Computing with noise: phase transitions in Boolean formulas

Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Existing bounds on their performance are straightforwardly retrieved, generalized, and identified as the corresponding typical-case phase transitions. Results on error rates, function depth, and sensitivity, and their dependence on the gate-type and noise model used are also obtained.

On reliable computation by noisy random Boolean formulas

We study noisy computation in randomly generated k-ary Boolean formulas. We establish bounds on the noise level above which the results of computation by random formulas are not reliable. This bound is saturated by formulas constructed from a single majority-like gate. We show that these gates can be used to compute any Boolean function reliably below the noise bound.

Comparison of two formulas used to calculate summarized retinal vessel calibers

PURPOSE: To compare the Parr-Hubbard and Knudtson formulas to calculate retinal vessel calibers and to examine the effect of omitting vessels on the overall result. METHODS: We calculated the central retinal arterial equivalent (CRAE) and central retinal venular equivalent (CRVE) according to the formulas described by Parr-Hubbard and Knudtson including the six largest retinal arterioles and venules crossing through a concentric ring segment (measurement zone) around the optic nerve head. Once calculated, we removed one arbitrarily selected artery and one arbitrarily selected vein and recalculated all outcome parameters again for (1) omitting one artery only, (2) omitting one vein only, and (3) omitting one artery and one vein. All parameters were compared against each other. RESULTS: Both methods showed good correlation (r for CRAE = 0.58; r for CRVE = 0.84), but absolute values for CRAE and CRVE were significantly different from each other when comparing both methods (p < 0.000001): CRAE had higher values for the Parr-Hubbard (165 [±16] μm) method compared with the Knudtson method (148 [±15] μm). In addition, CRAE and CRVE values dropped for both methods when omitting one arbitrarily selected vessel each (all p < 0.000001). Arteriovenous ratio (AVR) calculations showed a similar change for both methods when omitting one vessel each: AVR decreased when omitting one arteriole whereas it increased when omitting one venule. No change, however, was observed for AVR calculated with six or five vessel pairs each. CONCLUSIONS: Although the absolute value for CRAE and CRVE is changing significantly depending on the number of vessels included, AVR appears to be comparable as long as the same number of arterioles and venules is included.

Semi-supervised learning of hierarchical latent trait models for data visualisation

An interactive hierarchical Generative Topographic Mapping (HGTM) ¸iteH_GTM has been developed to visualise complex data sets. In this paper, we build a more general visualisation system by extending the HGTM visualisation system in 3 directions: bf (1) We generalize HGTM to noise models from the exponential family of distributions. The basic building block is the Latent Trait Model (LTM) developed in ¸iteKaban_pami. bf (2) We give the user a choice of initializing the child plots of the current plot in either em interactive, or em automatic mode. In the interactive mode the user interactively selects ``regions of interest'' as in ¸iteH_GTM, whereas in the automatic mode an unsupervised minimum message length (MML)-driven construction of a mixture of LTMs is employed. bf (3) We derive general formulas for magnification factors in latent trait models. Magnification factors are a useful tool to improve our understanding of the visualisation plots, since they can highlight the boundaries between data clusters. The unsupervised construction is particularly useful when high-level plots are covered with dense clusters of highly overlapping data projections, making it difficult to use the interactive mode. Such a situation often arises when visualizing large data sets. We illustrate our approach on a toy example and apply our system to three more complex real data sets.

Analysis and simulation of computer controlled cellular mobile radio systems

Cellular mobile radio systems will be of increasing importance in the future. This thesis describes research work concerned with the teletraffic capacity and the canputer control requirements of such systems. The work involves theoretical analysis and experimental investigations using digital computer simulation. New formulas are derived for the congestion in single-cell systems in which there are both land-to-mobile and mobile-to-mobile calls and in which mobile-to-mobile calls go via the base station. Two approaches are used, the first yields modified forms of the familiar Erlang and Engset formulas, while the second gives more complicated but more accurate formulas. The results of computer simulations to establish the accuracy of the formulas are described. New teletraffic formulas are also derived for the congestion in multi -cell systems. Fixed, dynamic and hybrid channel assignments are considered. The formulas agree with previously published simulation results. Simulation programs are described for the evaluation of the speech traffic of mobiles and for the investigation of a possible computer network for the control of the speech traffic. The programs were developed according to the structured progranming approach leading to programs of modular construction. Two simulation methods are used for the speech traffic: the roulette method and the time-true method. The first is economical but has some restriction, while the second is expensive but gives comprehensive answers. The proposed control network operates at three hierarchical levels performing various control functions which include: the setting-up and clearing-down of calls, the hand-over of calls between cells and the address-changing of mobiles travelling between cities. The results demonstrate the feasibility of the control netwvork and indicate that small mini -computers inter-connected via voice grade data channels would be capable of providing satisfactory control