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通过正则化核估计实现距离收缩与欧氏嵌入AI 解读

Summary Although recovering a Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based on the so-called regularized kernel estimate. We show that such an estimate can be characterized as simply applying a constant amount of shrinkage to all observed pairwise distances. This fact allows us to establish risk bounds for the…

通过似然基追踪进行变量选择与模型构建AI 解读

This article presents a nonparametric penalized likelihood approach for variable selection and model building, called likelihood basis pursuit (LBP). In the setting of a tensor product reproducing kernel Hilbert space, we decompose the log-likelihood into the sum of different functional components such as main effects and interactions, with each component represented by appropriate basis functions. Basis functions are chosen to be compatible with variable selection and model…

多元伯努利观测的平滑样条方差分析及眼科应用AI 解读

We combine a smoothing spline analysis of variance (SS-ANOVA) model and a log-linear model to build a partly flexible model for multivariate Bernoulli data. The joint distribution conditioning on the predictor variables is estimated. The log odds ratio is used to measure the association between outcome variables. A numerical scheme based on the block one-step successive over relaxation SOR–Newton-Ralphson algorithm is proposed to obtain an approximate solution for the…

大规模伯努利数据的平滑样条 ANOVA 模型与随机化 GACVAI 解读

We propose the randomized Generalized Approximate Cross Validation (ranGACV) method for choosing multiple smoothing parameters in penalized likelihood estimates for Bernoulli data. The method is intended for application with penalized likelihood smoothing spline ANOVA models. In addition we propose a class of approximate numerical methods for solving the penalized likelihood variational problem which, in conjunction with the ranGACV method allows the application of smoothing…

混合自适应样条AI 解读

An adaptive spline method for smoothing is proposed that combines features from both regression spline and smoothing spline approaches. One of its advantages is the ability to vary the amount of smoothing in response to the inhomogeneous “curvature” of true functions at different locations. This method can be applied to many multivariate function estimation problems, which is illustrated by an application to smoothing temperature data on the globe. The method's performance…

采用张量积薄板样条的半参数方差分析AI 解读

SUMMARY We consider fitting multivariate models to response data based on analysis-of-variance (ANOVA)-like decompositions of functions of several variables, f(t 1,..., td) = C + σα f α(ta + σα < β f αβ(t α, t β) + · · · · A theory for fitting (some components of) these models with polynomial smoothing splines exists when each tα is in a subset of the real line. In this case the various estimated components turn out to be certain tensor sums and products of polynomial…

半参数部分样条与广义样条模型中的参数零模型检验AI 解读

Cox and Koh (1986) considered the model $y_i = f(x(i)) + \varepsilon_i, \varepsilon_i$ i.i.d. $N(0, \sigma^2)$, with the (parametric) null hypothesis $f(x), x \in \lbrack 0, 1 \rbrack$, a polynomial of degree $m - 1$ or less, versus the alternative $f$ is "smooth," based on the Bayesian model for $f$ which leads to polynomial smoothing spline estimates for $f$. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We…

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广义样条平滑中选择平滑参数的 GCV 与 GML 比较AI 解读

The partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood (GML) estimate for the smoothing parameter. Then this estimate is compared with the generalized cross validation (GCV) estimate both analytically and by Monte Carlo methods. The comparison is based on a predictive mean square error criteria. It is shown that if the true, unknown function being estimated is smooth in a sense to be defined then the GML…

由二维截面估计三维肿瘤尺寸分布的交叉验证样条方法AI 解读

We study the problem of estimating the distribution of the three-dimensional radiuses of a collection of spheres, given measurements of the two-dimensional radiuses of a sample of planar cross sections. This problem arises in the estimation of the tumor size distribution of spherical microtumors induced in mouse livers following injection of a carcinogen. We first convert this problem to a form suitable for the application of cross-validated spline methods for the solution…

交叉验证平滑样条的贝叶斯“置信区间”AI 解读

SUMMARY We consider the model Y(ti)=g(ti) + ∈i, i = 1, 2, …, n, where g(t), t ∈ [0, 1] is a smooth function and the {∈i} are independent N(0, σ2) errors with σ2 unknown. The cross-validated smoothing spline can be used to estimate g non-parametrically from observations on Y(ti), i= 1, 2, …, n, and the purpose of this paper is to study confidence intervals for this estimate. Properties of smoothing splines as Bayes estimates are used to derive confidence intervals based on…

正交级数密度估计的数据驱动最优平滑AI 解读

Let $f$ be a density possessing some smoothness properties and let $X_1,\cdots, X_n$ be independent observations from $f$. Some desirable properties of orthogonal series density estimates $f_{n,m,\lambda}$ of $f$ of the form $f_{n,m,\lambda}(t) = \sum^n_{\nu = 1} \frac{\hat{f}_\nu}{(1 + \lambda\nu^{2m})} \phi_\nu(t)$ where $\{\phi_\nu\}$ is an orthonormal sequence and $\hat{f}_\nu = (1/n)\sum^n_{j=1} \phi_\nu(X_j)$ is an estimate of $f_\nu = \int \phi_\nu(t)f(t) dt$, are…

对数周期图的自动平滑AI 解读

We develop an objective optimum smoothing procedure for an estimate of the log spectral density, based on smoothing the log periodogram with a smoothing spline. We call the result the optimally smoothed spline (OSS) estimate. We show that an unbiased estimate [Rcirc](λ) of the expected integrated mean square error can be obtained as a function of the smoothing, or “bandwidth” parameter λ. The smoothing parameter then is chosen as that λ that minimizes [Rcirc](λ). The degree…

对数周期图的自动平滑AI 解读

We develop an objective optimum smoothing procedure for an estimate of the log spectral density, based on smoothing the log periodogram with a smoothing spline. We call the result the optimally smoothed spline (OSS) estimate. We show that an unbiased estimate [Rcirc](λ) of the expected integrated mean square error can be obtained as a function of the smoothing, or “bandwidth” parameter λ. The smoothing parameter then is chosen as that λ that minimizes [Rcirc](λ). The degree…

非正常先验、样条平滑与回归模型误差防范AI 解读

Summary Spline and generalized spline smoothing is shown to be equivalent to Bayesian estimation with a partially improper prior. This result supports the idea that spline smoothing is a natural solution to the regression problem when one is given a set of regression functions but one also wants to hedge against the possibility that the true model is not exactly in the span of the given regression functions. A natural measure of the deviation of the true model from the span…

以次序统计量为节点的直方样条密度估计AI 解读

Summary The histospline considered here is a quadratic spline density estimate analogous to the Boneva–Kendall–Stefanov (BKS) histospline, with their equally spaced knots replaced by knots at every knth-order statistic. This estimate can be expected to be relatively more “flexible” where the density is greatest. We obtain “mean square error at a point” convergence rates. The rates obtained are uniform over classes of densities which have first (alternatively second or third)…

密度估计的插值样条方法（一）：等距节点AI 解读

Statistical properties of a variant of the histospline density estimate introduced by Boneva-Kendall-Stefanov are obtained. The estimate we study is formed for $x$ in a finite interval, $x\in\lbrack a, b\rbrack = \lbrack 0, 1\rbrack$ say, by letting $\hat{F}_n(x), x \epsilon\lbrack 0, 1\rbrack$ be the unique cubic spline of interpolation to the sample cumulative distribution function $F_n(x)$ at equi-spaced points $x = jh, j = 0, 1,\cdots, l + 1, (l + 1)h = 1$, which…

可变结点、核与正交级数密度估计方法的最优收敛性质AI 解读

Let $W_p^{(m)}(M) = \{f: f^{(\nu)} \operatorname{abs}. \operatorname{cont}., \nu = 0, 1,\cdots, m - 1, f^{(m)} \in \mathscr{L}_p, \|f^{(m)}\|_p \leqq M\}$, where $\|\cdot\|_p$ is the norm in $\mathscr{L}_p, m$ is a positive integer and $p$ is a real number, $p \geqq 1$. Let $\{\hat{f}_n(x)\}, n = 1, 2,\cdots$ be any sequence of estimates of a density at the point $x$ where $\hat{f}_n(x)$ depends on $n$ independent observations from some density $f \in W_p^{(m)}(M)$. It is…

若干核等价类的回归设计AI 解读

Earlier results on asymptotically optimal sequences of regression designs for autoregressive stochastic processes are extended (nearly) to the equivalence classes of such processes.

平稳多元时间序列的若干独立性检验AI 解读

Summary Let X(t)=(X1(t)X2(t)⋮XP(t)),t=⋯−1,0,1…, be a p-dimensional zero-mean stationary Gaussian time series possessing a spectral density matrix F(ω) = {fij(ω)}, i, j = 1, 2, …, p, satisfying some mild regularity conditions. Let F^(ωl), l=1, 2, … ,M, F^(ωl)={f^ij(ωl)} be suitably defined sample spectral density matrices for M values of ω, based on a record of length T ≫ M. We consider the following (null) hypotheses: H1: Xi(t), Xi(s) independent if i ≠ j, all s, t, H2:…

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