灰色系統(tǒng)理論

1、灰色系統(tǒng)理論灰色系統(tǒng)理論 主要內(nèi)容主要內(nèi)容u 灰色系統(tǒng)u 灰色系統(tǒng)理論u 灰色預(yù)測(cè)方法u 灰色關(guān)聯(lián)度分析1.灰色系統(tǒng)灰色系統(tǒng)定義定義1.1 系統(tǒng)系統(tǒng)是客觀世界普遍存在的一種物質(zhì)運(yùn)動(dòng)形式,它和運(yùn)動(dòng)性一樣,是物質(zhì)存在的一種根本屬性.定義定義1.2 1.2 關(guān)于黑灰白關(guān)于黑灰白 在控制論中，人民常用顏色的深淺形容信息的明確程度，用“黑”表示信息未知，用“白”表示信息完全明確，用“灰”表示部分信息明確、部分信息不明確。相應(yīng)地，信息完全明確的系統(tǒng)稱為白色系白色系統(tǒng)統(tǒng)，信息未知的系統(tǒng)稱為黑色系統(tǒng)黑色系統(tǒng)，部分信息明確、部分信息不明確的系統(tǒng)稱為灰色系統(tǒng)灰色系統(tǒng)。黑色灰色白色1.灰色系統(tǒng)灰色系統(tǒng)1.灰色系統(tǒng)灰

2、色系統(tǒng)1.3灰色系統(tǒng)理論的產(chǎn)生灰色系統(tǒng)理論的產(chǎn)生 灰色系統(tǒng)理論，是在一般系統(tǒng)理論的基礎(chǔ)上產(chǎn)生的，它是系統(tǒng)科學(xué)思想發(fā)展的必然產(chǎn)物，是社會(huì)經(jīng)濟(jì)深入發(fā)展對(duì)科學(xué)刺激和需要的產(chǎn)物。當(dāng)我們認(rèn)識(shí)與研究自然和社會(huì)時(shí)，要從系統(tǒng)的角度出發(fā)，從宏觀上對(duì)其進(jìn)行深入的剖析和整體把握。在實(shí)際中，我們首先要對(duì)事物進(jìn)行系統(tǒng)性認(rèn)識(shí)，進(jìn)而對(duì)已有的系統(tǒng)進(jìn)行有效控制以及設(shè)計(jì)一些最優(yōu)系統(tǒng)來為人民服務(wù)。對(duì)系統(tǒng)進(jìn)行控制就要通過系統(tǒng)內(nèi)部和外部的信息和信息流來加以實(shí)施，通過對(duì)信息的控制進(jìn)而達(dá)到對(duì)系統(tǒng)本身的控制。 但是無論是現(xiàn)代控制理論還是經(jīng)典控制理論，它們都要依賴正確而精確的數(shù)學(xué)模型，否則，一切都很難取得滿意的結(jié)果。然而，在現(xiàn)實(shí)生活中，有許多

3、情況不大可能求得精確的數(shù)學(xué)模型，如工業(yè)系統(tǒng)、生物系統(tǒng)、經(jīng)濟(jì)系統(tǒng)、社會(huì)系統(tǒng)等。若得不出精確的數(shù)學(xué)模型，現(xiàn)代控制理論的方法和手段就無法施行，因而，現(xiàn)代控制理論對(duì)一些研究對(duì)象也鞭長莫及。 1.灰色系統(tǒng)灰色系統(tǒng)當(dāng)人們對(duì)這些問題進(jìn)行潛心研究時(shí)，查德于1965年首創(chuàng)模糊理論，第一次用精確的數(shù)學(xué)方式來分析和研究模糊量，取得了新的突破，隨后，模糊集合論迅速應(yīng)用于控制領(lǐng)域，收到了良好的效果。模糊控制能夠?qū)σ恍o法構(gòu)造數(shù)學(xué)模型的系統(tǒng)進(jìn)行控制，但模糊控制也表現(xiàn)出固有的弱點(diǎn)，即信息利用率低，控制粗糙、精度低等。因而，在要求高精度的情況下，這種控制難以勝任，并且它也未能對(duì)被控對(duì)象的運(yùn)動(dòng)規(guī)律作深刻的闡明，故模糊控制有它的

4、局限性，只適應(yīng)于一些特有的模糊系統(tǒng)。 經(jīng)典控制理論、現(xiàn)代控制理論和模糊控制理論都有一個(gè)共同點(diǎn)，那就是它們所研究的對(duì)象系統(tǒng)必須是白色系統(tǒng)（信息完全確知的系統(tǒng)），而事實(shí)上，無論是自然系統(tǒng)還是社會(huì)系統(tǒng)，宏觀系統(tǒng)還是微觀系統(tǒng)，無生命系統(tǒng)還是有生命系統(tǒng)，對(duì)我們認(rèn)識(shí)的主體來說，總是信息1.灰色系統(tǒng)灰色系統(tǒng)不完全的，艱難說明一個(gè)系統(tǒng)的內(nèi)部參數(shù)是完全的。毫無疑問，內(nèi)部參數(shù)不完全的系統(tǒng)具有極為普遍的意義。就像模糊理論的誕生一樣，灰色系統(tǒng)理論也應(yīng)運(yùn)而生了。 灰色系統(tǒng)理論是我國學(xué)者鄧聚龍教授于19世紀(jì)80年代初創(chuàng)立并發(fā)展的理論，它把一般系統(tǒng)論，信息論和控制論的觀點(diǎn)和方法延伸到社會(huì)，經(jīng)濟(jì)，生態(tài)等抽象系統(tǒng)，結(jié)合運(yùn)用數(shù)學(xué)

5、方法發(fā)展的一套解決灰色系統(tǒng)的理論和方法，20多年來，灰色系統(tǒng)理論引起了國內(nèi)外學(xué)者的廣泛關(guān)注?；疑到y(tǒng)理論已成功應(yīng)用到工業(yè)，農(nóng)業(yè)，社會(huì)，經(jīng)濟(jì)等眾多領(lǐng)域，解決了生產(chǎn)，生活和科學(xué)研究中的大量實(shí)際問題。 2.灰色系統(tǒng)理論灰色系統(tǒng)理論2.1 灰色系統(tǒng)理論的概念灰色系統(tǒng)理論的概念 灰色系統(tǒng)理論，是一種研究少數(shù)據(jù)、貧信息灰色系統(tǒng)理論，是一種研究少數(shù)據(jù)、貧信息不確定性問題的新方法不確定性問題的新方法。該理論以“部分信息已知，部分信息未知”的“小樣本”、“貧信息”不確定性系統(tǒng)為研究對(duì)象，主要通過對(duì)“部分”已知信息的生成、開發(fā)，提取有價(jià)值的信息，實(shí)現(xiàn)對(duì)系統(tǒng)運(yùn)行行為、演化規(guī)律的正確描述和有效控制。現(xiàn)實(shí)世界普遍存在的

6、“小樣本”、“貧信息”不確定系統(tǒng)，為灰色系統(tǒng)理論提供了十分豐富的研究資源。2.灰色系統(tǒng)理論灰色系統(tǒng)理論2.1灰色系統(tǒng)的基本原理灰色系統(tǒng)的基本原理l 差異信息原理：“差異”是信息，凡信息必有差異 l 解的非唯一性原理：信息不完全，不確定情況下的解是非唯一的l 最少信息原理：充分開發(fā)利用已占有的”最少信息”l 認(rèn)知根據(jù)原理：信息是認(rèn)知的根據(jù)l 新信息優(yōu)先原理：新信息對(duì)認(rèn)知的作用大于老信息l 灰性不滅原理：“信息不完全”（灰）是絕對(duì)的2.灰色系統(tǒng)理論灰色系統(tǒng)理論2.3灰色系統(tǒng)理論主要內(nèi)容灰色系統(tǒng)理論主要內(nèi)容 灰色系統(tǒng)理論經(jīng)過20多年的發(fā)展，已基本建立起了一門新興學(xué)科的結(jié)構(gòu)體系，其主要內(nèi)容包括：n以“

7、灰色朦朧集”為基礎(chǔ)的理論體系n以灰色關(guān)聯(lián)空間為依托的分析體系n以灰色序列生成為基礎(chǔ)的方法體系n以灰色模型（G，M）為核心的模型體系n以系統(tǒng)分析、評(píng)估、建模、預(yù)測(cè)、決策、控制、優(yōu)化為主體的技術(shù)體系2.灰色系統(tǒng)理論灰色系統(tǒng)理論2.4灰色系統(tǒng)理論應(yīng)用范疇灰色系統(tǒng)理論應(yīng)用范疇灰色系統(tǒng)的應(yīng)用范疇大致分為以下幾方面：u 灰色預(yù)測(cè)u 灰色關(guān)聯(lián)度分析u 灰色決策u 灰色統(tǒng)計(jì)u 灰色聚類u 灰色控制2.灰色系統(tǒng)理論灰色系統(tǒng)理論2.5灰色系統(tǒng)理論的的優(yōu)點(diǎn)灰色系統(tǒng)理論的的優(yōu)點(diǎn)不需要大量的樣本。樣本不需要有規(guī)律性分布。計(jì)算工作量小。定量定量分析結(jié)果與定性定性分析結(jié)果不會(huì)不一致。可用于近期、短期，和中長期預(yù)測(cè)。灰色預(yù)測(cè)

8、精準(zhǔn)度高。3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法3.1灰色預(yù)測(cè)灰色預(yù)測(cè) 灰色預(yù)測(cè)，是指對(duì)系統(tǒng)行為特征值的發(fā)展變化進(jìn)行的預(yù)測(cè)，對(duì)既含有已知信息又含有不確定信息的系統(tǒng)進(jìn)行的預(yù)測(cè)，也就是對(duì)在一定范圍內(nèi)變化的、與時(shí)間序列有關(guān)的灰過程進(jìn)行預(yù)測(cè)。盡管灰過程中所顯示的現(xiàn)象是隨機(jī)的、雜亂無章的，但畢竟是有序的、有界的，因此得到的數(shù)據(jù)集合具備潛在的規(guī)律?；疑A(yù)測(cè)是利用這種規(guī)律建立灰色模型對(duì)灰色系統(tǒng)進(jìn)行預(yù)測(cè)。 目前使用最廣泛的灰色預(yù)測(cè)模型就是關(guān)于數(shù)列預(yù)測(cè)的一個(gè)變量、一階微分的GM(1,1)模型。它是基于隨機(jī)的原始時(shí)間序列，經(jīng)按時(shí)間累加后所形成的新的時(shí)間序列呈現(xiàn)的規(guī)律可用一階線性微分方程的解來逼近。經(jīng)證明，經(jīng)一階線性微分方程

9、的解逼近所揭示的原始時(shí)間序列呈指數(shù)變化規(guī)律。因此，當(dāng)原始時(shí)間序列隱含著指數(shù)變化規(guī)律時(shí)，灰色模型GM(1,1)的預(yù)測(cè)是非常成功的。3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法3.2灰色系統(tǒng)預(yù)測(cè)模型灰色系統(tǒng)預(yù)測(cè)模型GM(1,1)3.2.1 GM(1,1)的一般形式 設(shè)有變量X(0)X(0)(i)，i=1,2，.，n為某一預(yù)測(cè)對(duì)象的非負(fù)單調(diào)原始數(shù)據(jù)列，為建立灰色預(yù)測(cè)模型：首先對(duì)X(0)進(jìn)行一次累加(1AGO, Acumulated Generating Operator)生成一次累加序列： X(1)X(1)(k)，k1，2，n其中 X(1)(k) X(0)(i) =X(1)(k1)+ X(0)(k)對(duì)X(1)可建立

10、下述白化形式的微分方程： 即GM(1,1)模型。ki 1uaXdtdX)1()1(3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法符號(hào)GM(1,1)的含義如下:G M (1, 1) Grey Model 1階方程 1個(gè)變量上述白化微分方程的解為(離散響應(yīng))：或者式中：k為時(shí)間序列，可取年、季或月aueauXXakk)()1()0()1()1(aueauXXkak)1()1()0()()1()(3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法3.2.2 辨識(shí)算法辨識(shí)算法 記參數(shù)序列為 , a,uT, 可用下式求解: (BTB)-1BTYn式中:B數(shù)據(jù)陣；Yn數(shù)據(jù)列 Yn(X(0)(2), X(0)(3), X(0)()Taaaa3.灰

11、色預(yù)測(cè)方法灰色預(yù)測(cè)方法3.2.3預(yù)測(cè)值的還原預(yù)測(cè)值的還原 由于GM模型得到的是一次累加量，k n+1,n+2,時(shí)刻的預(yù)測(cè)值，必須將GM模型所得數(shù)據(jù) (1)(k+1)(或 (1)(k)經(jīng)過逆生成即累減生成(IAGO)還原為 (0)(k+1)(或 (0)(k)，即：XXXX3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法因?yàn)?，所以 由上式，我們就可以得到預(yù)測(cè)值?；疑A(yù)測(cè)模型的特點(diǎn)在于根據(jù)自身數(shù)據(jù)建立動(dòng)態(tài)微分方程，再預(yù)測(cè)自身的發(fā)展。11)()0()1()1(kiikXX)1()1()()1()()0(kkkXXX3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法3.2.4 模型檢驗(yàn)?zāi)Ｐ蜋z驗(yàn)1.1.事中檢驗(yàn)事中檢驗(yàn) 通常采用殘差檢驗(yàn)、后驗(yàn)差

12、檢驗(yàn)、關(guān)聯(lián)度檢驗(yàn)與級(jí)比偏差檢驗(yàn)。通常采用殘差檢驗(yàn)、后驗(yàn)差檢驗(yàn)、關(guān)聯(lián)度檢驗(yàn)與級(jí)比偏差檢驗(yàn)。2.2.事后檢驗(yàn)事后檢驗(yàn) 事后檢驗(yàn)即預(yù)測(cè)檢驗(yàn)，主要為滾動(dòng)檢驗(yàn)，就是用時(shí)間存在軸上左事后檢驗(yàn)即預(yù)測(cè)檢驗(yàn)，主要為滾動(dòng)檢驗(yàn)，就是用時(shí)間存在軸上左邊的數(shù)據(jù)邊的數(shù)據(jù)( (前面的數(shù)據(jù)前面的數(shù)據(jù)) )建立模型，預(yù)測(cè)下一個(gè)數(shù)據(jù)建立模型，預(yù)測(cè)下一個(gè)數(shù)據(jù)( (后面一個(gè)數(shù)據(jù)后面一個(gè)數(shù)據(jù)) )，以了解其預(yù)測(cè)誤差。，以了解其預(yù)測(cè)誤差。3.2.5 預(yù)測(cè)預(yù)測(cè) 灰色預(yù)測(cè)是通過對(duì)原始數(shù)據(jù)列的處理和灰色模型的建立，對(duì)系統(tǒng)的未來狀態(tài)作出定量預(yù)測(cè)?；翌A(yù)測(cè)可分為五類：灰預(yù)測(cè)可分為五類：3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法1. 數(shù)列預(yù)測(cè)數(shù)列預(yù)測(cè)(Sequen

13、ce Grey Prediction) 級(jí)比級(jí)比 落于可容區(qū)的落于可容區(qū)的(大大)慣性序列慣性序列, 可以直接建立可以直接建立GM(1,1)模模型型, 以預(yù)測(cè)數(shù)據(jù)值的分布以預(yù)測(cè)數(shù)據(jù)值的分布, 稱為數(shù)列灰預(yù)測(cè)。概括的來說稱為數(shù)列灰預(yù)測(cè)。概括的來說, 即為對(duì)數(shù)即為對(duì)數(shù)據(jù)大小進(jìn)行的預(yù)測(cè)。據(jù)大小進(jìn)行的預(yù)測(cè)。2. 災(zāi)變?yōu)淖?異常值異常值)灰預(yù)測(cè)灰預(yù)測(cè)(Calamities Grey Prediction) 對(duì)于級(jí)比不是全部落于可容區(qū)的小慣性序列對(duì)于級(jí)比不是全部落于可容區(qū)的小慣性序列, 對(duì)跳變點(diǎn)時(shí)分布建對(duì)跳變點(diǎn)時(shí)分布建模以預(yù)測(cè)跳變點(diǎn)未來的時(shí)分布稱為災(zāi)變灰預(yù)測(cè)模以預(yù)測(cè)跳變點(diǎn)未來的時(shí)分布稱為災(zāi)變灰預(yù)測(cè), 或異

14、常值灰預(yù)測(cè)?；虍惓Ｖ祷翌A(yù)測(cè)。通俗的說通俗的說, 即為對(duì)一定時(shí)間內(nèi)是否發(fā)生災(zāi)變即為對(duì)一定時(shí)間內(nèi)是否發(fā)生災(zāi)變, 或某種異常的數(shù)據(jù)可或某種異常的數(shù)據(jù)可能發(fā)生在哪些年代的預(yù)測(cè)。能發(fā)生在哪些年代的預(yù)測(cè)。3. 季節(jié)災(zāi)變灰預(yù)測(cè)季節(jié)災(zāi)變灰預(yù)測(cè)(Seasonal Calamities Grey Prediction) 對(duì)發(fā)生在特定時(shí)區(qū)對(duì)發(fā)生在特定時(shí)區(qū)(季節(jié)季節(jié))的事件作時(shí)分布預(yù)測(cè)的事件作時(shí)分布預(yù)測(cè), 稱為季節(jié)災(zāi)變灰預(yù)稱為季節(jié)災(zāi)變灰預(yù)測(cè)。通俗的說測(cè)。通俗的說, 即為對(duì)一年或某個(gè)季節(jié)內(nèi)發(fā)生的災(zāi)變或異常值進(jìn)行即為對(duì)一年或某個(gè)季節(jié)內(nèi)發(fā)生的災(zāi)變或異常值進(jìn)行的預(yù)測(cè)。的預(yù)測(cè)。)()0(k3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法4. 拓?fù)?/p>

15、灰預(yù)測(cè)拓?fù)浠翌A(yù)測(cè)(Topological Grey Prediction) 對(duì)于大幅度擺動(dòng)序列對(duì)于大幅度擺動(dòng)序列, 按點(diǎn)集拓?fù)浠x取時(shí)分布序列按點(diǎn)集拓?fù)浠x取時(shí)分布序列, 作作GM(1,1)建模建模, 預(yù)測(cè)拓?fù)浠臅r(shí)分布預(yù)測(cè)拓?fù)浠臅r(shí)分布, 以達(dá)到預(yù)測(cè)擺動(dòng)序列未來發(fā)展態(tài)勢(shì)的目的以達(dá)到預(yù)測(cè)擺動(dòng)序列未來發(fā)展態(tài)勢(shì)的目的, 稱為拓?fù)浠翌A(yù)測(cè)。它是一種全波形預(yù)測(cè)稱為拓?fù)浠翌A(yù)測(cè)。它是一種全波形預(yù)測(cè), 是整體預(yù)測(cè)。是整體預(yù)測(cè)。5. 系統(tǒng)灰預(yù)測(cè)系統(tǒng)灰預(yù)測(cè)(Systematic Grey Prediction) 由多個(gè)行為變量形成的灰微分方程組由多個(gè)行為變量形成的灰微分方程組, 通過通過GM(1,1)嵌套的方法嵌

16、套的方法, 預(yù)測(cè)多個(gè)行為變量的發(fā)展變化預(yù)測(cè)多個(gè)行為變量的發(fā)展變化, 以避免解高階特征方程之繁以避免解高階特征方程之繁, 稱為系統(tǒng)稱為系統(tǒng)灰預(yù)測(cè)。灰預(yù)測(cè)。3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法3.2.6 灰預(yù)測(cè)的特點(diǎn)灰預(yù)測(cè)的特點(diǎn)(1)灰色模型是一種長期預(yù)測(cè)模型，將預(yù)測(cè)系統(tǒng)中的隨機(jī)元素作為灰色數(shù)據(jù)進(jìn)行處理，而找出數(shù)據(jù)的內(nèi)在規(guī)律。進(jìn)行預(yù)測(cè)所需原始數(shù)據(jù)量小，預(yù)測(cè)精度較高，無須像其它預(yù)測(cè)法要么需要數(shù)據(jù)量大且規(guī)律性強(qiáng)，要么需要憑經(jīng)驗(yàn)給出系數(shù)。(2)理論性強(qiáng)，計(jì)算方便，籍助計(jì)算機(jī)及其程序設(shè)計(jì)語言或相關(guān)軟件間接計(jì)算，使得數(shù)據(jù)處理簡便、快速、準(zhǔn)確性好。(3)用有限的表征系統(tǒng)行為特征的外部元素，分析系統(tǒng)的內(nèi)在規(guī)律。灰色系統(tǒng)

17、理論采用對(duì)系統(tǒng)的行為特征數(shù)據(jù)進(jìn)行生成的方法，對(duì)雜亂無章的系統(tǒng)的行為特征數(shù)據(jù)進(jìn)行處理，從雜亂無章的現(xiàn)像中發(fā)現(xiàn)系統(tǒng)的內(nèi)在規(guī)律，這是該方法的獨(dú)特之處。(4) 適用性強(qiáng)。用灰色模型既可對(duì)周期性變化的系統(tǒng)行為進(jìn)行預(yù)測(cè)，亦可對(duì)非周期性變化的系統(tǒng)行為進(jìn)行預(yù)測(cè)；既可進(jìn)行宏觀長期的預(yù)測(cè)，亦可用于微觀短期的預(yù)測(cè)。3.灰色預(yù)測(cè)方法灰色預(yù)測(cè)方法3.2.7 我所了解到灰預(yù)測(cè)的應(yīng)用4.灰色關(guān)聯(lián)度分析灰色關(guān)聯(lián)度分析4.1關(guān)聯(lián)度的意義 對(duì)兩個(gè)系統(tǒng)或兩個(gè)因素之間關(guān)聯(lián)性大小的度量，稱為關(guān)聯(lián)度。它描述系統(tǒng)發(fā)展過程中因素間相對(duì)變化的情況，也就是變化大小、方向及速度等指標(biāo)的相對(duì)性。如果兩者在系統(tǒng)發(fā)展過程中相對(duì)變化基本一致，則認(rèn)為兩者關(guān)

18、聯(lián)度大；反之，兩者關(guān)聯(lián)度就小。 可見，灰色關(guān)聯(lián)度分析是對(duì)于一個(gè)系統(tǒng)發(fā)展變化態(tài)勢(shì)的定量描述和比較。只有弄清楚系統(tǒng)或因素間的這種關(guān)聯(lián)關(guān)系，才能對(duì)系統(tǒng)有比較透徹的認(rèn)識(shí)，分清哪些是主導(dǎo)因素，哪些是潛在因素，哪些是優(yōu)勢(shì)而哪些又是劣勢(shì)。 所以，對(duì)一個(gè)灰色系統(tǒng)進(jìn)行分析研究時(shí)，首先要解決如何從隨機(jī)的時(shí)間序列中找到關(guān)聯(lián)性，計(jì)算關(guān)聯(lián)度，以便為因素判別、優(yōu)勢(shì)分析和預(yù)測(cè)精度檢驗(yàn)等提供依據(jù)，為系統(tǒng)決策打好基礎(chǔ)。因此說，灰色因素間的關(guān)聯(lián)度分析，實(shí)質(zhì)上是灰色系統(tǒng)分析、預(yù)測(cè)、決策的基礎(chǔ)。4.灰色關(guān)聯(lián)度分析灰色關(guān)聯(lián)度分析4.2 比較比較 一般的抽象系統(tǒng)都包含有許多影響因素，多種因素共同作用的結(jié)果決定了系統(tǒng)的發(fā)展態(tài)勢(shì)。我們希望從

19、眾多的因素中判斷出：哪些是主要因素、哪些是次要因素。這些都是系統(tǒng)分析的內(nèi)容，數(shù)理統(tǒng)計(jì)中的回歸分析、方差分析、主成分分析、相關(guān)分析等都可以用來進(jìn)行此類系統(tǒng)分析。這些方法的不足之處是：1、要求有大量的數(shù)據(jù)。 2、要求樣本服從某一種典型概率分布，各因素?cái)?shù)據(jù)與系統(tǒng)特征數(shù)據(jù)之間呈線性關(guān)系且各因素之間彼此無關(guān)。3、計(jì)算量大4、可能出現(xiàn)量化結(jié)果與定性分析結(jié)果不符的情況。4.灰色關(guān)聯(lián)度分析灰色關(guān)聯(lián)度分析4.3 灰色關(guān)聯(lián)度分析的基本特征灰色關(guān)聯(lián)度分析的基本特征n 建立的模型屬于非函數(shù)形式的序列模型n 計(jì)算方便易行n 對(duì)樣本數(shù)量多寡沒有嚴(yán)格要求n 不要求序列數(shù)據(jù)必須符合正態(tài)分布n 不會(huì)產(chǎn)生與定性分析大相徑庭的結(jié)論

20、4.灰色關(guān)聯(lián)度分析灰色關(guān)聯(lián)度分析4.4 灰色關(guān)聯(lián)度分析過程灰色關(guān)聯(lián)度分析過程 關(guān)聯(lián)度分析是灰色系統(tǒng)的分析和處理隨機(jī)量的一種方法也是數(shù)據(jù)到數(shù)據(jù)的“映射”。利用灰色理論識(shí)別機(jī)械故障的過程中所采用的判別函數(shù)即為未知模式與標(biāo)準(zhǔn)模式之間的關(guān)聯(lián)度計(jì)算式。其標(biāo)準(zhǔn)模式是在理論分析和經(jīng)驗(yàn)統(tǒng)計(jì)的基礎(chǔ)上參考前人工作建立的，適合于現(xiàn)場初步診斷，其主要優(yōu)點(diǎn)是：通用性較強(qiáng)，技術(shù)人員可在沒有任何先驗(yàn)經(jīng)驗(yàn)的條件下，利用標(biāo)準(zhǔn)故障模式對(duì)機(jī)械故障進(jìn)行診斷。從統(tǒng)計(jì)意義上講，有其可信度和準(zhǔn)確度。關(guān)聯(lián)度分析方法的優(yōu)點(diǎn)為：1）不追求大樣本量；2）不要求數(shù)據(jù)有特殊的分布；3）計(jì)算量比回歸分析小得多；4）可以得到較多的信息。4.灰色關(guān)聯(lián)度分析

21、灰色關(guān)聯(lián)度分析 在作關(guān)聯(lián)度分析之前，必須建立參考模式（標(biāo)準(zhǔn)故障模式），并且進(jìn)行數(shù)據(jù)化而形成標(biāo)準(zhǔn)故障模式序列矩陣 ，具體形式如下： 其中：n系統(tǒng)所要建立的標(biāo)準(zhǔn)故障模式的個(gè)數(shù)，對(duì)于簡易診斷n=2，對(duì)于精密診斷取 ；k每種標(biāo)準(zhǔn)模式所含的模式向量的個(gè)數(shù)。令 為待檢模式序列，)(0kXi3n )(),.,2(),1 ()(kXXXkXjjjjmj,.,2, 14.灰色關(guān)聯(lián)度分析灰色關(guān)聯(lián)度分析 其中：m待檢模式的個(gè)數(shù)，k每種待檢模式所含模式向量的個(gè)數(shù)，取值大小同前； -表示第j個(gè)待檢模式，則關(guān)聯(lián)系數(shù)的表達(dá)式如下： ：待檢模式序列Xj與標(biāo)準(zhǔn)模式序列X0j在第k點(diǎn)的關(guān)聯(lián)系數(shù)。 k1：分辨系數(shù)，在故障診斷領(lǐng)域根

22、據(jù)經(jīng)驗(yàn)取0.5左右，令 為 對(duì) 的關(guān)聯(lián)度，其計(jì)算式如下：)(kXj)(kjijijXiX0kkjijikk1)(14.灰色關(guān)聯(lián)度分析灰色關(guān)聯(lián)度分析在所求的關(guān)聯(lián)度排序中如果 則認(rèn)為X1狀態(tài)最危險(xiǎn)，X2次之，X3再次。iii3214.灰色關(guān)聯(lián)度分析灰色關(guān)聯(lián)度分析4.5 灰色關(guān)聯(lián)度分析實(shí)例灰色關(guān)聯(lián)度分析實(shí)例題外話題外話工作實(shí)例：戰(zhàn)區(qū)導(dǎo)彈技術(shù)室有三臺(tái)導(dǎo)彈檢測(cè)車，但是戰(zhàn)區(qū)下轄5個(gè)省，導(dǎo)彈部隊(duì)眾多，為了保證日常戰(zhàn)備與訓(xùn)練，保障裝備及時(shí)檢測(cè)，發(fā)現(xiàn)并排除故障，如何在現(xiàn)有的條件下解決檢測(cè)車不夠，檢測(cè)修理不及時(shí)的難題？THE SCIENTIFIC BACKGROUND FOR APPEARANCE OFGREY

23、SYSTEMS THEORY At the end of the 1940s, there appeared general systems theory,information theory, cybernetics and operations research.During the 1950s and the 1960s, systems dynamics and thetheory of dissipative structures have been put forwarded.During the 1970s, there appeared one by one such new

24、transfield and interfiled theories of systems science as thesynergetics, catastrophe theory, hypercycle theory, geneticalgorithms, chaos theory and fractal theory, etc. When people investigating a systems, due to both theexistence of internal and external disturbances and the limitation of our under

25、standing, the available information tends to contain various kinds of uncertainty and noises. Along with the development of science and technology and the progress of the mankind, our understanding of uncertainties of systems has been gradually deepened and the research of uncertain systems has reac

26、hed at a new height. During the second half of the 20th century, in the areas of systems science and systems engineering, the seemingly non-stoppable emergence of various theories and methodologies of unascertained systems has been a great scene. For instance, L. A. Zadeh established fuzzy mathemati

27、cs in the 1960s, Julong Deng developed grey systems theory andTHE SCIENTIFIC BACKGROUND FOR APPEARANCE OFGREY SYSTEMS THEORYZ. Pawlak advanced rough set theory in the 1980s, etc. All these works represent some of the most important efforts in the research of uncertain systems of this time period. Fr

28、om different angles, these works provide the theories and methodologies for describing and dealing with uncertain information. The grey systems theory, established by Julong Deng in1982, is a new methodology that focuses on the study of problems involving small samples and poor informaion. It deals

29、with uncertain systems with partially known information through generating, excavating, and extracting useful information from what is available. So, systems operational behaviors and their laws of evolution can be correctly described and effectively monitored. In the natural world, uncertain system

30、s with small samples and poor information exist commonly. That fact determines the wide range of applicability of grey systems theory.THE DEVELOPMENT HISTORY AND CURRENT STATE OFGREY SYSTEMS THEORY In 1982, Systems & Control Letters, an international journal by North-Holland, published the first pap

31、er in grey systems theory, The Control Problems of Grey Systems, by Julong Deng. In the same year, the Journal of Huazhong University ofScience and Technology published the first paper, also by Julong Deng, on grey systems theory in Chinese language. The publication of these papers signaled the offi

32、cial appearance of the cross disciplinary grey systems theory. As soon as these works appeared, they immediately caught the attention of many scholars and scientific practitioners from across the world. Numerous well-known scientists strongly supported the validity and livelihood of such research. M

33、any young scholars actively participated in the investigation of grey systems theory. With great enthusiasm these young men and women carried thetheoretical aspects of the theory to new heights and employed their exciting results to various fields of application. In particular, successful applicatio

34、ns in great many fields have won the attention of the international world of learning. Currently, a great number of scholars from China, United States, England, Romania, South Africa, Germany, Japan, Australia, Canada, Poland, Spain, Cuba, Korea, Russia, Turkey, the Netherlands, Iran, andTHE DEVELOP

35、MENT HISTORY AND CURRENT STATE OFGREY SYSTEMS THEORYothers, have been involved in the research and application of grey systems theory. In 1989, the British journal, The Journal of Grey System, was launched. Currently, this publication is indexed by INSPEC (formerly Science Abstracts) of England, Mat

36、hematical Review of the United States, Science Citation Index, and other important indexing agencies from around the world. In 1997, a Chinese publication, named Journal of Grey System, is launched in Taiwan. It is later in 2004 that this publication becomes all English. In 2011, a new journal, name

37、d Grey Systems: Theory and application, edited by the faculty of Institute for Grey Systems Studies, Nanjing University of Aeronautics and Astronautics, launched by Emerald of England. There are currently thousands of different professional journals in the world that have accepted and published pape

38、rs in grey systems theory. As of this writing, a journal of the Association for Computing Machinery (USA), Communications in Fuzzy Mathematics (Taiwan), Kybernetes: The International Journal of Systems & Cybernetics, have published special issues on grey systems theory respectively.CHARACTERISTICS O

39、F UNASCERTAINED SYSTEMS The fundamental characteristic of uncertain systems is the incompleteness and inadequacy in their information. Due to the dynamics of system evolutions, the biological limitations of the human sensing organs, and the constraints of relevant economic conditions and technologic

40、al availabilities, uncertain systems exist commonly.A. Incomplete InformationIncompleteness in information is one of the fundamental characteristics of uncertain systems. The situation involving incomplete system information can have the following four cases: The information about the elements (para

41、meters) is incomplete; The information about the structure of the system is incomplete; The information about the boundary of the system is incomplete; The information on the systems behaviors is incomplete.CHARACTERISTICS OF UNASCERTAINED SYSTEMSB. Inaccuracies in Data Another fundamental character

42、istic of uncertain systems is the inaccuracy naturally existing in the available data. The meanings of uncertain and inaccurate are roughly the same. They both stand for errors or deviations from the actual data values. From the essence of how uncertainties are caused, they can be categorized into t

43、hree types: the conceptual, level, and prediction types.1)The Conceptual Type Inaccuracies of the conceptual type item from the expression about a certain event, object, concept, or wish. For instance, all such frequently used concepts as large, small,“ many, few, high, low, fat, thin, good, bad,“ y

44、oung, beautiful, etc., are inaccurate due to the lack of clear defmition. It is very difficult to use exact quantities to express these concepts. As a second example, suppose that a job seeker with an MBA degree wishes to get an offer of an annual salary of no less than $150,000. A manufacturing fir

45、m plans to control its rate of deficient products to be less than 0.01%. These are all cases of inaccurate wishes.CHARACTERISTICS OF UNASCERTAINED SYSTEMS 2) The Level Type This kind of inaccuracy of data is caused by a change in the level of research or observation. The available data, when seen on

46、 the level of the system of concern, that is the macroscopic level, or on the level of the whole, or in broad outline level of cognitive, might be accurate. However, when they are seen on a lower level, that is a microscopic level or a partial localized level of the system, they generally become ina

47、ccurate. For example, the height of a person can be measured accurately to the unit of centimeters or millimeters. However, if the measurement has to be accurate to the level of one ten-thousandth micron, the earlier accurate reading will become extremely inaccurate.3) The Prediction Type (The Estim

48、ation Type)Because it is difficult to completely understand the laws of evolution, prediction of the future tends to be inaccurate. For instance, it is estimated that two years from now, the GDP of a certain specified area will surpass $10 billion; it is estimated thatCHARACTERISTICS OF UNASCERTAINE

49、D SYSTEMSa certain bank will attract as much savings from individual residents in an amount between $70 billion to $90 billion for the year in 2015; it is predicted that in the coming years the temperature in Nanjing during the month of October will not go beyond 30C; etc. All these examples provide

50、 the uncertain numbers of the prediction type. In statistics, it is often the case that samples are collected to estimate the whole. So, many statistical data are inaccurate. As a matter of fact, no matter what method is used, it is very difficult for anyone to obtain the absolutely accurate (estima

51、ted) value. When we draw out plans for the future and make decisions about what course of action to take, we in general have to rely on predictions and estimates which are not completely accurate.ELEMENTARY CONCEPTS AND FUNDAMENTALPRINCIPLES OF GREY SYSTEMS A. Elementary Concepts of Grey Systems Man

52、y social, economic, agricultural, industrial, ecological,biological, etc., systems are named by considering the features of classes of the research objects, while grey systems are labeled using the color of the systems of concern. In the theory of control, people often make use of colors to describe

53、 the degree of clearness of the available information.For instance, Ashby refers the objects with unknown internal information to as black boxes. This terminology has been widely accepted in the scientific community. For example, as a society moves toward democracy, the citizens gradually demand mor

54、e information regarding the formation of policies and more in depth meaning of the policies. That is, the citizenswant to have an increased degree of transparency. We use black to indicate unknown information, white the completely known information, and grey the partially known and partially unknown

55、 information. Accordingly, the systems with completely known information will be regarded as white, those systems with completely unknown information black, and the systems with partially known information and partially unknown information will be seen as grey.ELEMENTARY CONCEPTS AND FUNDAMENTALPRIN

56、CIPLES OF GREY SYSTEMS At this junction, we need to pay attention to the difference between systems and boxes. Usually, boxes are used when one does not pay much attention on or does not attempt to utilize the information regarding the interior while focusing on the external characteristics. In this

57、 case, the researcher generally investigates the properties and characteristics of the object through analyzing the input-output relation. Other the other hand, systems are employed to indicate the study of the objects structure and functions through analyzing the existing organic connections betwee

58、n the object, relevant factors, and its environment and the related laws of change. The research objects of grey systems theory consist of such uncertain systems that they are known only partially with small samples and poor information. The theory focuses on the generation and excavation of the par

59、tially known information to materialize the accurate description and understanding of the material world. Incompleteness in information is the fundamental meaning of being grey. From different angles and in varied situations, the meaning of grey can be expanded or stretched. For this end, see the de

60、tails in Table I.TABLE I. EXTENSIONS OF THE CONCEPT OF GREYELEMENTARY CONCEPTS AND FUNDAMENTALPRINCIPLES OF GREY SYSTEMSB. Fundamental Principles of Grey Systems In the process of establishing the grey systems theory, Professor Julong Deng discovered and extracted the following fundamental principle