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时空团状系数回归模型及其在海洋时空断面数据中的应用

时空团状系数回归模型及其在海洋时空断面数据中的应用
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(整理)第七章非参数回归模型与半参数回归模型

(整理)第七章非参数回归模型与半参数回归模型

第七章 非参数回归模型与半参数回归模型第一节 非参数回归与权函数法一、非参数回归概念前面介绍的回归模型，无论是线性回归还是非线性回归，其回归函数形式都是已知的，只是其中参数待定，所以可称为参数回归。

参数回归的最大优点是回归结果可以外延，但其缺点也不可忽视，就是回归形式一旦固定，就比较呆板，往往拟合效果较差。

另一类回归，非参数回归，则与参数回归正好相反。

它的回归函数形式是不确定的，其结果外延困难，但拟合效果却比较好。

设Y 是一维观测随机向量，X 是m 维随机自变量。

在第四章我们曾引进过条件期望作回归函数，即称g (X ) = E (Y |X ) （7.1.1）为Y 对X 的回归函数。

我们证明了这样的回归函数可使误差平方和最小，即22)]([min )]|([X L Y E X Y E Y E L-=-（7.1.2）这里L 是关于X 的一切函数类。

当然，如果限定L 是线性函数类，那么g (X )就是线性回归函数了。

细心的读者会在这里立即提出一个问题。

既然对拟合函数类L (X )没有任何限制，那么可以使误差平方和等于0。

实际上，你只要作一条折线(曲面)通过所有观测点(Y i ，X i )就可以了是的，对拟合函数类不作任何限制是完全没有意义的。

正象世界上没有绝对的自由一样，我们实际上从来就没有说放弃对L(X)的一切限制。

在下面要研究的具体非参数回归方法，不管是核函数法，最近邻法，样条法，小波法，实际都有参数选择问题(比如窗宽选择，平滑参数选择)。

所以我们知道，参数回归与非参数回归的区分是相对的。

用一个多项式去拟合(Y i ，X i )，属于参数回归；用多个低次多项式去分段拟合(Y i ，X i )，叫样条回归，属于非参数回归。

二、权函数方法非参数回归的基本方法有核函数法，最近邻函数法，样条函数法，小波函数法。

这些方法尽管起源不一样，数学形式相距甚远，但都可以视为关于Y i 的线性组合的某种权函数。

也就是说，回归函数g (X )的估计g n (X )总可以表为下述形式：∑==ni i i n Y X W X g 1)()(（7.1.3）其中｛W i (X )｝称为权函数。

潘省初计量经济学中级教程习题参考答案

潘省初计量经济学中级教程习题参考答案

计量经济学中级教程习题参考答案第一章 绪论一般说来，计量经济分析按照以下步骤进行：（1）陈述理论（或假说） （2）建立计量经济模型 （3）收集数据 （4）估计参数 （5）假设检验 （6）预测和政策分析我们在计量经济模型中列出了影响因变量的解释变量，但它（它们）仅是影响因变量的主要因素，还有很多对因变量有影响的因素，它们相对而言不那么重要，因而未被包括在模型中。

为了使模型更现实，我们有必要在模型中引进扰动项u 来代表所有影响因变量的其它因素，这些因素包括相对而言不重要因而未被引入模型的变量，以及纯粹的随机因素。

时间序列数据是按时间周期（即按固定的时间间隔）收集的数据，如年度或季度的国民生产总值、就业、货币供给、财政赤字或某人一生中每年的收入都是时间序列的例子。

横截面数据是在同一时点收集的不同个体（如个人、公司、国家等）的数据。

如人口普查数据、世界各国2000年国民生产总值、全班学生计量经济学成绩等都是横截面数据的例子。

估计量是指一个公式或方法，它告诉人们怎样用手中样本所提供的信息去估计总体参数。

在一项应用中，依据估计量算出的一个具体的数值，称为估计值。

如Y 就是一个估计量，1nii YYn==∑。

现有一样本，共4个数，100，104，96，130，则根据这个样本的数据运用均值估计量得出的均值估计值为5.107413096104100=+++。

第二章 经典线性回归模型判断题（说明对错；如果错误，则予以更正） （1）对 （2）对 （3）错只要线性回归模型满足假设条件（1）～（4），OLS 估计量就是BLUE 。

（4）错R 2 =ESS/TSS 。

（5）错。

我们可以说的是，手头的数据不允许我们拒绝原假设。

（6）错。

因为∑=22)ˆ(tx Var σβ，只有当∑2t x 保持恒定时，上述说法才正确。

应采用（1），因为由（2）和（3）的回归结果可知，除X 1外，其余解释变量的系数均不显着。

（检验过程略） (1) 斜率系数含义如下:: 年净收益的土地投入弹性, 即土地投入每上升1%, 资金投入不变的情况下, 引起年净收益上升%.733: 年净收益的资金投入弹性, 即资金投入每上升1%, 土地投入不变的情况下, 引起年净收益上升%.拟合情况:92.0129)94.01(*811)1)(1(122=----=-----=k n R n R ,表明模型拟合程度较高.(2) 原假设 0:0=αH备择假设 0:1≠αH检验统计量 022.2135.0/273.0)ˆ(ˆ===ααSe t 查表，447.2)6(025.0=t 因为t=<)6(025.0t ,故接受原假设,即α不显着异于0, 表明土地投入变动对年净收益变动没有显着的影响. 原假设 0:0=βH备择假设 0:1≠βH检验统计量 864.5125.0/733.0)ˆ(ˆ===ββSe t 查表，447.2)6(025.0=t 因为t=>)6(025.0t ,故拒绝原假设,即β显着异于0,表明资金投入变动对年净收益变动有显着的影响. （3） 原假设 0:0==βαH备择假设 1H : 原假设不成立 检验统计量查表，在5%显着水平下14.5)6,2(=F 因为F=47>，故拒绝原假设。

Covariance selection and estimation via penalised normal likelihood

Covariance selection and estimation via penalised normal likelihood
1
Covariance selection and estimation via penalised normal likelihood
By JIANHUA Z. HUANG Department of Statistics, Texas A & M University, College Station, TX 77843-3143, USA
2 1. Introduction The sample covariance matrix, the most commonly used estimator of a covariance matrix is known to be positive-definite and unbiased, but highly unstable for large covariance matrices (Lin & Perlman, 1985; Wong et al. 2003; Ledoit & Wolf, 2004). Recently, structured covariance matrices, with few parameters, reflecting characteristics such as compound symmetry, and autoregression of order one, have become popular in longitudinal studies and related areas, though using a structure far from the true covariance could lead to severe bias. Between these two extremes lies a wealth of structures that might yield data-driven methods that strike a balance between the variance and bias of the covariance estimator. Estimation of covariance matrices is difficult because the number of unknown elements in the covariance matrix grows quadratically with the size of the matrix and because of the positive-definiteness constraint. Many existing methods deal directly with the individual elements of the covariance matrix; for a review of some of these methods see Diggle & Verbyla (1998), Diggle et al. (2002), Boik (2002) and Wong et al. (2003). Dempster (1972) was the first to recognise the inverse covariance matrix as the canonical parameter of a multivariate normal distribution. His covariance selection method which identifies zeros in the inverse covariance matrix offers parsimony, but does not guarantee positive-definiteness of the estimator. The positive-definiteness was taken care of by Leonard & Hsu (1992) and Chiu et al. (1996) who modelled the matrix logarithm of a covariance matrix, and by Pourahmadi (1999, 2000), who considered generalised linear models for covariances using components of the modified Cholesky decomposition of the inverse covariance matrix whose nonredundant entries are unconstrained and enjoy statistical interpretation as certain regression coefficients and variances. In this paper, we develop a nonparametric and data-driven method in the spirit of Dempster’s covariance selection to identify parsimony in the covariance matrix through the unit triangular factor T of its modified Cholesky decomposition. The nonredundant entries of the rows of this matrix are the regression coefficients of one variable based on its predecessors, so that the nonintuitive task of modelling a covariance matrix can be reduced to that of modelling several regression models (Wu & Pourahmadi, 2003). Thus, familiar regression techniques such as ridge regression and variable selection can be used to shrink the off-diagonal elements of T , and identify any existing structural zeros. To this end, we use a penalised normal likelihood function with an Lp penalty for the nonredundant entries of T . Since the matrix T , in essence, gauges the degrees of ‘dependence’ in the vector of responses, imposing such a penalty will reduce the risk of using too many parameters to capture the dependence. Our approach is more flexible than but related to the recent work of Wu & Pourahmadi (2003) and the unpublished 2004 University of Pennsylvania Ph. D. thesis of N. Liu. They applied nonparametric smoothing, by local polynomials and splines, respectively, to the first few subdiagonals of the Cholesky factor and set to

SVM_VS_LSSVM

SVM_VS_LSSVM

Abstract
We study the relationship between Support Vector Machines (SVM) and Least Squares SVM (LS-SVM). Our main result shows that under mild conditions, LS-SVM for binaryclass classifications is equivalent to the hard margin SVM based on the well-known Mahalanobis distance measure. We further study the asymptotics of the hard margin SVM when the data dimensionality tends to infinity with a fixed sample size. Using recently developed theory on the asymptotics of the distribution of the eigenvalues of the covariance matrix, we show that under mild conditions, the equivalence result holds for the traditional Euclidean distance measure. These equivalence results are further extended to the multi-class case. Experimental results confirm the presented theoretical analysis.

改进的鲁棒迭代最小二乘平面拟合算法

改进的鲁棒迭代最小二乘平面拟合算法

儿坐标系， 进一步拟合局部曲面． 文献［ —８ 更为详 ７ ］ 细地 阐述 了研究 散乱 点云 的平 面拟合 算 法在 其他 方 关键词 ： 平面拟合 ； 最小平方 中位数法； 移动最小二乘法 ； 迭 面的 应用 ， 比如 利 用平 面 拟 合 的方 法测 试 墙 面 的平
的精确性 ， 同时提高了迭代 收敛速度 ．
摘要 ： 针对迭代特征值最小二 乘法不具备鲁 棒性 ， 提出一种 合． 首先 由移动最小二 乘法拟合 抽样 点 的近邻域平 面 ， 采用 最小平方中位数 法选择拟合模型 ， 将该模 型作为初 始模 型调
础上 ． 献［ 文 １—２利用 局部 平 面拟合 计算 法 向量 ； ］ 文 献 ［ —４ 拟合 得到局 部平 面 ， 点 云投影 到 该平 面 ， ３ ］ 将 然 后采用 二维 Ｄ ｌｕａ 分 网格 ； 献 ［ ｅａｎｙ剖 文 ５—６ 则 采 ］
（ Ｇｕ ｎ ｈ ｕ Ｕｒ ａ ａ ｎｎ ＆ Ｄｅ ｉｎ Ｓ ｒ ｅ ｓａ ｃ ｎ ｉｕｅ， １． ａ ｇｚ ｏ ｂ ｎ Ｐｌｎ ｉｇ ｓｇ ｕ ｖ ｙ Ｒｅ ｅ ｒｈ Ｉ ｓｔ ｔ Ｇｕ ｎ ｈ ｕ ５ ０ ６ ａ ｇｚｏ １ ０ ０， Ｃｈｎ ； ２ Ｄｅ ａ ｔ ｎｔ ｆ ｕ ｖ ｙｎ ａ ｄ ｉａ ｐ ｒｍｅ ｏ Ｓ ｒ ｅ ｉｇ ｎ Ｇｅ － ｏ
和鲁棒性常相互抵触 ， 而笔者通过对文献Ｅ］ ８ 的迭代 特征 值最 小 二乘 法 的改进 ， 出 了一种 既 不失 原 算 提 法 的精 确性又 具有鲁 棒性 的平 面拟合 方法 ．
本文 涉及 的统计 方面 的概 念和方 法定义 如下 ：
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CompStat


2 Iteration and Convergence for Nonlinear Equations 2.1 2.2 2.3 2.4 2.5 The Bisection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed-point Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Newton and Secant Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newton methods for multidimensional problems . . . . . . . . . . . . . . . . . . . . . . . Project 1. Maximum likelihood estimate by finding root . . . . . . . . . . . . . . . . . . .
2 4.5 4.6 4.7
CONTENTS
Gauss-Legendre quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singular integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Project 3. Alzheimer’s disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 46 47

自适应迭代重加权法

自适应迭代重加权法（Adaptive Iterative Reweighted Method，简称AIRW）是一种用于解决非线性最小二乘问题的数值优化算法。

该方法通过迭代的方式不断更新参数和权重，以逐步接近最优解。

本文将详细介绍AIRW方法的原理、流程和应用，希望可以帮助您更好地理解这一优化算法。

1. 算法原理AIRW方法的核心思想是通过迭代的重加权过程，不断调整参数和权重，以逐步逼近最优解。

其原理可以概括为以下几个步骤：1.1 权重更新在每一次迭代中，AIRW方法会根据当前的参数估计和观测数据的残差来更新每个样本的权重。

通常情况下，残差较大的样本将被赋予较小的权重，而残差较小的样本将被赋予较大的权重，以便更加关注对参数估计有重要影响的样本。

1.2 参数更新在得到新的权重之后，AIRW方法将使用加权最小二乘法来重新估计模型参数。

这意味着对于每一次迭代，参数都将根据当前的权重来进行调整，以使得模型更好地拟合观测数据。

1.3 收敛判据AIRW方法通常会设定一个收敛准则，当满足该准则时停止迭代。

这个准则可以是参数的变化小于某个阈值，或者观测数据的残差小于某个阈值等。

2. 算法流程AIRW方法的具体流程可以总结如下：2.1 初始化首先，需要对模型参数进行初始化，同时给定每个样本的初始权重。

2.2 迭代更新接下来，进入迭代更新过程。

在每一次迭代中，按照权重更新和参数更新的顺序进行，直至满足停止条件。

2.3 结果输出最终，输出收敛后的模型参数作为最优解，以及相应的权重信息。

3. 应用领域AIRW方法在实际应用中具有广泛的用途，特别是在医学成像、信号处理、机器学习等领域。

例如，在医学图像重建中，AIRW方法可以用于优化逆问题的求解，提高图像重建的质量；在机器学习中，AIRW方法可以用于非线性回归、分类等问题的求解，提高模型的拟合效果。

4. 总结通过对AIRW方法的原理、流程和应用进行介绍，我们可以发现该方法在非线性最小二乘问题的求解中具有重要的作用。

2023年计量经济学英文重点知识点考试必备

第一章1.Econometrics（计量经济学）：the social science in which the tools of economic theory, mathematics, and statistical inference are applied to the analysis of economic phenomena.the result of a certain outlook on the role of economics, consists of the application of mathematical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results.2.Econometric analysis proceeds along the following lines计量经济学分析环节1)Creating a statement of theory or hypothesis.建立一种理论假说2)Collecting data.搜集数据3)Specifying the mathematical model of theory.设定数学模型4)Specifying the statistical, or econometric, model of theory.设置记录或经济计量模型5)Estimating the parameters of the chosen econometric model.估计经济计量模型参数6)Checking for model adequacy : Model specification testing.核查模型旳合用性：模型设定检查7)Testing the hypothesis derived from the model.检查自模型旳假设8)Using the model for prediction or forecasting.运用模型进行预测●Step2：搜集数据➢Three types of data三类可用于分析旳数据1)Time series(时间序列数据):Collected over a period of time, are collected at regular intervals.准时间跨度搜集得到2)Cross-sectional截面数据:Collected over a period of time, are collected at regular intervals.准时间跨度搜集得到3)Pooled data合并数据（上两种旳结合）●Step3：设定数学模型1.plot scatter diagram or scattergram2.write the mathematical model●Step4：设置记录或经济计量模型➢C LFPR is dependent variable应变量➢C UNR is independent or explanatory variable独立或解释变量（自变量）➢W e give a catchall variable U to stand for all these neglected factors➢In linear regression analysis our primary objective is to explain the behavior of the dependent variable in relation to the behavior of one or more other variables, allowing for the data that the relationship between them is inexact.线性回归分析旳重要目旳就是解释一种变量（应变量）与其他一种或多种变量（自变量）只见旳行为关系，当然这种关系并非完全对旳●Step5：估计经济计量模型参数➢In short, the estimated regression line gives the relationship between average CLFPR and CUNR 简言之，估计旳回归直线给出了平均应变量和自变量之间旳关系➢That is, on average, how the dependent variable responds to a unit change in the independent variable.单位因变量旳变化引起旳自变量平均变化量旳多少。

PLS-regression

Ž.Chemometrics and Intelligent Laboratory Systems 582001109–130 r locate r chemometricsPLS-regression:a basic tool of chemometricsSvante Wold a,),Michael Sjostrom a ,Lennart Eriksson b¨¨aResearch Group for Chemometrics,Institute of Chemistry,Umea Uni Õersity,SE-90187Umea,Sweden˚˚bUmetrics AB,Box 7960,SE-90719Umea,Sweden˚AbstractŽ.ŽPLS-regression PLSR is the PLS approach in its simplest,and in chemistry and technology,most used form two-block.predictive PLS .PLSR is a method for relating two data matrices,X and Y ,by a linear multivariate model,but goes beyond traditional regression in that it models also the structure of X and Y .PLSR derives its usefulness from its ability to analyze data with many,noisy,collinear,and even incomplete variables in both X and Y .PLSR has the desirable property that the precision of the model parameters improves with the increasing number of relevant variables and observations.This article reviews PLSR as it has developed to become a standard tool in chemometrics and used in chemistry and engineering.The underlying model and its assumptions are discussed,and commonly used diagnostics are reviewed together with the interpretation of resulting parameters.Ž.Two examples are used as illustrations:First,a Quantitative Structure–Activity Relationship QSAR r Quantitative Struc-Ž.ture–Property Relationship QSPR data set of peptides is used to outline how to develop,interpret and refine a PLSR model.Second,a data set from the manufacturing of recycled paper is analyzed to illustrate time series modelling of process data by means of PLSR and time-lagged X-variables.q 2001Elsevier Science B.V.All rights reserved.Keywords:PLS;PLSR;Two-block predictive PLS;Latent variables;Multivariate analysis1.IntroductionIn this article we review a particular type of mul-tivariate analysis,namely PLS-regression,which uses the two-block predictive PLS model to model the re-lationship between two matrices,X and Y .In addi-tion PLSR models the A structure B of X and of Y ,which gives richer results than the traditional multi-ple regression approach.PLSR and similar ap-proaches provide quantitati Õe multivariate modelling methods,with inferential possibilities similar to mul-tiple regression,t -tests and ANOVA.)Corresponding author.Tel.:q 46-90-786-5563;fax:q 46-90-13-88-85.Ž.E-mail address:svante.wold@ S.Wold .The present volume contains numerous examples of the use of PLSR in chemistry,and this article is merely an introductory review,showing the develop-ment of PLSR in chemistry until,around,the year 1990.1.1.General considerationsŽ.PLS-regression PLSR is a recently developed Ž.generalization of multiple linear regression MLR w x 1–6.PLSR is of particular interest because,unlike MLR,it can analyze data with strongly collinear Ž.correlated ,noisy,and numerous X-variables,and also simultaneously model several response vari-ables,Y ,i.e.,profiles of performance .For the mean-ing of the PLS acronym,see Section 1.2.0169-7439r 01r $-see front matter q 2001Elsevier Science B.V.All rights reserved.Ž.PII:S 0169-74390100155-1本页已使用福昕阅读器进行编辑。

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Iteratively reweighted least squares1

Iteratively reweighted least squares

Regression analysis

Models•Linear regression•Simple regression•Ordinary least squares•Polynomial regression•General linear model

•Generalized linear model•Discrete choice•Logistic regression•Multinomial logit•Mixed logit•Probit•Multinomial probit•Ordered logit•Ordered probit•Poisson•Multilevel model•Fixed effects•Random effects•Mixed model

•Nonlinear regression•Nonparametric•Semiparametric•Robust•Quantile•Isotonic•Principal components•Least angle•Local•Segmented•Errors-in-variablesEstimation•Least squares•Ordinary least squares•Linear (math)•PartialIteratively reweighted least squares2

•Total•Generalized•Weighted•Non-linear•Iteratively reweighted•Ridge regression•LASSO•Least absolute deviations•Bayesian•Bayesian multivariate

Background•Regression model validation•Mean and predicted response•Errors and residuals•Goodness of fit•Studentized residual•Gauss–Markov theorem• Statistics portal

The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems. It solvesobjective functions of the form:

by an iterative method in which each step involves solving a weighted least squares problem of the form:IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression tofind an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set.For example, by minimizing the least absolute error rather than the least square error.Although not a linear regression problem, Weiszfeld's algorithm for approximating the geometric median can also beviewed as a special case of iteratively reweighted least squares, in which the objective function is the sum ofdistances of the estimator from the samples.One of the advantages of IRLS over linear and convex programming is that it can be used with Gauss–Newton andLevenberg–Marquardt numerical algorithms.Iteratively reweighted least squares3

ExamplesL1 minimization for sparse recoveryIRLS can be used for 1 minimization and smoothed p minimization, p < 1, in the compressed sensing problems.It has been proved that the algorithm has a linear rate of convergence for 1 norm and superlinear for t with t < 1,under the restricted isometry property, which is generally a sufficient condition for sparse solutions.[1][2]

Lp norm linear regressionTo find the parameters β = (β1, …,βk)T which minimize the Lp norm for the linear regression problem,

the IRLS algorithm at step t+1 involves solving the weighted linear least squares[3] problem:[4]where W(t) is the diagonal matrix of weights with elements:In the case p = 1, this corresponds to least absolute deviation regression (in this case, the problem would be betterapproached by use of linear programming methods).[citation needed]

Notes[3]http://toolserver.org/%7Edispenser/cgi-bin/dab_solver.py?page=Iteratively_reweighted_least_squares&editintro=Template:Disambiguation_needed/editintro&client=Template:Dn

References•University of Colorado Applied Regression lecture slides (http://amath.colorado.edu/courses/7400/2010Spr/lecture23.pdf)•Stanford Lecture Notes on the IRLS algorithm by Antoine Guitton (http://sepwww.stanford.edu/public/docs/sep103/antoine2/paper_html/index.html)•Numerical Methods for Least Squares Problems by Åke Björck (http://www.mai.liu.se/~akbjo/LSPbook.html) (Chapter 4: Generalized Least Squares Problems.)•Practical Least-Squares for Computer Graphics. SIGGRAPH Course 11 (http://graphics.stanford.edu/~jplewis/lscourse/SLIDES.pdf)Article Sources and Contributors4

Article Sources and ContributorsIteratively reweighted least squares Source: http://en.wikipedia.org/w/index.php?oldid=536800201 Contributors: 3mta3, BD2412, BenFrantzDale, Benwing, Colbert Sesanker, DavidEppstein, Giggy, Grumpfel, Kiefer.Wolfowitz, Lambiam, Lesswire, LutzL, Melcombe, Michael Hardy, Oleg Alexandrov, RainerBlome, Salix alba, Serg3d2, Stpasha, Wesleyyin, 13 anonymousedits

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