Pattern recognition based on scale invariant discriminant functions

Pukkila, Tarmo; Rao, C. Radhakrishna

doi:10.1016/0020-0255(88)90012-6

Cited by 20 publications

(20 citation statements)

References 3 publications

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“…In addition to p‐values and the critical value, we are also be interested in calculating the power

P_{1} false(R > r false) = P_{1} false(\tilde{y} \in {double-struck𝕋}_{r} false)

, for a critical value r , with

P_{1} false(\cdot false)

a probability calculated under some alternative hypothesis. By arguments due to Pukkila and Rao (), under alternative hypotheses,

\overset{boldy}{true}

follows an angular Gaussian distribution on

double-struck𝕊

, which means that

P_{1} false(R > r false) = \int_{{double-struck𝕋}_{r}} f normald ς

, where

f : 𝕊 \to ℝ

is the density of the angular Gaussian density (of course, under the null hypothesis

f \equiv 1

. )…”

Section: Methodsmentioning

confidence: 99%

Likelihood Ratio Tests for a Dose-Response Effect Using Multiple Nonlinear Regression Models

Gutjahr

Bornkamp

2016

Biometrics

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We consider the problem of testing for a dose-related effect based on a candidate set of (typically nonlinear) dose-response models using likelihood-ratio tests. For the considered models this reduces to assessing whether the slope parameter in these nonlinear regression models is zero or not. A technical problem is that the null distribution (when the slope is zero) depends on non-identifiable parameters, so that standard asymptotic results on the distribution of the likelihood-ratio test no longer apply. Asymptotic solutions for this problem have been extensively discussed in the literature. The resulting approximations however are not of simple form and require simulation to calculate the asymptotic distribution. In addition, their appropriateness might be doubtful for the case of a small sample size. Direct simulation to approximate the null distribution is numerically unstable due to the non identifiability of some parameters. In this article, we derive a numerical algorithm to approximate the exact distribution of the likelihood-ratio test under multiple models for normally distributed data. The algorithm uses methods from differential geometry and can be used to evaluate the distribution under the null hypothesis, but also allows for power and sample size calculations. We compare the proposed testing approach to the MCP-Mod methodology and alternative methods for testing for a dose-related trend in a dose-finding example data set and simulations.

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“…In addition to p‐values and the critical value, we are also be interested in calculating the power

P_{1} false(R > r false) = P_{1} false(\tilde{y} \in {double-struck𝕋}_{r} false)

, for a critical value r , with

P_{1} false(\cdot false)

a probability calculated under some alternative hypothesis. By arguments due to Pukkila and Rao (), under alternative hypotheses,

\overset{boldy}{true}

follows an angular Gaussian distribution on

double-struck𝕊

, which means that

P_{1} false(R > r false) = \int_{{double-struck𝕋}_{r}} f normald ς

, where

f : 𝕊 \to ℝ

is the density of the angular Gaussian density (of course, under the null hypothesis

f \equiv 1

. )…”

Section: Methodsmentioning

confidence: 99%

Likelihood Ratio Tests for a Dose-Response Effect Using Multiple Nonlinear Regression Models

Gutjahr

Bornkamp

2016

Biometrics

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“…Pukkila and Rao (1988). The g(u|x + tu) can be shown to be an even function around a symmetry point given by t min = (μ f − x) −1 f u/u −1 f u.…”

Section: Proposals Q With Unit Acceptance Ratementioning

confidence: 97%

“…The resulting g(u|x) is referred to as the Angular Gaussian Distribution, see e.g. Watson (1983) and Pukkila and Rao (1988). The simplest way to draw the auxiliary variable is to use a fixed density for z, independent of x.…”

Section: Choices For H(z | X) and G(u | X)mentioning

confidence: 99%

On directional Metropolis–Hastings algorithms

Eidsvik¹,

Tjelmeland²

2006

Stat Comput

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New Metropolis-Hastings algorithms using directional updates are introduced in this paper. Each iteration of a directional Metropolis-Hastings algorithm consists of three steps (i) generate a line by sampling an auxiliary variable, (ii) propose a new state along the line, and (iii) accept/reject according to the Metropolis-Hastings acceptance probability. We consider two classes of directional updates. The first uses a point in R n as auxiliary variable, the second an auxiliary direction vector. The proposed algorithms generalize previous directional updating schemes since we allow the distribution of the auxiliary variable to depend on properties of the target at the current state. By letting the proposal distribution along the line depend on the density of the auxiliary variable, we identify proposal mechanisms that give unit acceptance rate. When we use direction vector as auxiliary variable, we get the advantageous effect of large moves in the Markov chain and hence the autocorrelation length of the samples is small. We apply the directional Metropolis-Hastings algorithms to a Gaussian example, a mixture of Gaussian densities, and a Bayesian model for seismic data.

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“…Since this density for w A is Gaussian, it is possible to calculate the density for the auxiliary unit direction vector u (Pukkila and Rao, 1988). We denote this density by g…”

Section: Directional Metropolis-hastingsmentioning

confidence: 99%

Directional Metropolis : Hastings Updates for Posteriors with Nonlinear Likelihoods

Tjelmeland¹,

Eidsvik

2005

Geostatistics Banff 2004

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In this paper we consider spatial problems modeled by a Gaussian random field prior and a nonlinear likelihood linking the hidden variables to the data. We define a directional block Metropolis-Hastings algorithm to explore the posterior. The method is applied to seismic data from the North Sea. Based on our results we believe it is important to assess the actual posterior in order to understand possible shortcomings of linear approximations. POSTERIORS 95 2004, 95-104. © 2005 Springer. Printed in the Netherlands. tics O. Leuangthong and C. V. Deutsch (eds.), Geostatis Banff : DIRECTIONAL METROPOLIS HASTINGS UPDATES FOR

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Pattern recognition based on scale invariant discriminant functions

Cited by 20 publications

References 3 publications

Likelihood Ratio Tests for a Dose-Response Effect Using Multiple Nonlinear Regression Models

Likelihood Ratio Tests for a Dose-Response Effect Using Multiple Nonlinear Regression Models

On directional Metropolis–Hastings algorithms

Directional Metropolis : Hastings Updates for Posteriors with Nonlinear Likelihoods

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