系統(tǒng)的描述與模型建立

1、第三章 系統(tǒng)的描述與模型建立Chapter 3. Description of Systems and Modeling3.1 系統(tǒng)的描述3.1 Description of Systems- 在對系統(tǒng)進行分析時, 要對所研究的系統(tǒng)進行描述. 即為從開始確定問題時所寫的問題剖析報告到具體的數(shù)學模型公式或計算機程序. 目的是使我們確立對系統(tǒng)的認識, 便于和其他人進行交流和解釋.- 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.- 自然語言能夠形象生動地說明問題, 特別是一詞多意, 一意多詞, 豐富多彩, 富于表現(xiàn)力. 但隨著對事物認識從表面到本質, 由定性到定量, 要求對描述事物的語言有更高的精度,

3、自然語言很難勝任. 同一句話可能有不同的理解, 造成表達上的混亂. - 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.- 科學術語用精練濃縮的形式集中表現(xiàn)大量知識, 比自然語言有更大的精確 單義性. 術語和它表達的思維內容有唯一確定的對應關系. 但是精確性和明確性仍然受

5、到限制. - 符號語言, 或代碼語言用來代替科學術語, 使科學語言更加簡明和抽象, 但是信息承載量大大增加. 通用性好, 一經(jīng)約定, 不同國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(自學)(self review)3.3 數(shù)據(jù)的管理3.3 Management of Data(自學)(self review)3.4 指標和指標體系3.4 Targets and Target Systems(自學)(self review)3.5 不確定性描述 (一) 隨機性3.5 Description of Uncertainty (1) Randomness 3.5.1 隨機性描

8、述3.5.1 Description of Randomness - 不確定性描述分為: - Uncertainty description may be categorized into:1) 隨機性描述, 包含 i) 統(tǒng)計分析 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.- 隨機性描述有關概念和

9、定義.- Some concepts and definitions for randomness description.1) 總體: 所調查對象(事物, 現(xiàn)象等)的全體.1) Collection: all the elements under investigation. 2) 個體: 構成總體的元素, 其發(fā)生具有偶然性/隨機性, 為隨機變量.2) component/element: the parts that compose the collection, they occur randomly or accidently. 3) 樣本: 總體中抽取的部分個體的集合. 3) Sam

10、ple: a set taken out from the collection.由于總體的元素數(shù)量過多, 所以采用抽樣觀測, 才能呈現(xiàn)出統(tǒng)計的規(guī)律性.Due to the large number of elements/components in the collection, use sampled observation to observe the statistic rules對每一次觀測, 取p個樣本, 得出一次觀測 (x1, x2, x3 , xp), 它是一個p維隨機向量.用樣本的數(shù)字特征來推斷總體的特征.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. 樣本的數(shù)字特征包括: i) 樣本均值 ii) 樣本方差 iii) 樣本標準偏差The value characteristics of the samples include: i) mean ii) variance iii) standard devi

12、ation. 均值: (xi為個體元素.)Mean: (xi is element.): 方差: Variance: 標準偏差: Standard deviation: 樣本的數(shù)字特征也可用數(shù)字的分布表示, 通常是正態(tài)(高斯)分布.The value characteristics of samples may be expressed using value distribution, normally use normal distribution or Gaussian distribution. 概率密度函數(shù)為: The probability density function (pd

13、f) is: ()其中 為方差, 為平均值.Where is variance, and is mean. 分布函數(shù)為:The distribution function is通常記為 Normally denoted as . 注: 當 , 即 時, 稱為標準正態(tài)分布.Note: when , namely, , it is called standard normal distribution. ()在復雜系統(tǒng)中, 做為結果的應變量通常是多個自變量的函數(shù), 而這些自變量為彼此相關的隨機變量. 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. 多元變量可用一個p維向量表示:Multiple variables may be expressed using a p-dimension vector.x=x1, x2 xpT注: xi 為若干值, 本身是一個變量, 例如溫度等.Note: xi is more than just one value, it itself is a variable, such as temperatur

15、e. X的平均值為The mean of X is 其中 是xi 的均值.Where is the mean of xi .對應于單變量的方差, 多變量采用協(xié)方差Corresponding to the single-variable variable, the multiple-variable adopts co-variance .其中E 為數(shù)學期望.Where E is the expectation. 多變量的正態(tài)分布形式為The normal distribution of the multiple-variable takes the form of其中 是的行列式.Where

16、is the determinant of . 以上是涉及單/多變量樣本的數(shù)字特征.The description afore involves the value characteristics of single-/mult-variables. 3.5.2 統(tǒng)計分析: 多元線性回歸法3.5.2 Statistics analysis: Multiple Linear Regression (MLR)尋求應變量與自變量之間的關系(因果關系), 可用回歸分析法.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 : 觀測或響應. y : Observation or response. bj : 部分回歸系數(shù), j=0,1, 2, k bj : Partial regression coefficient, j=0,1, 2, k xj : 回歸變量, j=1, 2, kxj : Regressor variable, j=1, 2, ke: 隨機誤差, 均值為零, 方差為 s2.e: Random error with zero mean and variance s2.Y=Xb+e 其中 WhereY