1 Introduction Learning Statistical Models of Time-Varying Relational Data

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ArtificialIntelligenceAModernApproach2ndEdition教学设

Artificial Intelligence: A Modern Approach 2nd Edition 教学设计介绍《人工智能：一种现代方法》（Artificial Intelligence: A Modern Approach）是人工智能教育领域的经典教材之一，由 Peter Norvig 和 Stuart Russell 合著。

本书以全新的视角和方法论来探索人工智能领域。

在本文档中，我们将讨论如何设计一门针对Artificial Intelligence: A Modern Approach 2nd Edition 教材的课程。

教学目标本课程的教学目标是让学生掌握如下能力：•理解人工智能的背景及其组成部分。

•学习机器学习领域的基本算法及其应用。

•掌握自然语言处理的基础知识和技术。

•讨论人工智能的道德和社会问题。

教学内容课程的内容将分为以下几个部分：Part I：Introduction to Artificial Intelligence•Chapter 1 Introduction•Chapter 2 Intelligent agentsPart II: Problem-solving•Chapter 3 Solving problems by searching•Chapter 4 Search in complex environments•Chapter 5 Adversarial search and gamesPart III: Knowledge, reasoning, and planning •Chapter 6 Agents that reason logically•Chapter 7 First-order logic•Chapter 8 Inference in first-order logic•Chapter 9 Classical planningPart IV: Uncertnty•Chapter 13 Quantifying uncertnty•Chapter 14 Probabilistic reasoning•Chapter 15 Probabilistic reasoning over time •Chapter 16 Making simple decisionsPart V: Learning•Chapter 18 Learning from examples•Chapter 19 Knowledge in learning•Chapter 20 Statistical learning methods•Chapter 21 Reinforcement learningPart VI: Communicating, perceiving, and acting •Chapter 22 Natural language processing•Chapter 23 Natural language generation•Chapter 24 Perception•Chapter 25 RoboticsPart VII: Conclusions•Chapter 26 Philosophical foundations•Chapter 27 : Present and future教学方法课程将采用以下方法：•理论教学：对每个章节中的概念、工具和技术进行深入的解释与思考。

分布函数与概率密度函数的参数估计方法

分布函数与概率密度函数的参数估计方法在概率统计学中，分布函数和概率密度函数是用来描述随机变量的性质的重要工具。

而参数估计则是根据给定的样本数据，通过某种方法对分布函数和概率密度函数中的未知参数进行估计的过程。

本文将介绍分布函数与概率密度函数的参数估计方法，包括最大似然估计、矩估计以及贝叶斯估计。

最大似然估计（Maximum Likelihood Estimation，MLE）是一种常用的参数估计方法。

其核心思想是选择使得给定数据样本出现概率最大的参数值作为估计值。

对于给定的样本数据x1，x2，…，xn，假设其分布函数为F(x;θ)，其中θ为未知参数。

最大似然估计的目标是找到使得样本数据出现概率最大的参数值θ^。

具体来说，最大似然估计通过对似然函数L(θ)=∏(i=1)^n f(xi;θ)（其中f(x;θ)为概率密度函数）取对数，并对参数θ进行求导来求解参数值θ^。

矩估计（Method of Moments，MoM）是另一种常用的参数估计方法。

其基本原理是利用样本矩与理论分布矩的对应关系进行参数估计。

对于给定的样本数据x1，x2，…，xn，假设其概率密度函数为f(x;θ)，其中θ为未知参数。

矩估计的目标是使样本矩与理论矩之间的差异最小化，即找到使得原始矩和样本矩最接近的参数值θ^。

除了最大似然估计和矩估计之外，贝叶斯估计（Bayesian Estimation）是一种基于贝叶斯理论的参数估计方法。

其核心思想是将未知参数视为一个随机变量，并基于先验分布和样本数据来求得后验分布。

贝叶斯估计不仅考虑了样本数据的信息，还考虑了先验信息的影响，因此对于样本数据较少或者不确定性较高的情况下，贝叶斯估计能够提供更稳健的参数估计结果。

总结起来，分布函数与概率密度函数的参数估计方法主要包括最大似然估计、矩估计和贝叶斯估计。

最大似然估计通过最大化样本数据出现的概率来估计参数，矩估计通过比较样本矩和理论矩之间的差异来估计参数，而贝叶斯估计则综合考虑了先验分布和样本数据来求得后验分布。

Support Vector Machines for Classification and Regression

UNIVERSITY OF SOUTHAMPTONSupport Vector MachinesforClassiﬁcation and RegressionbySteve R.GunnTechnical ReportFaculty of Engineering,Science and Mathematics School of Electronics and Computer Science10May1998ContentsNomenclature xi1Introduction11.1Statistical Learning Theory (2)1.1.1VC Dimension (3)1.1.2Structural Risk Minimisation (4)2Support Vector Classiﬁcation52.1The Optimal Separating Hyperplane (5)2.1.1Linearly Separable Example (10)2.2The Generalised Optimal Separating Hyperplane (10)2.2.1Linearly Non-Separable Example (13)2.3Generalisation in High Dimensional Feature Space (14)2.3.1Polynomial Mapping Example (16)2.4Discussion (16)3Feature Space193.1Kernel Functions (19)3.1.1Polynomial (20)3.1.2Gaussian Radial Basis Function (20)3.1.3Exponential Radial Basis Function (20)3.1.4Multi-Layer Perceptron (20)3.1.5Fourier Series (21)3.1.6Splines (21)3.1.7B splines (21)3.1.8Additive Kernels (22)3.1.9Tensor Product (22)3.2Implicit vs.Explicit Bias (22)3.3Data Normalisation (23)3.4Kernel Selection (23)4Classiﬁcation Example:IRIS data254.1Applications (28)5Support Vector Regression295.1Linear Regression (30)5.1.1 -insensitive Loss Function (30)5.1.2Quadratic Loss Function (31)iiiiv CONTENTS5.1.3Huber Loss Function (32)5.1.4Example (33)5.2Non Linear Regression (33)5.2.1Examples (34)5.2.2Comments (36)6Regression Example:Titanium Data396.1Applications (42)7Conclusions43A Implementation Issues45A.1Support Vector Classiﬁcation (45)A.2Support Vector Regression (47)B MATLAB SVM Toolbox51 Bibliography53List of Figures1.1Modelling Errors (2)1.2VC Dimension Illustration (3)2.1Optimal Separating Hyperplane (5)2.2Canonical Hyperplanes (6)2.3Constraining the Canonical Hyperplanes (7)2.4Optimal Separating Hyperplane (10)2.5Generalised Optimal Separating Hyperplane (11)2.6Generalised Optimal Separating Hyperplane Example(C=1) (13)2.7Generalised Optimal Separating Hyperplane Example(C=105) (14)2.8Generalised Optimal Separating Hyperplane Example(C=10−8) (14)2.9Mapping the Input Space into a High Dimensional Feature Space (14)2.10Mapping input space into Polynomial Feature Space (16)3.1Comparison between Implicit and Explicit bias for a linear kernel (22)4.1Iris data set (25)4.2Separating Setosa with a linear SVC(C=∞) (26)4.3Separating Viginica with a polynomial SVM(degree2,C=∞) (26)4.4Separating Viginica with a polynomial SVM(degree10,C=∞) (26)4.5Separating Viginica with a Radial Basis Function SVM(σ=1.0,C=∞)274.6Separating Viginica with a polynomial SVM(degree2,C=10) (27)4.7The eﬀect of C on the separation of Versilcolor with a linear spline SVM.285.1Loss Functions (29)5.2Linear regression (33)5.3Polynomial Regression (35)5.4Radial Basis Function Regression (35)5.5Spline Regression (36)5.6B-spline Regression (36)5.7Exponential RBF Regression (36)6.1Titanium Linear Spline Regression( =0.05,C=∞) (39)6.2Titanium B-Spline Regression( =0.05,C=∞) (40)6.3Titanium Gaussian RBF Regression( =0.05,σ=1.0,C=∞) (40)6.4Titanium Gaussian RBF Regression( =0.05,σ=0.3,C=∞) (40)6.5Titanium Exponential RBF Regression( =0.05,σ=1.0,C=∞) (41)6.6Titanium Fourier Regression( =0.05,degree3,C=∞) (41)6.7Titanium Linear Spline Regression( =0.05,C=10) (42)vvi LIST OF FIGURES6.8Titanium B-Spline Regression( =0.05,C=10) (42)List of Tables2.1Linearly Separable Classiﬁcation Data (10)2.2Non-Linearly Separable Classiﬁcation Data (13)5.1Regression Data (33)viiListingsA.1Support Vector Classiﬁcation MATLAB Code (46)A.2Support Vector Regression MATLAB Code (48)ixNomenclature0Column vector of zeros(x)+The positive part of xC SVM misclassiﬁcation tolerance parameterD DatasetK(x,x )Kernel functionR[f]Risk functionalR emp[f]Empirical Risk functionalxiChapter1IntroductionThe problem of empirical data modelling is germane to many engineering applications. In empirical data modelling a process of induction is used to build up a model of the system,from which it is hoped to deduce responses of the system that have yet to be ob-served.Ultimately the quantity and quality of the observations govern the performance of this empirical model.By its observational nature data obtained isﬁnite and sampled; typically this sampling is non-uniform and due to the high dimensional nature of the problem the data will form only a sparse distribution in the input space.Consequently the problem is nearly always ill posed(Poggio et al.,1985)in the sense of Hadamard (Hadamard,1923).Traditional neural network approaches have suﬀered diﬃculties with generalisation,producing models that can overﬁt the data.This is a consequence of the optimisation algorithms used for parameter selection and the statistical measures used to select the’best’model.The foundations of Support Vector Machines(SVM)have been developed by Vapnik(1995)and are gaining popularity due to many attractive features,and promising empirical performance.The formulation embodies the Struc-tural Risk Minimisation(SRM)principle,which has been shown to be superior,(Gunn et al.,1997),to traditional Empirical Risk Minimisation(ERM)principle,employed by conventional neural networks.SRM minimises an upper bound on the expected risk, as opposed to ERM that minimises the error on the training data.It is this diﬀerence which equips SVM with a greater ability to generalise,which is the goal in statistical learning.SVMs were developed to solve the classiﬁcation problem,but recently they have been extended to the domain of regression problems(Vapnik et al.,1997).In the literature the terminology for SVMs can be slightly confusing.The term SVM is typ-ically used to describe classiﬁcation with support vector methods and support vector regression is used to describe regression with support vector methods.In this report the term SVM will refer to both classiﬁcation and regression methods,and the terms Support Vector Classiﬁcation(SVC)and Support Vector Regression(SVR)will be used for speciﬁcation.This section continues with a brief introduction to the structural risk12Chapter1Introductionminimisation principle.In Chapter2the SVM is introduced in the setting of classiﬁca-tion,being both historical and more accessible.This leads onto mapping the input into a higher dimensional feature space by a suitable choice of kernel function.The report then considers the problem of regression.Illustrative examples re given to show the properties of the techniques.1.1Statistical Learning TheoryThis section is a very brief introduction to statistical learning theory.For a much more in depth look at statistical learning theory,see(Vapnik,1998).Figure1.1:Modelling ErrorsThe goal in modelling is to choose a model from the hypothesis space,which is closest (with respect to some error measure)to the underlying function in the target space. Errors in doing this arise from two cases:Approximation Error is a consequence of the hypothesis space being smaller than the target space,and hence the underlying function may lie outside the hypothesis space.A poor choice of the model space will result in a large approximation error, and is referred to as model mismatch.Estimation Error is the error due to the learning procedure which results in a tech-nique selecting the non-optimal model from the hypothesis space.Chapter1Introduction3Together these errors form the generalisation error.Ultimately we would like toﬁnd the function,f,which minimises the risk,R[f]=X×YL(y,f(x))P(x,y)dxdy(1.1)However,P(x,y)is unknown.It is possible toﬁnd an approximation according to the empirical risk minimisation principle,R emp[f]=1lli=1Ly i,fx i(1.2)which minimises the empirical risk,ˆf n,l (x)=arg minf∈H nR emp[f](1.3)Empirical risk minimisation makes sense only if,liml→∞R emp[f]=R[f](1.4) which is true from the law of large numbers.However,it must also satisfy,lim l→∞minf∈H nR emp[f]=minf∈H nR[f](1.5)which is only valid when H n is’small’enough.This condition is less intuitive and requires that the minima also converge.The following bound holds with probability1−δ,R[f]≤R emp[f]+h ln2lh+1−lnδ4l(1.6)Remarkably,this expression for the expected risk is independent of the probability dis-tribution.1.1.1VC DimensionThe VC dimension is a scalar value that measures the capacity of a set offunctions.Figure1.2:VC Dimension Illustration4Chapter1IntroductionDeﬁnition1.1(Vapnik–Chervonenkis).The VC dimension of a set of functions is p if and only if there exists a set of points{x i}pi=1such that these points can be separatedin all2p possible conﬁgurations,and that no set{x i}qi=1exists where q>p satisfying this property.Figure1.2illustrates how three points in the plane can be shattered by the set of linear indicator functions whereas four points cannot.In this case the VC dimension is equal to the number of free parameters,but in general that is not the case;e.g.the function A sin(bx)has an inﬁnite VC dimension(Vapnik,1995).The set of linear indicator functions in n dimensional space has a VC dimension equal to n+1.1.1.2Structural Risk MinimisationCreate a structure such that S h is a hypothesis space of VC dimension h then,S1⊂S2⊂...⊂S∞(1.7) SRM consists in solving the following problemmin S h R emp[f]+h ln2lh+1−lnδ4l(1.8)If the underlying process being modelled is not deterministic the modelling problem becomes more exacting and consequently this chapter is restricted to deterministic pro-cesses.Multiple output problems can usually be reduced to a set of single output prob-lems that may be considered independent.Hence it is appropriate to consider processes with multiple inputs from which it is desired to predict a single output.Chapter2Support Vector ClassiﬁcationThe classiﬁcation problem can be restricted to consideration of the two-class problem without loss of generality.In this problem the goal is to separate the two classes by a function which is induced from available examples.The goal is to produce a classiﬁer that will work well on unseen examples,i.e.it generalises well.Consider the example in Figure2.1.Here there are many possible linear classiﬁers that can separate the data, but there is only one that maximises the margin(maximises the distance between it and the nearest data point of each class).This linear classiﬁer is termed the optimal separating hyperplane.Intuitively,we would expect this boundary to generalise well as opposed to the other possible boundaries.Figure2.1:Optimal Separating Hyperplane2.1The Optimal Separating HyperplaneConsider the problem of separating the set of training vectors belonging to two separateclasses,D=(x1,y1),...,(x l,y l),x∈R n,y∈{−1,1},(2.1)56Chapter 2Support Vector Classiﬁcationwith a hyperplane, w,x +b =0.(2.2)The set of vectors is said to be optimally separated by the hyperplane if it is separated without error and the distance between the closest vector to the hyperplane is maximal.There is some redundancy in Equation 2.2,and without loss of generality it is appropri-ate to consider a canonical hyperplane (Vapnik ,1995),where the parameters w ,b are constrained by,min i w,x i +b =1.(2.3)This incisive constraint on the parameterisation is preferable to alternatives in simpli-fying the formulation of the problem.In words it states that:the norm of the weight vector should be equal to the inverse of the distance,of the nearest point in the data set to the hyperplane .The idea is illustrated in Figure 2.2,where the distance from the nearest point to each hyperplane is shown.Figure 2.2:Canonical HyperplanesA separating hyperplane in canonical form must satisfy the following constraints,y i w,x i +b ≥1,i =1,...,l.(2.4)The distance d (w,b ;x )of a point x from the hyperplane (w,b )is,d (w,b ;x )= w,x i +bw .(2.5)Chapter 2Support Vector Classiﬁcation 7The optimal hyperplane is given by maximising the margin,ρ,subject to the constraints of Equation 2.4.The margin is given by,ρ(w,b )=min x i :y i =−1d (w,b ;x i )+min x i :y i =1d (w,b ;x i )=min x i :y i =−1 w,x i +b w +min x i :y i =1 w,x i +b w =1 w min x i :y i =−1 w,x i +b +min x i :y i =1w,x i +b =2 w (2.6)Hence the hyperplane that optimally separates the data is the one that minimisesΦ(w )=12 w 2.(2.7)It is independent of b because provided Equation 2.4is satisﬁed (i.e.it is a separating hyperplane)changing b will move it in the normal direction to itself.Accordingly the margin remains unchanged but the hyperplane is no longer optimal in that it will be nearer to one class than the other.To consider how minimising Equation 2.7is equivalent to implementing the SRM principle,suppose that the following bound holds,w <A.(2.8)Then from Equation 2.4and 2.5,d (w,b ;x )≥1A.(2.9)Accordingly the hyperplanes cannot be nearer than 1A to any of the data points and intuitively it can be seen in Figure 2.3how this reduces the possible hyperplanes,andhence thecapacity.Figure 2.3:Constraining the Canonical Hyperplanes8Chapter2Support Vector ClassiﬁcationThe VC dimension,h,of the set of canonical hyperplanes in n dimensional space is bounded by,h≤min[R2A2,n]+1,(2.10)where R is the radius of a hypersphere enclosing all the data points.Hence minimising Equation2.7is equivalent to minimising an upper bound on the VC dimension.The solution to the optimisation problem of Equation2.7under the constraints of Equation 2.4is given by the saddle point of the Lagrange functional(Lagrangian)(Minoux,1986),Φ(w,b,α)=12 w 2−li=1αiy iw,x i +b−1,(2.11)whereαare the Lagrange multipliers.The Lagrangian has to be minimised with respect to w,b and maximised with respect toα≥0.Classical Lagrangian duality enables the primal problem,Equation2.11,to be transformed to its dual problem,which is easier to solve.The dual problem is given by,max αW(α)=maxαminw,bΦ(w,b,α).(2.12)The minimum with respect to w and b of the Lagrangian,Φ,is given by,∂Φ∂b =0⇒li=1αi y i=0∂Φ∂w =0⇒w=li=1αi y i x i.(2.13)Hence from Equations2.11,2.12and2.13,the dual problem is,max αW(α)=maxα−12li=1lj=1αiαj y i y j x i,x j +lk=1αk,(2.14)and hence the solution to the problem is given by,α∗=arg minα12li=1lj=1αiαj y i y j x i,x j −lk=1αk,(2.15)with constraints,αi≥0i=1,...,llj=1αj y j=0.(2.16)Chapter2Support Vector Classiﬁcation9Solving Equation2.15with constraints Equation2.16determines the Lagrange multi-pliers,and the optimal separating hyperplane is given by,w∗=li=1αi y i x ib∗=−12w∗,x r+x s .(2.17)where x r and x s are any support vector from each class satisfying,αr,αs>0,y r=−1,y s=1.(2.18)The hard classiﬁer is then,f(x)=sgn( w∗,x +b)(2.19) Alternatively,a soft classiﬁer may be used which linearly interpolates the margin,f(x)=h( w∗,x +b)where h(z)=−1:z<−1z:−1≤z≤1+1:z>1(2.20)This may be more appropriate than the hard classiﬁer of Equation2.19,because it produces a real valued output between−1and1when the classiﬁer is queried within the margin,where no training data resides.From the Kuhn-Tucker conditions,αiy iw,x i +b−1=0,i=1,...,l,(2.21)and hence only the points x i which satisfy,y iw,x i +b=1(2.22)will have non-zero Lagrange multipliers.These points are termed Support Vectors(SV). If the data is linearly separable all the SV will lie on the margin and hence the number of SV can be very small.Consequently the hyperplane is determined by a small subset of the training set;the other points could be removed from the training set and recalculating the hyperplane would produce the same answer.Hence SVM can be used to summarise the information contained in a data set by the SV produced.If the data is linearly separable the following equality will hold,w 2=li=1αi=i∈SV sαi=i∈SV sj∈SV sαiαj y i y j x i,x j .(2.23)Hence from Equation2.10the VC dimension of the classiﬁer is bounded by,h≤min[R2i∈SV s,n]+1,(2.24)10Chapter2Support Vector Classiﬁcationx1x2y11-133113131-12 2.513 2.5-143-1Table2.1:Linearly Separable Classiﬁcation Dataand if the training data,x,is normalised to lie in the unit hypersphere,,n],(2.25)h≤1+min[i∈SV s2.1.1Linearly Separable ExampleTo illustrate the method consider the training set in Table2.1.The SVC solution is shown in Figure2.4,where the dotted lines describe the locus of the margin and the circled data points represent the SV,which all lie on the margin.Figure2.4:Optimal Separating Hyperplane2.2The Generalised Optimal Separating HyperplaneSo far the discussion has been restricted to the case where the training data is linearly separable.However,in general this will not be the case,Figure2.5.There are two approaches to generalising the problem,which are dependent upon prior knowledge of the problem and an estimate of the noise on the data.In the case where it is expected (or possibly even known)that a hyperplane can correctly separate the data,a method ofChapter2Support Vector Classiﬁcation11Figure2.5:Generalised Optimal Separating Hyperplaneintroducing an additional cost function associated with misclassiﬁcation is appropriate. Alternatively a more complex function can be used to describe the boundary,as discussed in Chapter2.1.To enable the optimal separating hyperplane method to be generalised, Cortes and Vapnik(1995)introduced non-negative variables,ξi≥0,and a penalty function,Fσ(ξ)=iξσiσ>0,(2.26) where theξi are a measure of the misclassiﬁcation errors.The optimisation problem is now posed so as to minimise the classiﬁcation error as well as minimising the bound on the VC dimension of the classiﬁer.The constraints of Equation2.4are modiﬁed for the non-separable case to,y iw,x i +b≥1−ξi,i=1,...,l.(2.27)whereξi≥0.The generalised optimal separating hyperplane is determined by the vector w,that minimises the functional,Φ(w,ξ)=12w 2+Ciξi,(2.28)(where C is a given value)subject to the constraints of Equation2.27.The solution to the optimisation problem of Equation2.28under the constraints of Equation2.27is given by the saddle point of the Lagrangian(Minoux,1986),Φ(w,b,α,ξ,β)=12 w 2+Ciξi−li=1αiy iw T x i+b−1+ξi−lj=1βiξi,(2.29)12Chapter2Support Vector Classiﬁcationwhereα,βare the Lagrange multipliers.The Lagrangian has to be minimised with respect to w,b,x and maximised with respect toα,β.As before,classical Lagrangian duality enables the primal problem,Equation2.29,to be transformed to its dual problem. The dual problem is given by,max αW(α,β)=maxα,βminw,b,ξΦ(w,b,α,ξ,β).(2.30)The minimum with respect to w,b andξof the Lagrangian,Φ,is given by,∂Φ∂b =0⇒li=1αi y i=0∂Φ∂w =0⇒w=li=1αi y i x i∂Φ∂ξ=0⇒αi+βi=C.(2.31) Hence from Equations2.29,2.30and2.31,the dual problem is,max αW(α)=maxα−12li=1lj=1αiαj y i y j x i,x j +lk=1αk,(2.32)and hence the solution to the problem is given by,α∗=arg minα12li=1lj=1αiαj y i y j x i,x j −lk=1αk,(2.33)with constraints,0≤αi≤C i=1,...,llj=1αj y j=0.(2.34)The solution to this minimisation problem is identical to the separable case except for a modiﬁcation of the bounds of the Lagrange multipliers.The uncertain part of Cortes’s approach is that the coeﬃcient C has to be determined.This parameter introduces additional capacity control within the classiﬁer.C can be directly related to a regulari-sation parameter(Girosi,1997;Smola and Sch¨o lkopf,1998).Blanz et al.(1996)uses a value of C=5,but ultimately C must be chosen to reﬂect the knowledge of the noise on the data.This warrants further work,but a more practical discussion is given in Chapter4.Chapter2Support Vector Classiﬁcation13x1x2y11-133113131-12 2.513 2.5-143-11.5 1.5112-1Table2.2:Non-Linearly Separable Classiﬁcation Data2.2.1Linearly Non-Separable ExampleTwo additional data points are added to the separable data of Table2.1to produce a linearly non-separable data set,Table2.2.The resulting SVC is shown in Figure2.6,for C=1.The SV are no longer required to lie on the margin,as in Figure2.4,and the orientation of the hyperplane and the width of the margin are diﬀerent.Figure2.6:Generalised Optimal Separating Hyperplane Example(C=1)In the limit,lim C→∞the solution converges towards the solution obtained by the optimal separating hyperplane(on this non-separable data),Figure2.7.In the limit,lim C→0the solution converges to one where the margin maximisation term dominates,Figure2.8.Beyond a certain point the Lagrange multipliers will all take on the value of C.There is now less emphasis on minimising the misclassiﬁcation error, but purely on maximising the margin,producing a large width margin.Consequently as C decreases the width of the margin increases.The useful range of C lies between the point where all the Lagrange Multipliers are equal to C and when only one of them is just bounded by C.14Chapter2Support Vector ClassiﬁcationFigure2.7:Generalised Optimal Separating Hyperplane Example(C=105)Figure2.8:Generalised Optimal Separating Hyperplane Example(C=10−8)2.3Generalisation in High Dimensional Feature SpaceIn the case where a linear boundary is inappropriate the SVM can map the input vector, x,into a high dimensional feature space,z.By choosing a non-linear mapping a priori, the SVM constructs an optimal separating hyperplane in this higher dimensional space, Figure2.9.The idea exploits the method of Aizerman et al.(1964)which,enables the curse of dimensionality(Bellman,1961)to be addressed.Figure2.9:Mapping the Input Space into a High Dimensional Feature SpaceChapter2Support Vector Classiﬁcation15There are some restrictions on the non-linear mapping that can be employed,see Chap-ter3,but it turns out,surprisingly,that most commonly employed functions are accept-able.Among acceptable mappings are polynomials,radial basis functions and certain sigmoid functions.The optimisation problem of Equation2.33becomes,α∗=arg minα12li=1lj=1αiαj y i y j K(x i,x j)−lk=1αk,(2.35)where K(x,x )is the kernel function performing the non-linear mapping into feature space,and the constraints are unchanged,0≤αi≤C i=1,...,llj=1αj y j=0.(2.36)Solving Equation2.35with constraints Equation2.36determines the Lagrange multipli-ers,and a hard classiﬁer implementing the optimal separating hyperplane in the feature space is given by,f(x)=sgn(i∈SV sαi y i K(x i,x)+b)(2.37) wherew∗,x =li=1αi y i K(x i,x)b∗=−12li=1αi y i[K(x i,x r)+K(x i,x r)].(2.38)The bias is computed here using two support vectors,but can be computed using all the SV on the margin for stability(Vapnik et al.,1997).If the Kernel contains a bias term, the bias can be accommodated within the Kernel,and hence the classiﬁer is simply,f(x)=sgn(i∈SV sαi K(x i,x))(2.39)Many employed kernels have a bias term and anyﬁnite Kernel can be made to have one(Girosi,1997).This simpliﬁes the optimisation problem by removing the equality constraint of Equation2.36.Chapter3discusses the necessary conditions that must be satisﬁed by valid kernel functions.16Chapter2Support Vector Classiﬁcation 2.3.1Polynomial Mapping ExampleConsider a polynomial kernel of the form,K(x,x )=( x,x +1)2,(2.40)which maps a two dimensional input vector into a six dimensional feature space.Apply-ing the non-linear SVC to the linearly non-separable training data of Table2.2,produces the classiﬁcation illustrated in Figure2.10(C=∞).The margin is no longer of constant width due to the non-linear projection into the input space.The solution is in contrast to Figure2.7,in that the training data is now classiﬁed correctly.However,even though SVMs implement the SRM principle and hence can generalise well,a careful choice of the kernel function is necessary to produce a classiﬁcation boundary that is topologically appropriate.It is always possible to map the input space into a dimension greater than the number of training points and produce a classiﬁer with no classiﬁcation errors on the training set.However,this will generalise badly.Figure2.10:Mapping input space into Polynomial Feature Space2.4DiscussionTypically the data will only be linearly separable in some,possibly very high dimensional feature space.It may not make sense to try and separate the data exactly,particularly when only aﬁnite amount of training data is available which is potentially corrupted by noise.Hence in practice it will be necessary to employ the non-separable approach which places an upper bound on the Lagrange multipliers.This raises the question of how to determine the parameter C.It is similar to the problem in regularisation where the regularisation coeﬃcient has to be determined,and it has been shown that the parameter C can be directly related to a regularisation parameter for certain kernels (Smola and Sch¨o lkopf,1998).A process of cross-validation can be used to determine thisChapter2Support Vector Classiﬁcation17parameter,although more eﬃcient and potentially better methods are sought after.In removing the training patterns that are not support vectors,the solution is unchanged and hence a fast method for validation may be available when the support vectors are sparse.Chapter3Feature SpaceThis chapter discusses the method that can be used to construct a mapping into a high dimensional feature space by the use of reproducing kernels.The idea of the kernel function is to enable operations to be performed in the input space rather than the potentially high dimensional feature space.Hence the inner product does not need to be evaluated in the feature space.This provides a way of addressing the curse of dimensionality.However,the computation is still critically dependent upon the number of training patterns and to provide a good data distribution for a high dimensional problem will generally require a large training set.3.1Kernel FunctionsThe following theory is based upon Reproducing Kernel Hilbert Spaces(RKHS)(Aron-szajn,1950;Girosi,1997;Heckman,1997;Wahba,1990).An inner product in feature space has an equivalent kernel in input space,K(x,x )= φ(x),φ(x ) ,(3.1)provided certain conditions hold.If K is a symmetric positive deﬁnite function,which satisﬁes Mercer’s Conditions,K(x,x )=∞ma mφm(x)φm(x ),a m≥0,(3.2)K(x,x )g(x)g(x )dxdx >0,g∈L2,(3.3) then the kernel represents a legitimate inner product in feature space.Valid functions that satisfy Mercer’s conditions are now given,which unless stated are valid for all real x and x .1920Chapter3Feature Space3.1.1PolynomialA polynomial mapping is a popular method for non-linear modelling,K(x,x )= x,x d.(3.4)K(x,x )=x,x +1d.(3.5)The second kernel is usually preferable as it avoids problems with the hessian becoming zero.3.1.2Gaussian Radial Basis FunctionRadial basis functions have received signiﬁcant attention,most commonly with a Gaus-sian of the form,K(x,x )=exp−x−x 22σ2.(3.6)Classical techniques utilising radial basis functions employ some method of determining a subset of centres.Typically a method of clustering isﬁrst employed to select a subset of centres.An attractive feature of the SVM is that this selection is implicit,with each support vectors contributing one local Gaussian function,centred at that data point. By further considerations it is possible to select the global basis function width,s,using the SRM principle(Vapnik,1995).3.1.3Exponential Radial Basis FunctionA radial basis function of the form,K(x,x )=exp−x−x2σ2.(3.7)produces a piecewise linear solution which can be attractive when discontinuities are acceptable.3.1.4Multi-Layer PerceptronThe long established MLP,with a single hidden layer,also has a valid kernel represen-tation,K(x,x )=tanhρ x,x +(3.8)for certain values of the scale,ρ,and oﬀset, ,parameters.Here the SV correspond to theﬁrst layer and the Lagrange multipliers to the weights.。

机器学习经典书目汇总

机器学习经典书目汇总本文总结了机器学习的经典书籍，包括数学基础和算法理论的书籍。

入门书单《数学之美》作者吴军大家都很熟悉。

以极为通俗的语言讲述了数学在机器学习和自然语言处理等领域的应用。

《Programming Collective Intelligence》（《集体智慧编程》）作者Toby Segaran也是《BeautifulData : The Stories Behind Elegant Data Solutions》（《数据之美：解密优雅数据解决方案背后的故事》）的作者。

这本书最大的优势就是里面没有理论推导和复杂的数学公式，是很不错的入门书。

目前中文版已经脱销，对于有志于这个领域的人来说，英文的pdf是个不错的选择，因为后面有很多经典书的翻译都较差，只能看英文版，不如从这个入手。

还有，这本书适合于快速看完，因为据评论，看完一些经典的带有数学推导的书后会发现这本书什么都没讲，只是举了很多例子而已。

《Algorithms of the Intelligent Web》（《智能web算法》）作者Haralambos Marmanis、Dmitry Babenko。

这本书中的公式比《集体智慧编程》要略多一点，里面的例子多是互联网上的应用，看名字就知道。

不足的地方在于里面的配套代码是BeanShell而不是python或其他。

总起来说，这本书还是适合初学者，与上一本一样需要快速读完，如果读完上一本的话，这一本可以不必细看代码，了解算法主要思想就行了。

《统计学习方法》作者李航，是国内机器学习领域的几个大家之一，曾在MSRA 任高级研究员，现在华为诺亚方舟实验室。

书中写了十个算法，每个算法的介绍都很干脆，直接上公式，是彻头彻尾的“干货书”。

每章末尾的参考文献也方便了想深入理解算法的童鞋直接查到经典论文；本书可以与上面两本书互为辅助阅读。

《Machine Learning》（《机器学习》）作者Tom Mitchell是CMU的大师，有机器学习和半监督学习的网络课程视频。

统计学习精要

统计学习精要(The Elements of Statistical Learning)课堂笔记系列课程教材：The Elements of StatisticalLearning /~tibs/ElemStatLearn/授课人：复旦大学计算机学院吴立德教授≪统计学习精要(The Elements of Statistical Learning)≫课堂笔记（一）前两天微博上转出来的，复旦计算机学院的吴立德吴老师在开?统计学习精要(The Elements of Statistical Learning)?这门课，还在张江...大牛的课怎能错过，果断请假去蹭课...为了减轻心理压力，还拉了一帮同事一起去听，eBay浩浩荡荡的十几人杀过去好不壮观！总感觉我们的人有超过复旦本身学生的阵势，五六十人的教室坐的满满当当，壮观啊。

这本书正好前阵子一直在看，所以才会屁颠屁颠的跑过去听。

确实是一本深入浅出讲data mining models的好书。

作者网站上提供免费的电子版下载，爽！/~tibs/ElemStatLearn/从这周开始，如无意外我会每周更新课堂笔记。

另一方面，也会加上自己的一些理解和实际工作中的感悟。

此外，对于data mining感兴趣的，也可以去coursera听课~貌似这学期开的machine learning评价不错。

我只在coursera上从众选了一门「Model Thinking」，相对来说比较简单，但是相当的优雅！若有时间会再写写这门课的上课感受。

笔记我会尽量用全部中文，但只是尽量...------------课堂笔记开始--------第一次上课，主要是导论，介绍这个领域的关注兴趣以及后续课程安排。

对应本书的第一章。

1. 统计学习是？从数据中学习知识。

简单地说，我们有一个想预测的结果(outcome)，记为Y，可能是离散的也可能是连续的。

同时，还有一些观察到的特征(feature)，记为X，X 既可能是一维的也可能是多维的。

数据分析英语报告模版(3篇)

第1篇Executive SummaryThis report presents the findings from a comprehensive data analysis of [Subject/Industry/Company Name]. The analysis was conducted using a variety of statistical and analytical techniques to uncover trends, patterns, and insights that are relevant to the decision-making process within the [Subject/Industry/Company Name]. The report is structured as follows:1. Introduction2. Methodology3. Data Overview4. Data Analysis5. Findings6. Recommendations7. Conclusion1. Introduction[Provide a brief overview of the report's purpose, the subject of the analysis, and the context in which the data was collected.]The objective of this report is to [state the objective of the analysis, e.g., identify market trends, assess customer satisfaction, or optimize business processes]. The data used in this analysis was sourced from [describe the data sources, e.g., internal databases, surveys, external market research reports].2. MethodologyThis section outlines the methods and techniques used to analyze the data.a. Data Collection- Describe the data collection process, including the sources of thedata and the methods used to collect it.b. Data Cleaning- Explain the steps taken to clean the data, such as removing duplicates, handling missing values, and correcting errors.c. Data Analysis Techniques- List the statistical and analytical techniques used, such asregression analysis, clustering, time series analysis, and machine learning algorithms.d. Tools and Software- Mention the tools and software used for data analysis, such as Python, R, Excel, and Tableau.3. Data OverviewIn this section, provide a brief overview of the data, including the following:- Data sources and types- Time period covered- Key variables and measures- Sample size and demographics4. Data AnalysisThis section delves into the detailed analysis of the data, using visualizations and statistical tests to illustrate the findings.a. Descriptive Statistics- Present descriptive statistics such as mean, median, mode, standard deviation, and variance for the key variables.b. Data Visualization- Use charts, graphs, and maps to visualize the data and highlight key trends and patterns.c. Hypothesis Testing- Conduct hypothesis tests to determine the statistical significance of the findings.d. Predictive Modeling- If applicable, build predictive models to forecast future trends or outcomes.5. FindingsThis section summarizes the key findings from the data analysis.- Highlight the most important trends, patterns, and insights discovered.- Discuss the implications of these findings for the[Subject/Industry/Company Name].- Compare the findings to industry benchmarks or past performance.6. RecommendationsBased on the findings, provide actionable recommendations for the [Subject/Industry/Company Name].- Outline specific strategies or actions that could be taken to capitalize on the insights gained from the analysis.- Prioritize the recommendations based on potential impact and feasibility.7. ConclusionConclude the report by summarizing the key points and reiterating the value of the data analysis.- Reiterate the main findings and their significance.- Emphasize the potential impact of the recommendations on the [Subject/Industry/Company Name].- Suggest next steps or future areas of analysis.Appendices- Include any additional information or data that supports the report but is not essential to the main narrative.References- List all the sources of data and any external references used in the report.---Note: The following is an example of how the report might be structured with some placeholder content.---Executive SummaryThis report presents the findings from a comprehensive data analysis of the e-commerce sales trends for XYZ Corporation over the past fiscal year. The analysis aimed to identify key patterns in customer behavior, sales performance, and market dynamics to inform strategic decision-making. The report utilizes a variety of statistical and analytical techniques, including regression analysis, clustering, and time series forecasting. The findings suggest several opportunities for improving sales performance and customer satisfaction.1. Introduction[Placeholder: Provide an introduction to the report, including the purpose and context.]2. Methodologya. Data CollectionData for this analysis was collected from XYZ Corporation's internal sales database, which includes transactional data for all online sales over the past fiscal year. The dataset includes information on customer demographics, purchase history, product categories, and sales performance metrics.b. Data CleaningThe data was cleaned to ensure accuracy and consistency. This involved removing duplicate entries, handling missing values, and correcting any inconsistencies in the data.c. Data Analysis TechniquesStatistical techniques such as regression analysis were used to identify correlations between customer demographics and purchase behavior. Clustering was employed to segment customers based on their purchasing patterns. Time series forecasting was used to predict future sales trends.d. Tools and SoftwarePython and R were used for data analysis, with Excel and Tableau for data visualization.3. Data OverviewThe dataset covers a total of 10 million transactions over the past fiscal year, involving over 1 million unique customers. The data includes information on over 5,000 product categories.4. Data Analysisa. Descriptive Statistics[Placeholder: Present descriptive statistics for key variables, such as average order value, customer acquisition cost, and customer lifetime value.]b. Data Visualization[Placeholder: Include visualizations such as line graphs for sales trends over time, bar charts for product category performance, and pie charts for customer segmentation.]c. Hypothesis Testing[Placeholder: Describe the hypothesis testing conducted, such as testing the relationship between customer age and spending habits.]d. Predictive Modeling[Placeholder: Outline the predictive models developed, such as a model to forecast sales based on historical data and external market indicators.]5. FindingsThe analysis revealed several key findings:- Customers aged 25-34 are the highest spenders.- The product category with the highest growth rate is electronics.- The company's customer acquisition cost is higher than the industry average.6. RecommendationsBased on the findings, the following recommendations are made:- Target marketing efforts towards the 25-34 age group.- Invest in marketing campaigns for the electronics product category.- Reduce customer acquisition costs by optimizing marketing channels.7. ConclusionThe data analysis provides valuable insights into XYZ Corporation's e-commerce sales performance and customer behavior. By implementing the recommended strategies, the company can improve its sales performance and enhance customer satisfaction.Appendices[Placeholder: Include any additional data or information that supports the report.]References[Placeholder: List all the sources of data and any external references used in the report.]---This template serves as a guide for structuring a comprehensive data analysis report. Adjust the content and format as needed to fit the specific requirements of your analysis and audience.第2篇Executive Summary:This report presents a comprehensive analysis of customer purchase behavior on an e-commerce platform. By examining various data points and employing advanced analytical techniques, we aim to uncover trends, patterns, and insights that can inform business strategies, enhance customer experience, and drive sales growth. The report is structuredinto several sections, including an overview of the dataset, methodology, results, and recommendations.1. Introduction1.1 Background:The rapid growth of e-commerce has transformed the retail landscape, offering businesses unprecedented opportunities to reach a global audience. Understanding customer purchase behavior is crucial for e-commerce platforms to tailor their offerings, improve customer satisfaction, and increase profitability.1.2 Objectives:The primary objectives of this analysis are to:- Identify key trends in customer purchase behavior.- Understand the factors influencing customer decisions.- Propose strategies to enhance customer satisfaction and drive sales.2. Dataset Overview2.1 Data Sources:The dataset used for this analysis is a combination of transactional data, customer demographics, and product information obtained from an e-commerce platform.2.2 Data Description:The dataset includes the following variables:- Customer demographics: Age, gender, location, income level.- Purchase history: Product categories purchased, purchase frequency, average order value.- Product information: Product category, price, brand, rating.- Transactional data: Purchase date, time, payment method, shipping address.3. Methodology3.1 Data Cleaning:Prior to analysis, the dataset was cleaned to address missing values, outliers, and inconsistencies.3.2 Data Exploration:Initial data exploration was conducted to identify patterns, trends, and relationships within the dataset.3.3 Statistical Analysis:Descriptive statistics were used to summarize the dataset and identify key characteristics of customer purchase behavior.3.4 Predictive Modeling:Advanced predictive models, such as regression analysis and clustering, were employed to identify factors influencing customer purchase decisions.3.5 Visualization:Data visualization techniques were used to present the results in an easily interpretable format.4. Results4.1 Customer Demographics:Analysis revealed that the majority of customers are between the ages of 25-34, with a slight male majority. Customers from urban areas tend to have higher average order values.4.2 Purchase Behavior:The dataset showed a strong preference for electronics and fashion products, with a significant number of repeat purchases in these categories. The average order value was highest during festive seasons and weekends.4.3 Influencing Factors:Several factors were identified as influential in customer purchase decisions, including product price, brand reputation, and customer reviews.4.4 Predictive Models:Predictive models accurately predicted customer purchase behavior based on the identified influencing factors.5. Discussion5.1 Key Findings:The analysis confirmed that customer demographics, product categories, and influencing factors play a significant role in shaping purchase behavior on the e-commerce platform.5.2 Limitations:The analysis was limited by the availability of data and the scope of the study. Further research could explore the impact of additional factors, such as marketing campaigns and social media influence.6. Recommendations6.1 Enhancing Customer Experience:- Implement personalized product recommendations based on customer purchase history.- Offer targeted promotions and discounts to encourage repeat purchases.6.2 Improving Marketing Strategies:- Allocate marketing budgets to products with high customer demand and positive reviews.- Develop targeted marketing campaigns for different customer segments.6.3 Product Development:- Invest in product development based on customer preferences and feedback.- Monitor market trends to stay ahead of the competition.7. ConclusionThis report provides valuable insights into customer purchase behavior on an e-commerce platform. By understanding the factors influencing customer decisions, businesses can tailor their strategies to enhance customer satisfaction and drive sales growth. The recommendations outlined in this report can serve as a roadmap for businesses looking to capitalize on the e-commerce market.References:- Smith, J., & Johnson, L. (2020). "Customer Purchase Behavior in E-commerce: A Review." Journal of E-commerce Studies, 15(2), 45-60.- Brown, A., & White, M. (2019). "The Role of Customer Demographics inE-commerce Success." International Journal of Marketing Research, 12(3), 78-95.Appendix:- Detailed data visualization plots and tables.- Code snippets for predictive modeling.---This template provides a comprehensive structure for an English reporton data analysis. You can expand on each section with specific data, insights, and recommendations tailored to your dataset and analysis objectives.第3篇---Executive SummaryThis report presents the findings of a comprehensive data analysis conducted on [Subject of Analysis]. The analysis aimed to [State the objective of the analysis]. The report outlines the methodology employed, the key insights derived from the data, and the recommendations based on the findings.---1. Introduction1.1 BackgroundProvide a brief background on the subject of analysis, including any relevant historical context or industry trends.1.2 ObjectiveClearly state the objective of the data analysis. What specificquestions or problems are you trying to address?1.3 ScopeDefine the scope of the analysis. What data sources were used? What time frame is covered?---2. Methodology2.1 Data CollectionExplain how the data was collected. Describe the data sources, data collection methods, and any limitations associated with the data.2.2 Data ProcessingDetail the steps taken to process the data. This may include data cleaning, data transformation, and data integration.2.3 Analytical TechniquesDescribe the analytical techniques used. This could include statistical analysis, predictive modeling, machine learning, or other relevant methods.2.4 Tools and SoftwareList the tools and software used in the analysis. For example, Python, R, SAS, SPSS, Excel, etc.---3. Data Analysis3.1 Descriptive StatisticsPresent descriptive statistics such as mean, median, mode, standard deviation, and variance to summarize the central tendency and spread of the data.3.2 Data VisualizationUse charts, graphs, and maps to visualize the data. Explain what each visualization represents and how it contributes to understanding the data.3.3 Hypothesis TestingIf applicable, discuss the hypothesis testing conducted. State the null and alternative hypotheses, the test statistics, and the p-values.3.4 Predictive ModelingIf predictive modeling was part of the analysis, describe the model built, the evaluation metrics used, and the model's performance.---4. Key Insights4.1 Major FindingsSummarize the major findings of the analysis. What trends, patterns, or relationships were discovered?4.2 ImplicationsDiscuss the implications of the findings for the business, industry, or research question at hand.4.3 LimitationsAcknowledge any limitations of the analysis. How might these limitations affect the validity or generalizability of the findings?---5. RecommendationsBased on the findings, provide actionable recommendations. These should be practical, specific, and tailored to the context of the analysis.5.1 Short-term RecommendationsOffer recommendations that can be implemented in the near term to address immediate issues or opportunities.5.2 Long-term RecommendationsProvide recommendations for strategies that can be developed over a longer period to support sustainable outcomes.---6. ConclusionReiterate the main findings and their significance. Emphasize the value of the analysis and how it contributes to the understanding of the subject matter.---7. AppendicesInclude any additional material that supports the report but is not essential to the main body. This could be detailed data tables, code snippets, or additional visualizations.---ReferencesList all the sources cited in the report, following the appropriate citation style (e.g., APA, MLA, Chicago).---8. About the AuthorProvide a brief biography of the author(s) of the report, including their qualifications and relevant experience.---9. Contact InformationInclude the contact information for the author(s) or the organization responsible for the report.---This template is designed to be flexible, allowing you to tailor the content to the specific requirements of your data analysis project. Remember to ensure that the report is clear, concise, and accessible to the intended audience.。

统计学习理论导论(清华大学张学工讲义)-1

• How to decide the structure of the MLP?
(How many hidden layers and nodes?)
– Ask God, or guess then pray
• How to choose the neuron function?
– Usually Sigmoid (S-shaped) function
– the effort to approach mathematic models for natural nervous systems
– the effort to implement man-made intelligence
• Three types of NN:
– Feedforward NN – Feedback NN – Competitive Learning (Self-organizing) NN
Xuegong Zhang
27
Tsinghua University
学习过程的应用分析与理论分析学派
• 关于感知器学习能力的若干结论： – 关于收敛性的结论 – 关于收敛以后的测试错误率（推广能力）的结论
[Novikoff, 1962] [Aizerman, Braverman, and Rozonoer, 1964]
• 学习过程的应用分析学派：
– 最小化训练错误数是不言而喻的归纳原则，学习的主要问题在于寻找同时构造所有神经元的系数的方法，使所形成的分类面能达
到最小的训练错误率，（这样即可得到好的推广性）
• 学习过程的理论分析学派：
Xuegong Zhang
14
Tsinghua University

多元回归模型的方差

多元回归模型的方差多元回归模型的方差1. 引言多元回归模型是统计学中一种重要的建模方法，可以用于分析多个自变量与一个因变量之间的关系。

在实际应用中，我们常常关注模型的预测能力和可解释性，而模型的方差则是评估其预测能力的一个重要指标。

本文将从深度和广度两个维度，探讨多元回归模型的方差。

2. 深度分析2.1 方差的定义和意义方差是统计学中常用的一个概念，用来衡量随机变量的离散程度。

在多元回归模型中，我们关注的是模型的方差，即预测结果与实际观测值之间的差异。

方差越大，说明模型的预测结果的离散程度越大，预测的准确性越低。

2.2 方差的计算方法在多元回归模型中，方差可以通过计算预测值与观测值的残差平方和来估计。

残差是指预测值与观测值之间的差异，残差平方和则反映了预测误差的总体离散程度。

方差的计算可以用以下公式表示：方差 = 残差平方和 / (样本量 - 1)2.3 影响方差的因素多元回归模型的方差受多个因素的影响，其中最常见的是以下三个因素：- 自变量的选择：自变量的选择对模型方差有较大影响。

如果自变量之间高度相关，那么模型的方差可能会增加，因为这些变量可能提供了相似的信息，增加了预测的不确定性。

- 样本量的大小：样本量的大小也会对模型方差产生影响。

当样本量较小时，模型很可能过拟合，导致方差较大。

而当样本量较大时，模型更有可能准确地捕捉到真实的关系，从而减小方差。

- 模型的复杂度：模型的复杂度是指模型中涉及的自变量个数和形式的多样性。

当模型过于复杂时，可能会产生过拟合现象，导致方差增大。

而当模型过于简单时，可能会漏掉一些重要的预测变量，导致方差减小。

3. 广度探讨3.1 方差的影响因素除了上述深度分析中提到的因素外，还有其他一些因素也会对多元回归模型的方差产生影响。

- 数据的噪声：如果数据中存在大量的噪声，即观测误差较大，则模型的方差会增大。

因为模型将尝试去适应这些噪声，从而导致预测误差的增加。

- 基础假设的违反：多元回归模型建立在一系列基础假设的基础上，比如线性独立性、方差齐性等。

博士必读数学

博士必读数学前面几篇谈了一些对数学的粗浅看法。

其实，如果对某门数学有兴趣，最好的方法就是走进那个世界去学习和体验。

这里说说几本我看过后觉得不错的数学教科书。

1. 线性代数(Linear Algebra)：我想国内的大学生都会学过这门课程，但是，未必每一位老师都能贯彻它的精要。

这门学科对于Learning 是必备的基础，对它的透彻掌握是必不可少的。

我在科大一年级的时候就学习了这门课，后来到了香港后，又重新把线性代数读了一遍，所读的是Introduction to Linear Algebra (3rd Ed.) by Gilbert Strang.这本书是MIT的线性代数课使用的教材，也是被很多其它大学选用的经典教材。

它的难度适中，讲解清晰，重要的是对许多核心的概念讨论得比较透彻。

我个人觉得，学习线性代数，最重要的不是去熟练矩阵运算和解方程的方法——这些在实际工作中MATLAB可以代劳，关键的是要深入理解几个基础而又重要的概念：子空间(Subspace)，正交(Orthogonality)，特征值和特征向量(Eigenvalues and eigenvectors)，和线性变换(Linear transform)。

（如果你能理解傅立叶变化究竟做了一件什么事情，你才能说你知道了子空间！学线性代数一定要理解MATLAB能为你做的事情之外其他的东西，这才是精髓。

而很遗憾，很多高校的线性代数考试只测试学生的计算能力。

有几个数学老师能告诉学生：我们为什么要计算特征值？）从我的角度看来，一本线代教科书的质量，就在于它能否给这些根本概念以足够的重视，能否把它们的联系讲清楚。

Strang 的这本书在这方面是做得很好的。

而且，这本书有个得天独厚的优势。

书的作者长期在MIT讲授线性代数课(18.06)，课程的video在MIT 的Open courseware网站上有提供。

有时间的朋友可以一边看着名师授课的录像，一边对照课本学习或者复习。

专题四--直线模型总结

专题四--直线模型总结直线模型是统计学中一个重要的概念，被广泛应用于回归分析和预测模型中。

本文对直线模型进行总结，并提供了一些使用直线模型的简单策略。

直线模型概述直线模型是指通过一条直线来描述变量之间的关系。

在统计学中，通常使用线性回归分析来建立直线模型。

直线模型的形式可以表示为：y = β₀ + β₁x，其中y是因变量，x是自变量，β₀和β₁是直线模型的系数。

直线模型的应用直线模型在许多领域中都有广泛的应用，并可以解决各种问题。

以下是一些使用直线模型的常见策略：1. 数据趋势分析：通过拟合一条直线到数据点上，可以分析数据的趋势，并预测未来的值。

2. 预测模型：直线模型可以用于建立预测模型，通过输入自变量的值，可以预测因变量的值。

3. 关系建模：直线模型可以用于建立变量之间的关系模型，从而探索变量之间的相关性。

4. 参数估计：直线模型可以通过最小二乘法来估计模型的系数，从而求解模型的参数。

使用直线模型的简单策略在使用直线模型时，可以考虑以下简单的策略来获取准确的结果：1. 数据准备：确保数据的质量和完整性，包括数据清洗、缺失值处理等。

2. 模型选择：根据实际问题选择合适的直线模型，可以考虑线性回归、多变量线性回归等。

3. 模型评估：通过拟合优度、残差分析等方法评估直线模型的好坏，并进行必要的调整。

4. 结果解释：在报告结果时，应清晰地解释直线模型的系数和统计显著性，以及模型的解释能力和局限性。

总之，直线模型是一个简单而强大的工具，可以用于数据分析、建模和预测。

通过合理地使用直线模型和相关策略，可以获得准确和有效的结果。

以上是对直线模型的总结，希望对您有所帮助。

参考文献：- Smith, J. (2010). Introduction to regression analysis. Wiley.- James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning. Springer.。

统计系本科生参考书整理

统计系本科⽣参考书整理前⾔：推荐的书单包括统计系本科⽣课程密切相关的中⽂书籍或者中译本：统计历史，统计学⼊门（⾮数学专业的统计书），数学分析，线性代数，概率论，数理统计，随机过程，R语⾔，⼤数据，⾦融统计，⾦融数学，⽣存分析，寿险精算，精算与风险模型。

希望国内的初学者多看⼀些国内外⼤师（国内是院⼠级别，国外的学者是资深会⼠级别）或者国内⼀些教学名师（⼀般来说得有20-30年教学经验）写的书，能在这些好书中取其精华，精华包括教学和研究⽅⾯的思想。

先引⽤⽹友的⼀段话：选⼀本适合⾃⼰的好的教材对⾃⼰以后的学习是决定性的重要–这是学数学的⼈⾸先必须明⽩的不仅是对概率统计⽅向，对数学的各个分⽀都是如此。

⼤⼀的时候齐名友⽼师跟我特别提到过这⼀点，可惜我当时不以为然，结果⾛了很多弯路，到研究⽣以后才慢慢明⽩这个道理。

⼀本⼭寨⼩学校的⽼师七拼⼋凑编写的烂书，常常对学习（特别是⾃学）不仅⽆益反⽽有害，因为你往往浪费了时间却只能得到这个⼀些⽀离破碎的印象，这样你会遗忘得很快，很可能到头来你还得重新学⼀遍；另⼀些时候，你选择了众⼈推荐的名著，但你如果当前的⽔平达不到⼀定的层次，它往往会打击你的信⼼让你灰⼼丧⽓，甚⾄会让你不再有学下去的欲望。

这两种情形显然都是⼈们应该尽量避免的。

要是英⽂好过了6级的同学，有英⽂版尽量看英⽂版。

统计历史：1.《⼥⼠品茶—21世纪统计学怎样变⾰了科学》 Salsburg David（是美国统计学家萨尔斯伯格以“⼥⼠品茶问题”为切⼊点所著的⼀部关于统计学历史与变⾰的书,以⼀种全新全新的视⾓带领读者进⼊统计学的世界，体会统计学带给哲学观、宇宙观的变⾰。

英⽂版：The Lady Tasting Tea—How Statistics Revolutionized Science in the Twentieth Century）2.《统计与真理—怎样运⽤偶然性》 C.R.Rao(本书是当代国际最著名的统计学家之⼀C.R.Rao的⼀部统计学哲理论著，也是他毕⽣统计学术思想的总结，同时还是⼀本通俗的关于统计学原理的普及教科书。

回归模型交互项系数的解释

回归模型交互项系数的解释引言回归模型是统计学中常用的分析工具，用于研究自变量和因变量之间的关系。

在回归模型中，交互项系数扮演着重要的角色，用于描述自变量之间的相互作用效应。

本文将详细介绍回归模型交互项系数的解释方法。

为什么需要交互项当我们研究自变量对因变量的影响时，可能会忽略了自变量之间的相互作用。

自变量之间的相互作用可能导致自变量的效应发生变化，因此在建立回归模型时，考虑自变量的交互项可以提高模型的准确性和解释力。

交互项系数的定义与表达在回归模型中，我们可以通过引入交互项来描述自变量之间的相互作用。

假设我们有两个自变量X1和X2，回归模型可以表示为：Y = β0 + β1*X1 + β2*X2 + β3*X1*X2 + ε其中，β3为交互项系数，表示自变量X1和X2的交互作用对因变量Y的影响。

交互作用的效应可以是增强或减弱的，具体取决于β3的正负。

交互项系数的解释为了解释交互项系数的含义，我们可以通过以下方法来理解它的影响：1. 系数的正负交互项系数的正负可以告诉我们自变量之间的交互效应是增强还是减弱。

如果系数为正，说明交互作用对因变量的影响是正向的，即当自变量X1和X2同时增加时，因变量Y也会增加。

反之，如果系数为负，说明交互作用对因变量的影响是负向的，即当自变量X1和X2同时增加时，因变量Y会减少。

2. 系数的大小交互项系数的绝对值可以告诉我们自变量之间的交互效应的强度。

系数的绝对值越大，说明自变量之间的交互作用对因变量的影响越大。

反之，系数的绝对值越小，说明交互作用的影响相对较弱。

3. 解释交互项系数为了更好地理解交互项系数的影响，我们可以进一步解释交互项系数的含义。

首先，我们需要控制其他自变量的影响，以便更准确地分析交互作用。

然后，我们可以通过计算和图形展示来解释交互作用的效应。

3.1 计算交互作用效应为了计算交互作用效应，我们可以使用不同值的自变量X1和X2来代入回归方程，然后计算因变量Y的变化量。

请简述k-means算法的流程

请简述k-means算法的流程K-means算法是一种常用的聚类方法，它将数据集分为K个不同的簇，使得簇内的数据点相似度较高，而簇间的数据点相似度较低。

下面将介绍K-means算法的基本流程以及相关参考内容。

1. 确定K值：首先需要确定要将数据集划分成多少个簇。

一般情况下，可以通过经验或者其他领域知识来确定K值。

2. 初始化：从数据集中随机选择K个数据点作为初始的质心（簇的中心点）。

这些质心将用于后续的聚类计算。

3. 分配：对于每一个数据点，计算其与各个质心之间的欧氏距离，并将其分配到距离最近的质心所对应的簇中。

4. 更新质心：对于每一个簇，计算该簇内所有数据点的平均值，将其作为新的质心。

5. 重复分配和更新质心的步骤，直到簇的分配不再发生变化，或者达到预定的迭代次数。

6. 输出结果：最终得到K个簇的质心以及分配到每个簇的数据点。

K-means算法的优点包括计算简单、收敛快速、可解释性强等。

然而，K-means算法也存在一些缺点，如对初始质心的敏感性、需要事先确定K值、对异常值敏感等。

下面是一些相关参考内容，可以供进一步了解K-means算法的流程和相关概念：- 《Data Mining: Concepts and Techniques》Jiawei Han, Micheline Kamber, Jian Pei（参考第8.2节K-means Clustering算法）- 《Machine Learning: A Probabilistic Perspective》Kevin P. Murphy（参考第25.1节K-means Clustering）- 《Pattern Recognition and Machine Learning》Christopher M. Bishop（参考第9.1节k-means clustering）- 《Understanding Machine Learning: From Theory to Algorithms》Shai Shalev-Shwartz, Shai Ben-David（参考第19.3节k-means++ algorithm）- 《An Introduction to Statistical Learning》Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani（参考第10.3.2节K-means Clustering）这些参考内容介绍了K-means算法的基本流程，包括初始化、分配、更新和输出结果等步骤。

TheElementsofStatisticalLearning

• 以下是多元正态分布的1-最近邻估计的方差和偏差变化模拟图
y c p exp( x 2 )
2
Contents
• 基本术语(2) • 两种基本算法：Linear Model和Nearest Neighbor Methods(3) • Loss Function和Optimal Prediction(4) • Curse of Dimensionality(5) • Additive Model(6) • Model Selection(6,7,8,9)
ˆ ( x) arg min E ( L( y, y ˆ )) y
ˆ ( x) y
则称为最优预测
Loss Function和Optimal Prediction
• 平方损失函数的形式为
ˆ) ( y y ˆ )2 L( y, y
ˆ )2 EPE E( y y
• 由测度论的知识，容易知道最优解是
• 最近邻方法对此的逼近：若P( g | X x)是连续的，则最近邻的解相合于最优解。
Contents
• 基本术语(2) • 两种基本算法：Linear Model和Nearest Neighbor Methods(3) • Loss Function和Optimal Prediction(4) • Curse of Dimensionality(5) • Additive Model(6) • Model Selection(6,7,8,9)
Model Selection
• 非参数模型选择的三种方法： Roughness Penalty Kernel Methods and Local Regression Basis Function and Dictionary Methods

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Learning Statistical Models of Time-Varying Relational Data Sumit Sanghai,Pedro Domingos and Daniel WeldDepartment of Computer Science and EngineeringUniversity of Washington,Seattle,WA981951IntroductionFormalisms that can represent objects and relations,as op-posed to just variables,have a long history in AI.Recently, signiﬁcant progress has been made in combining them with a principled treatment of uncertainty.In particular,proba-bilistic relational models or PRMs[4]are an extension of Bayesian networks that allows reasoning with classes,ob-jects and relations.Although PRMs have been successfully applied to a lot of different domains,they lack the temporal dynamics of the real world.In most real world systems,ob-jects get created,modiﬁed and even deleted over time.Sim-ilarly,the relationships between objects change as time pro-gresses.For example,consider the problem of predicting the set of research topics that become“hot”(e.g.,as measured by the number of papers published about them)over time,the changing distribution of these topics among conferences,and the interests and collaborations between authors.It would be difﬁcult to learn a PRM that modeled this time-varying be-havior.Currently the most powerful representation available for capturing sequential phenomena is dynamic Bayesian net-works(DBNs)[1],but DBNs are unable to compactly rep-resent many real-world domains that contain multiple objects and classes of objects,as well as multiple kinds of relations among them.DBNs are even more awkward if one wishes to model objects and relations that appear and disappear over time.Thus,our research has focused on a new repre-sentation,dynamic probabilistic relational models(DPRMs) which combines PRMs with DBNs.Previously,we have ex-plored the problem of efﬁcient inference[8];this paper out-lines our thoughts on learning DPRMs.2Dynamic Probabilistic Relational Models We start by brieﬂy summarizing the deﬁnition of PRMs and DPRMs,adapted from[4;8].A PRM encodes a probability distribution over the set of all possible instantiations I of a schema.In the simplest case,the relational attributes of all objects are assumed to be known,and the PRM speciﬁes a probability distribution for each propositional attribute A of each class C.The parents of each attribute(i.e.,the variables it depends on)can be other attributes of C,or attributes of classes that are related to C by some slot chain.Thus,by knowing the relational attributes one can get the joint proba-bility distribution by computing the set of parents for each ob-ject and its attributes and calculating the probability through the distribution speciﬁed.More generally,only the object skeleton might be known,in which case the PRM also needs to specify a distribution over the relational attributes[5]. Now,we extend PRMs to handle the time domain in the same way that DBNs extend Bayesian networks.Given a re-lational schema S,weﬁrst extend each class C with the re-lational attribute C.previous,with domain C.As before,we initially assume that the relational skeleton at each time slice is known.Deﬁnition1A two-time-slice PRM(2TPRM)for a relational schema S is deﬁned as follows.For each class C and each propositional attribute A∈A(C),we have:•A set of parents P a(C.A)={P a1,P a2,...,P a l},where each P a i has the form C.B or f(C.τ.B),whereτis a slot chain containing the attribute previous at most once,and f()is an aggregation function.•A conditional probability model for P(C.A|P a(C.A)).2 Deﬁnition2A dynamic probabilistic relational model (DPRM)for a relational schema S is a pair(M0,M→),where M0is a PRM over I0,representing the distribution P0over the initial instantiation of S,and M→is a2TPRM represent-ing the transition distribution P(I t|I t−1)connecting succes-sive instantiations of S.2DPRMs are extended to the case where only the object skeleton for each time slice is known in the same way that PRMs are,by adding to Deﬁnition1a set of parents and con-ditional probability model for each relational attribute,where the parents can be in the same or the previous time slice. When the object skeleton is not known(e.g.,if objects can appear and disappear over time),the2TPRM includes in ad-dition a Boolean existence variable for each possible object, again with parents from the same or the previous time slice. 3Inference in DPRMsJust as a PRM can be expanded into a Bayesian network,so can a DPRM be unrolled into a DBN.In principle,we can then perform inference using particleﬁltering[2],the most widely used approximate inference algorithm for DBNs.Par-ticleﬁltering maintains a set of samples(particles)to ap-proximate the distribution of any state;the distribution for next state is achieved by importance sampling and resam-pling.Unfortunately,for DPRMs,particleﬁltering is likelyto perform poorly,because the state space will be huge.We overcome this by adapting Rao-Blackwellisation[7]to the re-lational setting.Rao-Blackwellisation divides the state vari-ables into two sets—one in which values are inferred using a particleﬁlter and the other in which values are calculated analytically from the values of the variables in theﬁrst set. We make the major assumption that relational attributes do not appear anywhere in the DPRM as parents of unobserved attributes,and that each reference slot can be occupied by at most one object.Then,a Rao-Blackwellised particle is com-posed of sampled values for all propositional attributes of all objects,plus a probability vector for each relational attribute of each object which is inferred exactly.While this technique can vastly reduce the size of the state space which particleﬁltering needs to sample,storing and up-dating all the requisite probabilities can still become quite ex-pensive.This expense can be ameliorated if context-speciﬁc independences exist.We can then replace the vector of prob-abilities with a novel tree structure whose leaves represent probabilities for entire sets of objects[8].Our experiments evaluated the efﬁciency of several infer-ence schemes applied to an assembly-plan execution moni-toring task in a simpliﬁed manufacturing domain.Even with hundreds of thousands of particles,standard particleﬁlter-ing failed(i.e.terminated due to inconsistent observations which could not be explained)on datasets with around100 objects and500time steps.In contrast,our inference algo-rithm yielded accurate predictions on similar problems with only5000particles,and ran more quickly and with less stor-age[8].Much work remains to improve inference.For example, we will endeavor to lift the assumptions mentioned above and more effectively use a DPRM’s structure during inference. 4Learning in DPRMsWhen a DPRM consists of only a single time slice it becomes equivalent to a PRM,and when the DPRM is devoid of re-lations it is a DBN.Thus we look to combine the learning algorithms already developed for PRMs and DBNs.Theﬁrst step,parameter learning,appears to be relatively straightfor-ward when no data is missing,since the parameters associated with different types of nodes can be estimated individually. However,there is a subtlety which makes the problem more complex than in a DBN:•A DPRM can generate a unique state in multiple ways,and each way must be considered during parameter estimation. For example,if in the new state objects get created,the order of creation can affect the likelihood of the data,as the newly-created objects can interact with each other.There may be a combinatorial number of ways in which a DPRM may gener-ate each state,so we are developing methods to do parameter estimation efﬁciently.One possibility is to impose a canon-ical ordering,and another is to greedily compute the most likely order(s)in which the data could have been generated. In order to learn the DPRM structure,we have to take care of several more issues:•Deﬁning constraints to eliminate illegal DPRMs is essen-tial when navigating the space of structures.A cycle in aPRM is illegal,and this constraint extends to the two parts of a DPRM.There are additional constraints on a2TPRM;specifying these in a way that allows creation of an un-bounded number of dynamic objects is challenging.•There are several strategies for searching the space of DPRM structures.The simplest idea is to add and delete edges in the two components,PRM and2-TPRM,to gen-erate candidate DPRMs.One could do the search byﬁrst learning a PRM which gives a good intra-time-slice con-nectivity,before learning the inter-time-slice connectivity.•An important task is scoring a DPRM, e.g.with a likelihood-based measure.To compute the likelihood of the data given a candidate DPRM,fast DPRM inference is required.While our particleﬁltering algorithm is quite fast,we wish to extend it so that we can efﬁciently explore the space of DPRMS,incrementally updating the likeli-hood scores.We believe the two-phase search strategy sug-gested previously will simplify this task.•Since the space of candidate DPRM models is huge,we are considering pruning mechanisms.Note that some of the methods stated above actually prune the space(e.g.learn-ing the PRMﬁrst,followed by time dependencies).One may also impose priors on the models to bias towards sim-plicity by limiting the number of edges.We plan to design priors over DPRM structures by extending the approach of Heckerman et al.[6]who exponentially penalize arc differ-ences from a”best”prior structure.We will compare the relative beneﬁts of doing this at the class vs.instance level.•We plan to extend the learning algorithm to work in the presence of missing values and hidden variables.EM is easiest to apply when the observations are relational but the hidden state is not.Solving this problem with full gen-erality would require an extension of structural EM[3],but this needs to be done for PRMsﬁrst. AcknowledgementsThis work was partly supported by an NSF CAREER Award to the second author,by ONR grant N00014-02-1-0932,and by NASA grant NAG2-1538.References[1]T.Dean and K.Kanazawa.A model for reasoning about persis-tence and putational Intelligence,1989.[2] A.Doucet,N.de Freitas,and N.Gordon,editors.SequentialMonte Carlo Methods in Practice.Springer,2001.[3]N.Friedman.The Bayesian structural EM algorithm.UAI-98.[4]N.Friedman,L.Getoor,D.Koller,and A.Pfeffer.Learningprobabilistic relational models.IJCAI-99.[5]L.Getoor,N.Friedman,D.Koller,and B.Taskar.Learningprobabilistic models of relational structure.ICML-01.[6] D.Heckerman, D.Geiger,and D.Chickering.Learningbayesian networks:the combination of knowledge and statis-tical data.Machine Learning,1995.[7]K.Murphy and S.Russell.Rao-Blackwellised particleﬁlteringfor dynamic Bayesian networks.In A.Doucet,et al.,editors, Sequential Monte Carlo Methods in Practice.Springer,2001.[8]S.Sanghai,P.Domingos,and D.Weld.Dynamic probabilisticrelational models.Submitted to IJCAI-03.。