ON ALMOST LINEARITY OF LOW-DIMENSIONAL PROJECTIONS FROM HIGH-DIMENSIONAL DATA
ON ALMOST LINEARITY OF LOW-DIMENSIONAL PROJECTIONS FROM HIGH-DIMENSIONAL DATA
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DOI：
10.1214/aos/1176349155
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发表时间：
1993-06-01
影响因子：
4.5
通讯作者：
LI, KC
中科院分区：
文献类型：
--
作者：
HALL, P;LI, KC
本文研究了高维数据的低维投影形状。标准化后，设x为均值为零且协方差相同的p维随机变量。对于投影β 'x，\\beta\\ = 1，找到另一个方向B，使b' x对β 'x的回归曲线尽可能非线性。我们证明了当x的维数很大时，对于大多数方向β，即使是最非线性的回归仍然是接近线性的.我们的方法依赖于一对p维随机变量w1，w2的构造，称为旋转孪生，以及它相对于标准正态密度的密度函数.有了这个，我们能够获得封闭形式的表达式，用于测量偏离正态分布和偏离线性在适当的意义上的平均。作为一个有趣的副产品，从给定的一组数据中，我们可以找到E（f（beta 'x）（t）/phi 1（t）-1）2和E[（\\E（x\beta，beta' x = t）\\2-t2）f（beta 'x）2（t）/phi 1（2）t）]的简单无偏估计，其中phi 1是标准正态密度，f（beta' x）是beta 'x的密度，“E”是相对于均匀分布的β取的。这是在没有任何平滑的情况下实现的，也没有诉诸任何费力的投影程序，例如大图尔斯。我们的结果与Diaconis和Freedman的工作有关，并讨论了我们的结果对数据分析的几个方面的影响。例如，当回归模型的链接函数可能严重错误时，它有助于建立回归分析的有效性。进一步的推广，用B =（beta1，.，beta（k）），对于k个随机选择的标准正交向量（beta（i），i = 1，.，k），拓宽了切片逆回归（SIR）的应用范围。
This paper studies the shapes of low dimensional projections from high dimensional data. After standardization, let x be a p-dimensional random variable with mean zero and identity covariance. For a projection beta'x, \\beta\\ = 1, find another direction b so that the regression curve of b'x against beta'x is as nonlinear as possible. We show that when the dimension of x is large, for most directions beta even the most nonlinear regression is still nearly linear.Our method depends on the construction of a pair of p-dimensional random variables, w1, w2, called the rotational twin, and its density function with respect to the standard normal density. With this, we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average. As an interesting by-product, from a given set of data we can find simple unbiased estimates of E(f(beta'x)(t)/phi1(t)-1)2 and E[(\\E(x\beta, beta'x = t)\\2-t2)f(beta'x)2(t)/phi1(2)t)], where phi1 is the standard normal density, f(beta'x) is the density for beta'x and the ''E'' is taken with respect to the uniformly distributed beta. This is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours. Our result is related to the work of Diaconis and Freedman.The impact of our result on several fronts of data analysis is discussed. For example, it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong. A further generalization, which replaces beta'x by B'x with B = (beta1,...,beta(k)) for k randomly selected orthonormal vectors (beta(i), i = 1,...,k), helps broaden the scope of application of sliced inverse regression (SIR).