MCS: A Method for Finding the Number of Clusters

This paper proposes a maximum clustering similarity (MCS) method for determining the number of clusters in a data set by studying the behavior of similarity indices comparing two (of several) clustering methods. The similarity between the two clusterings is calculated at the same number of clusters, using the indices of Rand (R), Fowlkes and Mallows (FM), and Kulczynski (K) each corrected for chance agreement. The number of clusters at which the index attains its maximum is a candidate for the optimal number of clusters. The proposed method is applied to simulated bivariate normal data, and further extended for use in circular data. Its performance is compared to the criteria discussed in Tibshirani, Walther, and Hastie (2001). The proposed method is not based on any distributional or data assumption which makes it widely applicable to any type of data that can be clustered using at least two clustering algorithms.     

Asymptotic properties of univariate sample k-means clusters

A random sample of sizeN is divided intok clusters that minimize the within clusters sum of squares locally. Some large sample properties of this k-means clustering method (ask approaches withN) are obtained. In one dimension, it is established that the sample k-means clusters are such that the within-cluster sums of squares are asymptotically equal, and that the sizes of the cluster intervals are inversely proportional to the one-third power of the underlying density at the midpoints of the intervals. The difficulty involved in generalizing the results to the multivariate case is mentioned.This research was supported in part by the National Science Foundation under Grant MCS75-08374. The author would like to thank John Hartigan and David Pollard for helpful discussions and comments.     

Meaningfulness: A Third Dimension of the Good Life

This paper argues that an adequate conception of a good life should recognize, in addition to happiness and morality, a third dimension of meaningfulness. It further proposes that we understand meaningfulness as involving both a subjective and an objective condition, suitably linked. Meaning arises when subjective attraction meets objective attractiveness. In other words one’s life is meaningful insofar as one is gripped or excited by things worthy of one’s love, and one is able to do something positive about it. The paper concludes with some speculations about how this conception of meaningfulness might help to explain the conditions under which social volunteering can be especially rewarding.     

A Run Length Transformation for Discriminating Between Auto Regressive Time Series

We describe a simple time series transformation to detect differences in series that can be accurately modelled as stationary autoregressive (AR) processes. The transformation involves forming the histogram of above and below the mean run lengths. The run length (RL) transformation has the benefits of being very fast, compact and updatable for new data in constant time. Furthermore, it can be generated directly from data that has already been highly compressed. We first establish the theoretical asymptotic relationship between run length distributions and AR models through consideration of the zero crossing probability and the distribution of runs. We benchmark our transformation against two alternatives: the truncated Autocorrelation function (ACF) transform and the AR transformation, which involves the standard method of fitting the partial autocorrelation coefficients with the Durbin-Levinson recursions and using the Akaike Information Criterion stopping procedure. Whilst optimal in the idealized scenario, representing the data in these ways is time consuming and the representation cannot be updated online for new data. We show that for classification problems the accuracy obtained through using the run length distribution tends towards that obtained from using the full fitted models. We then propose three alternative distance measures for run length distributions based on Gower’s general similarity coefficient, the likelihood ratio and dynamic time warping (DTW). Through simulated classification experiments we show that a nearest neighbour distance based on DTW converges to the optimal faster than classifiers based on Euclidean distance, Gower’s coefficient and the likelihood ratio. We experiment with a variety of classifiers and demonstrate that although the RL transform requires more data than the best performing classifier to achieve the same accuracy as AR or ACF, this factor is at worst non-increasing with the series length, m, whereas the relative time taken to fit AR and ACF increases with m. We conclude that if the data is stationary and can be suitably modelled by an AR series, and if time is an important factor in reaching a discriminatory decision, then the run length distribution transform is a simple and effective transformation to use.     

Generation of Random Clusters with Specified Degree of Separation

We propose a random cluster generation algorithm that has the desired features: (1) the population degree of separation between clusters and the nearest neighboring clusters can be set to a specified value, based on a separation index; (2) no constraint is imposed on the isolation among clusters in each dimension; (3) the covariance matrices correspond to different shapes, diameters and orientations; (4) the full cluster structures generally could not be detected simply from pair-wise scatterplots of variables; (5) noisy variables and outliers can be imposed to make the cluster structures harder to be recovered. This algorithm is an improvement on the method used in Milligan (1985).     

文化嵌入与集群发展的共演机制研究

作为影响集群发展的重要"软环境",文化嵌入通过影响集群创新系统的结构和行为进而影响集群的竞争优势。从动态视角看,这些关系之间存在着共演,共演可能出现耦合或锁定两种不同结果,共演耦合一般出现在集群发展初期;而共演锁定则更可能在集群发展的后期出现,即最初导致集群发展的文化因素也往往是集群衰亡的原因。共演锁定表明了文化过度嵌入,从内在根源分析,文化的过度嵌入来自于文化功能和性质形成的路径依赖导致的锁定。因此,为了保持集群长期可持续发展,必须实现文化的适度嵌入。     

The Art of Living with NZT and ICT: Dialectics of an Artistic Case Study

I wholeheartedly sympathize conceptually with Coeckelbergh’s paper. The dialectical relationship between vulnerability and technology constitutes the core of Hegel’s Master and Slave (the primal scene of contemporary philosophy). Yet, the empirical dimension is underdeveloped and Coeckelbergh’s ideas could profit from exposure to case studies. Building on a movie/novel (Limitless) devoted to vulnerability coping and living with ICT, I challenge the claim that modern heroism entails overcoming vulnerability with the help of enhancement and computers.     

意识“难问题”的本质及其深层次问题研究

意识"难问题"充分说明了现象意识,即我们心理生活主观的、定性的维度。它要求我们解释带有主观性和定性的状态为什么以及如何存在。一般来说,尽管认知神经科学在意识问题的解释中起着重要作用,但其却不能充当意识"难问题"的充分条件或必要条件。查尔莫斯正是立足于这一认识论困境,提出了以信息为基础的意识的非还原解释方案,以此为迎合"难问题"的解释性挑战提供了适当理由;同时着眼于此,我们可以揭示出意识"难问题"的深层次问题。然而,要想阐释这样一个更大的难题,其间必然会面临现象概念策略与表征主义的质疑。     

s-Consensus index method: An additional axiom

A consensus index method is an ordered pair consisting of a consensus method and a consensus index Day and McMorris (1985) have specified two minimal axioms, one which should be satisfied by the consensus method and the other by the consensus index The axiom for consensus indices is not satisfied by the s-consensus index In this paper, an additional axiom, which states that a consensus index equal to one implies profile unanimity, is proposed The s-consensus method together with a modification of the s-consensus index (i e, normalized by the number of distinct nontrivial clusters in the profile) is shown to satisfy the two axioms proposed by Day and McMorris and the new axiom     

从希尔伯特规划到数学的地图

希尔伯特规划的原初目的是为无穷数学辩护,然而为哥德尔不完全性定理所挫。反推数学的根本目标是为数学命题找寻能够证明它的下限公理,而其中相当一部分工作可以看作为对希尔伯特规划的部分实现。本文在梳理有关工作的基础上试图为希尔伯特规划提供一个新的视角,即在绕开哲学负担之后,希尔伯特规划或许可以推进为为数学绘制地图。     

Fixed points approach to clustering

Assume that a dissimilarity measure between elements and subsets of the set being clustered is given. We define the transformation of the set of subsets under which each subset is transformed into the set of all elements whose dissimilarity to it is not greater than a given threshold. Then a cluster is defined as a fixed point of this transformation. Three well-known clustering strategies are considered from this point of view: hierarchical clustering, graph-theoretic methods, and conceptual clustering. For hierarchical clustering generalizations are obtained that allow for overlapping clusters and/or clusters not forming a cover. Three properties of dissimilarity are introduced which guarantee the existence of fixed points for each threshold. We develop the relation to the theory of quasi-concave set functions, to help give an additional interpretation of clusters.     

On the Equivalence of Cohen’s Kappa and the Hubert-Arabie Adjusted Rand Index

It is shown that one can calculate the Hubert-Arabie adjusted Rand index by first forming the fourfold contingency table counting the number of pairs of objects that were placed in the same cluster in both partitions, in the same cluster in one partition but in different clusters in the other partition, and in different clusters in both, and then computing Cohen’s κ on this fourfold table. The author thanks Willem Heiser, Mark de Rooij, Marian Hickendorff and three anonymous reviewers for their helpful comments and valuable suggestions on earlier versions of this article. Published online xx, xx, xxxx.     

Abduction and Estimation in Animals

One of the most pressing issues in understanding abduction is whether it is an instinct or an inference. For many commentators find it paradoxical that new ideas are products of an instinct and products of an inference at the same time. Fortunately, Lorenzo Magnani’s recent discussion of animal abduction sheds light on both instinctual and inferential character of Peircean abduction. But, exactly for what reasons are Peirce and Magnani so convinced that animal abduction can provide us with a novel perspective? Inspired by Peirce’s and Magnani’s discussions of animal abduction, I propose to compare Peirce’s and Magnani’s views of animal abduction with the estimative power of non-human animals and humans, which was one of the internal senses in medieval psychology.     

Philosophy of chemistry and the image of science

The philosophical analysis of chemistry has advanced at such a pace during the last dozen years that the existence of philosophy of chemistry as an autonomous discipline cannot be doubted any more. The present paper will attempt to analyse the experience of philosophy of chemistry at the, so to say, meta-level. Philosophers of chemistry have especially stressed that all sciences need not be similar to physics. They have tried to argue for chemistry as its own type of science and for a pluralistic understanding of science in general. However, when stressing the specific character of chemistry, philosophers do not always analyse the question ‘What is science?’ theoretically. It is obvious that a ‘monistic’ understanding of science should not be based simply on physics as the epitome of science, regarding it as a historical accident that physics has obtained this status. The author’s point is that the philosophical and methodological image of science should not be chosen arbitrarily; instead, it should be theoretically elaborated as an idealization (theoretical model) substantiated on the historical practice of science. It is argued that although physics has, in a sense, justifiably obtained the status of a paradigm of science, chemistry, which is not simply a physical science, but a discipline with a dual character, is also relevant for elaborating a theoretical model of science. The theoretical model of science is a good tool for examining various issues in philosophy of chemistry as well as in philosophy of science or science studies generally.     

Intelligent Choice of the Number of Clusters in K-Means Clustering: An Experimental Study with Different Cluster Spreads

The issue of determining “the right number of clusters” in K-Means has attracted considerable interest, especially in the recent years. Cluster intermix appears to be a factor most affecting the clustering results. This paper proposes an experimental setting for comparison of different approaches at data generated from Gaussian clusters with the controlled parameters of between- and within-cluster spread to model cluster intermix. The setting allows for evaluating the centroid recovery on par with conventional evaluation of the cluster recovery. The subjects of our interest are two versions of the “intelligent” K-Means method, ik-Means, that find the “right” number of clusters by extracting “anomalous patterns” from the data one-by-one. We compare them with seven other methods, including Hartigan’s rule, averaged Silhouette width and Gap statistic, under different between- and within-cluster spread-shape conditions. There are several consistent patterns in the results of our experiments, such as that the right K is reproduced best by Hartigan’s rule – but not clusters or their centroids. This leads us to propose an adjusted version of iK-Means, which performs well in the current experiment setting.     

从“天命”到“民主”--论芬伯格的技术变革之路

传统技术理论无论是工具理论还是实体理论,都视技术为一种天命,认为人类无法干预技术,只能"接受它或抛弃它",因此人类无法解决技术问题。在芬伯格看来,任何一种技术都是特定社会的产物,技术具有社会性。技术的社会性决定了技术不是天命,人们完全可以通过对技术的民主转化来变革技术,从而解决技术所存在的问题。芬伯格通过技术的变革来解决技术问题,不仅为我们思考技术的本质和解决技术问题提供了新思路,而且也为技术的未来发展指明了新方向。     

Expanding Mediation Theory

In his article In Between Us, Yoni van den Eede expands existing theories of mediation into the realm of the social and the political, focusing on the notions of opacity and transparency. His approach is rich and promising, but two pitfalls should be avoided. First, his concept of ‘in-between’ runs the risk to conceptualize mediation as a process ‘between’ pre-given entities. On the basis of current work in postphenomenology and actor-network theory, though, mediation should rather be seen as the origin of entities, not as an intermediary between them. Second, Van den Eede’s separate discussion of transparency and opacity in ‘use’ and in ‘context’ runs the risk to make invisible the complementarity of the two dimensions. While transparency of use embodies an experiential form of the distinction between transparency and opacity, transparency of context embodies a more cognitive dimension of the distinction. Only by linking the two it becomes possible to take responsibility for the impact that technological mediations can have. Users and designers need a ‘double vision’ to simultaneously see the transparency of both use and context.     

de Broglie Waves as the “Bridge of Becoming” Between Quantum Theory and Relativity

It is hypothesized that de Broglie’s ‘matter waves’ provide a dynamical basis for Minkowski spacetime in an antisubstantivalist or relational account. The relativity of simultaneity is seen as an effect of the de Broglie oscillation together with a basic relativity postulate, while the dispersion relation from finite rest mass gives rise to the differentiation of spatial and temporal axes. Thus spacetime is seen as not fundamental, but rather as emergent from the quantum level. A result by Solov’ev which demonstrates that time is not an applicable concept at the quantum level is adduced in support of this claim. Finally, it is noted that de Broglie waves can be seen as the “bridge of becoming” discussed by (2005).     

Towards a Theory of Mathematical Argument

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.