Bent-Cable Regression Theory and Applications

Chiu, Grace S.; Lockhart, Richard; Routledge, Richard

doi:10.1198/016214505000001177

Cited by 66 publications

(91 citation statements)

References 25 publications

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“…To avoid this problem, we chose to estimate the slopes with a "bent-cable" regressive method (22). The bent-cable model assumes a linear regime, followed by a quadratic transition component, followed by another linear regime, described by the equation:…”

Section: Measurements Of the Correlation Dimensionmentioning

confidence: 99%

The dimensionality of discourse

Doxas¹,

Dennis²,

Oliver³

2010

Proc. Natl. Acad. Sci. U.S.A.

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The paragraph spaces of five text corpora, of different genres and intended audiences, in four different languages, all show the same two-scale structure, with the dimension at short distances being lower than at long distances. In all five cases the short-distance dimension is approximately eight. Control simulations with randomly permuted word instances do not exhibit a low dimensional structure. The observed topology places important constraints on the way in which authors construct prose, which may be universal.correlation dimension | language | latent semantic analysis A s we transition from paragraph to paragraph in written discourse, one can think of the path through which one passes as a trajectory through a semantic space. Understanding the discourse is, in some sense, a matter of understanding this trajectory. Although it is difficult to predict exactly what will follow from a given discourse context, we can ask a broader question. Does the cognitive system impose any fundamental constraints on the way in which the discourse evolves?To investigate this issue, we constructed semantic spaces for five corpora, of different genres and intended audiences, in four different languages and then calculated the intrinsic dimensionality of the paragraph trajectories through these corpora. Each trajectory can be embedded in an arbitrary number of dimensions, but it has an intrinsic dimensionality, independent of the embedding space. For instance, in latent semantic analysis (LSA) applications, it is typical to use a 300-dimensional space (1). However, the points that represent the paragraphs in this 300-dimensional space do not fill the embedding space; rather they lie on a subspace with a dimension lower than the embedding dimension. The fact that a lowdimensional structure can be embedded in a higher dimensional space is routinely used in the study of nonlinear dynamic systems, in which the embedding theorem (2) relates the dimensionality of the dataset under study to the dimensionality of the dynamics that describes it.Historically, the dimensionality of the discourse trajectory has been implicitly assumed to be very large, but it has never been calculated. Here we show that the dimensionality of the trajectory is surprisingly low (approximately eight at short distances) and that its structure is probably universal across human languages. Although the question of dimensionality has not generally been considered before, it can be used to guide the development of new models of prose, which are constrained to reproduce the observed dimensional structure. Modeling SemanticsThe first step toward being able to calculate the dimensionality of text is to create a vector representation of the semantics conveyed by each paragraph. Recent years have seen increasing interest in automated methods for the construction of semantic representations of paragraphs [e.g., LSA (3), the topics model (4, 5), nonnegative matrix factorization (6), and the constructed semantics model (7)]. These methods were originally developed for use...

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Section: Measurements Of the Correlation Dimensionmentioning

confidence: 99%

The dimensionality of discourse

Doxas¹,

Dennis²,

Oliver³

2010

Proc. Natl. Acad. Sci. U.S.A.

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“…First, it assumes that the functional response of an organism to decreasing P O2 is biphasic, that is, it consists of two linear elements with a clear break between these two phases. Above P c , V O2 is assumed to be characterised by a linear function that is completely flat, while below P c , V O2 decreases linearly with P O2 with an abrupt transition between these two functions (Chiu et al, 2006). Of course, in reality, rates of respiration are likely to be a continuous function between these two phases, and furthermore, concentration-dependent reaction kinetics make a linear relationship between V O2 and P O2 highly unlikely.…”

Section: Introductionmentioning

confidence: 99%

Estimating physiological tolerances - a comparison of traditional approaches to nonlinear regression techniques

Marshall

Bode

White

2013

Journal of Experimental Biology

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SUMMARYTraditionally, physiologists have estimated the ability of organisms to withstand lower partial pressures of oxygen by estimating the partial pressure at which oxygen consumption begins to decrease (known as the critical P O2 or P c ). For almost 30years, the principal way in which P c has been estimated has been via piecewise ʻbroken stickʼ regression (BSR). BSR was a useful approach when more sophisticated analyses were less available, but BSR makes a number of unsupported assumptions about the underlying form of the relationship between the rate of oxygen consumption and oxygen availability. The BSR approach also distils a range of values into a single point with no estimate of error. In accordance with more general calls to fit functions to continuous data, we propose the use of nonlinear regression (NLR) to fit various curvilinear functions to oxygen consumption data in order to estimate P c . Importantly, our approach is back-compatible so that estimates using traditional methods in earlier studies can be compared with data estimates from our technique. When we compared the performance of our approach relative to the traditional BSR approach for real world and simulated data, we found that under realistic circumstances, NLR was more accurate and provided more powerful hypothesis tests. We recommend that future studies make use of NLR to estimate P c , and also suggest that this approach might be more appropriate for a range of physiological studies that use BSR currently.

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“…Therefore, the appropriate statistical method to analyze these data should involve a longitudinal model which can characterize a trend change over time. Chiu, Lockhart and Routledge (2006) and Chiu and Lockhart (2010) developed a special changepoint model to analyze a single profile that exhibits a trend change. The model is parsimonious with respect to the number of parameters, and appealing due to its simple structure, flexibility and interpretability.…”

Section: Introductionmentioning

confidence: 99%

A Statistical Investigation to Monitor and Understand Atmospheric CFC Decline with the Spatial-longitudinal Bent-cable Model

Khan¹,

Rana²,

Li³

et al. 2012

IJSP

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Concerns about the concentrations of chlorofluorocarbons (CFCs) in the atmosphere are based on their effects on the ozone layer by catalytically destroying ozone. The recent steady decline in atmospheric concentration of CFC could be a direct result of the Montréal Protocol's ban on CFC products, in effect since 1989. However, CFCs have long atmospheric residence times because of their low chemical reactivity, and as a consequence have already been distributed globally. To study the spatial effects and extent of the decline, we apply the proposed spatial-longitudinal bent-cable model to CFC data observed over a global detection network. The bent cable is a parametric regression model to study data that exhibits a trend change. It comprises two linear segments to describe the incoming and the outgoing phases, joined by a quadratic bend to model the transition period. For spatial longitudinal data, measurements taken over time are nested within spatially dependent locations. Here, it is useful to extend the existing longitudinal bent-cable regression to handle spatial effects. We do so in a hierarchical Bayesian framework by allowing the error terms to be correlated across space. The methodology is illustrated with applications to CFC-11 and 12 data. Our analysis reveals that: (a) there is a strong spatial relationship among all the monitoring locations across the globe; (b) both CFC-11 and CFC-12 increased significantly before entering into a transition zone; (c) after completing the transition, CFC-11 has been decreasing significantly from the atmosphere, but a slow (insignificant) decrease for CFC-12 is observed; and (d) it may take almost 5 times longer to diminish CFC-12 from the atmosphere compared to CFC-11.

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Bent-Cable Regression Theory and Applications

Cited by 66 publications

References 25 publications

The dimensionality of discourse

The dimensionality of discourse

Estimating physiological tolerances - a comparison of traditional approaches to nonlinear regression techniques

A Statistical Investigation to Monitor and Understand Atmospheric CFC Decline with the Spatial-longitudinal Bent-cable Model

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