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SW1PerS: Sliding Windows and 1Persistence Scoring; Discovering Periodicity in Gene Expression Time Series Data, preprint
, 2013
"... Motivation: Identifying periodically expressed genes across different processes such as the cell cycle, circadian rhythms, and metabolic cycles, is a central problem in computational biology. Biological time series data may contain (multiple) unknown signal shapes, have imperfections such as noise ..."
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Motivation: Identifying periodically expressed genes across different processes such as the cell cycle, circadian rhythms, and metabolic cycles, is a central problem in computational biology. Biological time series data may contain (multiple) unknown signal shapes, have imperfections such as noise, damping, and trending, or have limited sampling density. While many methods exist for detecting periodicity, their design biases can limit their applicability in one or more of these situations. Methods: We present in this paper a novel method, SW1PerS, for quantifying periodicity in time series data. The measurement is performed directly, without presupposing a particular shape or pattern, by evaluating the circularity of a highdimensional representation of the signal. SW1PerS is compared to other algorithms using synthetic data and performance is quantified under varying noise levels, sampling densities, and signal shapes. Results on biological data are also analyzed and compared; this data includes different periodic processes from various organisms: the cell and metabolic cycles in S. cerevisiae, and the circadian rhythms in M. musculus.
Approximating Persistent Homology in Euclidean Space Through Collapses
, 2014
"... The Čech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the Čech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at sma ..."
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The Čech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the Čech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at smaller scales than what the application calls for. In this paper we propose two methods to approximate the Čech persistence module. Both constructions are built on the level of spaces, i.e. as sequences of simplicial complexes induced by nerves. We also show how the bottleneck distance between such persistence modules can be understood by how tightly they are sandwiched on the level of spaces. In turn, this implies the correctness of our approximation methods. Finally, we implement our methods and apply them to some example point clouds in Euclidean space. 1
SUPPLEMENTARY INFORMATION SW1PerS: Sliding Windows and 1Persistence Scoring; Discovering Periodicity in Gene Expression Time Series Data
"... Methods We present in this paper a novel method, SW1PerS, for quantifying periodicity in time series data. The measurement is performed directly, without presupposing a particular shape or pattern, by evaluating the circularity of a highdimensional representation of the signal. SW1PerS is compared ..."
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Methods We present in this paper a novel method, SW1PerS, for quantifying periodicity in time series data. The measurement is performed directly, without presupposing a particular shape or pattern, by evaluating the circularity of a highdimensional representation of the signal. SW1PerS is compared to other algorithms using synthetic data and performance is quantified under varying noise levels, sampling densities, and signal shapes. Results on biological data are also analyzed and compared; this data includes different periodic processes from various organisms: the cell and metabolic cycles in S. cerevisiae, and the circadian rhythms in M. musculus. Results On the task of periodic/notperiodic classification, on synthetic data, SW1PerS performs on par with successful methods in periodicity detection. Moreover, it outperforms LombScargle and JTKcycle in the highnoise/lowsampling range. SW1PerS is shown to be the most shapeagnostic of the evaluated methods, and the only one to consistently classify damped signals as highly periodic. On biological data, and for several experiments, the lists of top 10 % genes ranked with SW1PerS recover up to 67 % of those generated with other popular algorithms. Moreover, lists of genes which are highlyranked only by SW1PerS contain noncosine patterns (e.g. ECM33, CDC9, SAM1,2 and MSH6 in the Yeast metabolic
tracking using Multiple Hypothesis Tracking, Topological Data
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Combining
"... persistent homology and invariance groups for shape comparison Patrizio Frosini · Grzegorz Jab loński This paper is dedicated to the memory of Marcello D’Orta and Jerry Essan Masslo. Abstract In many applications concerning the comparison of data expressed by Rmvalued functions defined on a topolo ..."
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persistent homology and invariance groups for shape comparison Patrizio Frosini · Grzegorz Jab loński This paper is dedicated to the memory of Marcello D’Orta and Jerry Essan Masslo. Abstract In many applications concerning the comparison of data expressed by Rmvalued functions defined on a topological space X, the invariance with respect to a given group G of selfhomeomorphisms of X is required. While persistent homology is quite efficient in the topological and qualitative comparison of this kind of data when the invariance group G is the group Homeo(X) of all selfhomeomorphisms of X, this theory is not tailored to manage the case in which G is a proper subgroup of Homeo(X), and its invariance appears too general for several tasks. This paper proposes a way to adapt persistent homology in order to get invariance just with respect to a given group of selfhomeomorphisms of X. The main idea consists in a dual approach, based on considering the set of all Ginvariant nonexpanding operators defined on the space of the admissible filtering functions on X. Some theoretical results concerning this approach are proven and two experiments are presented. An experiment illustrates the application of the proposed technique to compare 1Dsignals, when the invariance is expressed by the group of affinities, the group of orientationpreserving affinities, the group of isometries, the group of translations and the identity group. Another experiment shows how our technique can be used for image comparison.
DOI 10.1186/s1285901506456 METHODOLOGY ARTICLE Open Access SW1PerS: Sliding windows and
"... 1persistence scoring; discovering periodicity in gene expression time series data ..."
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1persistence scoring; discovering periodicity in gene expression time series data
John Harer, Supervisor
, 2014
"... We work on constructing mathematical models of gene regulatory networks for periodic processes, such as the cell cycle in budding yeast, using biological data sets and applying or developing analysis methods in the areas of mathematics, statistics, and computer science. We identify genes with perio ..."
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We work on constructing mathematical models of gene regulatory networks for periodic processes, such as the cell cycle in budding yeast, using biological data sets and applying or developing analysis methods in the areas of mathematics, statistics, and computer science. We identify genes with periodic expression and then the interactions between periodic genes, which defines the structure of the network. This network is then translated into a mathematical model, using Ordinary Differential Equations (ODEs), to describe these entities and their interactions. The models currently describe gene regulatory interactions, but we are expanding to capture other events, such as phosphorylation and ubiquitination. To model the behavior, we must then find appropriate parameters for the mathematical model that allow its dynamics to approximate the biological data. This pipeline for model construction is not focused on a specific algorithm or data set for each step, but instead on leveraging several sources of data and analysis from several algorithms. For example, we are incorporating data from multiple time series
Topological Data Analysis of Biological Aggregation Models
, 2015
"... We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describ ..."
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We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in positionvelocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.
Combining persistent homology and invariance groups for shape comparison
, 2014
"... This paper is dedicated to the memory of Marcello D’Orta and Jerry Essan Masslo. Abstract In many applications concerning the comparison of data expressed by Rmvalued functions defined on a topological space X, the invariance with respect to a given group G of selfhomeomorphisms of X is required. ..."
Abstract
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This paper is dedicated to the memory of Marcello D’Orta and Jerry Essan Masslo. Abstract In many applications concerning the comparison of data expressed by Rmvalued functions defined on a topological space X, the invariance with respect to a given group G of selfhomeomorphisms of X is required. While persistent homology is quite efficient in the topological and qualitative comparison of this kind of data when the invariance group G is the group Homeo(X) of all selfhomeomorphisms of X, this theory is not tailored to manage the case in which G is a proper subgroup of Homeo(X), and its invariance appears too general for several tasks. This paper proposes a way to adapt persistent homology in order to get invariance just with respect to a given group of selfhomeomorphisms of X. The main idea consists in a dual approach, based on considering the set of all Ginvariant nonexpanding operators defined on the space of the admissible filtering functions on X. Some theoretical results concerning this approach are proven and two experiments are presented. An experiment illustrates the application of the proposed technique to compare 1Dsignals, when the invariance is expressed by the group of affinities, the group of orientationpreserving affinities, the group of isometries, the group of translations and the identity group. Another experiment shows how our technique can be used for image comparison.