorado at Boulder.Uses the Hessian (transpose to Jacobian) definition of vector and matrix derivatives.•Matrix calculus (/hp/staff/dmb/matrix/calculus.html) Matrix Reference Manual , Imperial College London.•The Matrix Cookbook (), with a derivatives chapter. Uses the Hessian definition.Article Sources and Contributors5Article Sources and ContributorsMatrix calculus Source: /w/index.php?oldid=408981406 Contributors: Ahmadabdolkader, Albmont, Altenmann, Arthur Rubin, Ashigabou, AugPi, Blaisorblade,Bloodshedder, CBM, Charles Matthews, Cooli46, Cs32en, Ctacmo, DJ Clayworth, DRHagen, Dattorro, Dimarudoy, Dlohcierekim, Enisbayramoglu, Eroblar, Esoth, Excirial, Fred Bauder,Freddy2222, Gauge, Geometry guy, Giftlite, Giro720, Guohonghao, Hu12, Immunize, Jan mei118, Jitse Niesen, Lethe, Michael Hardy, MrOllie, NawlinWiki, Oli Filth, Orderud, Oussjarrouse, Ozob, Pearle, RJFJR, Rich Farmbrough, SDC, Sanchan89, Stpasha, TStein, The Thing That Should Not Be, Vgmddg, Willking1979, Xiaodi.Hou, Yuzisee, 170 anonymous editsLicenseCreative Commons Attribution-Share Alike 3.0 Unported/licenses/by-sa/3.0/。
中山大学计算机学院离散数学基础教学大纲(2019)
中山大学本科教学大纲Undergraduate Course Syllabus学院（系）：数据科学与计算机学院School (Department)：School of Data and Computer Science课程名称：离散数学基础Course Title：Discrete Mathematics二〇二〇年离散数学教学大纲Course Syllabus: Discreate Mathematics（编写日期：2020 年12 月）(Date: 19/12/2020)一、课程基本说明I. Basic Information二、课程基本内容 II. Course Content（一）课程内容i. Course Content1、逻辑与证明（22学时） Logic and Proofs (22 hours)1.1 命题逻辑的语法和语义(4学时) Propositional Logic (4 hours)命题的概念、命题逻辑联结词和复合命题，命题的真值表和命题运算的优先级，自然语言命题的符号化Propositional Logic, logic operators (negation, conjunction, disjunction, implication, bicondition), compound propositions, truth table, translating sentences into logic expressions1.2 命题公式等值演算(2学时) Logical Equivalences (2 hours)命题之间的关系、逻辑等值和逻辑蕴含，基本等值式，等值演算Logical equivalence, basic laws of logical equivalences, constructing new logical equivalences1.3 命题逻辑的推理理论（2学时）论断模式，论断的有效性及其证明，推理规则，命题逻辑中的基本推理规则（假言推理、假言易位、假言三段论、析取三段论、附加律、化简律、合取律）,构造推理有效性的形式证明方法Argument forms, validity of arguments, inference rules, formal proofs1.4 谓词逻辑的语法和语义 (4学时) Predicates and Quantifiers (4 hours)命题逻辑的局限，个体与谓词、量词、全程量词与存在量词，自由变量与约束变量，谓词公式的真值，带量词的自然语言命题的符号化Limitations of propositional logic, individuals and predicates, quantifiers, the universal quantification and conjunction, the existential quantification and disjunction, free variables and bound variables, logic equivalences involving quantifiers, translating sentences into quantified expressions.1.4 谓词公式等值演算(2学时) Nested Quantifiers (2 hours)谓词公式之间的逻辑蕴含与逻辑等值，带嵌套量词的自然语言命题的符号化，嵌套量词与逻辑等值Understanding statements involving nested quantifiers, the order of quantifiers, translating sentences into logical expressions involving nested quantifiers, logical equivalences involving nested quantifiers.1.5谓词逻辑的推理规则和有效推理(4学时) Rules of Inference (4 hours)证明的基本含和证明的形式结构，带量词公式的推理规则（全程量词实例化、全程量词一般化、存在量词实例化、存在量词一般化），证明的构造Arguments, argument forms, validity of arguments, rules of inference for propositional logic (modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplication, conjunction), using rules of inference to build arguments, rules of inference for quantified statements (universal instantiation, universal generalization, existential instantiation, existential generalization)1.6 数学证明简介(2学时) Introduction to Proofs (2 hours)数学证明的相关术语、直接证明、通过逆反命题证明、反证法、证明中常见的错误Terminology of proofs, direct proofs, proof by contraposition, proof by contradiction, mistakes in proofs1.7 数学证明方法与策略初步(2学时) Proof Methods and Strategy (2 hours)穷举法、分情况证明、存在命题的证明、证明策略（前向与后向推理）Exhaustive proof, proof by cases, existence proofs, proof strategies (forward and backward reasoning)2、集合、函数和关系（18学时）Sets, Functions and Relations(18 hours)2.1 集合及其运算(3学时) Sets (3 hours)集合与元素、集合的表示、集合相等、文氏图、子集、幂集、笛卡尔积Set and its elements, set representations, set identities, Venn diagrams, subsets, power sets, Cartesian products.集合基本运算（并、交、补）、广义并与广义交、集合基本恒等式Unions, intersections, differences, complements, generalized unions and intersections, basic laws for set identities.2.2函数(3学时) Functions (3 hours)函数的定义、域和共域、像和原像、函数相等、单函数与满函数、函数逆与函数复合、函数图像Functions, domains and codomains, images and pre-images, function identity, one-to-one and onto functions, inverse functions and compositions of functions.2.3. 集合的基数（1学时）集合等势、有穷集、无穷集、可数集和不可数集Set equinumerous, finite set, infinite set, countable set, uncountable set.2.4 集合的归纳定义、归纳法和递归（3学时）Inductive sets, inductions and recursions (3 hours)自然数的归纳定义，自然数上的归纳法和递归函数；数学归纳法（第一数学归纳法）及应用举例、强归纳法（第二数学归纳法）及应用举例；集合一般归纳定义模式、结构归纳法和递归函数。
matrix_cookbook
Kaare Brandt Petersen Michael Syskind Pedersen Version: October 3, 2005
What is this? These pages are a collection of facts (identities, approximations, inequalities, relations, ...) about matrices and matters relating to them. It is collected in this form for the convenience of anyone who wants a quick desktop reference . Disclaimer: The identities, approximations and relations presented here were obviously not invented but collected, borrowed and copied from a large amount of sources. These sources include similar but shorter notes found on the internet and appendices in books - see the references for a full list. Errors: Very likely there are errors, typos, and mistakes for which we apologize and would be grateful to receive corrections at cookbook@2302.dk. Its ongoing: The project of keeping a large repository of relations involving matrices is naturally ongoing and the version will be apparent from the date in the header. Suggestions: Your suggestion for additional content or elaboration of some topics is most welcome at cookbook@2302.dk. Acknowledgements: We would like to thank the following for discussions, proofreading, extensive corrections and suggestions: Esben Hoegh-Rasmussen and Vasile Sima. Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, differentiate a matrix.
SPSS词汇中英文对照
SPSS词汇(中英文对照) Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 一般线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换。
label matrix
label matrixLabel Matrix: An Ultimate GuideIntroductionIn the world of data analysis and machine learning, label matrix is an essential concept that plays a crucial role in various tasks such as classification, clustering, and supervised learning. This guide aims to delve into the depths of label matrix, explaining its definition, properties, and applications.What is a Label Matrix?A label matrix, also known as a target matrix or classification matrix, is a table-like data structure that represents the labels or categories to which data points belong. It is often used in supervised machine learning tasks, where the objective is to predict the label or category of unseen data based on a trained model.Properties of a Label Matrix1. Dimensions: A label matrix has two dimensions - rows and columns. The rows represent the unique data points or instances, while the columns represent the distinct labels or categories.2. Binary or Multiclass: A label matrix can be binary or multiclass, depending on the nature of the classification task. In binary classification, there are only two possible labels, often denoted as 0 or 1. On the other hand, multiclass classification involves more than two labels.3. Sparse or Dense: Label matrices can be sparse, meaning that a majority of the entries are empty or zero, or dense, where most of the entries have non-zero values. The sparsity of a label matrix depends on the distribution of the labels in the dataset.4. Class Imbalance: Class imbalance refers to the scenario where one or more labels have significantly more instances compared to others. This property is common in real-world datasets and can affect the model's performance. Handling class imbalance is crucial in machine learning tasks.Applications of Label Matrix1. Classification: The primary application of label matrix is in classification tasks. Given a labeled dataset, a model is trained using an algorithm such as logistic regression, support vector machines, or deep learning techniques. The label matrix is used during model training and evaluation, allowing the model to learn the relationship between the input features and the corresponding labels.2. Evaluation Metrics: Label matrix is essential in evaluating the performance of a classification model. Various evaluation metrics such as accuracy, precision, recall, and F1 score are calculated based on the values in the label matrix. These metrics provide insights into the model's predictive power and its ability to correctly classify different labels.3. Imbalanced Data Analysis: As mentioned earlier, label matrices often exhibit class imbalance. This property requires special attention to ensure the model performs well on minority classes. Various techniques such as oversampling, undersampling, and cost-sensitive learning can be applied to tackle class imbalance.4. Interpretation and Visualization: Label matrices can also be used for visualization and interpretation purposes.Techniques such as confusion matrices, heatmaps, and precision-recall curves can provide insights into the model's strengths and weaknesses in classifying different labels. These visualizations aid in identifying patterns and making informed decisions.Tips for Working with Label Matrices1. Preprocessing: Before working with label matrices, it is essential to preprocess the data. This may involve handling missing values, encoding categorical variables, and scaling numerical features. The quality of the label matrix greatly depends on the preprocessing steps performed.2. Model Selection: Choosing an appropriate model for a classification task is critical. Consider factors such as the dataset size, label imbalance, and the complexity of the problem. Different algorithms have their strengths and weaknesses, and it is crucial to select the one that suits the problem at hand.3. Cross-validation: To ensure the robustness of the model, it is advisable to use techniques like cross-validation. Cross-validation involves splitting the data into training and validation sets, allowing the model's performance to beevaluated on multiple partitions of the dataset. This technique helps in estimating the model's ability to generalize to unseen data.ConclusionLabel matrix is a fundamental concept in the field of data analysis and machine learning. It provides a structured representation of labels or categories associated with data points. Understanding the properties and applications of a label matrix is crucial for successful classification tasks, model evaluation, and handling class imbalance. By utilizing label matrix techniques, practitioners and researchers can enhance the accuracy and effectiveness of their machine learning models.。
SPSS中英文对照词典
SPSS中英文对照词典----------------------------------------------------------------------------- ---发表日期：2005年11月22日出处：社会学吧Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automa tic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 一般线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换SPSS新手速成作者：张佳转自中国统计网随着速度越来越快，计算机的功能越来越多，计算统计功能反而已经成为了计算机的一个次要部分。
2014美国大学生数学建模特等奖优秀论文
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Best all time college coach Summary
In order to select the “best all time college coach” in the last century fairly, We take selecting the best male basketball coach as an example, and establish the TOPSIS sort - Comprehensive Evaluation improved model based on entropy and Analytical Hierarchy Process. The model mainly analyzed such indicators as winning rate, coaching time, the time of winning the championship, the number of races and the ability to perceive .Firstly ， Analytical Hierarchy Process and Entropy are integratively utilized to determine the index weights of the selecting indicators Secondly，Standardized matrix and parameter matrix are combined to construct the weighted standardized decision matrix. Finally, we can get the college men's basketball composite score, namely the order of male basketball coaches, which is shown in Table 7. Adolph Rupp and Mark Few are the last century and this century's "best all time college coach" respectively. It is realistic. The rank of college coaches can be clearly determined through this method. Next, ANOVA shows that the scores of last century’s coaches and this century’s coaches have significant difference, which demonstrates that time line horizon exerts influence upon the evaluation and gender factor has no significant influence on coaches’ score. The assessment model, therefore, can be applied to both male and female coaches. Nevertheless, based on this, we have drawn coaches’ coaching ability distributing diagram under ideal situation and non-ideal situation according to the data we have found, through which we get that if time line horizon is chosen reasonably, it will not affect the selecting results. In this problem, the time line horizon of the year 2000 will not influence the selecting results. Furthermore, we put the data of the three types of sports, which have been found by us, into the above Model, and get the top 5 coaches of the three sports, which are illustrated in Table10, Table 11, Table12 and Table13 respectively. These results are compared with the results on the Internet[7], so as to examine the reasonableness of our results. We choose the sports randomly which undoubtedly shows that our model can be applied in general across both genders and all possible sports. At the same time, it also shows the practicality and effectiveness of our model. Finally, we have prepared a 1-2 page article for Sports Illustrated that explains our results and includes a non-technical explanation of our mathematical model that sports fans will understand. Key words: TOPSIS Improved Model; Entropy; Analytical Hierarchy Process; Comprehensive Evaluation Model; ANOVA
利用STATA创建空间权重矩阵及空间杜宾模型计算命令
** 创建空间权重矩阵介绍*设置默认路径cd C:\Users\xiubo\Desktop\F182013.v4\F101994\sheng**创建新文件*shp2dta:reads a shape (.shp) and dbase (.dbf) file from disk and converts them into Stata datasets.*shp2dta:读取CHN_adm1文件*CHN_adm1：为已有的地图文件*database (chinaprovince)：表示创建一个名称为“chinaprovince”的dBase数据集*database(filename)：Specifies filename of new dBase dataset*coordinates(coord)：创建一个名称为“coord”的坐标系数据集*coordinates(filename)：Specifies filename of new coordinates dataset*gencentroids(stub)：Creates centroid variables*genid(newvarname)：Creates unique id variable for database.dtashp2dta using CHN_adm1,database (chinaprovince) coordinates(coord) genid(id) gencentroids(c)**绘制2016年中國GDP分布圖*spmap:Visualization of spatial data*clnumber(#):number of classes*id(idvar):base map polygon identifier(识别符，声明变量名，一般以字母或下划线开头，包含数字、字母、下划线)*_2016GDP：变量*coord:之前创建的坐标系数据集spmap _2016GDP using coord, id(id) clnumber(5)*更改变量名rename x_c longituderename y_c latitude*spmat:用于定义与管理空间权重矩阵*Spatial-weighting matrices are stored in spatial-weighting matrix objects (spmat objects).*spmat objects contain additional information about the data used in constructing spatial-weighting matrices.*spmat objects are used in fitting spatial models; see spreg (if installed) and spivreg (if installed).*idistance:(产生距离矩阵)create an spmat object containing an inverse-distance matrix W*或contiguity:create an spmat object containing a contiguity matrix W*idistance_jingdu:命名名称为“idistance_jingdu”的距離矩陣*longitude:使用经度*latitude:使用纬度*id(id):使用id*dfunction(function[, miles]):(设置计算距离方法)specify the distance function.*function may be one of euclidean (default), dhaversine, rhaversine, or the Minkowski distance of order p, where p is an integer greater than or equal to 1.*normalize(row):（行标准化）specifies one of the three available normalization techniques: row, minmax, and spectral.*In a row-normalized matrix, each element in row i is divided by the sum of row i's elements.*In a minmax-normalized matrix, each element is divided by the minimum of the largest row sum and column sum of the matrix.*In a spectral-normalized matrix, each element is divided by the modulus of the largest eigenvalue of the matrix.spmat idistance idistance_jingdu longitude latitude, id(id) dfunction(euclidean) normalize(row)**保存stata可读文件idistance_jingdu.spmatspmat save idistance_jingdu using idistance_jingdu.spmat**将刚刚保存的idistance_jingdu.spmat文件转化为txt文件spmat export idistance_jingdu using idistance_jingdu.txtspmat contiguity contiguity_jingdu using coord, id(id) normalize(row)spmat save contiguity_jingdu using contiguity_jingdu.spmatspmat export contiguity_jingdu using contiguity_jingdu.txt**计算Moran’s I*安装spatwmat*spatwmat:用于定义空间权重矩阵*spatwmat:imports or generates the spatial weights matrices required by spatgsa, spatlsa, spatdiag, and spatreg.*As an option, spatwmat also generates the eigenvalues matrix required by spatreg.*name(W):读取空间权重矩阵W*name(W):使用生成的空间权重矩阵W*xcoord:x坐标*ycoord:y坐标*band(0 8):宽窗介绍*band(numlist) is required if option using filename is not specified.*It specifies the lower and upper bounds of the distance band within which location pairs must be considered "neighbors" (i.e., spatially contiguous)*and, therefore, assigned a nonzero spatial weight.*binary:requests that a binary weights matrix be generated. To this aim, all nonzero spatial weights are set to 1.spatwmat, name(W) xcoord(longitude) ycoord(latitude) band(0 8)*安装绘制Moran’s I工具:splagvar*splagvar --- Generates spatially lagged variables, constructs the Moran scatter plot,*and calculates global Moran's I statistics.*_2016GDP：使用变量_2016GDP*wname(W):使用空间权重矩阵W*indicate the name of the spatial weights matrix to be used*wfrom(Stata):indicate source of the spatial weights matrix*wfrom(Stata | Mata) indicates whether the spatial weights matrix is a Stata matrix loaded in memory or a Mata file located in the working directory.*If the spatial weights matrix had been created using spwmatrix it should exist as a Stata matrix or as a Mata file.*moran(_2016GDP):计算变量_2016GDP的Moran's I值*plot(_2016GDP):构建变量_2016GDPMoran散点图splagvar _2016GDP, wname(W) wfrom(Stata) moran(_2016GDP) plot(_2016GDP)**使用距离矩阵计算空间计量模型*设置默认路径cd D:\软件学习软件资料\stata\stata指导书籍命令\陈强高级计量经济学及stata应用（第二版）全部数据*使用product.dta数据集（陈强的高级计量经济学及其stata应用P594）*将数据集product.dta存入当前工作路径use product.dta , clear*创建新变量，对原有部分变量取对数gen lngsp=log(gsp)gen lnpcap=log(pcap)gen lnpc=log(pc)gen lnemp=log(emp)*将空间权重矩阵usaww.spat存入当前工作路径spmat use usaww using usaww.spmat*使用聚类稳健的标准误估计随机效应的SDM模型xsmle lngsp lnpcap lnpc lnemp unemp,wmat(usaww) model(sdm)robust nolog*使用选择项durbin(lnemp)，不选择不显著的变量，使用聚类稳健的标准误估计随机效应的SDM模型xsmle lngsp lnpcap lnpc lnemp unemp,wmat(usaww) model(sdm) durbin(lnemp) robust nolog noeffects*使用选择项durbin(lnemp)，不选择不显著的变量，使用聚类稳健的标准误估计固定效应的SDM模型xsmle lngsp lnpcap lnpc lnemp unemp,wmat(usaww) model(sdm) durbin(lnemp) robust nolog noeffects fe*存储随机效应和固定效应结果qui xsmle lngsp lnpcap lnpc lnemp unemp,wmat(usaww) model(sdm) durbin(lnemp) r2 nolog noeffects reest sto requi xsmle lngsp lnpcap lnpc lnemp unemp,wmat(usaww) model(sdm) durbin(lnemp) r2 nolog noeffects feest sto fe*esttab:将保存的结果汇总到一张表格中*b(fmt):specify format for point estimates*beta[(fmt)]:display beta coefficients instead of point est's*se[(fmt)]:display standard errors instead of t statistics*star( * 0.1 ** 0.05 *** 0.01):标记不同显著性水平对应的P值*r2|ar2|pr2[(fmt)]:display (adjusted, pseudo) R-squared*p[(fmt)]:display p-values instead of t statistics*label:make use of variable labels*title(string):specify a title for the tableesttab fe re , b se r2 star( * 0.1 ** 0.05 *** 0.01)*hausman检验*进行hausman检验前，回归中没有使用稳健标准误(没用“r”),*是因为传统的豪斯曼检验建立在同方差的前提下*constant:include estimated intercepts in comparison; default is to exclude*df(#):use # degrees of freedom*sigmamore:base both (co)variance matrices on disturbance variance estimate from efficient estimator*sigmaless:base both (co)variance matrices on disturbance variance estimate from consistent estimatorhausman fe re**有时我们还会得到负的chi2值,即chi2<0，表明模型不能满足Hausman检验的渐近假设。
初中生样看待负重前行英语作文
负重前行：初中生的视角As a junior high student, life often seems like a constant struggle, a journey laden with responsibilities and challenges. Unlike the carefree days of childhood, this phase of our lives is marked by the weight of expectations, academic pressures, and the need to forge our own identities.In this journey, we often find ourselves shouldering more than we can bear. The backpacks we carry are not just filled with textbooks and notebooks, but also with the hopes and dreams of our parents, teachers, and society. The weight of these expectations can be crushing, but it is also what drives us to push forward, to strive for excellence, and to become the best version of ourselves.However, this constant state of being burdened can also be debilitating. It can lead to stress, anxiety, and even burnout. It is important, therefore, to learn how to balance our load and to take periodic breaks to recharge our batteries. This could mean engaging in hobbies that we enjoy, spending time with friends and family, or simply taking a moment to appreciate the small joys of life.Moreover, we need to understand that this journey isnot just about us. It is about contributing to society, about making a positive impact on the world. The challenges we face and the weights we carry are not just personal;they are also collective. By helping each other, we can lighten the load and make the journey easier for everyone.In conclusion, while the road ahead may be fraught with challenges and weighted down with responsibilities, it is also filled with opportunities for growth and learning. By embracing the weight of our loads and finding the balance that works for us, we can turn these challenges into stepping stones for success. After all, it is only through struggle and perseverance that we truly discover ourstrength and潜能。
空间计量SpatialEconometrics.ppt
Spatial Dependence
Positive spatial autocorrelation: high or low values of a variable cluster in space Negative spatial autocorrelation: locations are surrounded by neighbors with very dissimilar values of the same variable
Hence, to estimate the model with a spatially lagged dependent variable, we apply the maximum likelihood estimation (MLE) method The log-likelihood function of the model is,
p p p ... p ij i 1 i2 ij w /( ) 2 2 ij d d ij ij
Travel time of freight vehicles between the centers of regions (Tiiu Paas, Friso Schlitte ‘Spatial effects of regional income disparities and growth in the EU countries and regions ’). The matrix W is calculated as follows:
3
Modelling Space
Spatial dependence modelling requires an appropriate representation of spatial arrangement Solution: relative spatial positions are represented by spatial weights matriceght Matrix
Hermitian matrix
Hermitian matrixIn mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose – that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:If the conjugate transpose of a matrix is denoted by , then the Hermitian property can be written concisely asHermitian matrices can be understood as the complex extension of real symmetric matrices.Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real.ExamplesFor example,Well-known families of Pauli matrices, Gell-Mann matrices and various generalizations are Hermitian. In theoretical physics such Hermitian matrices usually are multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices (see below).PropertiesThe entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix.Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A are real, and that A has n linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of C n consisting of n eigenvectors of A.The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible Hermitian matrix is Hermitian as well. However, the product of two Hermitian matrices A and B will only be Hermitian if they commute, i.e., if AB = BA. Thus A n is Hermitian if A is Hermitian and n is an integer.The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, since the identity matrix is Hermitian, but is not. However the complex Hermitian matrices do form a vector space over the real numbers. In the 2n2R dimensional vector space of complex n×n matrices, the complex Hermitian matrices formdenotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis a subspace of dimension n2. If Ejkcan be described as follows:for (n matrices)together with the set of matrices of the formfor ((n2−n)/2 matrices)and the matricesfor ((n2−n)/2 matrices)where denotes the complex number , known as the imaginary unit.If n orthonormal eigenvectors of a Hermitian matrix are chosen and written as the columns of thematrix U, then one eigendecomposition of A is where and therefore,where are the eigenvalues on the diagonal of the diagonal matrix .Additional facts related to Hermitian matrices include:•The sum of a square matrix and its conjugate transpose is Hermitian.•The difference of a square matrix and its conjugate transpose is skew-Hermitian (also calledantihermitian).•This implies that commutator of two Hermitian matrices is skew-Hermitian.•An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:•The determinant of a Hermitian matrix is real:Proof:Therefore ifReferences[1]Frankel, Theodore (2004). The geometry of physics: an introduction (http://books.google.ru/books?id=DUnjs6nEn8wC&lpg=PA652&dq="Lie algebra" physics "skew-Hermitian"&pg=PA652#v=onepage&q&f=false). Cambridge University Press. p. 652. ISBN 0521539277. .[2]Physics 125 Course Notes (/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf)at California Institute of TechnologyExternal links•Visualizing Hermitian Matrix as An Ellipse with Dr. Geo (/~ckhung/b/la/hermitian.en.php), by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.•"Hermitian Matrices" (/home/kmath306/kmath306.htm) at .Article Sources and Contributors3Article Sources and ContributorsHermitian matrix Source: /w/index.php?oldid=461372150 Contributors: 3mta3, Arlia101, AxelBoldt, Banus, BenFrantzDale, Bruguiea, Caesura, Chutzpan, Connelly, Dan Granahan, Drevicko, Dudegalea, Evanpw, Fropuff, Gene.arboit, Giftlite, Haseldon, Incnis Mrsi, Isnow, Janm67, Jasonevans, Jitse Niesen, Looxix, Lzur, Magister Mathematicae, Marra,MaxEnt, Mct mht, Mhss, Michael Hardy, Milez, Mmakin, Myasuda, Nappyrash, Neparis, Obersachse, Octahedron80, Oleg Alexandrov, Oliphaunt, Oyz, Pupdike, Qjqflash3, Qutezuce, RDBury, Randomblue, RexNL, Rschwieb, Scot.parker, Simonjwright, Smmurphy, Sofeshue, Soumya m, Stpasha, Tbsmith, TedPavlic, Tercer, Ugha, 48 ,קאלדנב הנוי anonymous editsLicenseCreative Commons Attribution-Share Alike 3.0 Unported///licenses/by-sa/3.0/。
学习WGCNA总结
6 7 8 9 10 11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 5556 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73dim(fpkm)names(fpkm)datExpr0=as.data.frame(t(fpkm[,-c(1)]));names(datExpr0)=fpkm$trans;rownames(datExpr0)=names(fpkm)[-c(1)];#data<-log10(date[,-1]+0.01)gsg = goodSamplesGenes(datExpr0, verbose = 3);gsg$allOKsampleTree = hclust(dist(datExpr0), method = "average")#sizeGrWindow(12,9)par(cex = 0.6)par(mar = c(0,4,2,0))plot(sampleTree, main = "Sample clustering to detect outliers", sub="", xlab="", b = 1.5,cex.axis = 1.5, cex.main = 2)abline(h = 80000, col = "red");clust = cutreeStatic(sampleTree, cutHeight = 80000, minSize = 10)table(clust)keepSamples = (clust==1)datExpr = datExpr0[keepSamples, ]nGenes = ncol(datExpr)nSamples = nrow(datExpr)save(datExpr, file = "AS-green-FPKM-01-dataInput.RData")#2. 选择合适的阀值powers = c(c(1:10), seq(from = 12, to=20, by=2))# Call the network topology analysis functionsft = pickSoftThreshold(datExpr, powerVector = powers, verbose = 5)# Plot the results:##sizeGrWindow(9, 5)par(mfrow = c(1,2));cex1 = 0.9;# Scale-free topology fit index as a function of the soft-thresholding powerplot(sft$fitIndices[,1], -sign(sft$fitIndices[,3])*sft$fitIndices[,2],xlab="Soft Threshold (power)",ylab="Scale Free Topology Model Fit,signed R^2",type="n",main = paste("Scale independence"));text(sft$fitIndices[,1], -sign(sft$fitIndices[,3])*sft$fitIndices[,2],labels=powers,cex=cex1,col="red");# this line corresponds to using an R^2 cut-off of habline(h=0.90,col="red")# Mean connectivity as a function of the soft-thresholding powerplot(sft$fitIndices[,1], sft$fitIndices[,5],xlab="Soft Threshold (power)",ylab="Mean Connectivity", type="n",main = paste("Mean connectivity"))text(sft$fitIndices[,1], sft$fitIndices[,5], labels=powers, cex=cex1,col="red")#===================================================================================== # ⽹络构建有两种⽅法，One-step和Step-by-step；# 第⼀种：⼀步法进⾏⽹络构建#===================================================================================== #3. ⼀步法⽹络构建：One-step network construction and module detectionnet = blockwiseModules(datExpr, power = 6, maxBlockSize = 6000,TOMType = "unsigned", minModuleSize = 30,reassignThreshold = 0, mergeCutHeight = 0.25,numericLabels = TRUE, pamRespectsDendro = FALSE,saveTOMs = TRUE,saveTOMFileBase = "AS-green-FPKM-TOM",verbose = 3)table(net$colors)#4. 绘画结果展⽰# open a graphics window#sizeGrWindow(12, 9)# Convert labels to colors for plottingmergedColors = labels2colors(net$colors)# Plot the dendrogram and the module colors underneathplotDendroAndColors(net$dendrograms[[1]], mergedColors[net$blockGenes[[1]]],"Module colors",dendroLabels = FALSE, hang = 0.03,addGuide = TRUE, guideHang = 0.05)#5.结果保存moduleLabels = net$colorsmoduleColors = labels2colors(net$colors)table(moduleColors)MEs = net$MEs;geneTree = net$dendrograms[[1]];save(MEs, moduleLabels, moduleColors, geneTree,file = "AS-green-FPKM-02-networkConstruction-auto.RData")#6. 导出⽹络到Cytoscape# Recalculate topological overlap if neededTOM = TOMsimilarityFromExpr(datExpr, power = 6);# Read in the annotation file# annot = read.csv(file = "GeneAnnotation.csv");# Select modules需要修改，选择需要导出的模块颜⾊modules = c("turquoise", "blue");# Select module probes选择模块探测probes = names(datExpr)inModule = is.finite(match(moduleColors, modules));modProbes = probes[inModule];#modGenes = annot$gene_symbol[match(modProbes, annot$substanceBXH)];# Select the corresponding Topological OverlapmodTOM = TOM[inModule, inModule];73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101102103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122123 124 125 126 127 128 129 130 131 132 133 134 135136modTOM = TOM[inModule, inModule];dimnames(modTOM) = list(modProbes, modProbes)# Export the network into edge and node list files Cytoscape can readcyt = exportNetworkToCytoscape(modTOM,edgeFile = paste("AS-green-FPKM-One-step-CytoscapeInput-edges-", paste(modules, collapse="-"), ".txt", s ep=""),nodeFile = paste("AS-green-FPKM-One-step-CytoscapeInput-nodes-", paste(modules, collapse="-"), ".txt", s ep=""),weighted = TRUE,threshold = 0.02,nodeNames = modProbes,#altNodeNames = modGenes,nodeAttr = moduleColors[inModule]);#=====================================================================================# 分析⽹络可视化，⽤heatmap可视化权重⽹络，heatmap每⼀⾏或列对应⼀个基因，颜⾊越深表⽰有较⾼的邻近#=====================================================================================options(stringsAsFactors = FALSE);lnames = load(file = "AS-green-FPKM-01-dataInput.RData");lnameslnames = load(file = "AS-green-FPKM-02-networkConstruction-auto.RData");lnamesnGenes = ncol(datExpr)nSamples = nrow(datExpr)#1. 可视化全部基因⽹络# Calculate topological overlap anew: this could be done more efficiently by saving the TOM# calculated during module detection, but let us do it again here.dissTOM = 1-TOMsimilarityFromExpr(datExpr, power = 6);# Transform dissTOM with a power to make moderately strong connections more visible in the heatmapplotTOM = dissTOM^7;# Set diagonal to NA for a nicer plotdiag(plotTOM) = NA;# Call the plot function#sizeGrWindow(9,9)TOMplot(plotTOM, geneTree, moduleColors, main = "Network heatmap plot, all genes")#随便选取1000个基因来可视化nSelect = 1000# For reproducibility, we set the random seedset.seed(10);select = sample(nGenes, size = nSelect);selectTOM = dissTOM[select, select];# There's no simple way of restricting a clustering tree to a subset of genes, so we must re-cluster.selectTree = hclust(as.dist(selectTOM), method = "average")selectColors = moduleColors[select];# Open a graphical window#sizeGrWindow(9,9)# Taking the dissimilarity to a power, say 10, makes the plot more informative by effectively changing# the color palette; setting the diagonal to NA also improves the clarity of the plotplotDiss = selectTOM^7;diag(plotDiss) = NA;TOMplot(plotDiss, selectTree, selectColors, main = "Network heatmap plot, selected genes")#=====================================================================================# 第⼆种：⼀步步的进⾏⽹络构建#=====================================================================================###################Step-by-step network construction and module detection#2.选择合适的阀值，同上#3. ⽹络构建：(1) Co-expression similarity and adjacencysoftPower = 6;adjacency = adjacency(datExpr, power = softPower);#(2) 邻近矩阵到拓扑矩阵的转换，Turn adjacency into topological overlapTOM = TOMsimilarity(adjacency);dissTOM = 1-TOM# (3) 聚类拓扑矩阵#Call the hierarchical clustering functiongeneTree = hclust(as.dist(dissTOM), method = "average");# Plot the resulting clustering tree (dendrogram)#sizeGrWindow(12,9)plot(geneTree, xlab="", sub="", main = "Gene clustering on TOM-based dissimilarity",labels = FALSE, hang = 0.04);#(4) 聚类分⽀的休整dynamicTreeCut# We like large modules, so we set the minimum module size relatively high:minModuleSize = 30;# Module identification using dynamic tree cut:dynamicMods = cutreeDynamic(dendro = geneTree, distM = dissTOM,deepSplit = 2, pamRespectsDendro = FALSE,minClusterSize = minModuleSize);table(dynamicMods)#4. 绘画结果展⽰# Convert numeric lables into colorsdynamicColors = labels2colors(dynamicMods)table(dynamicColors)# Plot the dendrogram and colors underneath#sizeGrWindow(8,6)plotDendroAndColors(geneTree, dynamicColors, "Dynamic Tree Cut",dendroLabels = FALSE, hang = 0.03,addGuide = TRUE, guideHang = 0.05,main = "Gene dendrogram and module colors")#5. 聚类结果相似模块的融合，Merging of modules whose expression profiles are very similar#在聚类树中每⼀leaf是⼀个短线，代表⼀个基因，#不同分之间靠的越近表⽰有⾼的共表达基因，将共表达极其相似的modules进⾏融合137 138 139 140141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203# Calculate eigengenesMEList = moduleEigengenes(datExpr, colors = dynamicColors)MEs = MEList$eigengenes# Calculate dissimilarity of module eigengenesMEDiss = 1-cor(MEs);# Cluster module eigengenesMETree = hclust(as.dist(MEDiss), method = "average");# Plot the result#sizeGrWindow(7, 6)plot(METree, main = "Clustering of module eigengenes",xlab = "", sub = "")#选择有75%相关性的进⾏融合MEDissThres = 0.25# Plot the cut line into the dendrogramabline(h=MEDissThres, col = "red")# Call an automatic merging functionmerge = mergeCloseModules(datExpr, dynamicColors, cutHeight = MEDissThres, verbose = 3)# The merged module colorsmergedColors = merge$colors;# Eigengenes of the new merged modules:mergedMEs = merge$newMEs;#绘制融合前(Dynamic Tree Cut)和融合后(Merged dynamic)的聚类图#sizeGrWindow(12, 9)#pdf(file = "Plots/geneDendro-3.pdf", wi = 9, he = 6)plotDendroAndColors(geneTree, cbind(dynamicColors, mergedColors),c("Dynamic Tree Cut", "Merged dynamic"),dendroLabels = FALSE, hang = 0.03,addGuide = TRUE, guideHang = 0.05)#dev.off()# 只是绘制融合后聚类图plotDendroAndColors(geneTree,mergedColors,"Merged dynamic",dendroLabels = FALSE, hang = 0.03,addGuide = TRUE, guideHang = 0.05)#5.结果保存# Rename to moduleColorsmoduleColors = mergedColors# Construct numerical labels corresponding to the colorscolorOrder = c("grey", standardColors(50));moduleLabels = match(moduleColors, colorOrder)-1;MEs = mergedMEs;# Save module colors and labels for use in subsequent partssave(MEs, moduleLabels, moduleColors, geneTree, file = "AS-green-FPKM-02-networkConstruction-stepByStep.RData")#6. 导出⽹络到Cytoscape# Recalculate topological overlap if neededTOM = TOMsimilarityFromExpr(datExpr, power = 6);# Read in the annotation file# annot = read.csv(file = "GeneAnnotation.csv");# Select modules需要修改modules = c("brown", "red");# Select module probesprobes = names(datExpr)inModule = is.finite(match(moduleColors, modules));modProbes = probes[inModule];#modGenes = annot$gene_symbol[match(modProbes, annot$substanceBXH)];# Select the corresponding Topological OverlapmodTOM = TOM[inModule, inModule];dimnames(modTOM) = list(modProbes, modProbes)# Export the network into edge and node list files Cytoscape can readcyt = exportNetworkToCytoscape(modTOM,edgeFile = paste("AS-green-FPKM-Step-by-step-CytoscapeInput-edges-", paste(modules, collapse="-"), ".tx t", sep=""),nodeFile = paste("AS-green-FPKM-Step-by-step-CytoscapeInput-nodes-", paste(modules, collapse="-"), ".tx t", sep=""),weighted = TRUE,threshold = 0.02,nodeNames = modProbes,#altNodeNames = modGenes,nodeAttr = moduleColors[inModule]);#=====================================================================================# 分析⽹络可视化，⽤heatmap可视化权重⽹络，heatmap每⼀⾏或列对应⼀个基因，颜⾊越深表⽰有较⾼的邻近#=====================================================================================options(stringsAsFactors = FALSE);lnames = load(file = "AS-green-FPKM-01-dataInput.RData");lnameslnames = load(file = "AS-green-FPKM-02-networkConstruction-stepByStep.RData");lnamesnGenes = ncol(datExpr)nSamples = nrow(datExpr)#1. 可视化全部基因⽹络# Calculate topological overlap anew: this could be done more efficiently by saving the TOM# calculated during module detection, but let us do it again here.dissTOM = 1-TOMsimilarityFromExpr(datExpr, power = 6);# Transform dissTOM with a power to make moderately strong connections more visible in the heatmapplotTOM = dissTOM^7;# Set diagonal to NA for a nicer plotdiag(plotTOM) = NA;# Call the plot function#sizeGrWindow(9,9)TOMplot(plotTOM, geneTree, moduleColors, main = "Network heatmap plot, all genes")203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243244245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264265 266 267TOMplot(plotTOM, geneTree, moduleColors, main = "Network heatmap plot, all genes")#随便选取1000个基因来可视化nSelect = 1000# For reproducibility, we set the random seedset.seed(10);select = sample(nGenes, size = nSelect);selectTOM = dissTOM[select, select];# There's no simple way of restricting a clustering tree to a subset of genes, so we must re-cluster.selectTree = hclust(as.dist(selectTOM), method = "average")selectColors = moduleColors[select];# Open a graphical window#sizeGrWindow(9,9)# Taking the dissimilarity to a power, say 10, makes the plot more informative by effectively changing# the color palette; setting the diagonal to NA also improves the clarity of the plotplotDiss = selectTOM^7;diag(plotDiss) = NA;TOMplot(plotDiss, selectTree, selectColors, main = "Network heatmap plot, selected genes")#此处画的是根据基因间表达量进⾏聚类所得到的各模块间的相关性图MEs = moduleEigengenes(datExpr, moduleColors)$eigengenesMET = orderMEs(MEs)sizeGrWindow(7, 6)plotEigengeneNetworks(MET, "Eigengene adjacency heatmap", marHeatmap = c(3,4,2,2), plotDendrograms = FALSE, xLab elsAngle = 90)果然，三者的length 不同，发现geneTree 少了⼀些，往回找geneTree 来源 geneTree = net$dendrograms[[1]]，net 来源于⽹络构建过程：所以，这是问题所在，继续察看⽂档发现blockwiseModules 函数默认最⼤maxBlockSize=5000，⽽我们的数据超过了这个值，所以函数⾃动做了拆分处理，⽽解决办法也很简单，设置maxBlockSize 参数⼤于我们的值即可。
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1 1 1 2 2 4 5 9 12
1 1 2 3 3 32 1 4 = 33 , 3 1 5 34 9 4 1 . . .. . . . . . ···
where the first column sequence A002212 in [13], which has two interpretations, the number of 3-Motzkin paths or the number of ways to assemble benzene rings into a tree [8]. Recall that a 3-Motzkin path is a lattice path from (0, 0) to (n−1, 0) that does not go below the x-axis and consists of up steps U = (1, 1), down steps D = (1, −1), and three types of horizontal steps H = (1, 0). The above matrix A = (ai,j ) is generated by the first column and the following recurrence relation ai,j = ai−1,j −1 + 3ai−1,j + ai−1,j +1 . We may prove the above identities (1.5) and (1.6) by using method of Riordan arrays. So the natural question is to find a matrix identity for the sequence (1, k, k 2 , k 3 , . . .). We need the combinatorial interpretation of the entries in the matrix in terms of weighted partial Motzkin paths, as given by Cameron and Nkwanta [4]. To be precise, a partial Motzkin path, also called a Motzkin path from (0, 0) to (n, k ) in [4], is just a Motzkin path but without the requirement of ending on the x-axis. A weighted partial Motkzin a partial Motzkin path with the weight assignment that the horizontal steps are endowed with a weight k and the down steps are endowed with a weight t, where k and t are regarded as positive integers. In this sense, our weighted Motzkin paths can be regarded as further generalization of k -Motzkin paths in the sense of 2-Motzkin paths and 3-Motkzin paths [1, 7, 13]. We introduce the notion of weighted free Motzkin paths which is a lattice path consisting of Motzkin steps without the restrictions that it has to end with a point on the x-axis and it does not go below the x-axis. We then give a bijection between weighted free Motzkin paths and weighted partial Motzkin paths with an elevation line, which leads to a matrix identity involving the number of weighted partial Motzkin paths and the sequence (1, k, k 2 , . . .). The idea of the elevation operation is also used by Cameron and Nkwanta in their combinatorial proof of the identity (1.1) in a more restricted form. By extending our argument to weighted partial Motzkin paths with multiple elevation lines, we obtain a combinatorial proof of an identity recently derived by Cameron and Nkwanta, in answer to their question. We also give a generalization of the matrix identity (1.1) and give a combinatorial proof by using colored Dyck paths.
Matrix Identities on Weighted Partial Motzkin Paths
arXiv:math/0509255v1 [math.CO] 12 Sep 2005
William Y.C. Chen1 , Nelson Y. Li2 , Louis W. Shapiro3 and Sherry H. F. Yan4
(1.5)
where the first column is the sequence of Motzkin numbers, and matrix A = (aij ) is generated by the following recurrence relation: ai,j = ai−1,j −1 + ai−1,j + ai−1,j 1 chen@, 2 nelsonli@, 3 lshapiro@,
Abstract. We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1, 4, 42, 43 , . . .) which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence (1, k, k 2 , k 3 , . . .) for any k ≥ 2. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A matrix identity on colored Dyck paths is also given, leading to a matrix identity for the sequence (1, t2 + t, (t2 + t)2 , . . .). Key words: Catalan number, Schr¨ oder number, Dyck path, Motzkin path, partial Motzkin path, free Motzkin path, weighted Motzkin path, Riordan array AMS Mathematical Subject Classifications: 05A15, 05A19. Corresponding Author: William Y. C. Chen, chen@
for j ≥ 2 and the ai,1 is the i-th little Schr¨ oder number si (sequence A001003 in [13]), which counts Schr¨ oder paths of length 2(i + 1). A Schr¨ oder path is a lattice path starting at (0, 0) and ending at (2n, 0) and using steps H = (2, 0), U = (1, 1) and D = (1, −1) such that no steps are below the x-axis and there are no peaks at level one. Imposing this last peak condition gives us little Schr¨ oder numbers while without it we would have the large Schr¨ oder numbers. For k = 3, we obtain the following matrix identity on Motzkin numbers: