Constructing localized non-linear projection manifolds in a principled way:hierarchical generative topographic mapping

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis (Bishop98a) in several directions: 1. We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. 2. We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. 3. Using tools from differential geometry we derive expressions for local directionalcurvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model.We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set andapply our system to two more complex 12- and 19-dimensional data sets.

Range of validity of weakly non-linear theory in Rayleigh-Bénard problem

In this paper we examine the equilibrium states of finite amplitude flow in a horizontal fluid layer with differential heating between the two rigid boundaries. The solutions to the Navier-Stokes equations are obtained by means of a perturbation method for evaluating the Landau constants and through a Newton-Raphson iterative method that results from the Fourier expansion of the solutions that bifurcate above the linear stability threshold of infinitesimal disturbances. The results obtained from these two different methods of evaluating the convective flow are compared in the neighborhood of the critical Rayleigh number. We find that for small Prandtl numbers the discrepancy of the two methods is noticeable. © 2009 The Physical Society of Japan.

Range of validity of weakly non-linear theory in Rayleigh-Bénard convection

In this paper we examine the equilibrium states of periodic finite amplitude flow in a horizontal channel with differential heating between the two rigid boundaries. The solutions to the Navier-Stokes equations are obtained by means of a perturbation method for evaluating the Landau coefficients and through a Newton-Raphson iterative method that results from the Fourier expansion of the solutions that bifurcate above the linear stability threshold of infini- tesimal disturbances. The results obtained from these two different methods of evaluating the convective flow are compared in the neighbourhood of the critical Rayleigh number. We find that for small Prandtl numbers the discrepancy of the two methods is noticeable.

The non-linear acousto-mechanical and thermodynamic response of spherical absorbers to laser pulses of various temporal profiles

This study is to theoretically investigate shockwave and microbubble formation due to laser absorption by microparticles and nanoparticles. The initial motivation for this research was to understand the underlying physical mechanisms responsible for laser damage to the retina, as well as the predict threshold levels for damage for laser pulses with of progressively shorter durations. The strongest absorbers in the retina are micron size melanosomes, and their absorption of laser light causes them to accrue very high energy density. I theoretically investigate how this absorbed energy is transferred to the surrounding medium. For a wide range of conditions I calculate shockwave generation and bubble growth as a function of the three parameters; fluence, pulse duration and pulse shape. In order to develop a rigorous physical treatment, the governing equations for the behavior of an absorber and for the surrounding medium are derived. Shockwave theory is investigated and the conclusion is that a shock pressure explanation is likely to be the underlying physical cause of retinal damage at threshold fluences for sub-nanosecond pulses. The same effects are also expected for non-biological micro and nano absorbers. ^

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

Non-linear oscillations assessment of the distribution static compensator operating in voltage control mode

This article deals with the non-linear oscillations assessment of a distribution static comensator ooperating in voltage control mode using the bifurcation theory. A mathematical model of the distribution static compensator in the voltage control mode to carry out the bifurcation analysis is derived. The stabiity regions in the Thevein equivalent plane are computed. In addition, the stability regions in the control gains space, as well as the contour lines for different Floquet multipliers are computed. The AC and DC capacitor impacts on the stability are analyzed through the bifurcation theory. The observations are verified through simulaation studies. The computation of the stability region allows the assessment of the stable operating zones for a power system that includes a distribution static compensator operating in the voltage mode.

Processing load and the use of concrete representations and strategies for solving linear equations

This paper is a report of students' responses to instruction which was based on the use of concrete representations to solve linear equations. The sample consisted of 21 Grade 8 students from a middle-class suburban state secondary school with a reputation for high academic standards and innovative mathematics teaching. The students were interviewed before and after instruction. Interviews and classroom interactions were observed and videotaped. A qualitative analysis of the responses revealed that students did not use the materials in solving problems. The increased processing load caused by concrete representations is hypothesised as a reason.

Validation Gating for Non-Linear Non-Gaussian Target Tracking

This paper develops a general theory of validation gating for non-linear non-Gaussian mod- els. Validation gates are used in target tracking to cull very unlikely measurement-to-track associa- tions, before remaining association ambiguities are handled by a more comprehensive (and expensive) data association scheme. The essential property of a gate is to accept a high percentage of correct associ- ations, thus maximising track accuracy, but provide a su±ciently tight bound to minimise the number of ambiguous associations. For linear Gaussian systems, the ellipsoidal vali- dation gate is standard, and possesses the statistical property whereby a given threshold will accept a cer- tain percentage of true associations. This property does not hold for non-linear non-Gaussian models. As a system departs from linear-Gaussian, the ellip- soid gate tends to reject a higher than expected pro- portion of correct associations and permit an excess of false ones. In this paper, the concept of the ellip- soidal gate is extended to permit correct statistics for the non-linear non-Gaussian case. The new gate is demonstrated by a bearing-only tracking example.

Asset life prediction and maintenance decision-making using a non-linear non-Gaussian state space model

Estimating and predicting degradation processes of engineering assets is crucial for reducing the cost and insuring the productivity of enterprises. Assisted by modern condition monitoring (CM) technologies, most asset degradation processes can be revealed by various degradation indicators extracted from CM data. Maintenance strategies developed using these degradation indicators (i.e. condition-based maintenance) are more cost-effective, because unnecessary maintenance activities are avoided when an asset is still in a decent health state. A practical difficulty in condition-based maintenance (CBM) is that degradation indicators extracted from CM data can only partially reveal asset health states in most situations. Underestimating this uncertainty in relationships between degradation indicators and health states can cause excessive false alarms or failures without pre-alarms. The state space model provides an efficient approach to describe a degradation process using these indicators that can only partially reveal health states. However, existing state space models that describe asset degradation processes largely depend on assumptions such as, discrete time, discrete state, linearity, and Gaussianity. The discrete time assumption requires that failures and inspections only happen at fixed intervals. The discrete state assumption entails discretising continuous degradation indicators, which requires expert knowledge and often introduces additional errors. The linear and Gaussian assumptions are not consistent with nonlinear and irreversible degradation processes in most engineering assets. This research proposes a Gamma-based state space model that does not have discrete time, discrete state, linear and Gaussian assumptions to model partially observable degradation processes. Monte Carlo-based algorithms are developed to estimate model parameters and asset remaining useful lives. In addition, this research also develops a continuous state partially observable semi-Markov decision process (POSMDP) to model a degradation process that follows the Gamma-based state space model and is under various maintenance strategies. Optimal maintenance strategies are obtained by solving the POSMDP. Simulation studies through the MATLAB are performed; case studies using the data from an accelerated life test of a gearbox and a liquefied natural gas industry are also conducted. The results show that the proposed Monte Carlo-based EM algorithm can estimate model parameters accurately. The results also show that the proposed Gamma-based state space model have better fitness result than linear and Gaussian state space models when used to process monotonically increasing degradation data in the accelerated life test of a gear box. Furthermore, both simulation studies and case studies show that the prediction algorithm based on the Gamma-based state space model can identify the mean value and confidence interval of asset remaining useful lives accurately. In addition, the simulation study shows that the proposed maintenance strategy optimisation method based on the POSMDP is more flexible than that assumes a predetermined strategy structure and uses the renewal theory. Moreover, the simulation study also shows that the proposed maintenance optimisation method can obtain more cost-effective strategies than a recently published maintenance strategy optimisation method by optimising the next maintenance activity and the waiting time till the next maintenance activity simultaneously.