(1+1)-dimensional separation of variables

微积分专有名词中英文对照absolutely convergent 绝对收敛absolute value 绝对值algebraic function 代数函数analytic geometry 解析几何antiderivative 不定积分approximate integration 近似积分approximation 近似法、逼近法arbitrary constant 任意常数arithmetic series/progression (AP)算数级数asymptotes (vertical and horizontal)(垂直/水平)渐近线average rate of change 平均变化率base 基数binomial theorem 二项式定理，二项展开式Cartesian coordinates 笛卡儿坐标(一般指直角坐标) Cartesian coordinates system 笛卡儿坐标系Cauch’s Mean Value Theorem 柯西均值定理chain rule 链式求导法则calculus 微积分学closed interval integral 闭区间积分coefficient 系数composite function 复合函数conchoid 蚌线continuity (函数的)连续性concavity (函数的)凹凸性conditionally convergent 有条件收敛continuity 连续性critical point 临界点cubic function 三次函数cylindrical coordinates 圆柱坐标decreasing function 递减函数decreasing sequence 递减数列definite integral 定积分derivative 导数determinant 行列式differential coefficient 微分系数differential equation 微分方程directional derivative 方向导数discontinuity 不连续性discriminant (二次函数)判别式disk method 圆盘法divergence 散度divergent 发散的domain 定义域dot product 点积double integral 二重积分ellipse 椭圆ellipsoid 椭圆体epicycloid 外摆线Euler's method (BC)欧拉法expected valued 期望值exponential function 指数函数extreme value heorem 极值定理factorial 阶乘finite series 有限级数fundamental theorem of calculus 微积分基本定理geometric series/progression (GP)几何级数gradient 梯度Green formula 格林公式half-angle formulas 半角公式harmonic series 调和级数helix 螺旋线higher derivative 高阶导数horizontal asymptote 水平渐近线horizontal line 水平线hyperbola 双曲线hyper boloid 双曲面implicit differentiation 隐函数求导implicit function 隐函数improper integral 广义积分、瑕积分increment 增量increasing function 增函数indefinite integral 不定积分independent variable 自变数inequality 不等式ndeterminate form 不定型infinite point 无穷极限infinite series 无穷级数infinite series 无限级数inflection point (POI) 拐点initial condition 初始条件instantaneous rate of change 瞬时变化率integrable 可积的integral 积分integrand 被积分式integration 积分integration by part 分部积分法intercept 截距intermediate value of Theorem ：中间值定理inverse function 反函数irrational function 无理函数iterated integral 逐次积分Laplace transform 拉普拉斯变换law of cosines 余弦定理least upper bound 最小上界left-hand derivative 左导数left-hand limit 左极限L'Hospital's rule 洛必达法则limacon 蚶线linear approximation 线性近似法linear equation 线性方程式linear function 线性函数linearity 线性linearization 线性化local maximum 极大值local minimum 极小值logarithmic function 对数函数MacLaurin series 麦克劳林级数maximum 最大值mean value theorem (MVT)中值定理minimum 最小值method of lagrange multipliers 拉格朗日乘数法modulus 绝对值multiple integral 多重积分multiple 倍数multiplier 乘子octant 卦限open interval integral 开区间积分optimization 优化法，极值法origin 原点orthogonal 正交parametric equation (BC)参数方程partial derivative 偏导数partial differential equation 偏微分方程partial fractions 部分分式piece-wise function 分段函数parabola 抛物线parabolic cylinder 抛物柱面paraboloid ：抛物面parallelepiped 平行六面体parallel lines 并行线parameter ：参数partial integration 部分积分partiton ：分割period ：周期periodic function 周期函数perpendicular lines 垂直线piecewise defined function 分段定义函数plane 平面point of inflection 反曲点point-slope form 点斜式polar axis 极轴polar coordinates 极坐标polar equation 极坐标方程pole 极点polynomial 多项式power series 幂级数product rule 积的求导法则quadrant 象限quadratic functions 二次函数quotient rule 商的求导法则radical 根式radius of convergence 收敛半径range 值域(related) rate of change with time (时间)变化率rational function 有理函数reciprocal 倒数remainder theorem 余数定理Riemann sum 黎曼和Riemannian geometry 黎曼几何right-hand limit 右极限Rolle's theorem 罗尔(中值)定理root 根rotation 旋转secant line 割线second derivative 二阶导数second derivative test 二阶导数试验法second partial derivative 二阶偏导数series 级数shell method (积分)圆筒法sine function 正弦函数singularity 奇点slant 母线slant asymptote 斜渐近线slope 斜率slope-intercept equation of a line 直线的斜截式smooth curve 平滑曲线smooth surface 平滑曲面solid of revolution 旋转体symmetry 对称性substitution 代入法、变量代换tangent function 正切函数tangent line 切线tangent plane 切(平)面tangent vector 切矢量taylor's series 泰勒级数three-dimensional analytic geometry 空间解析几何total differentiation 全微分trapezoid rule 梯形(积分)法则。

SPSS（统计）名词解释2007-11-13 16:29:16| 分类：学习| 标签：|举报|字号大中小订阅Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP 统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 通用线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数标准Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换文案。

Abel transformFrom Wikipedia, the free encyclopediaJump to: navigation, searchFor summation transformation, see summation by parts.In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by:Assuming f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given byIn image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the reverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center.In recent years, the inverse Abel transformation (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transformation are the Onion Peeling and BAsis Set Expansion (BASEX) methods of photoelectron and photoion image analysis.Contents∙ 1 Geometrical interpretation∙ 2 Verification of the inverse Abel transform∙ 3 Generalization of the Abel transform to discontinuous F(y)∙ 4 Relationship to other integral transformso 4.1 Relationship to the Fourier and Hankel transformso 4.2 Relationship to the Radon transform∙ 5 References∙ 6 External linksGeometrical interpretationA geometrical interpretation of the Abel transform in two dimensions. An observer (I) looks along a line parallel to the x-axis a distance y above the origin. What the observer sees is the projection (i.e. the integral) of the circularly symmetric function f(r) along the line of sight. The function f(r) is represented in gray in this figure. The observer is assumed to be located infinitely far from the origin so that the limits of integration are ±∞In two dimensions, the Abel transform F(y) can be interpreted as the projection of a circularly symmetric function f(r) along a set of parallel lines of sight which are a distance y from the origin. Referring to the figure on the right, the observer (I) will seewhere f(r) is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at x = ∞ so that the limits of integration are ±∞ and all lines of sight are parallel to the x-axis. Realizing that the radius r is related to x and y via r2 = x2 + y2, it follows thatThe path of integration in r does not pass through zero, and since both f(r) and the above expression for d x are even functions, we may write:Substituting the expression for dx in terms of r and rewriting the integration limits accordingly yields the Abel transform.The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function f(ρ,z) where ρ2 = x2 + y2is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the z axis. Without loss of generality, we can take that plane to be the yz-plane so thatwhich is just the Abel transform of f(ρ,z) in ρ and y.A particular type of axial symmetry is spherical symmetry. In this case, we have a function f(r) where r2 = x2 + y2 + z2. The projection onto, say, the yz-plane will then be circularly symmetric and expressible as F(s) where s2 = y2 + z2. Carrying out the integration, we have:which is again, the Abel transform of f(r) in r and s. Verification of the inverse Abel transform Assuming f is continuously differentiable and f, f' drop to zero faster than 1/r, we can set u = f(r) and . Integration by parts then yieldsDifferentiating formally,Now plug this into the inverse Abel transform formula:1r r y π∞∞∞'-=()f s dsdy 'By Fubini's theorem, the last integral equals()(1)()()r y rs ds f s ds f r ∞∞∞''=-=⎰⎰⎰ Generalization of the Abel transform todiscontinuous F (y )Consider the case where F (y ) is discontinuous at y = y Δ, where it abruptly changes its value by a finite amount ΔF . That is, y Δ and ΔF are defined by . Such a situation is encountered in tethered polymers (Polymer brush ) exhibiting a vertical phase separation, where F (y ) stands for the polymer density profile and f (r ) is related to the spatial distribution of terminal, non-tethered monomers of the polymers.The Abel transform of a function f (r ) is under these circumstances again given by:Assuming f (r ) drops to zero more quickly than 1/r , the inverse Abel transform is however given bywhere δ is the Dirac delta function and H (x ) the Heaviside step function . The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous F (y ), and it reduces to the classical Abel transform when ΔF = 0. If F (y ) has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform whichcontains n additional terms, each of them corresponding to one of the n discontinuities.Relationship to other integral transforms Relationship to the Fourier and Hankel transformsThe Abel transform is one member of the FHA cycle of integral operators. For example, in two dimensions, if we define A as the Abel transform operator, F as the Fourier transform operator and H as the zeroth-order Hankel transform operator, then the special case of the Projection-slice theorem for circularly symmetric functions states that:In other words, applying the Abel transform to a 1-dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.Relationship to the Radon transformThe Abel transform is a projection of f(r) along a particular axis. The two-dimensional Radon transform gives the Abel transform as a function of not only the distance along the viewing axis, but of the angle of the viewing axis as well.ReferencesBracewell, R. (1965). The Fourier Transform and its Applications. New York: McGraw-Hill. ISBN0-07-007016-4.External linksRetrieved from "/wiki/Abel_transform"Categories: Integral transforms | Image processing | Niels Henrik AbelPersonal tools∙Log in / create account Namespaces∙Article∙Discussion。

Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOV A （analysis of variance）, 方差分析ANOV A Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 通用线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOV A , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS 统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性V ARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转V olume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换。

模糊集与粗糙集的简单入门1. 刖言Zadeh在1965年创立了模糊集理论[1],Pawlak在1982年又给出了粗糙集的概念[2],模糊集理论和粗糙集理论都是研究信息系统中只是不完全，不确定问题的两种方法，是经典集合论的推广，它们各自具有优点和特点，并且分别在许多领域都有成功的应用，如模式识别、机器学习、决策分析、决策支持、知识获取、知识发现等•模糊理论是简历集合的子集边缘的病态定义模型，隶属函数多数是凭经验给出的，带有明显的主观性；粗糙集理论基于集合中对象间的不可分辨行的思想，作为一种刻画不完整想和不确定性的数学工具，它无需任何先验信息，能邮箱分析处理不精确、不完整等不完备信息，对不确定集合的分析方法是客观的• 两种理论之间有着密切的关系和很强的互补性，同事粗糙集理论和模糊集理论可以进行结合，产生粗糙模糊集理论和模糊粗糙集理论，并且发挥着不同的优势•本文在已有的模糊集理论和粗糙集理论的基础之上，分析和总结了模糊集和粗糙集理论，对二者进行了全面的比较.2. 基本概念这部分将集中介绍模糊集和粗糙集的基本概念及其性质•2.1模糊集模糊理论[3][4]是一种用以数学模型来描述语意式的模糊信息的方法•模糊概念也是没有明确外延的概念.根据普通集合论的要求，一个对象对应于一个集合,要么属于，要么不属于，二者必居其一；而模糊集则通常用隶属函数表示模糊概念.2.1.1模糊集合的基本定义定义1设X是有限非空集合，称为论域，X上的模糊集A用隶属函数表示如下：A: X > [0,1], x > A(x)其中A(x)表示元素x隶属于模糊集合A的程度,记X上的模糊集合全体为F(X).模糊集合的数学表示方式为A ={( x, A(x)) | x X}, where A(x) [0,1]2.1.2模糊集合的运算设代B为X上的两个模糊集，它们的并集，交集和余集都是模糊集，且其隶属函数分别定义为A B = max{ A(x), B(x)} - x XA B 二min{ A(x), B(x)} ~x X_A =1 _ A2.1.3模糊集合的关系模糊集合之间关系的表示方式,是以集合所存在的隶属函数A(x), B(x)作为集合之间的关系表示的.(1) 模糊集合之间的相等：rw nr rvA = B= A(x)二B(x) —x X(2) 模糊集合之间的包含：A B= A(x)乞B(x) —x X2.1.4截集与支集定义2 对于 A F(X)和任意■ ■ [0,1],定义={x A(x) > 人}A； = {x A(x) > 人}分别为A的■截集和A的■强截集.特别的,当，=1时,A为A的核；当‘ =0 时,A为A的支集.表示为如下：core( A) = A ={x A(x) =l}support (A)=兀={x A(x) =0}则根据上面截集的概念，模糊子集通过■截集就变成了普通集合.截集就是将模糊集合转化为普通集合的方法，截集的概念是联系模糊集合与普通集合之间的桥梁.2.2粗糙集2.2.1粗糙集合的基本定义(1) 粗糙集合提出的背景由于经典逻辑只有真假二值之分，而在现实生活中存在许多含糊的现象，并不能简单的用真假值来表示.于是，在1904年,谓词逻辑的创始人G.frege提出了含糊(vague) 一词，他把含糊现象归结到边界线上.1965年丄.A. Zadeh 提出Fuzzy Sets 的概念，试图通过这一理论解决G.frege 的含糊概念.Zadeh的FS方法是利用隶属函数描述边界上的不确定对象.1982年，波兰华沙理工大学乙Pawlak教授针对G. frege的边界线区域思想提出了Rough Sets理论.Pawlak的RS方法：把无法确认的个体都归属于边界区域,把边界区域定义为上近似集和下近似集的差集.(2) 粗糙集合的定义粗糙集理论特点是不需要预先给定默写特征或属性的数量描述，直接从给定的问题的描述集合出发，通过不可分辨关系和不可分辨类确定给定问题的近似域找出问题内在规律•定义2设K =(X,A,V, f)是一个知识库,其中X是一个非空集合,称为论域.A=C D是属性的非空有限集合，C为D的决策属性，C D-：，，V a是属性a A 的值域，f : X A > V是一个信息函数,它为每个对象赋予一个信息值•定义3设X是一个有限的非空论域，R为X上的等价关系,等价关系R把集合X划分为多个互不相交的子集，每个子集称为一个等价类，用[X]R来表示，[X]R二{y・X xRy},其中x・X ,称x,y为关于R的等价关系或者不可分辨关系.论域X上的所有等价类的集合用X / R来表示.2.2.2上、下近似集,粗糙度(1) 上下近似集的定义定义4对于任意的丫 X,Y的R上、下近似集分别定义为R(Y) = {Z X / R|Z 丫=门}R(Y)二{Z X / R| Z Y}集合posR(Y)称为集合丫的正域，posR(Y) =R(Y);集合negR(Y) = X -貝X) 称为集合Y的负域；集合bnR(Y)二R(Y) -R(Y)称为Y的R边界域.集合的不确定性是由于边界域的存在，集合的边界域越大，精确性越低,粗糙度越大.当R(Y)=R(Y)时，称Y为R的精确集；当R(Y)=R(Y)时，称Y为R的粗糙集, 粗糙集可以近似使用精确集的两个上下近似集来描述.(2) 粗糙度粗糙度是表示知识的不完全程度，由等价关系R定义的集合X的粗糙度为：门|RX|：R(X)十一RX其中X学①,X表示集合X的基数.3研究对象、应用领域及研究方法3.1模糊集的研究对象、应用领域及研究方法(1) 模糊集的研究对象模糊集研究不确定性问题，主要着眼于知识的模糊性，强调的是集合边界的不分明性•(2) 模糊集的应用领域模糊集理论⑸广泛应用与现代社会与生活中，主要有以下几个方面：消费电子产品、工业控制器、语音辨识、影像处理、机器人、决策分析、数据探勘、数学规划以及软件工程等等•(3) 研究方法模糊集理论的计算方法是知识的表达和简化.从知识的“粒度”的描述上来看,模糊集是通过计算对象关于集合的隶属程度来近似描述不确定性；从集合的关系来看,模糊集强调的是集合边界上的病态定义，也即集合边界的不分明性；从研究的对象来看，模糊集研究属于同一类的不同对象间的隶属关系，强调隶属程度；从隶属函数来看，模糊集的隶属函数反映了概念的模糊性，而且模糊集的隶属函数大多是专家凭经验给出的，带有强烈的主观意志.3.2粗糙集的研究对象、应用领域及研究方法(1) 粗糙集的研究对象[6]粗糙集理论研究不确定性问题，基于集合中对象间的不可分辨性思想，建立集合的子集边缘的病态定义模型•(2) 粗糙集的应用领域粗糙集理论在近些年得到飞速发展，在数据挖掘，模式识别,粗糙逻辑方面取得较大进展•与粗糙集理论相关的学科主要有以下几方面：人工智能，离散数学, 概率论,模糊集理论，神经网络,计算机控制，专家系统等等[7].(3) 粗糙集的研究方法粗糙集理论的研究方法就是对知识的含糊度的一个刻画，其计算方法主要是连续特征函数的产生•粗糙集理论研究认知能力产生的集合对象之间的不可分辨性,通过引入一对上下近似集合，用它们的差集来描述不确定的对象•从集合的关系来看,粗糙集强调的是对象间的不可分辨性，与集合上的等价关系相联系；从研究的对象来看，粗糙集研究的是不同类对象组成的集合关系，强调分类；从隶属函数来看,粗糙集的粗糙隶属函数的计算是从被分析的数据中直接获得，是客观的[8] .4.基本研究内容4.1模糊集理论研究的主要内容模糊集理论研究的内容很广泛，主要包括以下几方面：模糊控制，模糊聚类分析,模糊模式识别，模糊综合评判，模糊集的扩展•4.1.1模糊控制自从Zadeh发展出模糊集理论之后，对于不明确系统的控制有极大的贡献，自七十年代以后，便有一些实用的模糊控制器相继的完成，使得我们在控制领域中又向前迈进了一大步，在此将对模糊控制理论做一番浅介[6].模糊控制利用模糊集理论的基本思想和理论的控制方法•在传统的控制领域里，控制系统动态模式的精确与否是影响控制优劣的最主要关键，系统动态的信息越详细,则越能达到精确控制的目的•然而,对于复杂的系统，由于变量太多，往往难以正确的描述系统的动态，于是工程师便利用各种方法来简化系统动态，以达成控制的目的，但却不尽理想.换言之，传统的控制理论对于明确系统有强而有力的控制能力，但对于过于复杂或难以精确描述的系统，则显得无能为力了.所以,模糊集理论便被用来处理这些控制问题.4.1.2模糊聚类分析模糊聚类分析的研究是基于模糊等价关系和以及模糊分类上的[4].主要有以下的定理以及定义.定理1令R是一个模糊等价关系,并且0 —〉：：：: -1,则对- y • X有[y]"=[y]R：..定义5设数据集X二{X1,X2，…,X n},且A1,A2,…，A c是其一个分类,若该分类满足以下条件：⑴ 对- k ,存在i 使得X k 三A ；(2) 对所以i 均有A =门;则称该分类是X 的一个模糊划分.基于上面的理论，我们可以用一个划分矩阵D = (d ik )cn 来刻画数据集的分类 其中0 , X k 「A 定义6对于上面的矩阵 (1) d ik ©"；c⑵' dik =1, -k ;i ni(3) ' d ik 0, -i ；k 4则称D 是X 上的一个精确的 定义7设c 和n 时两个给定的正整数若模糊矩阵 D = (d ik )cn 满足以下三个条件：(1) d ik〔0,1〕； c(2) 二.d ik = 1, _ k ;i 吕 n(3) 0 ；二 d ik :: n, -i ;k 丝则称D 为X 上的一个模糊的c-划分矩阵.定义 8 设 X 二{X 1,X 2, ,X n }R m , V 二{V 1,V 2, ,V c } R m , D=(d ik )cn (CE n) 是X 上的一个模糊的c-划分矩阵,则c n J(D,V) 乂乂 [d ik 卩 M —X k 彳(p R )^4 k z!称为模糊划分上的一个聚类准则函数，这里m 2丄|x |[送(X ⑴)]2i 7 定义9如果对于任意的X ={X ,,X 2，…,X n }R m ,存在V\{v ；,v 2,…，v ；}5 R m 以及模糊的c-划分矩阵D *使得J(D,V^i J(D *,V *)对所有的X 二{X 1,X 2，…，X n } R m 以及模糊的c-划分矩阵D 都成立，则称D *为最 优模糊c-划分矩阵，V *为一个模糊聚类中心.d ik 二D,若其满足以下三个条件：c-划分矩阵.4.1.3模糊模式识别模糊模式识别是利用模糊集理论对行为的识别 .根据识别模式的性质，可以 将模式识别分为两类：具体事物的识别，如对文字，音乐,语言等周围事物的识别； 抽象事物的识别，如对已知的一个论点或者一个问题的理解等 .下面介绍一些基 本的定理及定义.定义10清晰度增强因子：令A F(X)是X 上的一个模糊集,定义另外一个模糊集I ⑵(A) • F(X),其中I ⑵(A)(x) 称I ⑵(A)(x)为清晰度增强因子4.1.4模糊综合评判模糊综合评判是利用模糊集理论对一个事物进行评价.具体的过程为：将评 价目标看成是由多种因素组成的模糊集合 X,再设定这些因素所能选取的评审 等级,组成评语的模糊集合(称为评判集V ),分别求出各单一因素对各个评审等 级的归属程度(称为模糊矩阵D),然后根据各个因素在评价目标中的权重分配，通过计算(称为模糊矩阵合成),求出评价的定量解值.定义11设f ：[0,1]n > [0,1]满足以下几个条件：(1) 花=X 2 二 二X n =X= f(X 1,X 2, ,X n )二 X ；(2) X j ⑴兰 X j ⑵二 f (捲，…,X ij , X (1),X+"^ 必)兰 f (洛，…,X i 』，X (2),Xy , X n ),W ；(3) f(X 1,X 2/ ,X n )对每个变量都是连续的；则称f 为n-维综合函数.常用的n-维综合函数主要有加权平均函数，几何平均函数，单因素决策函数 显著因素准则函数等等.4.2粗糙集理论研究的主要内容粗糙集理论作为一种数据分析处理理论 ，无论是在理论方面还是在应用实践方面都取得2A(x)2 , 1-2(1-A(x)) A(x) [0,0.5] A(x) (0.5,1]了很大的进展，展示了它光明的前景，因而其研究内容以及领域也是非常广泛的，主要包括以下几方面：变精度粗糙集，集值信息系统，粗糙集理论的应用，支持向量基等.421变精度粗糙集变精度粗糙集模型［9］是Pawlak粗糙集模型的扩充,它是在基本粗糙集模型的基础上引入了'(^ - :：: 0.5),即允许一定的错误分类率存在，这一方面完善了近似空间的概率，另一方面也有利于用粗糙集理论从认为不相关的数据集中发现相关的数据.当然,变精度粗糙集模型的主要任务是解决属性间无函数或不确定关系的数据分类问题.当1 =0时,Pawlak粗糙集模型是变精度粗糙集模型的一个特例.4.2.2集值信息系统集值信息系统⑸是信息系统的一般化模型，在实际应用中信息系统随着对象的变化而不断地动态变化.S = (X ,AT)是信息系统,其中X是对象的非空有限集合,AT是属性的非空有限集合,对于每个AT有a:X > V a,其中V a称为a的值域.每个属性子集A AT决定了一个不可区分关系ind(A):ind(A)={(x, y) X X|~a・代a(x)二a(y)}.关系ind (A)( AT)构成了X的划分,用X/i nd (A)来表示.对于一个对象，一些属性值可能是缺省的.为了表明这种情况，通常给定一个区分值(即空值null value)给出这些属性定义12如果至少有一个属性a・AT使得V a含有空值,则称S是一个不完备信息系统［5］,否则称它是完备的，我们用*表示空值.设S是一个不完备信息系统，a • AT使得V a含有空值*时，并且该空值*的取值为一个集合，该集合的元素是这个属性中其他所有可能值的集合，则S就是集值信息系统.423支持向量基支持向量机(Support Vector Machine,SVM)[10][11] 是Corinna Cortes 和Vapnik8等于1995年首先提出的.SVM起初是广泛应用在神经信息处理系统(Neural Information Processing Systems,NIPS),但是，现今,SVM 已经在所有的机器学习研究领域中起着重要作用•SVM 是一种学习系统，他利用高维空间中的线性分类器，在这个空间中建立一个最大的间隔超平面，这里的最大是基于最优化理论的.广义的SVM起源于统计学习理论[12].5.模糊集与粗糙集的结合由上面的讨论可知，模糊集理论与粗糙集理论各具特点，两种理论有着很强的联系与互补性，因此将两者的特点结合起来形成研究不完全数据集的有效方法.此外,通过模糊聚类和粗糙集两种方法进行属性的对象约简和属性约简，可以使数据得到横向和纵向两个方向上的约简，对象约简是引入了相似性的概念进行模糊聚类的过程，对象约简改变了标准粗糙集模型的不可分辨关系的确定条件由于粗糙集所处理的都是离散数据，所以在数据分析中需要应用模糊聚类或隶属函数离散化，进而应用粗糙集理论属性约简、提取规则.所以结合模糊集、粗糙集理论能够有效地分析数据，提高生成规则的可信性和和合理性，倒出可信的规则集.5.1模糊粗糙集及粗糙模糊集结合模糊集和粗糙集两种理论可以得到模糊粗糙集及粗糙模糊集模型，当知识库中的知识模块是清晰的概念，而被描述的概念是一个模糊的概念，人们建立粗糙模糊集模型来解决此类问题的近似推理；当知识库中的知识模块是模糊知识而被近似的概念是模糊概念时，则需要建立模糊粗糙集模型，也有人将普通关系推广称模糊关系或者模糊划分而获得模糊粗糙集模型•定义13设R是X上的一个等价关系,A F(X) , • • [0,1],模糊集A、A以fij 及A s的上下近似分别为：R(A,)={X・X|[X]R A, *}, R(A,)二{x X|[X]R A,}— s c s . S sR(A )二{x X |[X]R A, -：」}, R(AJ ={X X |[X]R A,} R(A)二{x X |[X]R A—：，},R(A)二{X X|[X]R』A}可以验证，当A是X上的经典集合时，上面所介绍的上下近似就是Pawlak意义下的上下近似.定义14设R是X上的等价关系，A是X的一个模糊集合，A- F(X),则A 关于R 的上下近似分别定义如下：A R(X) =sup{A(y) | y [X]R},企(x) =inf{ A(y)|y [X]R}可以看出，模糊集A F(X)关于等价关系R的上下近似仍为模糊集合，若A R =A R ,则称A是可定义的，否则称A是粗糙集，称A是A关于近似空间(X , R) 的正域，称~ A R是A关于(X , R)的负域，称A R (~ A R)为A的边界• A R可以理解为对象X 肯定属于模糊集A的隶属程度；A R理解为对象X可能属于模糊集A的隶属程度，同样可以验证，当A时X上的经典集合时，就是Pawlak意义下的上下近似•在标准粗糙集模型中引入变精度，提高了相对近似精度，而在粗糙模糊集引入变精度,得到新定义：A R(X)二sup{A(y) | y [X]R A(y) 1 - ：}A R(X)，inf{A(y)|y [X]R A(y) 一J这样下近似集合中元素隶属度降低，而上近似的隶属度提高，提高了相对精度•5.2粗糙隶属函数粗糙隶属函数式借助模糊理论来研究粗糙集理论的方法，通过粗糙隶属度函数可以将粗糙集理论与模糊集理论联系起来，建立一种粗糙集理论与模糊集理论的关系,并得到一些性质.定义15设R是论域X上的一个相似关系，若A是X上的一个模糊集合，则A关于R的一个下近似R(A)和上近似R(A)分别定义为X上的一个模糊集合，称隶属函数A(x)表示的是x 的等价类[X ]R 隶属于A 的程度.由定义14和定义15可以得到：模糊集A 的下近似且关于等价关系R 的等价 类隶属于A 的程度为1;模糊集A 的上近似且关于等价关系R 的等价类隶属于A 的程度为大于0小于1,因此有：性质 1 Core(A)二 & ={x| A(x) =1,x X/R}二RAs support(A) =A 0 ={x| A(x)》0,x E X / R}bnR(A)二R A —RA 二{x|0 ：： A(x) ：：：1,x X / R}negR(A) =X — R A ={X | A(x) =0,x X / R} 性质 2 y [X ]R 二 A(x)二 A(y)[X ]R 』A= A (x ) =1[X ]R A - '」一A(x) = 0[X ]R 二 A a nd [x]R 二 A(x) (0,1)6总结本文系统的介绍了模糊集理论与粗糙集理论，二者研究的主要内容，以及二者 的结合的相关理论•是对本学期所学的模糊计算和粗糙计算的一个简单的小结，也 是我本人对该学科的一个简单的入门•为粗糙隶属度函数⑸，定义为A(x) 粗糙隶属函数表示的是一个模糊概念 |A [X ]R |般不是Zadeh 意义下的隶属函数.粗糙参考文献[1] L.A.Zadeh, Fuzzy sets[J], I nformation and Con trol, 1965,8:338-353.[2] Pawlak Z, Rough sets[J], I ntern ati onal Jour nal of Computer andIn formation scie nee, 1982,1(11):341-356.[3] 胡宝清,模糊理论基础，武汉:武汉大学出版社,2010.[4] 张文修,模糊数学基础，西安:西安交通大学出版社，1984.⑸张文修,粗糙集理论与方法，北京:科学出版社,2001[6] http://baike.baidu.eom/view/87377.htm[7] K. Y. Cha n, C.K. Kwo ng, B.Q. Hu, Market segme ntation and ideal poi ntidentification for newproduet design using fuzzy data eompression andfuzzy clusteri ng methods[J], Applied Soft Computi ng, 2012, 12, 1371-1378.[8] Z.Pawlak, Rough sets and fuzzy sets [J], Fuzzy sets and Systems,1985,17,99-102.[9] Beynon M.Reducts within the variable precision rough sets model: afurther investigation[J], European Journal of Operational Research,2001,134:592-605.[10] 邓乃扬，田英杰，数据挖掘中的新方法：支持向量基，北京：科学出版社,2004.[11] 邓乃扬，田英杰，支持向量基-理论、算法与拓展，北京：科学出版社,2009.[12] V.Vap nik, Statistical Learni ng Theory, Joh n Wiley & Son s, 1998.。

微积分专有名词中英文对照absolutely convergent 绝对收敛absolute value 绝对值algebraic function 代数函数analytic geometry 解析几何antiderivative 不定积分approximate integration 近似积分approximation 近似法、逼近法arbitrary constant 任意常数arithmetic series/progression (AP)算数级数asymptotes (vertical and horizontal)(垂直/水平）渐近线average rate of change 平均变化率base 基数binomial theorem 二项式定理，二项展开式Cartesian coordinates 笛卡儿坐标(一般指直角坐标) Cartesian coordinates system 笛卡儿坐标系Cauch's Mean Value Theorem 柯西均值定理chain rule 链式求导法则calculus 微积分学closed interval integral 闭区间积分coefficient 系数composite function 复合函数conchoid 蚌线continuity (函数的)连续性concavity (函数的）凹凸性conditionally convergent 有条件收敛continuity 连续性critical point 临界点cubic function 三次函数cylindrical coordinates 圆柱坐标decreasing function 递减函数decreasing sequence 递减数列definite integral 定积分derivative 导数determinant 行列式differential coefficient 微分系数differential equation 微分方程directional derivative 方向导数discontinuity 不连续性discriminant (二次函数)判别式disk method 圆盘法divergence 散度divergent 发散的domain 定义域dot product 点积double integral 二重积分ellipse 椭圆ellipsoid 椭圆体epicycloid 外摆线Euler’s method (BC)欧拉法expected valued 期望值exponential function 指数函数extreme value heorem 极值定理factorial 阶乘finite series 有限级数fundamental theorem of calculus 微积分基本定理geometric series/progression （GP)几何级数gradient 梯度Green formula 格林公式half-angle formulas 半角公式harmonic series 调和级数helix 螺旋线higher derivative 高阶导数horizontal asymptote 水平渐近线horizontal line 水平线hyperbola 双曲线hyper boloid 双曲面implicit differentiation 隐函数求导implicit function 隐函数improper integral 广义积分、瑕积分increment 增量increasing function 增函数indefinite integral 不定积分independent variable 自变数inequality 不等式ndeterminate form 不定型infinite point 无穷极限infinite series 无穷级数infinite series 无限级数inflection point （POI）拐点initial condition 初始条件instantaneous rate of change 瞬时变化率integrable 可积的integral 积分integrand 被积分式integration 积分integration by part 分部积分法intercept 截距intermediate value of Theorem ：中间值定理inverse function 反函数irrational function 无理函数iterated integral 逐次积分Laplace transform 拉普拉斯变换law of cosines 余弦定理least upper bound 最小上界left-hand derivative 左导数left-hand limit 左极限L'Hospital's rule 洛必达法则limacon 蚶线linear approximation 线性近似法linear equation 线性方程式linear function 线性函数linearity 线性linearization 线性化local maximum 极大值local minimum 极小值logarithmic function 对数函数MacLaurin series 麦克劳林级数maximum 最大值mean value theorem （MVT)中值定理minimum 最小值method of lagrange multipliers 拉格朗日乘数法modulus 绝对值multiple integral 多重积分multiple 倍数multiplier 乘子octant 卦限open interval integral 开区间积分optimization 优化法,极值法origin 原点orthogonal 正交parametric equation （BC）参数方程partial derivative 偏导数partial differential equation 偏微分方程partial fractions 部分分式piece—wise function 分段函数parabola 抛物线parabolic cylinder 抛物柱面paraboloid ：抛物面parallelepiped 平行六面体parallel lines 并行线parameter ：参数partial integration 部分积分partiton ：分割period ：周期periodic function 周期函数perpendicular lines 垂直线piecewise defined function 分段定义函数plane 平面point of inflection 反曲点point—slope form 点斜式polar axis 极轴polar coordinates 极坐标polar equation 极坐标方程pole 极点polynomial 多项式power series 幂级数product rule 积的求导法则quadrant 象限quadratic functions 二次函数quotient rule 商的求导法则radical 根式radius of convergence 收敛半径range 值域(related) rate of change with time （时间）变化率rational function 有理函数reciprocal 倒数remainder theorem 余数定理Riemann sum 黎曼和Riemannian geometry 黎曼几何right—hand limit 右极限Rolle's theorem 罗尔(中值）定理root 根rotation 旋转secant line 割线second derivative 二阶导数second derivative test 二阶导数试验法second partial derivative 二阶偏导数series 级数shell method （积分)圆筒法sine function 正弦函数singularity 奇点slant 母线slant asymptote 斜渐近线slope 斜率slope-intercept equation of a line 直线的斜截式smooth curve 平滑曲线smooth surface 平滑曲面solid of revolution 旋转体symmetry 对称性substitution 代入法、变量代换tangent function 正切函数tangent line 切线tangent plane 切（平)面tangent vector 切矢量taylor’s series 泰勒级数three—dimensional analytic geometry 空间解析几何total differentiation 全微分trapezoid rule 梯形（积分)法则。

一维数据分形维数variogram method -回复什么是一维数据分形维数variogram method？一维数据分形维数variogram method 是一种用于分析和描述一维数据集的方法。

它基于分形理论，通过计算数据集中数据点之间的差异和变化来评估其分形特征。

这个方法常用于地质学、地球物理学、地质工程和环境科学等领域中，用于研究和理解地质和地貌过程以及空间数据的分布和变化。

为了理解一维数据分形维数variogram method，首先需要了解什么是分形。

分形是指具有自相似性和尺度不变性的图形或结构。

自相似性是指一个对象的部分与整体之间具有相似的结构或特征。

尺度不变性是指对象的形态和特征在不同尺度上保持不变。

分形理论认为自然界中的很多现象都具有分形特征，如云朵、树枝、河流网络等。

在一维数据分形维数variogram method 中，我们关注的是数据集中数据点之间的差异和变化。

为了计算分形维数，需要使用半变异函数（semivariogram function）来描述数据点之间的差异程度。

半变异函数是用来衡量数据点的变异性，并且可以通过计算数据点之间的差异值的均方差来定义。

它可以通过以下公式表示：γ(h) = 1/2N(h)∑[z(xi) - z(xi + h)]^2其中，γ(h) 是半变异函数的值，N(h) 是距离为h 的数据对的数量，z(xi) 和z(xi + h) 是数据集中两个点的值，差异之和的平方即为半变异函数的值。

计算半变异函数的值之后，可以通过拟合该函数来获取分形维数。

常用的拟合方法包括线性模型、指数模型和高斯模型等。

根据具体的数据集和研究目的，选择最合适的拟合模型来计算分形维数。

分形维数代表了数据集中数据点之间的分形特征的程度，可以提供有关数据集自相似性和尺度不变性的信息。

一维数据分形维数variogram method 在地质学和地球物理学中有广泛的应用。

它可以帮助研究人员理解地质和地貌过程的变化和演化，从而提供对地下水资源、矿产资源和地震活动等的预测和评估。