unit 9 随机误差的统计学基本分析.doc

Unit 9 Basic Statistical Analysis of Random Errors （随机误差的统计学基本分析）Random errors are those variables that remain after mistakes are detected and eliminated and all systematic errors have been removed or corrected from the measured values.（随机误差是在错误被察觉【detect】和消除【eliminate】后，并且所有系统误差被从测量值中移除或修正后，保留下的那些变量【variable变量、变化n.】）They are beyond the control of the observer.（它们是观测者无法控制的）So the random errors are errors the occurrence of which does not follow a deterministic pattern.（因此随机误差是不遵循某个确定性【deterministic确定性的】模式【pattern】而发生的误差）In mathematical statistics, they are considered as stochastic variables, and despite their irregular behavior, the study of random errors in any well-conducted measuring process or experiment has indicated that random errors follow the following empirical rules:（在数理统计【mathematical statistics】中，它们被当成随机变量【stochastic variable】，尽管它们的行为无规律，在任一正确的【well-conducted原意为品行端正的，这里指测量实验和活动是无误的】测量活动和实验中，对的随机误差的研究显示【indicate】随机误差遵循以下经验法则【empirical rule】：）A random error will not exceed a certain amount.（随即误差不会超过一个确定的值）Positive and negative random errors may occur at the same frequency.（正负误差出现的频率相同）Errors that are small in magnitude are more likely to occur than those that are larger in magnitude.（误差数值【magnitude量值、大小】小的比数值大的误差出现可能性大【be likely to 可能】）The mean of random errors tends to zero as the sample size tends to infinite.（当【as】样本大小【sample size】趋近于无穷【infinite】时，随机误差的平均值趋近于0）In mathematical statistics, random errors follow statistical behavioral laws such as the laws of probability.（在数理统计中，随机误差遵循统计学的【statistical】行为【behavioral行为的】规律，如概率法则）A characteristic theoretical pattern of error distribution occurs upon analysis of a large number of repeated measurements of a quantity, which conform to normal or Gaussian distribution.一个误差分布的典型理论模式出现于对某一量的大量重复观测中，这些重复观测遵从【conform to遵照】正态分布或高斯分布【在对一个量进行大量重复观测分析后，得到一个误差分布的理论特征正态或高斯分布】The plot of error sizes versus probabilities would approach a smooth curve of the characteristic bell-shape.（误差大小与【versus与、与的关系、与相对】概率的关系图，接近一条光滑的特有的【characteristic特有的】钟形曲线。）This curve is known as the normal error distribution curve.（这条曲线被称为正态分布曲线）It is also called the probability density function of a normal random variable.（也叫做正态随机变量【normal random variable】的概率密度【probability density】函数）It is important to notice that the total area of the vertical bars for each plot equals 1.（需特别注意的是，每个图的条形图总面积为1。）This is true no matter the value of n (the number of single combined measurements), and thus the area under the smooth normal error distribution curve is equal to 1.无论【no matter】n（【独立观测数】）是多少，在光滑的误差正态分布曲线下的面积都是1。If an event has a probability of 1, it is certain to occur, and therefore the area under the curve represents the sum of all the probabilities of the occurrence of errors.（如果一件事的概率为1，它一定会发生，因此曲线下方的面积代表了所有误差发生的概率。）A number of properties that relate a random variable and its probability density function are useful in our understanding of its behavior.随机变量的大量属性和其概率密度函数，有助于我们对其行为的理解。Mean and standard deviation are two most popular statistical properties of a random variable.均值和标准差就是随机变量的两个最常用的统计属性Generally, a random variable which is normally distributed with a mean and standard deviation can be written in symbol form as N(,2).（一般地，一个通常由平均值和标准偏差描述的随机变量可以用符号【symbol】表示为N(,2)。They can be explained as follows.（【它们可以】解释如下） Mean: The most commonly used measure of central tendency is the mean of a set of data (a sample).（平均值：应用最普遍的中心趋向的估计【measure】就是一系列数据（一个样本）的平均值）The concept of mean refers to the most probable value of the random variable.（平均值的概念【concept】涉及到随机变量的最或是值）It is also called by any of the several termsexpectation, expected value, mean or average. （还可以由其它几个术语来称呼它期望、期望值、平均值或平均值）The mean is defined as （平均值定义为）公式Where xi are the observations, n is the sample size, or total number of observations in the sample, and x is the mean which is also called most probable value (MPV).（xi是观测值，n是样本大小，或者叫样本内观测值的总数，x是平均值，经常被称为最或是值（MPV）The MPV is the closest approximation to the true value that can be easily achieved from a set of data.（MPV是最接近真值的近似值【approximation】，可以很容易由一系列数据得到。）It can be shown that the arithmetic mean of a set of independent observations is an unbiased estimate of the meanof the population.（可以看出【It can be shown that】一系列独立【independent】观测值的算数平均值【arithmetic mean】是一个样本【population】的期望值的无偏估计【unbiased estimate】。）Standard deviation is a numerical value indicating the amount of variation about a central value.（标准偏差是一个数值【numerical value】，它指示【indicate】相对于中值的偏离）In order to appreciate the concept upon which indices【index的复数】 of precision devolve one must consider a measure that takes into account all the values in a set of data.（考虑一系列数据的所有值精度指标 必需顾及一个量，这个量考虑到【takes into account考虑】一组【a set of】数据的所有值）Such a measure is the deviation from the mean x of each observed value xi i.e. (xi- x), and the mean of the squares of the deviations may be used, and this is called the variance2,（这个量是每个观测值xi相对于平均值x 的离差【deviation】，也就是，(xi- x)，离差的平方的平均值被采用，称之为方差2）公式Where is the mean (expectation) of the population.（这里是对象总体【样本】的平均值（期望值）。）The square root of the variance is called standard deviation . （方差的平方根被称为标准差）Theoretically the standard deviation, which is the value on the X axis of the probability curve that occurs at the points of inflecxion【估计应为inflexions拐点】 of the curve, is obtained from an infinite number of variables known as the population.（理论上来说，标准差是概率曲线拐点的X轴坐标，由无穷多的变量（被称为样本）得到）In practice, however, only a sample of variables is available and S is used as an unbiased estimator.然而，实际上，变量的样本是有限的，且将S称为的无偏估计。Account is taken of the small number of variables in the sample by using (n-1) as the divisor, which is referred to in statistics as the Bessel correction; hence, variance is计算【Account is taken of】中使用有限【small小的】数量的变量，且将(n-1)作为除数，这就是统计学中的白赛尔公式，因此，方差可表述为：公式To obtain an index of precision in the same units as the original data, therefore the square root of the variance is used, and this also called the standard deviation S. （为了获得与源数据一样单位的精度指标，因此采用了方差的平方根，又叫做标准差S）The standard deviation is the measure of the dispersion or spread of the random variable.（标准差是随机变量分布的量度标准。）A survey measurement, such as a distance or angle, after mistakes are eliminated and systematic errors corrected, is a random variable.（一个测量值，例如距离或角度观测值，在粗差被去除、系统误差被修正后，就是一个随机变量。）If a distance is measured 20 times, it is not unusual to get values for each of the measurements that differ slightly from its true value that is never known.（如果一段距离被测了20次，每次的测量值与永远未知的真值有些微的差值是很正常的）So owing to random variability, an error was defined as the difference between a random variable, the measured value (observation) and the constant, the true value i.e. error= measured value.因此，由于【owing to】随机可变性，误差被定义为随机变量、测量值和常量、真值之间的差值，也就是，误差测量值真值And a correction (residual), which is the negative of the error in practice, was defined as correction between the MPV and measured value i.e. correction=MPV-measured value.（改正值，习惯上【in practice】是误差的负值，定义为MPV和测量值之间的修正值，也就是改正值MPV测量值）When the so-called true values are available to compare with calculated values, the mean square error (MSE) is given by （当所谓的真值可以与计算值相比较时，误差均方差（MSE）为：）公式In which xi is the measured value, x is the true value and n is the number of measurements.（其中xi是测量值，x是真值，n是观测数）Propagation of errors (or error propagation): Much data in surveying is obtained indirectly from various combinations of observations.误差传播：测量的许多数据是间接由各种测量值综合得到的【combination是名词，这里翻译时用到了词性转换】For instance the coordinates of a line are a function of its length and bearing.（例如，一条直线的坐标就是其长度和方位的函数）As each measurement contains an error, it is necessary to consider the combined effect of these errors on the derived quantity.（由于每项测量值都包含误差，必需考虑这些源数据的误差的联合影响）Error propagation is one of the many aspects of analyzing errors.（误差传播是各种误差分析方法的其中之一）It is the mathematical process used to estimate the expected random error in a computed or indirectly measured quantity, caused by one or more identified and estimated random errors in one or more identified variables that influence the precision of the quantity.它(误差传播)是一种数学方法【process或者译为 过程】，用来估计【estimate】在一个计算出的或间接测量的参量【quantity】的期望随机误差【或偶然误差】，该参量是由一个或多个独立【identified】变量的偶然误差引起的，从而影响该量的精度。）The general procedure is to differentiate with respect to each of the observed quantities in turn and