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外文文献资料翻译：李睿钦指导老师：刘文军Medical image registration with partial dataSenthil Periaswamy，Hany FaridThe goal of image registration is to find a transformation that aligns one image to another. Medical image registration has emerged from this broad area of research as a particularly active field. This activity is due in part to the many clinical applications including diagnosis, longitudinal studies, and surgical planning, and to the need for registration across different imaging modalities (e.g., MRI, CT, PET, X-ray, etc.). Medical image registration, however, still presents many challenges. Several notable difficulties are (1) the transformation between images can vary widely and be highly non-rigid in nature; (2) images acquired from different modalities may differ significantly in overall appearance and resolution; (3) there may not be a one-to-one correspondence between the images (missing/partial data); and (4) each imaging modality introduces its own unique challenges, making it difficult to develop a single generic registration algorithm.In estimating the transformation that aligns two images we must choose: (1) to estimate the transformation between a small number of extracted features, or between the complete unprocessed intensity images; (2) a model that describes the geometric transformation; (3) whether to and how to explicitly model intensity changes; (4) an error metric that incorporates the previous three choices; and (5) a minimization technique for minimizing the error metric, yielding the desired transformation.Feature-based approaches extract a (typically small) number of corresponding landmarks or features between the pair of images to be registered. The overall transformation is estimated from these features. Common features include corresponding points, edges, contours or surfaces. These features may be specified manually or extracted automatically. Fiducial markers may also be used as features;these markers are usually selected to be visible in different modalities. Feature-based approaches have the advantage of greatly reducing computational complexity. Depending on the feature extraction process, these approaches may also be more robust to intensity variations that arise during, for example, cross modality registration. Also, features may be chosen to help reduce sensor noise. These approaches can be, however, highly sensitive to the accuracy of the feature extraction. Intensity-based approaches, on the other hand, estimate the transformation between the entire intensity images. Such an approach is typically more computationally demanding, but avoids the difficulties of a feature extraction stage.Independent of the choice of a feature- or intensity-based technique, a model describing the geometric transform is required. A common and straightforward choice is a model that embodies a single global transformation. The problem of estimating a global translation and rotation parameter has been studied in detail, and a closed form solution was proposed by Schonemann. Other closed-form solutions include methods based on singular value decomposition (SVD), eigenvalue-eigenvector decomposition and unit quaternions. One idea for a global transformation model is to use polynomials. For example, a zeroth-order polynomial limits the transformation to simple translations, a first-order polynomial allows for an affine transformation, and, of course, higher-order polynomials can be employed yielding progressively more flexible transformations. For example, the registration package Automated Image Registration (AIR) can employ (as an option) a fifth-order polynomial consisting of 168 parameters (for 3-D registration). The global approach has the advantage that the model consists of a relatively small number of parameters to be estimated, and the global nature of the model ensures a consistent transformation across the entire image. The disadvantage of this approach is that estimation of higher-order polynomials can lead to an unstable transformation, especially near the image boundaries. In addition, a relatively small and local perturbation can cause disproportionate and unpredictable changes in the overall transformation. An alternative to these global approaches are techniques that model the global transformation as a piecewise collection of local transformations. For example, the transformation between each local region may bemodeled with a low-order polynomial, and global consistency is enforced via some form of a smoothness constraint. The advantage of such an approach is that it is capable of modeling highly nonlinear transformations without the numerical instability of high-order global models. The disadvantage is one of computational inefficiency due to the significantly larger number of model parameters that need to be estimated, and the need to guarantee global consistency. Low-order polynomials are, of course, only one of many possible local models that may be employed. Other local models include B-splines, thin-plate splines, and a multitude of related techniques. The package Statistical Parametric Mapping (SPM) uses the low-frequency discrete cosine basis functions, where a bending-energy function is used to ensure global consistency. Physics-based techniques that compute a local geometric transform include those based on the Navier–Stokes equilibrium equations for linear elastici and those based on viscous fluid approaches.Under certain conditions a purely geometric transformation is sufficient to model the transformation between a pair of images. Under many real-world conditions, however, the images undergo changes in both geometry and intensity (e.g., brightness and contrast). Many registration techniques attempt to remove these intensity differences with a pre-processing stage, such as histogram matching or homomorphic filtering. The issues involved with modeling intensity differences are similar to those involved in choosing a geometric model. Because the simultaneous estimation of geometric and intensity changes can be difficult, few techniques build explicit models of intensity differences. A few notable exceptions include AIR, in which global intensity differences are modeled with a single multiplicative contrast term, and SPM in which local intensity differences are modeled with a basis function approach.Having decided upon a transformation model, the task of estimating the model parameters begins. As a first step, an error function in the model parameters must be chosen. This error function should embody some notion of what is meant for a pair of images to be registered. Perhaps the most common choice is a mean square error (MSE), defined as the mean of the square of the differences (in either feature distance or intensity) between the pair of images. This metric is easy to compute and oftenaffords simple minimization techniques. A variation of this metric is the unnormalized correlation coefficient applicable to intensity-based techniques. This error metric is defined as the sum of the point-wise products of the image intensities, and can be efficiently computed using Fourier techniques. A disadvantage of these error metrics is that images that would qualitatively be considered to be in good registration may still have large errors due to, for example, intensity variations, or slight misalignments. Another error metric (included in AIR) is the ratio of image uniformity (RIU) defined as the normalized standard deviation of the ratio of image intensities. Such a metric is invariant to overall intensity scale differences, but typically leads to nonlinear minimization schemes. Mutual information, entropy and the Pearson product moment cross correlation are just a few examples of other possible error functions. Such error metrics are often adopted to deal with the lack of an explicit model of intensity transformations .In the final step of registration, the chosen error function is minimized yielding the desired model parameters. In the most straightforward case, least-squares estimation is used when the error function is linear in the unknown model parameters. This closed-form solution is attractive as it avoids the pitfalls of iterative minimization schemes such as gradient-descent or simulated annealing. Such nonlinear minimization schemes are, however, necessary due to an often nonlinear error function. A reasonable compromise between these approaches is to begin with a linear error function, solve using least-squares, and use this solution as a starting point for a nonlinear minimization.译文：部分信息的医学图像配准Senthil Periaswamy，Hany Farid图像配准的目的是找到一种能把一副图像对准另外一副图像的变换算法。
1.-ability表名词，“能…；性质”useability n 可用性（use用）inflammability n 易燃性（inflame点火+ablity）adaptability n 适应能力（adapt适应）可靠性dependability n 可靠性（depend依靠）variability n 变化性（vary变化）lovability n 可爱（love爱）2.-able 表形容词，“可…的，能…”knowable a 可知的inflammable a 易燃的（inflame点燃）trantable a 温顺的（travt 拉→拉得动→温顺的）conceivable a 想象得出的（conceive 设想，想象）desirable a 值得要的（desire 渴望）inviolable a 不右侵犯的（in 不+viol 冒犯+able ;参考：violate违反，冒犯impregnable a 攻不破的（im不+pregn拿住+able；参考：pregnant怀孕的）3.-ably 表副词，“能…地”suitably ab 恰当地（suit恰当）deoebdabky ab 可靠地（depend可靠）lovably ab 可爱地（love爱）4.-aceous表形容词，“具有…特征的”herbaceous a 草本植物的（herb草）pa[uraceous a 似纸的（papyr纸+aceous；参考：pspyrus莎草纸）curvacious a 有曲线美的（curve曲线）foliaceous a 叶状的（foli 叶+aceous;参考：foliage树叶）5.-acious表形容词，“有特征的，多…的”rapacious a 掠夺成性的（rape掠夺，强奸）sagacious a 睿智的（sage智者）capacious a 宽敞的（cap能→的+acious；参考：capble有能力的）fallacious a 错误的（fall错；参考：fallacy谬误）vivacious a 活泼的（viv活；参考：revive复活）audacious a 大胆的（aud大胆+acious；名词：audacity大胆）veracious a 真实的（ver真；参考：verify证实）lopuacious a 啰唆的（loqu讲话；参考：eloquent雄辩）salacious a 好色的（sal跳→见到女人就跳起来→好色）tenacious a 固执的（ten拿住；参考：tenable防守得住的）sequacious a 前后连贯的（sequ=secu跟随；参考：persecute迫害）voracious a 贪婪的（vor吃；参考：omnivorous无所不吃的）6.-acity表名词，“有…倾向”eapacity n 掠夺（rape掠夺）capacity n 容量；能力（cap容纳）loquavity n 健谈，多话（loqu讲话）ausacity n 大胆（auda大胆）veracity n 真实（ver真）tenacity n 固执（ten拿住）mendacity n 虚假；说谎（mend缺点；参考：amend弥补，改正）7.-acle 表名词，“…物品，状态”receptacle n 容器（recept接受）manacle n 手铐（man手；参考：manuscript手稿）miracle n 奇迹（mir惊奇；参考：mirror镜子）tentacle n 触角（tent触，摸；参考：attentive注意的，关心的）pinnacle n 尖塔(pinn尖+acle）obstacle n 障碍（ob在+sta{=stand 站}+acle→站在中间的物品→障碍）debacle n 解冻；崩溃（de去掉+bacle[=block]+acle→去掉大块东西→解冰块）8.-acy表名词，“→性质，状态”fallacy n 谬误，错误（fall错）supremacy n 至高无上（supreme崇高的）intimacy n 亲密（intim亲密；参考：intimate亲密的）conspiracy n 同谋（con共同+spir呼吸+acy→共同呼吸→同谋）celibacy n 独身（celib未婚；参考：celibate独身者）intricacy n 错综复杂（in进入+tric[=trifle琐碎事]+acy→进入琐碎之事）contumacy n 抗命，不服从（contum反叛+acy）9.ad表名词，“…东西，状态”doodad n 美观而无用之物myriad n 许多，无数（myri多+ad）ballad n 歌谣，歌曲（ball舞，歌）nomad n 流浪者，游牧人（nom流浪）10.表名词（1）表示“状态，物品”blockade n 封锁（block挡住）cannonade n 炮击（cannon大炮）decade n 十年（dec十；参考：decagon 十角形cockade n 帽章（cock帽边）lemonade n 柠檬水（lemon柠檬）arcade n 拱廊（arc弧形）brricade n 栅栏（barr栅栏+ic+adc）cascade n 小瀑布（casc下落+ade）（2）表示“个人或集体”brigade n 旅，队（brig战斗+ade）crusade n 十字军，改革运动（crus 十字+ade）cavalcade n 骑兵队(caval马+cade）renegade n 叛徒（renege背信弃义）11.-名词反缀（1）表示“状态，总称”tonnage n 吨位（ton 一吨）mileage n 英里数（mile英里）percentage n 百分比（percent百分之几）foliage n 树叶（foli树叶；参考：portfolio 文件夹）visage n 面貌（vis 看；参考：vision视力）advantage n 利益（ad+vant来到+age）verbiage n 冗词，赘语（verb词语）dotage n 老年昏愦（dot打盹+age）pilgrimage n 朝圣（pilgrim朝香客）marriage n 结婚（marry结婚）pillage n 掠夺（pill来自中世纪法语piller，意为“掠夺”）drainage n 排水（drain 排干）patronage n 赞助，惠客（patron赞助人）（2）表示“场所，物品”anchorage n 停泊地（anchor铁猫）village n 村庄（villa茅房，别墅）hermitage n 隐居处（hermit隐）士appendage n 附属物（append附属）heritage n 遗产（herit继承）beverage n 饮料（bever饮，喝+age）qreckage n 残骸，碎片（wreck残骸，破坏）（3）表示“费用”postage n 邮费（post邮局）railage n 铁路运费（rail铁路）towage n 拖（车、船）费（tow拖）waterage n 水运费（water水）12.-ain 表名词，“…人”captain n 船长（capt头+ain）chiefain n 酋长，头目（chief主要+tain人）villain n 恶棍（vill[=vile ]+ain）riverain n 住在河边的人（river河）13.-aire表名词millionaire n 百万富翁（million百万）koctrinaire n 教条者，空论家（doctrine教条）questionnaire n 调查表，问卷14。
Simultaneous Equation MethodIntroductionIn mathematics, simultaneous equations play a crucial role in solving real-world problems and modeling various phenomena. The simultaneous equation method is a powerful technique used to find solutions for a system of equations. This method involves solving multiple equations together to determine the values of unknown variables. In this article, we will explore the simultaneous equation method in detail and discuss its applications.Understanding Simultaneous EquationsDefinitionSimultaneous equations, also known as a system of equations, are a set of equations that share the same variables. The solutions of these equations simultaneously satisfy each equation in the system. The general form of simultaneous equations can be written as:a1x + b1y = c1a2x + b2y = c2Here, x and y are the variables, while a1, a2, b1, b2, c1, and c2 are constants.Types of Simultaneous EquationsSimultaneous equations can be classified into three types based on the number of solutions they have:1.Consistent Equations: These equations have a unique solution,meaning there is a specific set of values for the variables thatsatisfy all the equations in the system.2.Inconsistent Equations: This type of system has no solution. Theequations are contradictory and cannot be satisfied simultaneously.3.Dependent Equations: In this case, the system has infinitely manysolutions. The equations are dependent on each other and represent the same line or plane in geometric terms.To solve simultaneous equations, we employ various methods, with the simultaneous equation method being one of the most commonly used techniques.The Simultaneous Equation MethodThe simultaneous equation method involves manipulating and combining the given equations to eliminate one variable at a time. By eliminating one variable, we can reduce the system to a single equation with one variable, making it easier to find the solution.ProcedureThe general procedure for solving simultaneous equations using the simultaneous equation method is as follows:1.Identify the unknow n variables. Let’s assume we have n variables.2.Write down the given equations.3.Choose two equations and eliminate one variable by employingsuitable techniques such as substitution or elimination.4.Repeat step 3 until you have a single equation with one variable.5.Solve the single equation to determine the value of the variable.6.Substitute the found value back into the other equations to obtainthe values of the remaining variables.7.Verify the solution by substituting the found values into all theoriginal equations. The values should satisfy each equation.If the system is inconsistent or dependent, the simultaneous equation method will also lead to appropriate conclusions.Applications of Simultaneous Equation MethodThe simultaneous equation method finds applications in numerous fields, including:EngineeringSimultaneous equations are widely used in engineering to model and solve various problems. Engineers employ this method to determine unknown quantities in electrical circuits, structural analysis, fluid mechanics, and many other fields.EconomicsIn economics, simultaneous equations help analyze the relationship between different economic variables. These equations assist in studying market equilibrium, economic growth, and other economic phenomena.PhysicsSimultaneous equations are a fundamental tool in physics for solving complex problems involving multiple variables. They are used in areas such as classical mechanics, electromagnetism, and quantum mechanics.OptimizationThe simultaneous equation method is utilized in optimization techniques to find the optimal solution of a system subject to certain constraints. This is applicable in operations research, logistics, and resource allocation problems.ConclusionThe simultaneous equation method is an essential mathematical technique for solving systems of equations. By employing this method, we can find the values of unknown variables and understand the relationships between different equations. The applications of this method span across various fields, making it a valuable tool in problem-solving and modeling real-world situations. So, the simultaneous equation method continues to be akey topic in mathematics and its practical applications in diverse disciplines.。
Structural Equation ModelsJohn FoxJanuary 20021 IntrodutionStructural equation models(SEMs) , also called simultaneous equation models , are multivariate(i.e.,multi-equation) regression models . Unlike the more traditional multivariate linear model , however , the response variable in one regression equation in an SEM may influence one-another reciprocally , either directly or through other variables as intermediaries . These structural equations are meant to represent causal relationships among the variables in the model .A cynical view of SEMs is that their popularity in the social sciences reflects the legitimacy that the models appear to lend to causal interpretation of observational data , when in fact such interpretation is no less problematic than for other kinds of regression models applied to observational data . A more charitable interpretation is that SEMs are close to the kind of informal thinking about causal relationgships that is common in social-science theorizing , and that , therefore , these models facilitate translating such theories into data analysis . In economics , in contrast , structural-equation models may stem from formal theory .To my knowledge , the only facility in S for fitting structural equation models is my sem library , which at present is available for R but not for S-PLUS . The sem library includes functions for estimating structrual equations in observed-variables models by two-stage least squares , and for fitting general structural equation models with multinormal errors and latent variables by full-information maximum likelihood . These methods are covered (along with the associated terminology) in the subsequent sections of the appendix . As I write this appendix ,the sem library is in a preliminary form , and the capabilities that it provides are modest compared with specialized structural equation software .Structural equation models is a large subject . Relatively brief introductions may be found in Fox (1984:Ch. 4) and in Duncan (1975);Bollen (1989) is a standard book-length treatment , now slightly dated ; and most general econmetric texts (e.g., Greene, 1993: CH.20; Judge et al., 1985: Part 5) take up at least observed-variables structural equation models .2 Observed-Variables Models and Two-Stage Least-Squares Estimation2.1 An Example: Klein’s ModelKlein’s (1950) macroeconomic model of the U.S. economy often appears in econometrics texts (e.g., Greene, 1993) as a simple example of a structual equation model:·The variable on the left-hand side of the structual equations are endogenous variables——that is , variables whose values are determined by the model . There is , in general , one structrual equation for each endogenous variable in a SEM .·The ζ’s (Green zeta) are error variables , also called structural disturbances or errors in equations ; they play a role analogous to the error in a single-equation regression model . It is not general assumed that different disturbances are independent of one-another , although such assumptions are sometimes made in particular models .·The remaining variables on the right-hand side of the model are exogenous variables , whose values are treated as conditionally fixed ; an additional defining characteristic of exogenous variables is that they are assumed to be independent of the errors ( much as the predictors in a common regression model are taken to be independent of the error ) .·The γ’( Greek gamma ) are structural parameters ( regression coefficients ) relating the endogenous variables to the exogenous variables ( including an implicit constant regressor for each of the first three equations ) .·Similary , the β’s ( Greek beta ) are structural parameters relating the endogenous variables to one-another .·The last three equations have no error variables and no structural parameters . These equations are identities , and could be substituted out of the model . Our task is to estimate the first three equations , which contain unknown parameters .The variables in model (1) have the following definitions:C Consumption ( in year t )tI InvestmenttpW Private wagestX Equilibrium demandtP Private profitstK Capital stocktG Government non-wage spendingtT Indirect business taxes and net exportstgW Government wagestA Time trend , year – 1931tThe use of the subscript t for observations reflects the fact that Klein estimated the model with annual time-series data for the years 1921 through 1941 . Klein’s data are in the data frame Klein in the sem library .The data in Klein conform to my usual practice of using lower-case names forvariables . Some of the variables in Klein ’s model have to be constructed from the data :Notice , in particular how the lagged variables 1-t P and 1-t X are created by shifting t P and t X forward one time period — placing and NA at the beginning of each variable , and dropping the last observation . The first observation for 1-t P and 1-t X is missing because there are no data available for 0P and 0X .Estimating Klein ’s model is complicated by the presence of endogenous variables on the right-hand side of the structural equations . In general , we cannot assume that an endogenous predictor is uncorrelated with the error variable in a structural equation , and consequently ordinary least-squares ( OLS ) regression cannot be relied upon to produce consistent estimates of the parameters of the equation , for t C ; but t X is a component of t P , and t X , in turn , depends upon t C , one of whose components is the error t 1ζ . Thus , indirectly , t 1ζ is a component of t P , and the two are likely correlated . Similar reasoning applies to the other endogenous predictors in the model , as a consequence of the simultaneous determination of the endogenous variables .2.4 Recursive ModelsOutside of economics , it is common to specify a structural equation model in the form of a graph called a path diagram . A well known example , Blau and Duncan ’s (1967) basic stratification model , appears in Figure 1 .The following conventions , some of them familiar from Klein ’s macroeconomic model , are employed in drawing the path diagram :·Directly observable variables are enclosed in rectangular boxes .·Unobservable variables are enclosed in circles ( more generally , in ellipses ); in this model , the only unobservable variables are the disturbances .·Exogenous variables are represented by x ’s ; endogenous variables by y ’s ; and disturbances by ζ’s .·Directed ( i.e., single-headed) arrows represent structural parameters . The endogenous variables are distinguished from the exogenous variables by having directed arrows pointing towards them , while exogenous variables appear only at the tails of directed arrows .Figure 1 :Blau and Duncan recursive basic stratification model.·Bidirectional ( double-headed ) arrows represent non-causal , potentially nonzero , covariances between exogenous variables ( and , more generally , also between disturbances ) .·As before , γ’s are used for structural parameters relating an endogenous to an exogenous variable , while β’s are used for structural parameters relating one endogenous variable to another .·To the extent possible , horizontal ordering of the variables corresponds to their causal ordering: Thus , ‘causes ’ appear to the left of ‘effects.’The structural equations of the model may be read off the path diagram:i i i i x x y 1212111101ζγγγ+++=i i i i i y x x y 2121222121202ζβγγγ++++=i i i i i y y x y 2232131232303ζββγγ++++=Blau and Duncan ’s model is a menber of a special class of SEMs called recursive models . Recursive models have the following two defining characteristics:1. There are no reciprocal directed paths or feedback loops in the diagram.2. Different disturbances are independent of one-another ( and hence are unlinked by bidirectional arrows ).As a consequence of these two properties , the predictors in a structural equation of a recursive model are always independent of the error of that equation , and thestructural equation may be estimated by OLS regression . Estimating a recursive model is simply a sequence of OLS regressions . In S , we would of course use 1m to fit the regressions . This is a familiar operation , and therefor I will not pursue the example further .Structural equation models that are not recursive are sometimes termed nonrecursive ( an awkward and often-confused adjective ).Figure 2: Duncan , Haller , and Portes ’s general structural equation model for peer influences on aspirations.3 General Structural Equation ModelsGeneral structural equation model include unobservable exogenous or endogenous variables ( also termed factors or latent variables ) in addition to the unobservable disturbances . General structural equation models are sometimes called LISREL models , after the first widely available computer program capable of estimating this class of models ( Joreskog , 1973 ); LISREL is an acronym for linear structural relations .Figure 2 shows the path diagram for an illustrative general structural equation model , from path-breaking work by Duncan , Haller , and Portes ( 1968 ) concerning peer influences on the aspirations of male high-school students . The most striking new feature of this model is that two of the endogenous variables , Respondent ’s General Aspirations (1η) and Friend ’s General Aspirations (2η) , are unobserved variables . Each of these variables has two observed indicators: The occupational andeducational aspirations of each boy —1y and 2y for the respondent , and 3y and 4y for his best friend .3.1 The LISREL ModelIt is common in general structural equation models such as the peer-influences model to distinguish between two sub-models:1. A structural submodel , relating endogenous to exogenous variables and to one-another . In the peer-influences model , the endogenous variables and unobserved , while the exogenous variables are directly observed .2. A measurement submodel , relating latent variables ( here only latent endogenous variables ) to their indicators .I have used the following notation , associated with Joreskog ’s LISREL model , in drawing the path diagram in Figure 2:·x’s are used to represent obsercable exogenous variables . If there were latent exogenous variables in the model , these would be represented by s 'ξ(Greek xi) , and x ’s would be used to represent their observable indicators .·y’s are employed to represent the indicators of the latent endogenous variables , which are symbolized by η’s (Greek eta) . Were there directly observed endogenous variables in the model , then these too would be represented by y ’s .·As before , γ’s and β’s are used , respectively , for structural coefficients relating endogenous variables to exogenous variables and to one-another , and ζ’s are used for structural disturbances . The parameter 12ψ is the covariancebetween the disturbances 1ζ and 2ζ. The variances of the disturbances , 12ψand 22ψ, are not shown on the diagram .·In the measurement submodel , λ’s ( Greek lambda ) represent regression coefficient ( also called factor loadings ) relating observable indicators to latent variables . The superscript y in y λ indicates that the factor loading in this model pertain to indicators of latent endogenous variables .·The ε’s ( Greek epsilon ) represent measurement error in the endogenous indicators; if there were exogenous indicators indicators in the model ,then the measurement errors associated with them would be represented by δ’s ( Greek delta ) .We are swimming in notation , but we still require some more ( not all of whichis necessary for the peer-influences model ): We use ij σ (Greek sigma ) to representthe covariance between two observable variables; εθij to represent the covariancebetween two measurement-error variables for endogenous indicators , i ε and j ε; δθij to represent the covariance between two measurement-error variables for exogenous indicators , i δ and j δ; and ij φ to represent the covariance between two latent exogenous variables i ξ and j ξ .The LISREL notation for general structural equation models is summarized in Table 1. The structural and measurement submedels are written as follows:i i i i B ςξηη+Γ+=i i y i y εη+Λ=i i x i x δξ+Λ=In order to identify the model , many of the parameters in B,,,,,,ψΦΛΛΓy x εΘ,and δΘ must be constrained , typically by setting parameters to 0 or 1 , or by defining certain parameters to be equal .3.2 The RAM FormulationAlthough LISREL notation is most common , there are several equivalent ways to represent general structural equation models . The sem function uses the simpler RAM ( reticular action model – don ’t ask ! ) formulation of McArdle (1980) and McArdle and McDonald (1984); the notation that I employ below is from McDonald and Hartmann (1992) .The RAM model includes two vectors of variables: v , which contains the indicator variables , directly observed exogenous variables , and the latent exogenous and endogenous variables in the model; and u , which contains directly observed exogenous variables , measurement-error variables , and structural disturbances . The two sets of variables are related by the equationv=Av+uThus , the matrix A includes structural coefficients and factor loadings. For example , for the Duncan , Haller , and Portes model , we have ( using LISREL notation for the individual parameters ):It is typically the case that A is sparse , containing many 0’s . Notice the special treatment of the observed exogenous variables , x1 through x6 , which are specified to be measured without error , and which consequently appear both in v and u .The final component of the RAM formulation is the covariance matrix P of u .Assuming that all of the error variables have expectations of 0 , and that all other variables have been expressed as deviations from their expectations , P=E(uu ’) . For the illustrative model ,For convenience , I use a double-subscript notation for both covariances andvariances; thus , for example , 11σ is the variance of x1 (usually written 21σ); εθ11is the variance of 1ε; and 11ψ is the variance of 1ζ.The key to estimating the model is the connection between the covariances of the observed variables , which may be estimated directly from sample data , and the parameters in A and P . Let m denote the numble of variables in v , and ( without loss of generality ) let the first n of these be the observed variables in the model . Define the m ⨯m selection matrix J to pick out the observed variables; that isWhere n I is the order-n identity matrix , and the 0’s are zero matrices of appropriate orders . The model implies the following covariances among the observed variables:Let S denote the observed-variable covariances computed directly from the sample . Fitting the model to the data-that is , estimating the free parameters in A and P –entails selecting parameter values that make S as close as possible to the model-implied covariances C . Under the assumptions that the errors and latent variables are multivariately normally distributed , finding the maximum-likelihood estimates of the free parameters in A and P is equivalent to minimizing the criterion。
●供给廣泛的訓練及學習函式●供给Simulink的類神經網路block●可自動將MATLAB產生的類神經網路物件轉成Simulink 模型●供给前處理及後處理函式改良類神經網路訓練及效能●供给視覺化函式更轻易瞭解類神經網路之效能(Preceptron)、倒傳遞網路(Backpropagation)、自組織映射網路(SOM)、徑向基網路(Radial Basis Network)、學習向量量化網路(LVQN)..等.◎ Curve Fitting Toolbox(曲線契合对象箱)產品特点●可透過圖形化应用者介面或指令列操作各種功能●可作資料前置處理的例行法度榜样設定，例如資料排序、瓜分、腻滑化、及除去界外值(outlier)等●擁有線性或非線性參數契合模型的龐大年夜的函式庫，與最佳化的肇端點(starting points)以及非線性模型參數的解題器●多樣的線性和非線性契合方法，包含最小平方法、加權最小平方法、或強韌契合法度榜样(robustfitting procedures) (上述全部增援限制係數範圍或不限制係數範圍的功能)●客製化的線性或非線性模型發展●应用Splines或內插法(interpolants)進行非參數(Nonparametric)契合●增援內插法、外插法、微分以及積分計算簡介1.功能确实是可將客戶的資料畫成圖形，接著供给現有標準的一些數學式子來找出相符這圖形的方程式，例如y=ax+bx^2+cx^3。
2.裡面有供给許多的數學式子，也可讓User自定本身的數學函式，而此对象箱可幫客戶算出數學式子的係數(如a,b,c等)3.供给polynomial, exponential, Fourier, rational等數學model給客戶Note: 此套对象不克不及作曲面的契合，等于假如客戶要求要作y=ax1+bx2 找出a與b，是弗成以的，因為此对象箱只能找出單一變數x的係數。
建議他們購買OP and SP 解決他們的問題。
Simultaneous Equations Models (1)The Nature of Simultaneous Equations Models● In the previous study, we considered single equation models – one Y and one or more X ’s.● The cause-and-effect relationship is from X ’s to Y .● There may be a two-way, or simultaneous, relationship between Y and X ’s.● It is difficult to distinguish dependent and explanatory variables.● Set up simultaneous equations model where variables are jointly dependent or endogenous.● Can not estimate the parameters of a single equation without taking into account of theinformation provided by other equations.● OLS estimator for single equation in simultaneous model is biased and inconsistent.● i i i i ii i i u X Y Y u X Y Y 21211222021111212101+++=+++=γββγββY 1i and u 2i are correlated, Y 2i and u 1i are correlated, so OLS leads to inconsistent estimates.Examples of Simultaneous Equations Models● Example1: Demand –and-supply modelDemand function: 0 1110<++=αααt t d t u P Q Supply function:0 1210>++=βββt t s t u P QEquilibrium condition: s t d t Q Q = Wheretime t supplied, Q demanded,quantity s t ===quantityQ d t● Price P and quantity Q are determined by the intersection of the demand and supply curves.Demand and supply curves are linear.● P and Q are jointly dependent.● the demand curve will shift upward if u 1t is positive and downward if u 1t is negative. ● A shift in the demand curve changes both P and Q● A change in u 2t will shift supply curve then change both P and Q● So u 1t and P, u 2t and P are correlated – violate the important assumption of CLRM. ● Example 2: Keynesian model of income determination● Consumption function: 10 1 10<<++=βββt t t u Y CIncome identity:)(t t t t S I C Y =+=Where C = consumption expenditure, Y = income, I = investment (assumed exogenous), S = savings, t = time, u = stochastic disturbance term● β1 is marginal propensity to consume(MPC) lying between 0 and 1.● C and Y are interdependent and Y is not expected to be independent of the disturbance term. ● Because U i shifts, then the consumption function also shifts, which, in turn, affects Y.The simultaneous equation bias: Inconsistency of OLS estimator● Use simple Keynesian model of income determination to show OLS estimator is inconsistentin simultaneous model● We want to estimate consumption function10t t t u Y C ++=ββ● First show that Y t and u t are correlated.Substituting consumption function into income identity:t t t u I Y 111011111ββββ-+-+-=t t I Y E 11011)(ββββ-+-=so12t 1]E[u)]()][([),cov(β-=--=t t t t t t u E u Y E Y E u Y● Second show that the OLS estimator 1ˆβis an inconsistent estimator of 1β, because of the correlation of Y t and u t∑∑∑∑∑∑∑∑+=++==---=21210221)( )())((ˆtttt ttt t t t t t tyy u y y u Y yy C Y Y Y Y C C ββββSo)(11 )/lim()/lim()ˆlim(2211211Yt t t N y p N u y p p σσββββ-+=+=∑∑● Plim(1ˆβ) will be always be greater than 1βThe Identification Problem● Recall the demand and supply model, if we have time-series data on P and Q only and noadditional information, can we estimate the demand function?● Need to solve the identification problem.Notations and Definitions● Take income determination model as example:Consumption function : 10 1 10<<++=βββt t t u Y CIncome identity : )(t t t t S I C Y =+=● - Endogenous variables: determined within the model- Predetermined variables: determined outside the model.- Predetermined variables include current and lagged exogenous variables and lagged endogenous variables.- Lagged endogenous variable is nonstochastic, hence a predetermined variable. - Be careful to defend the classification.●β’s are known as the structural parameters or coefficients.● Solve for endogenous variables to derive the reduced-form equations.● Reduced-form equation is the one which expresses an endogenous variable solely in terms ofthe predetermined variables and the stochastic disturbances.● Substitute consumption function into income identity:t t t w I Y +∏+∏=10Where 1t 111001 w ,11,1ββββ-=-=∏-=∏t uSubstitute income identity into consumption functiont t t w I C +∏+∏=32Where 1t 1131021 w ,1 ,1βββββ-=-=∏-=∏t u● 31 and ∏∏ are impact multipliers● Reduced form equations give the equilibrium values of the relevant endogenous variables.● The OLS method can be applied to estimate the coefficients of the reduced –form equations● Structural coefficients can be “retrieved ” from the reduced form coefficients.The identification problem● The identification problem is whether numerical estimates of the parameters of a structuralequation can be obtained from the estimated reduced form coefficients.● Identified, underidentified, exactly identified and overidentified.Underidentified● Consider the demand and supply model, together with market clearing condition.(Insert equations)● There are four structural coefficients corresponding two reduced form coefficients. – modelcan not be solved.● What does “underidentified ” mean? See figures● An alternative way to looking at the identification problem. – “mongrel ” equations. Ifmongrel equation is observational indistinguishable with demand function, then demand function is underidentified.Just , or exact, identification● Demand function: 0 ,0211210><+++=αααααt t t t u I P Q Supply function: 0 1210>++=βββt t t u P Q● There is an additional variable in the demand equation● Derive reduced form equations● Five structural coefficients corresponding with four reduced form coefficients – remainunderidentified.● Demand curve is underidentified, but supply curve is identified.● “mongrel ” equation is distinguishable from supply function but not from demand function.● The presence of an additional variable in the demand function enables us to identify thesupply function.● ConsiderDemand function : 0,0 211210><+++=αααααt t t t u I P QSupply function:0 ,0 21 21210>>+++=-βββββt t t t u P P QExactly identified!Overidentified● Demand function : 13210t t t t t u R I P Q ++++=ααααSupply function: 21210t t t t u P P Q +++=-βββ● Solvefor the structural equations and get reduced form:t t t t t v P R I P +∏++∏+∏+∏=-13210t t t t t w P R I Q +∏+∏+∏+∏=-17654● 7 coefficients corresponding 8 equations. Will have multiple solutions. For example:151201 ∏∏=∏∏=ββ● The reason for the multiple solution is that we have “too much ” information to identify thesupply curve.● “too much ” reflects by the exclusion of two variables in the supply function. One should beenough.Rules for identification● Solve structural equations, then get reduced form, check how many structural coefficientsand how many reduced form coefficients – no need for this time-consuming process● Order conditions of identificationM – number of endogenous variables in the modelm – number of endogenous variables in a given equation K – number of predetermined variables in the modelk – number of predetermined variables in a given equationA equivalent explanation is, in a model of M simultaneous equations, in order for an equationto be identified, the number of predetermined variables excluded from the equation must not be less than the number of endogenous variables included in that equation less 1. that is: K-k>= m-1● Check the previous examples.。
