Linear Partitions Research Articles

Estimated beef cattle performance under intensive grazing systems in Uruguay

The objective of this study was to analyze the effect of the quantity and quality of the feed offered on the performance of male beef cattle, >6 months old, on intensive grazing systems in Uruguay. Two hundred and fifty-four results from different experimental trials (nexp=68) performed between 1991 and 2020 by different research institutions in Uruguay were compiled and analyzed. The experiments included feeding studies with cultivated pastures as a single nutritional source (nexp=61), as well as supplementation studies on the same type of pastures (nexp=49). Simple, quadratic, multiple linear regressions and recursive partitions were performed to synthesize the information. Quadratic relationships between forage allowance (FA) and average daily gain (ADG) were observed for the different animal categories and seasons of the year. Including other parameters such as initial body weight and supplementation with energy concentrates or silages allowed to predict the ADG of animals under a wide range of feeding alternatives. Briefly, low FA levels (≤1.75 %BW) limit animal performance to maintenance gain conditions (0.2 kg/day). To achieve adequate levels of ADG (0.6-0.7 kg/day), with a low to moderate FA (between 1.75 to 3.5 %BW), the nutritional value of pasture needs to reach at least level 3 of quality (>58 %ODM, >13 %CP, <50 %NDF, >21 %Leg and <19 %DR). These daily gains can be exceeded if the FA is >3.5 %BW (ADG in the range of 0.7 to 1.0 kg/day). The increased response in additional ADG (0.4-0.5 kg/day) by the effect of supplementation with energy concentrate was achieved with low FA levels (≤2.25 %BW). The prediction of additional ADG using a multiple linear regression parameter allows evaluating whether supplementation is appropriate.

A Laplacian SVM Based Semi‐Supervised Classification Using Multi‐Local Linear Model

Semi‐supervised learning considers a classification problem of learning from both labeled and unlabeled data. This paper proposes a semi‐supervised classification method, in which the potential separation boundary is detected and its information is ingeniously incorporated into a Laplacian support vector machine (LapSVM) in both kernel level and graph level. By applying a pseudo‐labeling approach, the input space is first divided into several linear separable partitions along the potential separation boundary. A multi‐local linear model is then built for the separation boundary, by interpolating multiple local linear models assigned to the local linear separable partitions. The multi‐local linear model is further formulated into a linear regression form with a new input vector in the spanned feature space, which contains the information of potential separation boundary. Then the linear parameters are estimated globally by a LapSVM algorithm. Furthermore, the input in the spanned feature space and pseudo labels are used to construct a label guided graph. Numerical experiments on various real‐world datasets and visual representation on toy example exhibit the effectiveness of the proposed method. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Optimizing sizing schedules of aircrew helmets using machine learning techniques

Introduction: Aircrew equipment assemblies are life critical equipment worn by the aircrew to ensure protection from various life-threatening environmental conditions that may occur in nominal or off-nominal flight conditions. It is important for the sizing system of body worn aircrew equipment to be divided into different size classes such that it is as representative of existing groups in the population as feasible. Machine learning techniques provide a powerful tool to develop efficient sizing systems that are representative of actual body types existing in the population. The objective of the study was to use machine learning clustering techniques to identify naturally existing body types to formulate an optimal sizing system for an aircrew helmet. Material and Methods: The standard sizing schedule of an aircrew helmet using head length and head breadth was studied. An iterative semi-supervised machine learning protocol called K-means clustering was used to identify naturally occurring clusters of head types within the population. The cluster boundaries were identified to develop the final sizing schedule. Analysis of variance (ANOVA) with post hoc analysis was carried out on the four head lengths and head breadths. Results: Four clusters of head type were identified using K-means clustering. This led to a sizing schedule which can be descriptively called “Short-Narrow,” “Short-Broad,” “Long-Narrow,” “Long-Broad.” ANOVA showed a statistically significant difference between the four clusters for both head length and head breadth. Conclusion: Consideration of several variables in sizing schedules often fails to meet desired fitment. Standard sizing methods use linear partitions on individual control parameters. In reality, the proportional variation between parameters is not linear. Machine learning tools help in identification of naturally occurring clusters within the database considering several variables at the same time. This study uses machine learning techniques to identify existing groups in a population and develop a final sizing schedule. Similar analysis for other aircrew equipment assemblies can be carried out to determine sizing schedules which assist in design and fitment.

A geometry‐based two‐step method for nonlinear classification using quasi‐linear support vector machine

This paper proposes a two‐step method to construct a nonlinear classifier consisting of multiple local linear classifiers interpolated with a basis function. In the first step, a geometry‐based approach is first introduced to detect local linear partitions and build local linear classifiers. A coarse nonlinear classifier can then be constructed by interpolating the local linear classifiers. In the second step, a support vector machine (SVM) formulation is used to further implicitly optimize the linear parameters of the nonlinear classifier. In this way, the nonlinear classifier is constructed in exactly the same way as a standard SVM, using a special data‐dependent quasi‐linear kernel composed of the information of the local linear partitions. Numerical experiments on several real‐world datasets demonstrate the effectiveness of the proposed classifier and show that, in cases where traditional nonlinear SVMs run into overfitting problems, the proposed classifier is effective in improving the classification performance. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A new segmented oversampling method for imbalanced data classification using quasi‐linear SVM

Data imbalance occurs on most real‐world classification problems and decreases the performance of classifiers. An oversampling method addresses the imbalance problem by generating synthetic samples to balance the data distribution. However, many of the existing oversampling methods have potential problems in generating wrong and unnecessary synthetic samples, which makes the learning tasks difficult. This paper proposes a new segmented oversampling method for imbalanced data classification. The input space is first partitioned into several linearly separable local partitions along the potential separation boundary by introducing a bottom‐up, minimal‐spanning‐tree‐based clustering method; an oversampling method is then applied within each local linear partition to prevent the generation of wrong and unnecessary synthetic samples; a quasi‐linear support vector machine is finally used to realize the classification by taking advantages of the local linear partitions. Simulation results on different real‐world datasets show that the proposed segmented oversampling method is effective for imbalanced data classifications. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

高维线性分拆理论及其应用——Ramsey数R(5,5;2)=47的注记

高维线性分拆理论及其应用——Ramsey数R(5,5;2)=47的注记

An Efficient Control-Driven Period Optimization Algorithm for Distributed Real-Time Systems

The sampling periods of real-time embedded control functions have a significant impact on control performance and system schedulability. Exploring period assignment for optimizing control performance while meeting schedulability constraints is very challenging, in particular for distributed systems where control loops share a network of computation and communication resources. In this work, we propose an efficient approach that approximates the performance of each control loop in the system with a piecewise linear function of its sampling period and end-to-end delay, and then optimizes the periods of tasks and messages by exploring the linear partitions of the approximated functions and solving a series of geometric programming (GP) formulations. Experiments on sample control models, an automotive industrial case study and a set of synthetic examples demonstrate the effectiveness and efficiency of our approach.

Statistical mechanics approach in the counting of integer partitions

The treatment of the number-theoretical problem of integer partitions within the approach of statistical mechanics is discussed. Historical overview is given and known asymptotic results for linear and plane partitions are reproduced. From numerical analysis of restricted plane partitions an asymptotic formula is conjectured for an intermediate number of parts.

Olp: An R package for optimal linear partitions of finite sets of points on the plane

Given a set of points on the plane and an assignment of values to them, an optimal linear partition is a division of the set into two subsets which are separated by a straight line and maximally contrast with each other in the values assigned to their points. We present a method for inspecting and rating all linear partitions of a finite set, and a package of three functions in the R language for executing the computations. One function is for finding the optimal linear partitions and corresponding separating lines, another for graphically representing the results, and a third for testing how well the data comply with the linear separability condition. We illustrate the method on possible data from a psychophysical experiment (concerning the size-weight illusion) and compare its performance with that of linear discriminant analysis and multiple logistic regression, adapted to dividing linearly a set of points on the plane.

A Helly Type Theorem for Abstract Projective Geometries

We prove that the lattice of linear partitions in a projective geometry of rank n has Helly number ⌊3n/2⌋.

On odd and semi-odd linear partitions of cubic graphs

On odd and semi-odd linear partitions of cubic graphs

On isomorphic linear partitions in cubic graphs

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and l a ( G ) is the minimum number of linear forests in a linear partition. It is well known that l a ( G ) = 2 when G is a cubic graph and Wormald [N. Wormald, Problem 13, Ars Combinatoria 23(A) (1987) 332–334] conjectured that if | V ( G ) | ≡ 0 (mod 4), then it is always possible to find a linear partition in two isomorphic linear forests. Here, we give some new results concerning this conjecture.

Flat transversals to flats and convex sets of a fixed dimension

Helly and Hadwiger type theorems for transversal m-flats to families of flats and, respectively, convex sets of dimension n are proved in the case of general position. The proofs rely on Helly type theorems for “linear partitions” and “convex partitions,” so that a general theory of Helly numbers is also developed.

The role of linear power partitioning in beam filamentation

We demonstrate through computer simulation that power partitioning due to linear diffraction can play a crucial role in the break up of high intensity apertured beams propagating in air. By investigating the way linear diffraction partitions power in these beams, we are able to predict the positions and relative powers of supercritical hot spots.

On isomorphic linear partitions in cubic graphs

Abstract A linear forest is a graph that connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that l a ( G ) = 2 when G is a cubic graph and Wormald [Wormald, N., Problem 13, Ars Combinatoria 23 (1987), pp. 332–334] conjectured that if | V ( G ) | ≡ 0 ( mod 4 ) , then it is always possible to find a linear partition in two isomorphic linear forests. We give here some new results concerning this conjecture.

On the number of multilinear partitions and the computing capacity of multiple-valued multiple-threshold perceptrons

We introduce the concept of multilinear partition of a point set V/spl sub/R/sup n/ and the concept of multilinear separability of a function f:Vtwo head right arrowK={0,...,k-1}. Based on well-known relationships between linear partitions and minimal pairs, we derive formulae for the number of multilinear partitions of a point set in general position and of the set K(2). The (n,k,s)-perceptrons partition the input space V into s+1 regions with s parallel hyperplanes. We obtain results on the capacity of a single (n,k,s)-perceptron, respectively, for V subset R(n) in general position and for V=K(2). Finally, we describe a fast polynomial-time algorithm for counting the multilinear partitions of K(2).

Generalizing a neuropsychological model of visual categorization to auditory categorization of vowels.

Twelve male listeners categorized 54 synthetic vowel stimuli that varied in second and third formant frequency on a Bark scale into the American English vowel categories [see text]. A neuropsychologically plausible model of categorization in the visual domain, the Striatal Pattern Classifier (SPC; Ashby & Waldron, 1999), is generalized to the auditory domain and applied separately to the data from each observer. Performance of the SPC is compared with that of the successful Normal A Posteriori Probability model (NAPP; Nearey, 1990; Nearey & Hogan, 1986) of auditory categorization. A version of the SPC that assumed piece-wise linear response region partitions provided a better account of the data than the SPC that assumed linear partitions, and was indistinguishable from a version that assumed quadratic response region partitions. A version of the NAPP model that assumed nonlinear response regions was superior to the NAPP model with linear partitions. The best fitting SPC provided a good account of each observer's data but was outperformed by the best fitting NAPP model. Implications for bridging the gap between the domains of visual and auditory categorization are discussed.

The Computing Capacity of Three-Input Multiple-Valued One-Threshold Perceptrons

In this paper, an exact and general formula is derived for the number of linear partitions of a given point set V in three-dimensional space, depending on the configuration formed by the points of V. The set V can be a multi-set, that is it may contain points that coincide. Based on the formula, we obtain an efficient algorithm for counting the number of k-valued logic functions simulated by a three-input k-valued one-threshold perceptron.

Linear Size Binary Space Partitions for Uncluttered Scenes

We describe a new and simple method for constructing binary space partitions (BSPs) in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O(n) for an uncluttered scene consisting of n objects. The construction time is O(n log n) . Because any set of disjoint fat objects is uncluttered, our result implies an efficient method to construct a linear size BSP for fat objects. We use our BSP to develop a data structure for point location in uncluttered scenes. The query time of our structure is O( log n) , and the amount of storage is O(n) . This result can in turn be used to perform range queries with not-too-small ranges in scenes consisting of disjoint fat objects or, more generally, in so-called low-density scenes.

A Combinatorial Interpretation of Punctured Partitions

We give a combinatorial interpretation of punctured partitions (i.e., n-tuples (p1, p2, …, pn) of natural numbers such that p1+p2+…+pk=k whenever pk≠0) in terms of linear partitions of linearly ordered sets. As an application we give an explicit expression of the permanent (determinant) of a particular kind of Hessenberg matrices in terms of punctured partitions (i.e., linear partitions). Then we show that for suitable choices of the Hessenberg matrix these permanents give the number of the enriched (linear) partitions of a finite (linearly ordered) set or more generally the associated polynomials forming a sequence of (Newjonian) binomial type. Instances of these polynomials are the exponential, rising factorial, Laguerre, Abel, inverse-Abel, Mittag–Leffler polynomials. A further application deals with formal series inversion; in particular we derive an expression of elementary symmetric functions in terms of complete symmetric functions and vice versa.