Parameter Estimation of Two-Dimensional Moving Average Random Fields

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概率与统计英语

《概率论与数理统计》基本名词中英文对照表英文中文Probability theory 概率论mathematical statistics 数理统计deterministic phenomenon 确定性现象random phenomenon 随机现象sample space 样本空间random occurrence 随机事件fundamental event 基本事件certain event 必然事件impossible event 不可能事件random test 随机试验incompatible events 互不相容事件frequency 频率classical probabilistic model 古典概型geometric probability 几何概率conditional probability 条件概率multiplication theorem 乘法定理Bayes's formula 贝叶斯公式Prior probability 先验概率Posterior probability 后验概率Independent events 相互独立事件Bernoulli trials 贝努利试验random variable 随机变量probability distribution 概率分布distribution function 分布函数discrete random variable 离散随机变量distribution law 分布律hypergeometric distribution 超几何分布random sampling model 随机抽样模型binomial distribution 二项分布Poisson distribution 泊松分布geometric distribution 几何分布probability density 概率密度continuous random variable 连续随机变量uniformly distribution 均匀分布exponential distribution 指数分布numerical character 数字特征mathematical expectation 数学期望variance 方差moment 矩central moment 中心矩n-dimensional random variable n-维随机变量two-dimensional random variable 二维离散随机变量joint probability distribution 联合概率分布joint distribution law 联合分布律joint distribution function 联合分布函数boundary distribution law 边缘分布律boundary distribution function 边缘分布函数exponential distribution 二维指数分布continuous random variable 二维连续随机变量joint probability density 联合概率密度boundary probability density 边缘概率密度conditional distribution 条件分布conditional distribution law 条件分布律conditional probability density 条件概率密度covariance 协方差dependency coefficient 相关系数normal distribution 正态分布limit theorem 极限定理standard normal distribution 标准正态分布logarithmic normal distribution 对数正态分布covariance matrix 协方差矩阵central limit theorem 中心极限定理Chebyshev's inequality 切比雪夫不等式Bernoulli's law of large numbers 贝努利大数定律statistics 统计量simple random sample 简单随机样本sample distribution function 样本分布函数sample mean 样本均值sample variance 样本方差sample standard deviation 样本标准差sample covariance 样本协方差sample correlation coefficient 样本相关系数order statistics 顺序统计量sample median 样本中位数sample fractiles 样本极差sampling distribution 抽样分布parameter estimation 参数估计estimator 估计量estimate value 估计值unbiased estimator 无偏估计unbiassedness 无偏性biased error 偏差mean square error 均方误差relative efficient 相对有效性minimum variance 最小方差asymptotic unbiased estimator 渐近无偏估计量uniformly estimator 一致性估计量moment method of estimation 矩法估计maximum likelihood method of estimation 极大似然估计法likelihood function 似然函数maximum likelihood estimator 极大似然估计值interval estimation 区间估计hypothesis testing 假设检验statistical hypothesis 统计假设simple hypothesis 简单假设composite hypothesis 复合假设rejection region 拒绝域acceptance domain 接受域test statistics 检验统计量linear regression analysis 线性回归分析1 概率论与数理统计词汇英汉对照表Aabsolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.(degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值Ffactor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代Kkernal 核Kolmogorov-Smirnov test柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function 损失函数Mmain effect 主效应matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数model 模型Monte Carle method 蒙特卡罗法moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equation 多元回归方程multiple response 多响应multivariate analysis 多元分析Nnegative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设number of cases 个案数Oone-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分数percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数probit analysis 概率分析proportion 比例Qqadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitative 数量的，度量的quartiles 四分位数Rrandom 随机的random number 随机数random number 随机数random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank 秩rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Ssample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归stimulus 刺激strong assumption 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum 和sum of squares 平方和summary 概括，综述Ttable 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的vector 向量Wweight 加权，权重weighted average 加权平均值within groups 组内的ZZ score Z分数。

心理学名词翻译

[名词委审定]汉英心理学名词5-羟色胺5-hydroxytryptamine, 5-HT, s erotoninA 型行为类型type A behavior patternB 型行为类型type B behavior patternC 型行为类型type C behavior pattern F 分布F distributionF 检验F testP 物质substance P, SPP300 波P300 wavePM 理论PM theoryT型迷津T mazeX 理论theory XY 理论theory YY型迷津Y mazeZ 理论theory Z[对]人知觉person perception[视觉]变形小屋distorted room[心理]咨询counselingt 分布t distributiont 检验t testz 分数z scorez 检验z testⅠ型错误type ⅠerrorⅡ型错误type Ⅱerror［人格］表面特质surface trait［人类］工效学ergonomics［婴儿的］单词语holophraseα波α waveβ波β waveγ-氨基丁酸γ-aminobutyric acid, GABA δ波δ waveθ波θ waveφ系数φ coefficientχ2分布chi-square distributionχ2检验chi-square testχ2可加性additivity of chi-squareψ现象ψ-phenomenon阿德勒心理学Adlerian psychology阿德勒心理治疗Adlerian psychotherap y艾宾豪斯错觉Ebbinghaus illusion艾森克人格问卷Eysenck Personality Q uestionnaire, EPQ爱德华兹个人爱好量表Edwards Person al Preference Schedule, EPPS爱好preference安全标准safety criterion安全分析safety analysis安全工程safety engineering安全评价safety evaluation安全心理学safety psychology安全训练safety training安慰剂placebo安慰剂效应placebo effect暗示suggestion暗示性suggestibility暗示治疗suggestive therapy暗适应dark adaptation暗适应曲线dark adaptation curve暗视觉scotopic vision奥地利学派Austrian school巴宾斯基反射Babinski reflex巴黎学派Paris school靶细胞target cell白板说theory of tabula rasa白质white matter百分等级percentile rank伴随性负电位变化contingent negative variation, CNV半规管semicircular canal剥夺deprivation保持retention保持曲线retention curve保持性复述maintenance rehearsal保健health care保守性聚焦conservative focusing饱和度saturation抱负水平aspiration level暴露疗法exposure therapy背根dorsal root背景音乐background music贝利婴儿发展量表Bayley Scales of Inf ant Development贝叶斯定理Bayes' theorem倍音overtone备择假设alternative hypothesis被动-攻击性人格passive-aggressive pe rsonality被害妄想persecutory delusion被试变量subject variable本能instinct本能冲动instinctive impulse本能论instinct theory本能行为instinctive behavior本体感受proprioception本我id本族语native language比较认知comparative cognition比较水平comparison level比较心理学comparative psychology 比率量表ratio scale比率智商ratio intelligence quotient比内-西蒙智力量表Binet-Simon Scale of Intelligence比赛心理定向mental orientation in co mpetition比赛型运动员competition type athlete 比赛自我意象self-image in competitio n毕生发展观life-span perspective闭环控制closed-loop control; 具有反馈信息的控制［过程］。

应用统计方法第二章参数估计

2
1 1
2 x 1 1
2 2
2 dx
1 12
(
2
1 ) 2
令
X
S 2
1 2
(1
2
)
1 12
( 2
1)
解上述关于
1
,
2
的方程得
1 2
X X
3S 3S
8
.
Example 2.4 贝努利试验中，事件 A 发生的频率是该事件发生概率的矩法估计。 Solution 此处，实际上我们视总体 X 为“唱票随机变量”，即 X 服从两点分布：
此,必须采用求极值的办法，即对对数似然函数关于 i 求导，再令之为 0，即得
ln L( ) i
(Xi
X )2
S2
记为ˆ 2 S 2 ．第三步等号再一次用到习题 1.4．
7
.
Example 2.3 设 X 为[1, 2 ]上的均匀分布， X1, X 2 ,, X n
为样本，求1, 2 的矩估计。
Solution
a 1
2 xdx
1 2 1
2 2
12
2( 2 1 )
1 2
(
1
2)
2
.
Chapter 2 参数估计
(Parameter Estimation)
1
.
§2.1 点估计(Point Estimation) §2.2 估计量的评价准则 §2.3 区间估计(Interval Estimation)
2
.
§2.1 点估计(Point Estimation)
一、矩估计法
若总体 X 的期望存在，E(X ) , X1, X 2 ,, X n 是出自X 的样本，则由柯尔莫哥洛夫强大数定律,以

Mathematical Modeling

Mathematical ModelingArnold NeumaierNovember6,2003Institut f¨u r Mathematik,Universit¨a t WienStrudlhofgasse4,A-1090Wien,Austriaemail:Arnold.Neumaier@univie.ac.atWWW:http://www.mat.univie.ac.at/∼neum/1Why mathematical modeling?Mathematical modeling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides in-sight,answers,and guidance useful for the originating application.Mathematical modeling•is indispensable in many applications•is successful in many further applications•gives precision and direction for problem solution•enables a thorough understanding of the system modeled•prepares the way for better design or control of a system•allows the eﬃcient use of modern computing capabilitiesLearning about mathematical modeling is an important step from a theoretical mathematical training to an application-oriented mathematical expertise,and makes the studentﬁt for mastering the challenges of our modern technological culture.2A list of applicationsIn the following,I give a list of applications whose modeling I understand,at least in some detail.All areas mentioned have numerous mathematical challenges.This list is based on my own experience;therefore it is very incomplete as a list of applications of mathematics in general.There are an almost endless number of other areas with interesting mathematical problems.Indeed,mathematics is simply the language for posing problems precisely and unambiguously (so that even a stupid,pedantic computer can understand it).1Anthropology•Modeling,classifying and reconstructing skulls Archeology•Reconstruction of objects from preserved fragments •Classifying ancient artiﬁcesArchitecture•Virtual realityArtiﬁcial intelligence•Computer vision•Image interpretation•Robotics•Speech recognition•Optical character recognition •Reasoning under uncertaintyArts•Computer animation(Jurassic Park)Astronomy•Detection of planetary systems •Correcting the Hubble telescope•Origin of the universe•Evolution of starsBiology•Protein folding•Humane genome project2•Population dynamics •Morphogenesis•Evolutionary pedigrees •Spreading of infectuous diseases(AIDS)•Animal and plant breeding(genetic variability) Chemical engineering•Chemical equilibrium•Planning of production unitsChemistry•Chemical reaction dynamics •Molecular modeling•Electronic structure calculationsComputer science•Image processing•Realistic computer graphics(ray tracing) Criminalistic science•Finger print recognition•Face recognitionEconomics•Labor data analysisElectrical engineering•Stability of electric curcuits •Microchip analysis•Power supply network optimizationFinance•Risk analysis•Value estimation of options3Fluid mechanics•Wind channel•TurbulenceGeosciences•Prediction of oil or ore deposits•Map production•Earth quake predictionInternet•Web search•Optimal routingLinguistics•Automatic translationMaterials Science•Microchip production•Microstructures•Semiconductor modelingMechanical engineering•Stability of structures(high rise buildings,bridges,air planes)•Structural optimization•Crash simulationMedicine•Radiation therapy planning•Computer-aided tomography•Blood circulation models4Meteorology•Weather prediction•Climate prediction(global warming,what caused the ozone hole?) Music•Analysis and synthesis of soundsNeuroscience•Neural networks•Signal transmission in nervesPharmacology•Docking of molecules to proteins•Screening of new compoundsPhysics•Elementary particle tracking•Quantumﬁeld theory predictions(baryon spectrum)•Laser dynamicsPolitical Sciences•Analysis of electionsPsychology•Formalizing diaries of therapy sessionsSpace Sciences•Trajectory planning•Flight simulation•Shuttle reentryTransport Science•Air traﬃc scheduling•Taxi for handicapped people•Automatic pilot for cars and airplanes53Basic numerical tasksThe following is a list of categories containing the basic algorithmic toolkit needed for ex-tracting numerical information from mathematical models.Due to the breadth of the subject,this cannot be covered in a single course.For a thorough education one needs to attend courses(or read books)at least on numerical analysis(which usually covers some numerical linear algebra,too),optimization,and numerical methods for partial diﬀerential equations.Unfortunately,there appear to be few good courses and books on(higher-dimensional)nu-merical data analysis.Numerical linear algebra•Linear systems of equations•Eigenvalue problems•Linear programming(linear optimization)•Techniques for large,sparse problemsNumerical analysis•Function evaluation•Automatic and numerical diﬀerentiation•Interpolation•Approximation(Pad´e,least squares,radial basis functions)•Integration(univariate,multivariate,Fourier transform)•Special functions•Nonlinear systems of equations•Optimization=nonlinear programming•Techniques for large,sparse problems6Numerical data analysis(=numerical statistics)•Visualization(2D and3D computational geometry)•Parameter estimation(least squares,maximum likelihood)•Prediction•Classiﬁcation•Time series analysis(signal processing,ﬁltering,time correlations,spectral analysis)•Categorical time series(hidden Markov models)•Random numbers and Monte Carlo methods•Techniques for large,sparse problemsNumerical functional analysis•Ordinary diﬀerential equations(initial value problems,boundary value problems,eigen-value problems,stability)•Techniques for large problems•Partial diﬀerential equations(ﬁnite diﬀerences,ﬁnite elements,boundary elements, mesh generation,adaptive meshes)•Stochastic diﬀerential equations•Integral equations(and regularization)Non-numerical algorithms•Symbolic methods(computer algebra)•Sorting•Compression•Cryptography•Error correcting codes74The modeling diagramThe nodes of the following diagram represent information to be collected,sorted,evaluated,and organized.MMathematical ModelSProblem StatementRReportTTheoryPProgramsNNumerical Methods..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................The edges of the diagram represent activities of two-way communication (ﬂow of relevantinformation)between the nodes and the corresponding sources of information.S.Problem Statement •Interests of customer/boss •Often ambiguous/incomplete •Wishes are sometimes incompatibleM.Mathematical Model •Concepts/Variables •Relations •Restrictions •Goals•Priorities/Quality assignments8T.Theory•of Application•of Mathematics•Literature searchN.Numerical Methods•Software libraries•Free software from WWW•Background informationP.Programs•Flow diagrams•Implementation•User interface•DocumentationR.Report•Description•Analysis•Results•Model validation•Visualization•Limitations•RecommendationsUsing the modeling diagram•The modeling diagram breaks the modeling task into16=6+10diﬀerent processes.•Each of the6nodes and each of the10edges deserve repeated attention,usually at every stage of the modeling process.9•The modeling is complete only if the’traﬃc’along all edges becomes insigniﬁcant.•Generally,working on an edge enriches both participating nodes.•If stuck along one edge,move to another one!Use the general rules below as a check list!•Frequently,the problem changes during modeling,in the light of the understanding gained by the modeling process.At the end,even a vague or contradictory initial problem description should have mutated into a reasonably well-deﬁned description, with an associated precisely deﬁned(though perhaps inaccurate)mathematical model.5General rules•Look at how others model similar situations;adapt their models to the present situa-tion.•Collect/ask for background information needed to understand the problem •Start with simple models;add details as they become known and useful or necessary.•Find all relevant quantities and make them precise.•Find all relevant relationships between quantities([diﬀerential]equations,inequalities, case distinctions).•Locate/collect/select the data needed to specify these relationships.•Find all restrictions that the quantities must obey(sign,limits,forbidden overlaps, etc.).Which restrictions are hard,which soft?How soft?•Try to incorporate qualitative constraints that rule out otherwise feasible results(usu-ally from inadequate previous versions).•Find all goals(including conﬂicting ones)•Play the devil’s advocate toﬁnd out and formulate the weak spots of your model.•Sort available information by the degree of impact expected/hoped for.•Create a hierarchy of models:from coarse,highly simplifying models to models with all known details.Are there useful toy models with simpler data?Are there limiting cases where the model simpliﬁes?Are there interesting extreme cases that help discover diﬃculties?10•First solve the coarser models(cheap but inaccurate)to get good starting points for theﬁner models(expensive to solve but realistic)•Try to have a simple working model(with report)after1/3of the total time planned for the e the remaining time for improving or expanding the model based on your experience,for making the programs more versatile and speeding them up,for polishing documentation,etc.•Good communication is essential for good applied work.•The responsibility for understanding,for asking the questions that lead to it,for recog-nizing misunderstanding(mismatch between answers expected and answers received), and for overcoming them lies with the mathematician.You cannot usually assume your customer to understand your scientiﬁc jargon.•Be not discouraged.Failures inform you about important missing details in your understanding of the problem(or the customer/boss)–utilize this information!•There are rarely perfect solutions.Modeling is the art ofﬁnding a satisfying compro-mise.Start with the highest standards,and lower them as the deadline approaches.If you have results early,raise your standards again.•Finish your work in time.Lao Tse:”People often fail on the verge of success;take care at the end as at the beginning, so that you may avoid failure.”6Conﬂicts•fast–slow•cheap–expensive•short term–long term•simplicity–complexity•low quality–high quality•approximate–accurate•superﬁcial–in depth•sketchy–comprehensive11•concise–detailed•short description–long descriptionEinstein:”A good theory”(or model)”should be as simple as possible,but not simpler.”•perfecting a program–need for quick results•collecting the theory–producing a solution•doing research–writing up•quality standards–deadlines•dreams–actual resultsThe conﬂicts described are creative and constructive,if one does not give in too easily.As a good material can handle more physical stress,so a good scientist can handle more stress created by conﬂict.”We shall overcome”–a successful motto of the black liberation movement,created by a strong trust in God.This generalizes to other situations where one has to face diﬃculties, too.Among other qualities it has,university education is not least a long term stress test–if you got your degree,this is a proof that you could overcome signiﬁcant barriers.The job market pays for the ability to persist.7Attitudes•Do whatever you do with love.Love(even in diﬃcult circumstances)can be learnt;it noticeably improves the quality of your work and the satisfaction you derive from it.•Do whatever you do as a service to others.This will improve your attention,the feedback you’ll get,and the impact you’ll have.•Take responsibility;ask if in doubt;read to conﬁrm your understanding.This will remove many impasses that otherwise would delay your work.Jesus:”Ask,and you will receive.Search,and you willﬁnd.Knock,and the door will be opened for you.”8ReferencesSee my home page,quoted on page1.12。

dSCFT Correspondence in Two Dimensions

Here we discuss the case of two-dimensional de Sitter space. The asymptotic boundary I−
is a circle, which upon a Wick rotation turns into time. rmal quantum mechanical model. We shall explicitly construct this model for the case of a scalar particle, obtain the generators of the conformal group, calculate the eigenvalues of the
then comment on what changes need to be made to turn φ periodic (ﬁnite temperature).
Consider a scalar ﬁeld Φ of mass m. It obeys the wave equation in de Sitter space
(14)
which diﬀers from [12] by a phase. The Green function for the modes Φ+k can be obtained from
G+(q, φ; q′, φ′) =
dk k
Φ+k (q,
−φ)Φ+k (q′
,
φ′).
(15)
After some algebra, we arrive at
The two-dimensional de Sitter space (dS2) may be parametrized as

统计学专业英语词汇

概率论与数理统计词汇英汉对照表Aabsolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.(degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值Ffactor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代Kkernal 核Kolmogorov-Smirnov test柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function 损失函数Mmain effect 主效应matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数model 模型Monte Carle method 蒙特卡罗法moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equation 多元回归方程multiple response 多响应multivariate analysis 多元分析Nnegative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设number of cases 个案数Oone-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分数percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数probit analysis 概率分析proportion 比例Qqadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitative 数量的，度量的quartiles 四分位数Rrandom 随机的random number 随机数random number 随机数random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank 秩rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Ssample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归stimulus 刺激strong assumption 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum 和sum of squares 平方和summary 概括，综述Ttable 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的vector 向量Wweight 加权，权重weighted average 加权平均值within groups 组内的ZZ score Z分数最优化方法词汇英汉对照表Aactive constraint 活动约束active set method 活动集法analytic gradient 解析梯度approximate 近似arbitrary 强制性的argument 变量attainment factor 达到因子Bbandwidth 带宽be equivalent to 等价于best-fit 最佳拟合bound 边界Ccoefficient 系数complex-value 复数值component 分量constant 常数constrained 有约束的constraint 约束constraint function 约束函数continuous 连续的converge 收敛cubic polynomial interpolation method 三次多项式插值法curve-fitting 曲线拟合Ddata-fitting 数据拟合default 默认的，默认的define 定义diagonal 对角的direct search method 直接搜索法direction of search 搜索方向discontinuous 不连续Eeigenvalue 特征值empty matrix 空矩阵equality 等式exceeded 溢出的Ffeasible 可行的feasible solution 可行解finite-difference 有限差分first-order 一阶GGauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient 梯度gradient method 梯度法Hhandle 句柄Hessian matrix 海色矩阵Iindependent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的initial feasible solution 初始可行解initialize 初始化inverse 逆invoke 激活iteration 迭代iteration 迭代JJacobian 雅可比矩阵LLagrange multiplier 拉格朗日乘子large-scale 大型的least square 最小二乘least squares sense 最小二乘意义上的Levenberg-Marquardt method列文伯格-马夸尔特法line search 一维搜索linear 线性的linear equality constraints 线性等式约束linear programming problem 线性规划问题local solution 局部解Mmedium-scale 中型的minimize 最小化mixed quadratic and cubic polynomial interpolation and extrapolation method 混合二次、三次多项式内插、外插法multiobjective 多目标的Nnonlinear 非线性的norm 范数Oobjective function 目标函数observed data 测量数据optimization routine 优化过程optimize 优化optimizer 求解器over-determined system 超定系统Pparameter 参数partial derivatives 偏导数polynomial interpolation method多项式插值法Qquadratic 二次的quadratic interpolation method 二次内插法quadratic programming 二次规划Rreal-value 实数值residuals 残差robust 稳健的robustness 稳健性，鲁棒性Sscalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method序列二次规划法simplex search method 单纯形法solution 解sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点step length 步长subspace trust region method 子空间置信域法sum-of-squares 平方和symmetric matrix 对称矩阵Ttermination message 终止信息termination tolerance 终止容限the exit condition 退出条件the method of steepest descent 最速下降法transpose 转置Uunconstrained 无约束的under-determined system 负定系统Vvariable 变量vector 矢量Wweighting matrix 加权矩阵样条词汇英汉对照表Aapproximation 逼近array 数组a spline in b-form/b-spline b样条a spline of polynomial piece /ppform spline分段多项式样条Bbivariate spline function 二元样条函数break/breaks 断点coefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条cubic spline interpolation三次样条插值/三次样条内插curve 曲线Ddegree of freedom 自由度dimension 维数Eend conditions 约束条件Iinput argument 输入参数interpolation 插值/内插interval 取值区间Kknot/knots 节点Lleast-squares approximation 最小二乘拟合Mmultiplicity 重次multivariate function 多元函数Ooptional argument 可选参数order 阶次output argument 输出参数Ppoint/points 数据点Rrational spline 有理样条rounding error 舍入误差（相对误差）Sscalar 标量sequence 数列（数组）spline 样条spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线spline interpolation 样条插值/样条内插spline surface 样条曲面smoothing spline 平滑样条Ttolerance 允许精度Uunivariate function 一元函数Vvector 向量Wweight/weights 权重4 偏微分方程数值解词汇英汉对照表Aabsolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格Bboundary condition 边界条件Ccontour plot 等值线图converge 收敛coordinate 坐标系Ddecomposed 分解的decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditionsDirichlet边界条件Eeigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解Ggeneralized Neumann boundary condition推广的Neumann边界条件geometry 几何形状geometry description matrix 几何描述矩阵geometry matrix 几何矩阵graphical user interface（GUI）图形用户界面Hhyperbolic 双曲线的Iinitial mesh 初始网格Jjiggle 微调LLagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值loop 循环Mmachine precision 机器精度mixed boundary condition 混合边界条件NNeuman boundary condition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量PParabolic 抛物线型的partial differential equation 偏微分方程plane strain 平面应变plane stress 平面应力Poisson's equation 泊松方程polygon 多边形positive definite 正定Qquality 质量Rrefined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual 残差residual norm 残差范数Ssingular 奇异的。

(完整word版)英文版概率论与数理统计重点单词

贝叶斯公式、逆概率公式
Bayes formula
后验概率
posterior probability
先验概率
prior probability
独立事件
independent event
独立随机事件
independent random event
独立实验
independent experiment
两两独立
classical probabilistic model
几何概率
geometric probability
乘法定理
product theorem
概率乘法
multiplication of probabilities
条件概率
conditional probability
全概率公式、全概率定理
formula of total probability
基本事件
elementary event
必然事件
certain event
不可能事件
impossible event
等可能事件
equally likely event
事件运算律
operational rules of events
事件的包含
implication of events
并事件
union events
似然方程
likelihood equation
似然函数
likelihood function
区间估计
interval estimation
置信区间
confidence interval
置信水平
confidence level

parameter estimation

Example 6 X1 , · · · , Xn ∼ Poisson(θ). Let T =
n i=1
Xi . Then, P (X n = xn and T = t) . P (T = t) if T (xn ) = t if T (X n ) = t
pX n |T (xn |t) = P(X n = xn |T (X n ) = t) = But
n P Xi t/n
]=
i
E [eXi (t/n) ]
α n
= [MX (t/n)] = This is the mgf of Γ(nα, β/n).
1 1 − βt/n
1 = 1 − β/nt
nα
.
Example 2 If X1 , . . . , Xn ∼ N (µ, σ 2 ) then X ∼ N (µ, σ 2 /n). Example 3 If X1 , . . . , Xn iid Cauchy(0,1), p(x) = for x ∈ R, then X ∼ Cauchy(0,1). Example 4 If X1 , . . . , Xn ∼ N (µ, σ 2 ) then (n − 1) 2 S ∼ χ2 (n−1) . 2 σ The proof is based on the mgf. 1 π (1 + x2 )
n
=
j =k n
P (exactly j of the X1 , . . . , Xn ≤ x) n [FX (x)]j [1 − FX (x)]n−j j
=
j =k
Diﬀerentiate to ﬁnd the pdf (See CB p. 229): pX(k) (x) = n! [FX (x)]k−1 p(x) [1 − FX (x)]n−k . (k − 1)!(n − k )!

《概率论与数理统计》基本名词中英文对照表

第九章参数估计Parameter'sestimation-

是相互矛盾的两个方面。
1
一、有关区间估计的几个概念 1. 置信区间：区间估计是求所谓置信区间的方法。置
信区间就是我们为了增加参数被估计到的信心而在点估计两边设置的估计区间。
2. 显著性水平：用置信区间来估计的不可靠程度。
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3. 置信度（水平）1 ：用置信区间估计的可靠性

在统计学中，常常用符号“S ” 来表示无偏估计量。数学上可以证明，对于随机样本而言S，2 才是
总体方差 2 的无偏估计量，它称为修正样本方差。
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4
[例]研究者要调查某社区居民家庭收入分布的差异情况，现随机抽查了10户，得到样本方差为＝200(元2)。试以此资料估计总体家庭收入分布的差异情况。
23
3.整群抽样
总体可看作是以群为单位的简单随机抽样。
群间方差（群内方差不进入）：
R
(Xi )2
2 i1
整群抽样平均误差：
R
(Xcl)

r
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第五节样本容量的确定
回置：
X
Z

n
不回置： p Z
pq n
n

Z 2

2 x
2
n

Z 2 pq
S n 1
＝52±2.064
12 ＝52±5.06 24
因此，置信水平95％的总体均值的置信区
间是从46.94到57.06。
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2. 大样本总体成数的估计
从总体的均值估计过渡到总体的成数估计，其方法和

思路完全相同，只要用 p代替 X

多元加权总体最小二乘新解法

多元加权总体最小二乘新解法汪奇生【摘要】将多元加权总体最小二乘模型进行变换,转化为加权总体最小二乘模型,推导构造新的系数矩阵和系数矩阵元素协因数阵的公式,研究多元加权总体最小二乘的解算流程.以Jazaeri加权总体最小二乘为例,给出多元总体最小二乘参数的解算过程.通过算例分析和比较,验证了该方法的有效性.%The new model of multivariate weighted total least squares by transposition processing,which is similar to the weighted total least squares model,is proposed in this paper.The formula for constructing the new coefficient matrix and its variance-covariance matrix is deduced and the solution flow multivariate weighted total least squares is studied.Applying this method,the solution process by the Jazaeri algorithm is deduced.The proposed method is proven to be effective and feasible through example analysis and comparison with other algorithms.【期刊名称】《大地测量与地球动力学》【年(卷),期】2017(037)012【总页数】5页(P1281-1284,1290)【关键词】多元加权总体最小二乘;EIV模型;参数估计;迭代算法【作者】汪奇生【作者单位】湖南软件职业学院,湘潭市开源路1号,411100【正文语种】中文【中图分类】P207近年来，总体最小二乘[1]在测量数据处理领域得到了众多学者的关注和研究。

Parameter estimation

EstimationEstimation: the process of inferring the value of a quantity of interest from indirect, inaccurate and uncertain observations. More rigorously, estimation can be viewed asthe process of selecting a point from a continuous space - the “best estimate”.In general, one can classify the variable that is to be estimated into the following two categories:• A parameter—a time-invariant quantity (a scalar, a vector, or a matrix)• The state of a dynamic system (usually a vector), which evolves in time according to a stochastic equationParameter EstimationBASIC CONCEPTS IN ESTIMATIONThe problem of parameter estimationParameter: used to designate a quantity (scalar or vector valued) that is assumed to be time-invariant.Given the measurementsz j=ℎj,x,w j j=1,2,⋯,kmade in the presence of the noises w j, find a function of the k observationsx k≜x k,Z kand the function is called the estimator.There are two models one can use in the estimation of a parameter:1.Nonrandom (“unknown constant”): There is an unknown true value x0.This is also called the non-Bayesian approach.2.Random: The parameter is a random variable with a prior pdf p(x)—a realization ofx according to p(x) is assumed to have occurred; this value then stays constant during the measurement process.This is also called the Bayesian approach.The Bayesian approachObtain posterior pdf of parameter using Bayes’ formula:p x Z =p Z x p xp Z=1cp Z x p xwhere c is the normalization constant, which does not depend on x .The non-Bayesian approachIn this case, one has the pdf of themeasurements conditioned on the parameter, called the likelihood function (LF) of the parameter:Λz x ≜p Z xorΛk x ≜p Z k xThe LF serves as a measure of the evidence from the data.Maximum Likelihood and Maximum A PosteriorestimatorsDefinition of ML estimatorsMaximum likelihood estimator (MLE):x ML Z=arg maxx Λz x=arg maxxp Z xThe MLE is the solution of the following equation:dΛz x dx =dp Z xdx=0Definition of MAP estimatorsMAP estimators:x MAP Z=arg maxxp x Z=arg maxxp Z x p xConsider the single measurementz=x+wwhere w~N0,σ2.First assume that x is an unknown constant (no prior information about it is available). The LF of x isΛx=p z x=N z;x,σ2=12πσe −z−x22σ2Thenx ML=arg maxxΛx=zNext assume that the prior information about the parameter is that x is Gaussian with mean x and variance σ02, that is,p x=N x;x ,σ02It is also assumed that x is independent of w.Then the posterior pdf of x conditioned on the observation z isp x z=p z x p xp z=1ce−z−x22σ2−x−x22σ02where c=2πσσ0p z.After rearranging the exponent in the previous page by completing the squares in x, the posterior pdf of x isp x z=N x;ξz,σ12=12πσ1e −x−ξz22σ12whereξz≜σ2σ02+σ2x +σ02σ02+σ2z=x +σ02σ02+σ2z−xσ12=2σ2σ02σ02+σ2∴x MAP=ξzx MAP=ξzNote that this MAP estimator for this (purely Gaussian) problem is a weighted combination of1. z, the MLE, which is the peak (or mode) of the LF;2. x , which is the peak of the prior pdf of the parameter to be estimated.MLE vs. MAP Estimator with Gaussian Prior Rewrite:x MAP=σ0−2+σ−2−1σ0−2x +σ0−2+σ−2−1σ−2z=σ0−2+σ−2−1xσ02+zσ2which indicates that the weightings of the prior mean and the measurement are inversely proportional to their variances.Similarly,σ1−2=σ0−2+σ−2which shows the inverse variances are additive.Consider the same problem as before except that the prior pdf of x is a one-sided exponential pdfp x=ae−ax x≥0The ML estimate is the same as beforex ML=zThe posterior pdf of x is nowp x z=c z e−z−x22σ2−ax x≥0where c z=a2πσp x . In view of the fact that it cannot be negative, then we have x MAP=max z−σ2a,0x ML=z x MAP=max z−σ2a,0Note that the MAP estimate in this case will always be smaller than the MLE as long as the latter is not negative because the prior attaches higher probability to smaller values of x.Assume a diffuse prior pdf for the parameterp x =ϵ for x <12ϵwhere ϵ>0 but small.p Z = p Z x p x ∞−∞dx =ϵp Z x 12ϵ−12ϵdx =ϵg Zwhere g does not depend on x .p x Z =p Z x p x p Z =p Z x ϵϵg Z =p Z xg Zsince ϵ≠0.p x Z=p Z x g ZA diffuse prior causes the posterior pdf of x to be proportional to its LF and, thus, the MAP estimate to coincide with the MLE.Bayesian vs. Non-Bayesian Philosophies:The non-Bayesian MLE is nothing but the Bayesian MAP estimate with complete prior ignorance, reflected by the diffuse prior. This provides a philosophically unifying view of the Bayesian and non-Bayesian approaches to estimation.Least Squares and Minimum Mean Square ErrorestimationDefinition of LS estimatorGiven the (scalar and nonlinear)measurementsz j=ℎj,x+w j j=1,⋯,kthe LS estimator(LSE) of x isx LS k=arg minxz j−ℎj,x2 kj=1If the function ℎ is linear in x, then one has the linear LS problem. Definition of MMSE estimator MMSE estimator:x MMSE Z=arg minxE x−x2|Z=E x Z=xp x Z dx∞−∞x LS k=arg minxz j−ℎj,x2 kj=1The above criterion makes no assumptions about the “measurement errors” or “noises” w(j). Ifw j~N0,σ2In this case,z j~Nℎj,x,σ2 j=1,⋯,kThe LF of x is thenΛk x ≜p Z k x =p z 1,z 2,⋯,z k x = N z j ;ℎj,x ,σ2kj=1=ce −12σ2 z j −ℎj,x 2k j=1Note that the LS method is a “disguised” ML approach.MMSE vs. MAP Estimator in Gaussian Noise In the single measurement example with a prior pdf on the parameter to be estimated, the posterior pdf of x was obtained asp x z=12πσ1e −x−ξz22σ12It is apparent that the mean of this Gaussian pdf is ξz, which is also the mode (peak) of this pdf. Thusx MMSE=E x Z=ξz=x MAPMain properties of the estimators1.Unbiasedness2.The Variance and MSE3.Consistency and efficiencyUnbiased estimatorsNon-Bayesian caseFor a nonrandom parameter, an estimator is said to be unbiased ifE x k,Z k=x0where x0 is the true value of the parameter. Bayesian caseIf x is a random variable with a prior pdf p(x), then the unbiasedness property is written asE x k,Z k=E xGeneral DefinitionThe above unbiasedness requirements can be unified by requiring that the estimation errorx≜x−xbe zero mean, that is, E x=0. An estimator is unbiased if holds for all k and is asymptotically unbiased if it holds in the limit as k→∞.Unbiasedness of an ML and a MAP Estimatorx ML=zE x ML=E z=E x0+w=x0+E w=x0x MAP=ξz≜σ2σ02+σ2x +σ02σ02+σ2zE x MAP=Eξz=σ2σ02+σ2x +σ02σ02+σ2E z=σ2σ02+σ2x +σ02σ02+σ2x +E w=x =E xNote that both of these estimates are unbiased.Non-Bayesian CaseFor a non-Bayesian estimator, x Z, (LS or ML) the variance of the estimator isvar x Z≜E x Z−E x Z2If this estimator is unbiased, that is, E x Z=x0, thenvar x Z=E x Z−x02If this estimator is biased, then the above formulation is its MSEMSE x Z=E x Z−x02Bayesian CaseFor a Bayesian estimator, the unconditional MSE isMSE x Z=E x Z−x2=E E x Z−x2|Z=E MSE x Z|Z For the MMSE estimator, the conditional MSE isE x MMSE Z−x2|Z=E x−E x Z2|Z=var x Z Averaging over Z yieldsE var x Z=E x−E x Z2which is the unconditional MSE of the estimate x MMSE.General definition:With the definition of the estimation errorx≜x−xone can say in a unified manner that the expected value of the square of the estimation error is the estimator’s variance or MSE:E x2=var x if x is unbiased and x is nonrandom MSE x in all casesComparison of Variances of an ML and a MAPEstimatorvar x ML=E x ML−x02=E z−x02≜σ2var x MAP=E x MAP−x2=Eσ2σ02+σ2x +σ02σ02+σ2x+w−x2=Eσ2σ02+σ2x −x+σ02σ02+σ2w2=σ02σ2σ02+σ2<σ2=var x MLIt can be seen that the variance of the MAP estimator is smaller than that of the MLE, this is due to the availability of prior information.Consistency of estimatorsNonrandom parameter case:limk→∞E x k,Z k−x02=0 is the condition for consistency in the mean square sense. Random parameter case: Similarly,limk→∞E x k,Z k−x2=0Consistency can be expressed as the requirement that the estimation error converge to zero, that islim k→∞x k,Z k=0in some stochastic (e.g., mean square) sense.Efficiency of estimators(scalar case)nonrandom parameter caseE x Z−x02≥J−1whereJ≜−E ð2lnΛxðx2x=x0=E ðlnΛxðx2x=x0random parameter caseE x Z−x2≥J−1whereJ≜−Eð2ln p Z,xðx2x=x0=Eðln p Z,xðx2x=x0If an estimator’s variance is equal to the CRLB, then such an estimator is called efficient.Efficiency of estimators(multidimensional case) For nonrandom vector parameters, the CRLB states that the covariance matrix of anunbiased estimator is bounded from below as follows:E x Z−x0x Z−x0′≥J−1J≜−E∇x∇x′lnΛxx=x0=E∇x lnΛx∇x lnΛx′x=x0A similar expression holds for the case of a multidimensional random parameter.Parameter EstimationLINEAR ESTIMATION IN STATIC SYSTEMSConsider two random vectors x and z that are jointly Gaussian distributed.Define the stacked vectory≜x zy~N y ,P yy wherey=xz ,P yy=P xx P xzP zx P zz,P xx=E x−x x−x′,P xz=E x−x z−z ′x MMSE≜E x z=x +P xz P zz−1z−zP xx|z≜E x−x x−x′|z=P xx−P xz P zz−1P zxx MMSE≜E x z=x +P xz P zz−1z−zThe MMSE estimate - the conditional mean - of a Gaussian random vector in terms of another Gaussian random vector (the measurement) is a linear combination of1.The prior (unconditional) mean of the variable to be estimated;2.The difference between the measurement and its prior mean.P xx|z≜E x−x x−x′|z=P xx−P xz P zz−1P zxThe conditional covariance of one Gaussian random vector given another Gaussian random vector (the measurement) is independent of the measurement.Linear MMSE Estimation for Zero-Mean Random VariablesThe linear MMSE estimator of a zero-mean random variable x in terms of z i ,i =1,⋯,n , is given by x = βi z i ni=1and has to be such that the norm of the estimation errorx ≜x −x is minimum.x 2=E x −x 2=E x − βi z i ni=12min x 2=E x −x 2=E x − βi z i n i=12−12ððβk x 2=E x − βi z i n i=1z k =E x z k =x ,z k =0 k =1,⋯,n The above formulation is equivalent to requiring the following orthogonal property:x ⊥z k ∀kPrinciple of orthogonality : In order for theerror to have minimum norm, it has to beorthogonal to the observations.Linear MMSE Estimation for Nonzero-Mean Random VariablesFor a random variable x with nonzero mean x , the best linear estimator is of the formx =β0+βi z i n i=1 Since E x 2=E x 2+var xin order to minimize it, the estimate should have the unbiasedness property: E x =0∴β0=x −βi z i n i=1where z i =E z i .x =x + βi z i −z i ni=1The error corresponding to the above estimate is x =x −x =x −x − βi z i −z i n i=1This has transformed the nonzero-mean case into the zero-mean case.Linear MMSE Estimation for Vector Random VariablesConsider the vector-valued random variables x and z, which are not necessarily Gaussian or zero-mean.The “best linear” estimate of x in terms of z is obtained as follows. Find the estimatorx=Az+bthat minimizes the scalar MSE criterion, which in the multidimensional case is the expected value of the squared norm of the estimation error,J=E x−x′x−xAccording to the previous discussion, the linear MMSE estimator is such that the estimation errorx=x−xis zero-mean (the estimate is unbiased) and orthogonal to the observation z.E x=x −Az +b=0⟹b=x −AzThe estimation error isx=x−x −A z−znow the orthogonality requirement can be written asE x z′=E x−x −A z−z z′=E x−x −A z−z z−z ′=P xz−AP zzThe solution for the weighting matrix A is thusA=P xz P zz−1The existence of the above requires the invertibility of P zz, i.e., no linear dependence between the measurement.Now we get the expression of the linear MMSE estimator for the multidimensional casex=x +P xz P zz−1z−zThe corresponding matrix MSEE x x′=P xx−P xz P zz−1P zx=P xx|zFundamental equations of linear estimationx=x +P xz P zz−1z−zE x x′=P xx−P xz P zz−1P zx=P xx|zFrom the above derivations, we can get•the best estimator (in the MMSE sense) for Gaussian random variablesis identical to•the best linear estimator for arbitrarily distributed random variables with the same 1st- and 2nd-order moments.The batch LS estimationFrom the linear observations z i =H i x +w i i =1,⋯,ksuch as to minimize the quadratic error J k = z i −H i x ′R i −1z i −H i x k i=1weighted with the inverses of the positive definite matrices R i .Rewrite in a compact form: J k =z k −H k x ′R k −1z k −H k xThe LS estimator is obtained by setting∇x J k=−2H k′R k−1z k−H k x=0which yieldsx k=H k′R k−1H k −1H k′R k−1z kNote that the above formulation is a batch estimator - the entire data have to be processed simultaneously for every k.Properties of the LS EstimatorWith the assumption that w(i) are uncorrelated, zero-mean random variables with covariance R(i), LS estimator is unbiased, that is,E x k =H k ′R k −1H k −1H k ′R k −1E H k x +w k =x The estimation error is x k =x −x k =−H k ′R k −1H k −1H k ′R k −1w kThus, the covariance matrix of the LS estimator is P k ≜E x k −E x k x k −E x k′=H k ′R k −1H k −1The Recursive LS EstimatorThe Recursion for the Inverse Covariance−1P k=H k′R k−1H kthe above formulation at k+1 can be expressed recursively asP k+1−1=P k−1+H k+1′R k+1−1H k+1This can be interpreted as follows:The information at k+1 equals the sum of the information at k and the new information about x obtained from z(k+1).P k+1=P k−1+H k+1′R k+1−1H k+1−1=P k−P k H k+1′H k+1P k H k+1′+R k+1−1H k+1P k The Residual Covariance and the Update GainDenote the matricesS k+1≜H k+1P k H k+1′+R k+1W k+1=P k H k+1′S k+1−1The Recursion for the Covariancecompact form: P k+1=I−W k+1H k+1P k, that isP k+1=P k−W k+1S k+1W k+1′Alternative Expression for the GainP k+1H k+1′R k+1−1=*P k H k+1′−P k H k+1′ ∙H k+1P k H k+1′+R k+1−1 ∙H k+1P k H k+1′+R k+1−1=P k H k+1′∙H k+1P k H k+1′+R k+1−1 ∙*H k+1P k H k+1′+R k+1−H k+1P k H k+1′+R k+1−1=P k H k+1′S k+1−1=W k+1∴W k+1=P k+1H k+1′R k+1−1The Recursion for the EstimateThe batch estimation equation for k+1x k+1=H k+1′R k+1−1H k+1−1H k+1′R k+1−1z k+1Rewrite:x k+1=P k+1H k+1′R k+1−1z k+1=P k+1H k′H k+1′R k00R k+1−1z kz k+1=P k+1H k′R k−1z k+P k+1H k+1′R k+1−1z k+1 =I−W k+1H k+1P k H k′R k−1z k+W k+1z k+1 =I−W k+1H k+1x k+W k+1z k+1Recursive parameter estimate updating equationx k+1=x k+W k+1z k+1−H k+1x k。

参数估计的评价方法

参数估计的评价方法英文回答：Evaluation Methods for Parameter Estimation.Parameter estimation plays a vital role in statistics and machine learning, as it provides insights into the underlying relationships within data. To ensure the reliability and accuracy of estimated parameters, it is essential to evaluate their performance. Several methods are commonly employed for this purpose:1. Mean Squared Error (MSE): This metric measures the average squared difference between the estimated parameter and its true value. A lower MSE indicates better estimation accuracy.2. Root Mean Squared Error (RMSE): The square root of the MSE provides a more interpretable measure of estimation error in the same units as the data.3. Mean Absolute Error (MAE): This metric calculates the average absolute difference between the estimated parameter and its true value. It is less sensitive to outliers than MSE and RMSE.4. Bias: Bias measures the systematic error in the parameter estimation. A biased estimator consistently overestimates or underestimates the true value. Bias can be estimated by comparing the mean of the estimated parameters to the true value.5. Variance: Variance quantifies the variability in the parameter estimates. A higher variance indicates that the estimates are more dispersed around the true value. Variance can be estimated by calculating the sample variance of the estimated parameters.6. Mean Average Percentage Error (MAPE): This metric calculates the average percentage difference between the estimated parameter and its true value. It is particularly useful when comparing estimates of different scales.7. Akaike Information Criterion (AIC): AIC is a statistical measure that combines the goodness of fit of a model with the model's complexity. A lower AIC indicates a better balance between accuracy and simplicity.8. Bayesian Information Criterion (BIC): Similar to AIC, BIC is a Bayesian measure that penalizes model complexity more strongly. It can be used to select the best model among several competing models.9. Cross-Validation: This technique divides the datainto multiple subsets (folds) and iteratively trains and evaluates the model on different combinations of these folds. Cross-validation provides a more reliable estimateof the model's performance on unseen data.10. Holdout: This approach reserves a portion of the data as a holdout set, which is not used for training the model. The holdout set is then used to evaluate the model's predictive performance.中文回答：参数估计的评价方法。

EP7.0 欧洲药典 2.9.38. Particle-size distribution 粒度分布_中文_翻译

2.9.38. 筛分法评估颗粒度大小分布(17)PARTICLE-SIZE DISTRIBUTION ESTIMATION BY ANALYTICAL SIEVING(17)筛分是对粉末和颗粒通过颗粒大小分布进行分级的最老的方法之一。

当使用织物筛布时，筛分基本上依据筛网的中间尺寸大小（即幅度或宽度）对颗粒进行分类。

如果大部分的颗粒大于75微米左右，采用机筛法是最合适的。

对于更小的颗粒，由于质量轻，在筛分过程中，它们不能克服表面凝聚力和粘附力，使颗粒相互粘在一起或粘在筛网上，从而导致那些本该能通过筛孔的颗粒而未能通过。

对于这样的材料，其它的振动方式，如喷气筛分或声波筛分器筛分可能更为合适。

然而，筛分有时可被用于一些中值颗粒尺寸小于75微米的粉末或颗粒，只要该方法经过验证。

在制药行业，对单一粉末或颗粒进行粗略等级分类时，筛分通常是被选择的方法。

这是一个尤其引人注目的方法，因为该方法仅依据粒径大小对粉末和颗粒进行颗粒大小分级，在大多数情况下，干粉状态下即可进行检测。

Sieving is one of the oldest methods of classifying powders and granules by particle-size distribution. When using a woven sieve cloth, the sieving will essentially sort the particles by their intermediate size dimension (i.e. breadth or width). Mechanical sieving is most suitable where the majority of the particles are larger than about 75 µm. For smaller particles, their light weight provides insufficient force during sieving to overcome the surface forces of cohesion and adhesion that cause the particles to stick to each other and to the sieve, and thus cause particles that would be expected to pass through the sieve to be retained. For such materials other means of agitation such as air-jet sieving or sonic-sifter sieving may be more appropriate. Nevertheless, sieving can sometimes be used for some powders or granules having median particle sizes smaller than 75 µm where the method can be validated. In pharmaceutical terms, sieving is usually the method of choice for classification of the coarser grades of single powders or granules. It is a particularly attractive method in that powders and granules are classified only on the basis of particle size, and in most cases the analysis can be carried out in the dry state.筛分方法的局限性包括：需要可观质量的样品（通常至少为25克，这取决于粉末或颗粒的密度和试验筛的直径）；以及在筛分那些容易堵塞筛孔的油性或有粘着力的粉末或颗粒时存在困难。

数学专业词汇及翻译

一、字母顺序表 (1)二、常用的数学英语表述 (7)三、代数英语（高端） (13)一、字母顺序表1、数学专业词汇Aabsolute value 绝对值 accept 接受 acceptable region 接受域additivity 可加性 adjusted 调整的 alternative hypothesis 对立假设analysis 分析 analysis of covariance 协方差分析 analysis of variance 方差分析 arithmetic mean 算术平均值 association 相关性 assumption 假设 assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的 band 带宽 bar chart 条形图beta-distribution 贝塔分布 between groups 组间的 bias 偏倚 binomial distribution 二项分布 binomial test 二项检验Ccalculate 计算 case 个案 category 类别 center of gravity 重心 central tendency 中心趋势 chi-square distribution 卡方分布 chi-square test 卡方检验 classify 分类cluster analysis 聚类分析 coefficient 系数 coefficient of correlation 相关系数collinearity 共线性 column 列 compare 比较 comparison 对照 components 构成，分量compound 复合的 confidence interval 置信区间 consistency 一致性 constant 常数continuous variable 连续变量 control charts 控制图 correlation 相关 covariance 协方差 covariance matrix 协方差矩阵 critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的 cubic term 三次项 cumulative distribution function 累加分布函数 curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计 deviations 差异 df.(degree of freedom) 自由度 diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等 effects of interaction 交互效应 efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布 extreme value 极值F factor 因素，因子 factor analysis 因子分析 factor score 因子得分 factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值 fixed model 固定模型 fixed variable 固定变量 fractional factorial design 部分析因设计 frequency 频数 F-test F检验 full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布 geometric mean 几何均值 group 组Hharmomic mean 调和均值 heterogeneity 不齐性histogram 直方图 homogeneity 齐性homogeneity of variance 方差齐性 hypothesis 假设 hypothesis test 假设检验Iindependence 独立 independent variable 自变量independent-samples 独立样本 index 指数 index of correlation 相关指数 interaction 交互作用 interclass correlation 组内相关 interval estimate 区间估计 intraclass correlation 组间相关 inverse 倒数的iterate 迭代Kkernal 核 Kolmogorov-Smirnov test柯尔莫哥洛夫-斯米诺夫检验 kurtosis 峰度Llarge sample problem 大样本问题 layer 层least-significant difference 最小显著差数 least-square estimation 最小二乘估计 least-square method 最小二乘法 level 水平 level of significance 显著性水平 leverage value 中心化杠杆值 life 寿命 life test 寿命试验 likelihood function 似然函数 likelihood ratio test 似然比检验linear 线性的 linear estimator 线性估计linear model 线性模型 linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数 logistic 逻辑的 lost function 损失函数Mmain effect 主效应 matrix 矩阵 maximum 最大值 maximum likelihood estimation 极大似然估计 mean squared deviation(MSD) 均方差 mean sum of square 均方和 measure 衡量 media 中位数 M-estimator M估计minimum 最小值 missing values 缺失值 mixed model 混合模型 mode 众数model 模型Monte Carle method 蒙特卡罗法 moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较 multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数 multiple regression analysis 多元回归分析multiple regression equation 多元回归方程 multiple response 多响应 multivariate analysis 多元分析Nnegative relationship 负相关 nonadditively 不可加性 nonlinear 非线性 nonlinear regression 非线性回归 noparametric tests 非参数检验 normal distribution 正态分布null hypothesis 零假设 number of cases 个案数Oone-sample 单样本 one-tailed test 单侧检验 one-way ANOVA 单向方差分析 one-way classification 单向分类 optimal 优化的optimum allocation 最优配制 order 排序order statistics 次序统计量 origin 原点orthogonal 正交的 outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计 partial correlation 偏相关partial correlation coefficient 偏相关系数 partial regression coefficient 偏回归系数 percent 百分数percentiles 百分位数 pie chart 饼图 point estimate 点估计 poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关 power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析 proability 概率 probability density function 概率密度函数 probit analysis 概率分析 proportion 比例Qqadratic 二次的 Q-Q plot Q-Q概率图 quadratic term 二次项 quality control 质量控制 quantitative 数量的，度量的 quartiles 四分位数Rrandom 随机的 random number 随机数 random number 随机数 random sampling 随机取样random seed 随机数种子 random variable 随机变量 randomization 随机化 range 极差rank 秩 rank correlation 秩相关 rank statistic 秩统计量 regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域 relationship 关系 reliability 可*性 repeated 重复的report 报告，报表 residual 残差 residual sum of squares 剩余平方和 response 响应risk function 风险函数 robustness 稳健性 root mean square 标准差 row 行 run 游程run test 游程检验Sample 样本 sample size 样本容量 sample space 样本空间 sampling 取样 sampling inspection 抽样检验 scatter chart 散点图 S-curve S形曲线 separately 单独地 sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验 significant 显著的，有效的 significant digits 有效数字 skewed distribution 偏态分布 skewness 偏度 small sample problem 小样本问题 smooth 平滑 sort 排序 soruces of variation 方差来源 space 空间 spread 扩展square 平方 standard deviation 标准离差 standard error of mean 均值的标准误差standardization 标准化 standardize 标准化 statistic 统计量 statistical quality control 统计质量控制 std. residual 标准残差 stepwise regression analysis 逐步回归 stimulus 刺激 strong assumption 强假设 stud. deleted residual 学生化剔除残差stud. residual 学生化残差 subsamples 次级样本 sufficient statistic 充分统计量sum 和 sum of squares 平方和 summary 概括，综述Ttable 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验 test of goodness of fit 拟合优度检验 test of homogeneity 齐性检验 test of independence 独立性检验 test rules 检验法则 test statistics 检验统计量 testing function 检验函数 time series 时间序列 tolerance limits 容许限total 总共，和 transformation 转换 treatment 处理 trimmed mean 截尾均值 true value 真值 t-test t检验 two-tailed test 双侧检验Uunbalanced 不平衡的 unbiased estimation 无偏估计 unbiasedness 无偏性 uniform distribution 均匀分布Vvalue of estimator 估计值 variable 变量 variance 方差 variance components 方差分量 variance ratio 方差比 various 不同的 vector 向量Wweight 加权，权重 weighted average 加权平均值 within groups 组内的ZZ score Z分数2. 最优化方法词汇英汉对照表Aactive constraint 活动约束 active set method 活动集法 analytic gradient 解析梯度approximate 近似 arbitrary 强制性的 argument 变量 attainment factor 达到因子Bbandwidth 带宽 be equivalent to 等价于 best-fit 最佳拟合 bound 边界Ccoefficient 系数 complex-value 复数值 component 分量 constant 常数 constrained 有约束的constraint 约束constraint function 约束函数continuous 连续的converge 收敛 cubic polynomial interpolation method三次多项式插值法 curve-fitting 曲线拟合Ddata-fitting 数据拟合 default 默认的，默认的 define 定义 diagonal 对角的 direct search method 直接搜索法 direction of search 搜索方向 discontinuous 不连续Eeigenvalue 特征值 empty matrix 空矩阵 equality 等式 exceeded 溢出的Ffeasible 可行的 feasible solution 可行解 finite-difference 有限差分 first-order 一阶GGauss-Newton method 高斯-牛顿法 goal attainment problem 目标达到问题 gradient 梯度 gradient method 梯度法Hhandle 句柄 Hessian matrix 海色矩阵Independent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的initial feasible solution 初始可行解initialize 初始化inverse 逆 invoke 激活 iteration 迭代 iteration 迭代JJacobian 雅可比矩阵LLagrange multiplier 拉格朗日乘子 large-scale 大型的 least square 最小二乘 least squares sense 最小二乘意义上的 Levenberg-Marquardt method 列文伯格-马夸尔特法line search 一维搜索 linear 线性的 linear equality constraints 线性等式约束linear programming problem 线性规划问题 local solution 局部解M medium-scale 中型的 minimize 最小化 mixed quadratic and cubic polynomialinterpolation and extrapolation method 混合二次、三次多项式内插、外插法multiobjective 多目标的Nnonlinear 非线性的 norm 范数Oobjective function 目标函数 observed data 测量数据 optimization routine 优化过程optimize 优化 optimizer 求解器 over-determined system 超定系统Pparameter 参数 partial derivatives 偏导数 polynomial interpolation method 多项式插值法Qquadratic 二次的 quadratic interpolation method 二次内插法 quadratic programming 二次规划Rreal-value 实数值 residuals 残差 robust 稳健的 robustness 稳健性，鲁棒性S scalar 标量 semi-infinitely problem 半无限问题 Sequential Quadratic Programming method 序列二次规划法 simplex search method 单纯形法 solution 解 sparse matrix 稀疏矩阵 sparsity pattern 稀疏模式 sparsity structure 稀疏结构 starting point 初始点 step length 步长 subspace trust region method 子空间置信域法 sum-of-squares 平方和 symmetric matrix 对称矩阵Ttermination message 终止信息 termination tolerance 终止容限 the exit condition 退出条件 the method of steepest descent 最速下降法 transpose 转置Uunconstrained 无约束的 under-determined system 负定系统Vvariable 变量 vector 矢量Wweighting matrix 加权矩阵3 样条词汇英汉对照表Aapproximation 逼近 array 数组 a spline in b-form/b-spline b样条 a spline of polynomial piece /ppform spline 分段多项式样条Bbivariate spline function 二元样条函数 break/breaks 断点Ccoefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式 cubic smoothing spline 三次平滑样条 cubic spline 三次样条cubic spline interpolation 三次样条插值/三次样条内插 curve 曲线Ddegree of freedom 自由度 dimension 维数Eend conditions 约束条件 input argument 输入参数 interpolation 插值/内插 interval取值区间Kknot/knots 节点Lleast-squares approximation 最小二乘拟合Mmultiplicity 重次 multivariate function 多元函数Ooptional argument 可选参数 order 阶次 output argument 输出参数P point/points 数据点Rrational spline 有理样条 rounding error 舍入误差（相对误差）Sscalar 标量 sequence 数列（数组） spline 样条 spline approximation 样条逼近/样条拟合spline function 样条函数 spline curve 样条曲线 spline interpolation 样条插值/样条内插 spline surface 样条曲面 smoothing spline 平滑样条Ttolerance 允许精度Uunivariate function 一元函数Vvector 向量Wweight/weights 权重4 偏微分方程数值解词汇英汉对照表Aabsolute error 绝对误差 absolute tolerance 绝对容限 adaptive mesh 适应性网格Bboundary condition 边界条件Ccontour plot 等值线图 converge 收敛 coordinate 坐标系Ddecomposed 分解的 decomposed geometry matrix 分解几何矩阵 diagonal matrix 对角矩阵 Dirichlet boundary conditions Dirichlet边界条件Eeigenvalue 特征值 elliptic 椭圆形的 error estimate 误差估计 exact solution 精确解Ggeneralized Neumann boundary condition 推广的Neumann边界条件 geometry 几何形状geometry description matrix 几何描述矩阵 geometry matrix 几何矩阵 graphical user interface（GUI）图形用户界面Hhyperbolic 双曲线的Iinitial mesh 初始网格Jjiggle 微调LLagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值 loop 循环Mmachine precision 机器精度 mixed boundary condition 混合边界条件NNeuman boundary condition Neuman边界条件 node point 节点 nonlinear solver 非线性求解器 normal vector 法向量PParabolic 抛物线型的 partial differential equation 偏微分方程 plane strain 平面应变 plane stress 平面应力 Poisson's equation 泊松方程 polygon 多边形 positive definite 正定Qquality 质量Rrefined triangular mesh 加密的三角形网格 relative tolerance 相对容限 relative tolerance 相对容限 residual 残差 residual norm 残差范数Ssingular 奇异的二、常用的数学英语表述1.Logic∃there exist∀for allp⇒q p implies q / if p, then qp⇔q p if and only if q /p is equivalent to q / p and q are equivalent2.Setsx∈A x belongs to A / x is an element (or a member) of Ax∉A x does not belong to A / x is not an element (or a member) of AA⊂B A is contained in B / A is a subset of BA⊃B A contains B / B is a subset of AA∩B A cap B / A meet B / A intersection BA∪B A cup B / A join B / A union BA\B A minus B / the diference between A and BA×B A cross B / the cartesian product of A and B3. Real numbersx+1 x plus onex-1 x minus onex±1 x plus or minus onexy xy / x multiplied by y(x - y)(x + y) x minus y, x plus yx y x over y= the equals signx = 5 x equals 5 / x is equal to 5x≠5x (is) not equal to 5x≡y x is equivalent to (or identical with) yx ≡ y x is not equivalent to (or identical with) yx > y x is greater than yx≥y x is greater than or equal to yx < y x is less than yx≤y x is less than or equal to y0 < x < 1 zero is less than x is less than 10≤x≤1zero is less than or equal to x is less than or equal to 1| x | mod x / modulus xx 2 x squared / x (raised) to the power 2x 3 x cubedx 4 x to the fourth / x to the power fourx n x to the nth / x to the power nx −n x to the (power) minus nx (square) root x / the square root of xx 3 cube root (of) xx 4 fourth root (of) xx n nth root (of) x( x+y ) 2 x plus y all squared( x y ) 2 x over y all squaredn! n factorialx ^ x hatx ¯ x barx ˜x tildex i xi / x subscript i / x suffix i / x sub i∑ i=1 n a i the sum from i equals one to n a i / the sum as i runs from 1 to n of the a i4. Linear algebra‖ x ‖the norm (or modulus) of xOA →OA / vector OAOA ¯ OA / the length of the segment OAA T A transpose / the transpose of AA −1 A inverse / the inverse of A5. Functionsf( x ) fx / f of x / the function f of xf:S→T a function f from S to Tx→y x maps to y / x is sent (or mapped) to yf'( x ) f prime x / f dash x / the (first) derivative of f with respect to xf''( x ) f double-prime x / f double-dash x / the second derivative of f with r espect to xf'''( x ) triple-prime x / f triple-dash x / the third derivative of f with respect to xf (4) ( x ) f four x / the fourth derivative of f with respect to x∂f ∂ x 1the partial (derivative) of f with respect to x1∂ 2 f ∂ x 1 2the second partial (derivative) of f with respect to x1∫ 0 ∞the integral from zero to infinitylim⁡x→0 the limit as x approaches zerolim⁡x→0 + the limit as x approaches zero from abovelim⁡x→0 −the limit as x approaches zero from belowlog e y log y to the base e / log to the base e of y / natural log (of) yln⁡y log y to the base e / log to the base e of y / natural log (of) y一般词汇数学mathematics, maths(BrE), math(AmE)公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.)被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.)被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.)被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.)被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to 大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor 等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth power n次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth root n次方根the root of n, the nth root集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.)无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral有理数rational number无理数irrational number实数real number虚数imaginary number复数complex number矩阵matrix行列式determinant几何geometry点point线line面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width附：在一个分数里，分子或分母或两者均含有分数。

Parameter Estimation Problem

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Adaptive Internet Intelligent Agent Lab
機器學習與網路代理人實驗室
Triple Jump Framework (1)
In iteration t, TJ selects the first candidate as θ (t) that satisfies L(θ(t) ) − L(θ(t−1) ) ≥ δ .
Parameterized EM (pEM)
Repeat (in iteration t)
θ (t) = Mη (θ (t−1) )
Until L(θ(t) ) − L(θ (t−1) ) < δ Likelihood increases monotonically in the neighborhood of θ∗ if 0 < η < 2 [Bauer et al. 1997] . pEM with η = 1 is EM. Local maximum: θ ∗ = M (θ ∗ ) = Mη (θ ∗ )
L(θ(t) ) ≥ L(θ(t−1) )
θ∗ = M (θ∗ )
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Adaptive Internet Intelligent Agent Lab
機器學習與網路代理人實驗室
Taylor Expansion of M
In the neighbor of θ∗ , we apply Taylor expansion to M [Dempster et al. 1977] :
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Adaptive Internet Intelligent Agent Lab
機器學習與網路代理人實驗室
Aitken’s Acceleration for EM (1)

二参数估计PPT课件

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2.1.1 无偏估计与渐进无偏估计
参数估计：由N个样本数据推出p个参数的值。

parameter词根

parameter的词根parameter是一个常用的英语单词，它在数学、物理、统计等领域有着重要的意义。

parameter的词根是para-和meter，意思是“在旁边”和“测量”。

parameter有多种同义词和反义词，也有多种派生词和复合词。

parameter的发音是pəˈræmɪtə(r)，它的复数形式是parameters。

parameter的来源parameter这个单词是由两个词根组成的：para-和meter。

para-有多种含义，其中一个是“在旁边”，“在周围”，“与……相比”。

例如：parallel：平行的，相同的paradox：悖论，自相矛盾的事物paragon：典范，模范parasite：寄生虫，寄生生物meter在是“测量”的意思。

例如：meter：米，计量器diameter：直径perimeter：周长thermometer：温度计parameter这个单词就是由para-和meter组合而成的，它最初是用来指代一种数学上的常量，表示一组曲线或方程的特征。

例如：y = ax + b 这个线性方程中，a和b就是参数，它们决定了直线的斜率和截距。

parameter的含义parameter这个单词后来逐渐扩展到其他领域，表示一种决定或限制某个系统或过程的因素或变量。

例如：在物理学中，parameter可以指代一个物理量，如温度、压力、密度等。

在统计学中，parameter可以指代一个描述总体特征的量，如均值、方差、相关系数等。

在计算机科学中，parameter可以指代一个传递给函数或程序的变量或值。

在语言学中，parameter可以指代一个影响语言结构或表达方式的规则或选项。

parameter这个单词还可以表示一种特征、元素、因素或特点。

例如：some of the parameters that determine the taste of a wine 决定葡萄酒口味的一些元素parameter这个单词还可以表示一种限制、范围、界限或边界。

第四讲 (3)参数反演-文档资料

第四讲（3）参数反演

采用参数反演的方法，会更快更容易得到含水层参数值。MODFLOW包含三个参数反演模块—MODFLOW2000PES、PEST和UCODE，其中MODFLOW2000PES为MODFLOW本身自有模块，PEST和UCODE为独立模块，其计算理论可参考有关指南。参数反演模块能够自动迭代运算，通过自动调整参数，达到计算值与观测值差的最小化。
第五步设定迭代次数第六步：保存、运行、观察2.试点法（PEST法）

Model-Independent Parameter Estimation
参数区划法对所分的区给出同一个值，在分区界限出现参数突变的现象，试点法采用一系列2D散点，通过插值的方法得到每个单元的参数值，反演的结果是参数值在计算地域范围内渐变，各处不一样。由于PEST 模块提供了一个附加调整模块，可以强制进行插值反演计算，在观测点少于散点情况下，也可以进行数值反演，所以定义较少的散点，也可以反演渗透系数变化复杂的情况，故散点法应用PEST计算引擎更适合。
1.分区法

分区法：设定各区参数一致，在界面处突变给每个区设定一个指针参数，是小于0 的整数，其实就是代号反演前要设定参数变化范围、初始值
例题：首先打开一个文件，有观测井
第二步：设定代号
第三步：参数选择设定

选择计算类型：
第四步：编辑参数

现在需要输入所要反演参数的最大值、最小值和初值。执行MODFLOW|Parameters命令