系统的描述与模型建立

1、第三章 系统的描述与模型建立Chapter 3. Description of Systems and Modeling3.1 系统的描述3.1 Description of Systems- 在对系统进行分析时, 要对所研究的系统进行描述. 即为从开始确定问题时所写的问题剖析报告到具体的数学模型公式或计算机程序. 目的是使我们确立对系统的认识, 便于和其他人进行交流和解释.- When analyzing a system, it is needed to describe the system of interest, that is, from writing the problem i

2、dentification and analysis to furnishing the concrete mathematic model or computer programs. The purpose of doing so is to better learn the system, and better communicate with other staff members in the project.- 自然语言能够形象生动地说明问题, 特别是一词多意, 一意多词, 丰富多彩, 富于表现力. 但随着对事物认识从表面到本质, 由定性到定量, 要求对描述事物的语言有更高的精度,

3、自然语言很难胜任. 同一句话可能有不同的理解, 造成表达上的混乱. - The natural language can be used to more vividly describe the problem, especially one word may contain/convey more-than-one implications, and is full of thrust. However, along with the deeper and better understanding the phenomena, from naturally to quantitatively

4、, one is required to add more accuracy to the subject, and natural language is not able to have such competency to do the job. This is because one word/sentence may be understood differently, and thus give rise to confusion.- 科学术语用精练浓缩的形式集中表现大量知识, 比自然语言有更大的精确 单义性. 术语和它表达的思维内容有唯一确定的对应关系. 但是精确性和明确性仍然受

5、到限制. - 符号语言, 或代码语言用来代替科学术语, 使科学语言更加简明和抽象, 但是信息承载量大大增加. 通用性好, 一经约定, 不同国家和民族的人都可以使用.- Science terminologies use very concise and condensed meaning to convey a large multitude of knowledge collectively, with much better improved accuracy, and with more concentrated definition. Terminologies has an one-

6、to-one match to the thinking in terms of contents, but the accuracy/precision is not satisfactory. To remedy this weakness, symbol language is created to replace the science terminology, on one hand, making it able to be even more concise and abstract, on the other hand, able to contain much more in

7、formation. Anyone with any nationality may understand it once agreed. 3.2 量化和尺度3.2 Quantify and Measure(自学)(self review)3.3 数据的管理3.3 Management of Data(自学)(self review)3.4 指标和指标体系3.4 Targets and Target Systems(自学)(self review)3.5 不确定性描述 (一) 随机性3.5 Description of Uncertainty (1) Randomness 3.5.1 随机性描

8、述3.5.1 Description of Randomness - 不确定性描述分为: - Uncertainty description may be categorized into:1) 随机性描述, 包含 i) 统计分析 ii) 主元分析 iii) 因子分析1) Description of randomness, including i) statistic analysis ii) principle component analysis, iii) factor analysis.2) 模糊性描述.2) Description of fuzziness.- 随机性描述有关概念和

9、定义.- Some concepts and definitions for randomness description.1) 总体: 所调查对象(事物, 现象等)的全体.1) Collection: all the elements under investigation. 2) 个体: 构成总体的元素, 其发生具有偶然性/随机性, 为随机变量.2) component/element: the parts that compose the collection, they occur randomly or accidently. 3) 样本: 总体中抽取的部分个体的集合. 3) Sam

10、ple: a set taken out from the collection.由于总体的元素数量过多, 所以采用抽样观测, 才能呈现出统计的规律性.Due to the large number of elements/components in the collection, use sampled observation to observe the statistic rules对每一次观测, 取p个样本, 得出一次观测 (x1, x2, x3 , xp), 它是一个p维随机向量.用样本的数字特征来推断总体的特征.For each observation, take p sample

11、s, obtain one observation (x1, x2, x3 , xp), it is a random vector of p order. Use the value characteristics of the samples to deduce the characteristic of the collection. 样本的数字特征包括: i) 样本均值 ii) 样本方差 iii) 样本标准偏差The value characteristics of the samples include: i) mean ii) variance iii) standard devi

12、ation. 均值: (xi为个体元素.)Mean: (xi is element.): 方差: Variance: 标准偏差: Standard deviation: 样本的数字特征也可用数字的分布表示, 通常是正态(高斯)分布.The value characteristics of samples may be expressed using value distribution, normally use normal distribution or Gaussian distribution. 概率密度函数为: The probability density function (pd

13、f) is: ()其中 为方差, 为平均值.Where is variance, and is mean. 分布函数为:The distribution function is通常记为 Normally denoted as . 注: 当 , 即 时, 称为标准正态分布.Note: when , namely, , it is called standard normal distribution. ()在复杂系统中, 做为结果的应变量通常是多个自变量的函数, 而这些自变量为彼此相关的随机变量. In a complex/intriguing system, as a responsive v

14、ariable, it is usually relates more-than-one variables, and these variables interrelated random ones. 多元变量可用一个p维向量表示:Multiple variables may be expressed using a p-dimension vector.x=x1, x2 xpT注: xi 为若干值, 本身是一个变量, 例如温度等.Note: xi is more than just one value, it itself is a variable, such as temperatur

15、e. X的平均值为The mean of X is 其中 是xi 的均值.Where is the mean of xi .对应于单变量的方差, 多变量采用协方差Corresponding to the single-variable variable, the multiple-variable adopts co-variance .其中E 为数学期望.Where E is the expectation. 多变量的正态分布形式为The normal distribution of the multiple-variable takes the form of其中 是的行列式.Where

16、is the determinant of . 以上是涉及单/多变量样本的数字特征.The description afore involves the value characteristics of single-/mult-variables. 3.5.2 统计分析: 多元线性回归法3.5.2 Statistics analysis: Multiple Linear Regression (MLR)寻求应变量与自变量之间的关系(因果关系), 可用回归分析法.建立模型, Seek the relation between the self-variables and response (c

17、ausal-effect relation), MLR may be applied to build a model. 其中,wherey : 观测或响应. y : Observation or response. bj : 部分回归系数, j=0,1, 2, k bj : Partial regression coefficient, j=0,1, 2, k xj : 回归变量, j=1, 2, kxj : Regressor variable, j=1, 2, ke: 随机误差, 均值为零, 方差为 s2.e: Random error with zero mean and variance s2.Y=Xb+e 其中 WhereY