A GENERALIZED CROSS VALIDATION METHOD FOR THE INVERSE PROBLEM OF 3-D MAXWELL'S EQUATION
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第三章地理加权回归模型介绍3.1 基本模型在地学空间分析中,n组观测数据通常是在n个不同地理位置上获取的样本数据,全局空间回归模型就是假定回归参数与样本数据的地理位置无关,或者说在整个空间研究区域内保持稳定一致,那么在n个不同地理位置上获取的样本数据,就等同于在同一地理位置上获取的n个样本数据,其回归模型与最小二乘法回归模型相同,采用最小二乘估计得到的回归参数户既是该点的最优无偏估计,也是研究区域内所有点上的最优无偏估计。
而在实际问题研究中我们经常发现回归参数在不同地理位置上往往表现为不同,也就是说回归参数随地理位置变化,这时如果仍然采用全局空间回归模型,得到的回归参数估计将是回归参数在整个研究区域内的平均值,不能反映回归参数的真实空间特征。
为了解决这一问题,国外有些学者提出了空间变参数回归模型(Spatially Varying-Coeffi Cient Regression Model)(Fosterand Gorr,1986; Gorrand Olligschlaeger,1994),将数据的空间结构嵌入回归模型中,使回归参数变成观测点地理位置的函数。
Fortheringham等(Brunsdonetal,1996;Fortheringham et al,1997;Brunsdon et al,1998)在空间变系数回归模型基础上利用局部光滑思想,提出了地理加权回归模型(Geographieally Weighted Regression Model-GWR)。
地理加权回归模型(GWR)是对普通线性回归模型(OLR)的扩展,将样点数据的地理位置嵌入到回归参数之中,即:式中:(u i,v i)为第i个样点的坐标(如经纬度);βk(u i,v i)是第i个样点的第k个回归参数;εi是第i个样点的随机误差。
为了表述方便,我们将上式简写为:若β1k=β2k=⋯=βnk,则地理加权回归模型(GWR)就退变为普通线性回归模型(OLR)。
最优模型选择中的交叉验证(Crossvalidation)方法很多时候,大家会利用各种方法建立不同的统计模型,诸如普通的cox回归,利用Lasso方法建立的cox回归,或者稳健的cox回归;或者说利用不同的变量建立不同的模型,诸如模型一只考虑了三个因素、模型二考虑了四个因素,最后对上述模型选择(评价)的时候,或者是参数择优的时候,通常传统统计学方法中会用AIC,BIC、拟合优度-2logL,或者预测误差最小等准则来选择最优模型;而最新的文献中都会提到一种叫交叉验证(Cross validation)的方法,或者会用到一种将原始数据按照样本量分为两部分三分之二用来建模,三分之一用来验证的思路(临床上有医生称为内部验证),再或者利用多中心数据,一个中心数据用来建模,另外一个中心数据用来验证(临床上称为外部验证),这些都是什么?总结一下自己最近看的文献和书籍,在这里简单介绍下,仅供参考。
一、交叉验证的概念交叉验证(Cross validation),有时亦称循环估计,是一种统计学上将数据样本切割成较小子集的实用方法。
于是可以先在一个子集上做建模分析,而其它子集则用来做后续对此分析的效果评价及验证。
一开始的子集被称为训练集(Train set)。
而其它的子集则被称为验证集(Validationset)或测试集(Test set)。
交叉验证是一种评估统计分析、机器学习算法对独立于训练数据的数据集的泛化(普遍适用性)能力(Generalize).例如下图文献中,原始数据集中449例观测,文献中将数据集分为了训练集(Primary Cohort)367例,验证集(Validation Cohort)82例。
二、交叉验证的原理及分类假设利用原始数据可以建立n个统计模型,这n 个模型的集合是M={M1,M2,…,Mn},比如我们想做回归,那么简单线性回归、logistic回归、随机森林、神经网络等模型都包含在M中。
回归分析是统计学中常用的一种分析方法,用来探索自变量和因变量之间的关系。
在回归分析中,广义加法模型(Generalized Additive Model, GAM)是一种常用的非参数回归方法,它可以灵活地处理非线性关系,同时可以控制其他变量的影响,使得模型更加准确和可解释。
本文将介绍回归分析中的广义加法模型的应用技巧,以帮助读者更好地理解和运用这一方法。
回归分析是一种用来探索变量之间关系的方法。
在实际应用中,通常会有多个自变量同时影响因变量,而且它们之间的关系可能是非线性的。
传统的线性回归模型可以很好地处理线性关系,但对于非线性关系的拟合能力有限。
这时,广义加法模型就能够发挥其优势。
广义加法模型是一种非参数回归方法,它通过对自变量的非线性部分进行平滑处理,从而能够更好地拟合非线性关系。
在GAM中,每个自变量的作用被建模为一个非参数的平滑函数,这使得模型能够更好地适应非线性关系。
此外,GAM还可以对连续变量、离散变量和交互作用进行灵活建模,从而更好地控制其他变量的影响。
在实际应用中,广义加法模型有一些应用技巧需要注意。
首先,对于连续型自变量,可以选择不同的平滑函数来对其建模。
常用的平滑函数包括自然样条、样条平滑和 LOESS 等。
选择适当的平滑函数可以使模型更准确地拟合数据。
其次,对于离散型自变量和交互作用,可以使用适当的转换方法来进行建模,比如使用虚拟变量对离散型自变量进行编码,使用乘积项来建模交互作用。
这些方法可以帮助模型更好地捕捉变量之间的复杂关系。
此外,广义加法模型的参数估计通常使用的是广义交叉验证(Generalized Cross Validation, GCV)或最小二乘交叉验证(Least Squares Cross Validation, LSCV)等方法,以选择适当的平滑参数。
在实际应用中,需要根据数据情况选择合适的交叉验证方法,并结合模型的拟合效果来进行参数的选择。
在应用广义加法模型时,还需要注意模型的解释和诊断。
第三章地理加权回归模型介绍3.1 基本模型在地学空间分析中,n组观测数据通常是在n个不同地理位置上获取的样本数据,全局空间回归模型就是假定回归参数与样本数据的地理位置无关,或者说在整个空间研究区域内保持稳定一致,那么在n个不同地理位置上获取的样本数据,就等同于在同一地理位置上获取的n个样本数据,其回归模型与最小二乘法回归模型相同,采用最小二乘估计得到的回归参数户既是该点的最优无偏估计,也是研究区域内所有点上的最优无偏估计。
而在实际问题研究中我们经常发现回归参数在不同地理位置上往往表现为不同,也就是说回归参数随地理位置变化,这时如果仍然采用全局空间回归模型,得到的回归参数估计将是回归参数在整个研究区域内的平均值,不能反映回归参数的真实空间特征。
为了解决这一问题,国外有些学者提出了空间变参数回归模型(Spatially Varying-Coeffi Cient Regression Model)(Fosterand Gorr,1986;Gorrand Olligschlaeger,1994),将数据的空间结构嵌入回归模型中,使回归参数变成观测点地理位置的函数。
Fortheringham等(Brunsdonetal,1996;Fortheringham et al,1997;Brunsdon et al,1998)在空间变系数回归模型基础上利用局部光滑思想,提出了地理加权回归模型(Geographieally Weighted Regression Model-GWR)。
地理加权回归模型(GWR)是对普通线性回归模型(OLR)的扩展,将样点数据的地理位置嵌入到回归参数之中,即:式中:(u i,v i)为第i个样点的坐标(如经纬度);βk(u i,v i)是第i个样点的第k个回归参数;i是第i个样点的随机误差。
为了表述方便,我们将上式简写为:若,则地理加权回归模型(GWR)就退变为普通线性回归模型(OLR)。
第三章地理加权回归模型介绍3.1 基本模型在地学空间分析中,n组观测数据通常是在n个不同地理位置上获取的样本数据,全局空间回归模型就是假定回归参数与样本数据的地理位置无关,或者说在整个空间研究区域内保持稳定一致,那么在n个不同地理位置上获取的样本数据,就等同于在同一地理位置上获取的n个样本数据,其回归模型与最小二乘法回归模型相同,采用最小二乘估计得到的回归参数户既是该点的最优无偏估计,也是研究区域内所有点上的最优无偏估计。
而在实际问题研究中我们经常发现回归参数在不同地理位置上往往表现为不同,也就是说回归参数随地理位置变化,这时如果仍然采用全局空间回归模型,得到的回归参数估计将是回归参数在整个研究区域内的平均值,不能反映回归参数的真实空间特征。
为了解决这一问题,国外有些学者提出了空间变参数回归模型(Spatially Varying-Coeffi Cient Regression Model)(Fosterand Gorr,1986; Gorrand Olligschlaeger,1994),将数据的空间结构嵌入回归模型中,使回归参数变成观测点地理位置的函数。
Fortheringham等(Brunsdonetal,1996;Fortheringham et al,1997;Brunsdon et al,1998)在空间变系数回归模型基础上利用局部光滑思想,提出了地理加权回归模型(Geographieally Weighted Regression Model-GWR)。
地理加权回归模型(GWR)是对普通线性回归模型(OLR)的扩展,将样点数据的地理位置嵌入到回归参数之中,即:式中:(u i,v i)为第i个样点的坐标(如经纬度);βk(u i,v i)是第i个样点的第k个回归参数;εi是第i个样点的随机误差。
为了表述方便,我们将上式简写为:若β1k=β2k=⋯=βnk,则地理加权回归模型(GWR)就退变为普通线性回归模型(OLR)。
基于GAM_Tweedie模型的车险定价研究摘要:广义线性模型作为车险费率厘定的主流方法,其假设协变量的影响为预测函数的线性形式,但在实际的情况下,许多对索賠频率、索賠强度或纯保费的影响因素不仅仅是表现成线性形式的,单纯地用线性估计会造成一些变量的不显著而丢失重要影响因素。
本文以一组汽车保险损失数据为样本,建立Tweedie广义加法模型,通过与Tweedie广义线性模型对比,表明Tweedie广义加法模型可以更好的解释各因素对索赔额的影响。
关键词:广义线性模型,车险费率厘定,Tweedie分布,广义加法模型一、引言车险定价实则是对索赔频率、索赔强度或纯保费进行预测。
在车险定价实务中,经常假设索赔频率与索赔强度相互独立,并分别建立索赔频率和索赔强度的广义线性模型。
在独立的假设下,可以把索赔频率与索赔强度的预测值相乘从而求得纯保费的预测值。
这种方法简单易行,在非寿险精算实务中得到广泛的应用,但其忽略了索赔频率与索赔强度之间可能存在的相依关系,从而造成预测的偏差。
而在纯保费的预测中,主要是应用Tweedie广义线性模型。
Tweedie广义线性模型,是假定保单的累积赔付额服从Tweedie分布,对赔付额的均值函数建立回归模型。
其要求协变量的影响为预测函数的线性形式,但在实际的情况下,许多对纯保费的影响因素不仅仅是表现成线性形式的,如空间协变量,大多数情况下其对响应变量均值函数的影响是非线性的,如果单纯地用线性估计会造成一些变量的不显著而丢失重要的影响因素。
为了更好的拟合数据,从而有必要对其进行优化推广,在广义线性模型中纳入平滑预测项,将其推广到广义加法模型。
从线性和非线性两个方面去分析各因素对预测函数不同的影响程度。
本文以一组汽车保险损失数据为样本,建立Tweedie广义加法模型,利用R软件对模型的参数进行估计检验。
通过与Tweedie广义线性模型对比,表明Tweedie 广义加法模型可以更好的解释各因素对索赔额的影响,从而改进了传统广义线性模型对纯保费的预测精度。
交叉验证--Cross validation交叉验证(Cross validation),有时亦称循环估计,是一种统计学上将数据样本切割成较小子集的实用方法。
于是可以先在一个子集上做分析,而其它子集则用来测试训练得到的模型(model),以此来做为评价分类器的性能指标。
一开始的子集被称为训练集。
而其它的子集则被称为验证集或测试集。
交叉验证是一种评估统计分析、机器学习算法对独立于训练数据的数据集的泛化能力(generalize)。
交叉验证一般要尽量满足:训练集的比例要足够多,一般大于一半训练集和测试集要均匀抽样其中第2点特别重要,均匀取样的目的是希望减少training/test set与完整集合之间的偏差(bias),但却也不易做到。
一般的作法是随机取样,当样本数量足够时,便可达到均匀取样的效果。
然而随机也正是此作法的盲点,也是经常是可以在数据上做手脚的地方。
举例来说,当辨识率不理想时,便重新取样一组training set与test set,直到test set的辨识率满意为止,但严格来说这样便算是作弊了交叉验证主要分成以下几类:1)Hold-Out Method将原始数据随机分为两组,一组做为训练集,一组做为验证集,利用训练集训练分类器,然后利用验证集验证模型,记录最后的分类准确率为此Hold-OutMethod下分类器的性能指标.此种方法的好处的处理简单,只需随机把原始数据分为两组即可,其实严格意义来说Hold-Out Method并不能算是CV,因为这种方法没有达到交叉的思想,由于是随机的将原始数据分组,所以最后验证集分类准确率的高低与原始数据的分组有很大的关系,所以这种方法得到的结果其实并不具有说服性.2)K-fold cross-validation:将原始数据分成K组(一般是均分),将每个子集数据分别做一次验证集,其余的K-1组子集数据作为训练集,这样会得到K个模型,用这K个模型最终的验证集的分类准确率的平均数作为此K-CV下分类器的性能指标。
Mathematical and Computational Applications, Vol. 15, No. 5, pp. 784-789, 2010.© Association for Scientific ResearchA GENERALIZED CROSS VALIDATION METHOD FOR THE INVERSEPROBLEM OF 3-D MAXWELL’S EQUATIONLiang Ding, Bo Han, Jiaqi LiuDepartment of Mathematics, Harbin Institute of Technology, 150001Harbin, P. R. China, iamashen@Abstract- The inverse problem of estimation of the electrical conductivity in the Maxwell’s equation is considered, which is reformulated as a nonlinear equation. The Generalized Cross Validation is used to estimate the global regularization parameter and the damped Gauss-Newton is applied to impose local regularization. The damped Gauss-Newton method requires no calculation of the Hessian matrix which is expensive for traditional Newton method. GCV method decreases the computational expense and overcomes the influence of nonlinearity and ill-posedness. The results of numerical simulation testify that this method is efficient.Key Words- Maxwell’s Equation, Nonlinear, Ill-posed, Inversion, Damped Gauss-Newton, GCV1. INTRODUCTIONIn this paper, we consider the estimation of the electrical conductivity in Maxwell’s equations. This problem can be reformulated as a nonlinear equation, which is ill-posed. Our work is devoted to a new numerical method for the inverse problem. There are two main computational difficulties in the solution of the inverse problem. Firstly, the regularization parameter is unknown, and secondly, a nonlinear functional has to be minimized. These impose special difficulties if the problem is large.There have been many new methods applied to this domain [1,2]. The first pioneering solution of the fully 3-D Maxwell’s equation inverse problem was presented by Eaton[3] more than 15 years ago. Yet in spite of this, until recently, the trial-and-error forward modeling was almost the only available tool to interpret the fully 3-D EM dataset. Today the situation has slightly been improved, and the methods of unconstrained nonlinear optimization [4] are gaining popularity to address the problem. In spite of these successes and the relatively high level of modern computing possibilities, the proper numerical solution of the 3-D inverse problem still remains a very difficult and computationally intense task for the following reasons. (1)It requires a fast, accurate and reliable forward 3-D problem solution. Approximate forward solutions [5,6,7,8,9] may deliver a rapid solution of the inverse problem (especially, for models with low conductivity contrasts), but the general reliability and accuracy of this solution are still open to question.(2)The problem is ill-posed in nature with nonlinear and extremely sensitive solutions. This means that due to the fact that data are limited and contaminated by noise there are many models that can equally fit the data within a given tolerance threshold. (3)We need to solve a new nonlinear problem for every new regularization parameter and therefore the algorithm is computationally expensive. The algorithm is substantially cheaper if we estimate βto be close to the optimal*β.L. Ding, B. Han and J. Liu785Therefore, the shortages of the above mentioned motive us to design a highly efficient, numerically stable and globally convergent algorithm.As mentioned above, the goal of this paper is to apply the damped Gauss-Newton method coupled with GCV (Generalized Cross-Validation) method to choose an adaptive regularization parameter. The paper is built as follows. We start with a review of the damped Gauss–Newton method as applied to a minimization problem with a constant regularization parameter. We then present the major ideas of our algorithm. Our algorithm uses the Generalized Cross Validation and we review it in section 3. Section 4 gives an example of inverting inductivity from Maxwell’s equations.2. DAMPED GAUSS-NEWTON METHODIn this section we present damped Gauss-Newton method. Our forward problem time-dependent Maxwell equations can be written asH E+0tμ∂∇×=∂ (1) E H E (r s t tσε)∂∇×−−=∂ (2) over a domain ,where and are the electric and magnetic fields, [0,]T Ω×E H σ is the conductivity, ε is the permittivity, μ is the permeability and is a source. Theequations are given with boundary and initial conditions:r s n H=0H(x,0)=0E(x,0)=0×⎧⎪⎨⎪⎩(3)By [10], we know an inverse electromagnetic problem can be formulated as the following nonlinear operator equation:()F d δσ=, (4)where is a nonlinear operator, d F δis the observed data. The goal of this paper is to reconstruct the inductivity from the observed data with noise. The optimal distribution of conductivity minimized the functional ()()22ref F d W δφσβσσ=−+−, (5) where ⋅ denotes 2-norm, W is positive weight function, ref σis reference model, β isregularization parameter. If β is fixed, we must solve the unconstrained optimization problem (5). This leads to the Euler-Lagrange system()()()()(0T T ref g W W J F d δ)φσβσσσσσ∂==−+−∂= (6) ()g σis the gradient of (5), ()J σis sensitive matrix ()F J σσ∂=∂. (7) In order to avoid the difficulty of calculating Hessian matrix, which is second derivative of , we linearize the operatorF FA Generalized Cross Validation Method for the Inverse Problem of 3-DMaxwell’s Equation786 ()()()(),F F J R σδσσσδσσδσ+=++ (8)where ()()2,R σδσοδσ=. From (6) and (8), we can obtain Gauss-Newton equation()()()()()0TT ref W W J F J d δβσδσσσσσδσ+−++−= (9) The minimization problem is solved by the iterative format as follows, at the k th step ()()()()()()()T TT T k k k k k r J J W W J d F W W δσσβδσσσβσσ+=−−−ef (10) Denoted 1k k σσδσ+=+, (10) can be written as ()()()()()()()1T T T T k k k k k k k J J W W J d F J W W δref σσβσσσσσβσ++=−+− (11) The iterative solution at k th step is the minimum norm solution of Tikhonov functional ()()()()221k k k k k ref F J d W δφσσσσβσσ+=+−−+−1+. (12) If the initial value is close to true inductivity, the algorithm is fast convergent. If the initial value is undesirable, the search step must become bigger. Consequently, the (),R σδσwhat we removed in the linear process became larger, this will affect accuracy. In order to keep the Gauss-Newton iteration in the descent direction and the iterative step is small enough, we adopt armijo method to amend the iterative step. If the iterative direction is no longer descent, we amend the step δσ to ωδσ, where 0.10.5ω<<. As discussed above, with small enough step, the nonlinear functional is always in the descent direction()()1,k ,k φβσφβσ+< (13)What we discussed above is in the case of regularization fixed. Next we introduce how to adaptively select regularization parameter, stopping criteria for iteration and general sketch of algorithm.3. GCV FUNCTION AND INVERSION ALGORITHMGeneralized Cross Validation (GCV) method is a deformation of Unbiased Predictive Risk Estimator (UPRE) method [11]. Even if we don’t know the variance of noise, this method can give the adaptive regularization parameter and distinguish the noise and nonlinearity. We denote 1k ref (σσσ+)k r r r J ef σσ=−()k J J , σ=, −,=()22J r W φσβ=−+σ . (14)And let ()()r J βσβ=, ()σβ is the solution of (number), the GCV function()()()()2GCV trace r r I C βββ−=− (15) where C J , ()()T 1T T J J W W J ββ−=+I is unit matrix. With regularization parameter βwhich minimizes the GCV function, we denote the regularization parameter k βof the k th iteration. Next we introduce some necessary condition and stopping criteria of damped Gauss-Newton method with adaptive regularization parameter. If we got k βandL. Ding, B. Han and J. Liu787k σat the k th iteration, we can obtain 1k σ+by(number). In the case of βfixed,()(1,k ),k φβσφβσ+< at every iterative step. But with the parameter changing, we demand()()111,k k k k ,φβσφβσ+++<, (16)namely ()()()()22221111k k k ref k k k ref F dW F d W δδσβσσσβσσ++++−+−<−+− (17) We select ()11max ,k kk k σσδσσ++−< (18)as stopping criteria.The damped Gauss-Newton algorithm on basis of GCV function:Select initial value 0σ and reference model ref σ, as 1,2k ="i) Calculate (k J )σand ()k r σ.ii) Calculate 1k σ+by (11) and evaluate k βby GCV function.iii) Use the regularization parameter to calculate 1k k δσσσ+=−,, ()11,k k m φβ++()1,k k m φβ+.iv) If , let m ()()111,,k k k k m m φβφβ+++≤1k k m += and go to i).v) If , let ()()111,,k k k k m m φβφβ+++>1k k m m m ωδ++ and go to iii).=vi) If stopping criteria()11max ,k k k k m m m m δ++−< is satisfied, stop iteration.4. NUMERICAL SINMULATIONSIn this section, we compare the efficiency of traditional damped Gauss-Newton method and GCV method which is based on the damped Gauss-Newton method in two experiments.The physical domain is Ω. For simplicity the inverse problem is considered on a uniform,grid. The source function is 343516××()(20sin t )f t t e t αω−=, where α=,002f ωπ=, is the centre frequency.0100MHz f =4.1. Experiment 1In this section we suppose there is a cavity in the homogeneous background, where permittivity , permeability 1256.6410F/m ε−=×0μ=. The relative conductivity of cavity and background are -3.2775 and -5.2983 respectively. Both methods select the same relative initial value 1.5σ=−. The inversion took approximately 20 minutes by GCV method against to 30 minutes by traditional damped Gauss-Newton method run on a 3.0 Ghz Pentium Dual-Core with 2Gb of RAM. The relative errorA Generalized Cross Validation Method for the Inverse Problem of 3-DMaxwell’s Equation 788 /inv true true σσσ−of a and b in Fig.1 is 35.76%, a and c is 36.75%, where inv σisiterative solution, true σis the true model.Fig.1 a)True model; b)Results of damped Gauss-Newton method; c)Results of GCV method; d)Cross-section of a at x=0; e) Cross-section of b at x=0; f) Cross-section of c at x=04.2. Experiment 2This model has two cavities in the homogeneous background, the relative conductivity of cavity and background are -3.2775 and -5.2983 respectively, other parameters are same as the experiment 1. The observed data were generated using the forward modeling algorithm (Finite Difference in Time Domain), these data were corrupted with 5% uniformly distributed random noise. The relative error of a and b in Fig.2 is 157.76%, a and c is 45.75%.Fig.2 a)True model; b)Results of damped Gauss-Newton method; c)Results of GCV method; d)Cross-section of a at Z=8; e) Cross-section of b at Z=8; f) Cross-section of c at Z=8L. Ding, B. Han and J. Liu 7895. CONCLUDING REMARKSThis paper has addressed the GCV method for solving inverse Maxwell’s equations. GCV method coupled with traditional damped Gauss-Newton methods have been used in the experiments. Results show that this new method is more available than damped Gauss-Newton method. It is a widely and fast convergent method even if the initial guess value is far away from the true coefficient.6. REFERENCES1. G. A. Newman, S. Recher, B. Tezkan and F. M.Neubauer, 3D inversion of a scalar radio magnetotelluric field data set, Geophysics68, 782-790, 2003.2. E. Haber, Quasi-Newton methods for large-scale electromagnetic inverse problems, Inverse Problem21, 305-323, 2005.3. P. A. 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