This paper proposes a new algorithm in the fuzzy-c-means family, which is designed to cluster time-series and is particularly suited for short time-series and those with unevenly spaced sampling points.

Summarizing the whole support of a random variable into minimum volume sets of its probability density function is studied in the paper. We prove that the level sets of a probability density function correspond to minimum volume sets and also determine the conditions for which the inverse proposition is verified. The distribution function of the level cuts of a density function is also introduced. It provides a di#erent visualization of the distribution of the probability for the random variable in question. It is also very useful to prove the above proposition. The volume # of the minimum volume sets varies according to its probability #: smaller volume implies lower probability and vice versa and larger volume implies larger probability and vice versa. In this context, 1 # is the error of an erroneously classification of a new observation inside of the minimum volume set or corresponding level set. To decide the volume and/or the error of the level set that will serve to summarize the support of the random variable, a # # plot is defined. We also study the relation of the minimum volume set approach with random set theory when # is a random variable and extend the most specific probability-possibility transformation proposed in [DPS93] to continuous piece-wise strictly monotone probability density functions. Keywords: Minimum volume set, level set, random set, probability-possibility transformation.

Contents 1 Biology for System Scientists 4 2 System Th:NJ for Biologists 8 3 Knowledge Representation 9 4 Reasoning about Data 18 5 Formalities 22 6 Conclusions 26 7 Appendix 28 ## ## # # Back View 3 Introduction Aims and Objectives # ConcexRxR Frame ork # # Working Me[] dology The Laboratory, Equipment #Syste TheR # Fuzzy Mathe000R) ## ## # # Back View Biologyfor System Scientists 4 1. Biology for System Scientists organism ce ge chromosome ger transcribe d re ORF non-transcribe d re re ryse promote teter ## ## # # Back View Biologyfor System Scientists 5 Gene Expression and Regulation: Context Genotype Expr ession Function Phenotype # ## ## # # Back View Biologyfor System Scientists 6 Gene Expression #Ce tral Dogma

With thestaggeramount of gene and prd( sequenceinfor( that is now available it is evident that the dynamics of biologicalr egulator mechanisms cannot beunder( od bymer identifying components, cataloguing pr ducts and ensembles. The aim of this paper is to outline afor mathematical fr wo r which maps sample sequence andder ed secondar data onto existing model genomes, metabolic,r egulator and developmental pathways in genome analysis.Apar fra a conceptual fr or (which wer efer to as fuzzy relati bii gy), the objective is to develop a wor methodologyfor the analysis of genomes in ter of gene expr( Ther( tation of biological knowledge (concepts, facts, andr( is based onr esults in the mathematics of fuzzy mappings. Keys ds--- Genome Analysis, Factor(- Models, Fuzzy Mathematics, System Theor . 1 Introducti Hestejz we conside life spe'zzk by genomes containing biologic al information to construct and maintain an organism. Genesare functional units ofthe geej controlling a multiple bio...