本文選題：局部影響分析 + 空間位移函數(shù)��； 參考：《云南財(cái)經(jīng)大學(xué)》2017年碩士論文

【摘要】：在多元非參數(shù)回歸問(wèn)題中,有可能存在這樣的情形:響應(yīng)變量?jī)H僅通過(guò)自變量的少數(shù)幾個(gè)線性組合與自變量發(fā)生關(guān)聯(lián).在這樣的情形下,找出這些自變量的線性組合可以降低回歸的維數(shù),從而帶來(lái)一些回歸結(jié)果的改善,比如:提高回歸曲線擬合的精度、可視化,等等。各種充分降維方法的提出正是為了解決這個(gè)問(wèn)題。在這些充分降維方法中,切片逆回歸和切片平均方差估計(jì)方法是較為常用的兩種方法。然而,當(dāng)逆條件均值和逆條件方差為常量時(shí),這兩種方法均會(huì)失效。切片平均三階矩估計(jì)方法的提出解決了這個(gè)問(wèn)題并受到了廣泛的關(guān)注。這種方法的使用需要估計(jì)自變量向量的條件三階矩,所以,研究該方法的敏感性問(wèn)題是有必要的。本文關(guān)注切片平均三階矩估計(jì)法下中心子空間估計(jì)量的局部影響分析。本文在切片平均三階矩估計(jì)法下提出的局部影響分析方法基于一個(gè)空間位移函數(shù),該函數(shù)用于度量模型被擾動(dòng)前后的中心子空間估計(jì)之間的差異。我們構(gòu)建了一個(gè)切片平均三階矩估計(jì)法局部影響分析的基本理論框架,這個(gè)框架下的所有關(guān)鍵量(如:擬曲率和強(qiáng)影響方向)的表達(dá)式都可以獲得。在此框架下,局部影響評(píng)價(jià)統(tǒng)計(jì)量——最強(qiáng)影響方向,可以通過(guò)最小化擬曲率輕易地獲取,因?yàn)楹笳呖梢员硎緸閿_動(dòng)方向的一個(gè)二次型。因此,這個(gè)方法的計(jì)算負(fù)擔(dān)較輕。為了評(píng)價(jià)各個(gè)樣本點(diǎn)對(duì)中心子空間估計(jì)的影響,我們?cè)O(shè)計(jì)了一個(gè)擾動(dòng)方案,并在這個(gè)擾動(dòng)方案下推導(dǎo)出了擬曲率和最強(qiáng)影響方向的具體表達(dá)式。為了說(shuō)明本文所提出的上述方法,我們將其應(yīng)用于一組模擬數(shù)據(jù),該數(shù)據(jù)從一個(gè)經(jīng)典的模型中產(chǎn)生,該模型中自變量向量的逆條件均值和逆條件方差均為常量。在這個(gè)模型下,切片平均三階矩估計(jì)表現(xiàn)良好,而切片逆回歸和切片平均方差估計(jì)方法均失效。模擬結(jié)果顯示,本文提出的局部影響分析方法可以成功地識(shí)別出人為設(shè)置的異常點(diǎn),此外,模擬結(jié)果還展示出了一些有趣的新發(fā)現(xiàn)。
[Abstract]:In multivariate nonparametric regression problems, it is possible that response variables are associated with independent variables only through a few linear combinations of independent variables. In this case, finding out the linear combination of these independent variables can reduce the dimension of regression and bring about some improvement of regression results, such as improving the precision of regression curve fitting, visualization, and so on. All kinds of sufficient dimensionality reduction methods are proposed to solve this problem. Among these sufficient dimensionality reduction methods, slice inverse regression and slice mean variance estimation are two common methods. However, when the inverse conditional mean and inverse conditional variance are constant, both methods will fail. The method of slice average third order moment estimation solves this problem and is paid more and more attention. The use of this method requires the estimation of conditional third-order moments of independent variable vectors, so it is necessary to study the sensitivity of the method. This paper focuses on the analysis of the local influence of the central subspace estimator under the third order moment estimation of slice average. In this paper, the local impact analysis method based on slice average third-order moment estimation is based on a spatial displacement function, which is used to measure the difference between the central subspace estimates before and after the model is disturbed. We construct a basic theoretical framework for local impact analysis of slice average third-order moment estimation, in which expressions of all key quantities (such as quasi curvature and strong influence direction) can be obtained. In this framework, the local impact evaluation statistics-the strongest direction of influence-can be easily obtained by minimizing quasi curvature, which can be expressed as a quadratic form of the direction of disturbance. Therefore, the computational burden of this method is relatively light. In order to evaluate the influence of each sample point on the estimation of the central subspace, we design a perturbation scheme and derive the specific expressions of the quasi curvature and the direction of the strongest influence under the perturbation scheme. In order to illustrate the above method proposed in this paper we apply it to a set of simulation data which is generated from a classical model in which the inverse conditional mean and inverse conditional variance of the independent variable vector are constant. In this model, slice average third-order moment estimation is good, while slice inverse regression and slice average variance estimation are invalid. The simulation results show that the proposed local impact analysis method can successfully identify the artificial outliers. In addition, the simulation results also show some interesting new findings.
【學(xué)位授予單位】：云南財(cái)經(jīng)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O212.1
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本文編號(hào)：1921626