Topic
Permutation
About: Permutation is a research topic. Over the lifetime, 9021 publications have been published within this topic receiving 156897 citations. The topic is also known as: arrangement & rearrangement.
Papers published on a yearly basis
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Papers
TL;DR: The standard nonparametric randomization and permutation testing ideas are developed at an accessible level, using practical examples from functional neuroimaging, and the extensions for multiple comparisons described.
Abstract: Requiring only minimal assumptions for validity, nonparametric permutation testing provides a flexible and intuitive methodology for the statistical analysis of data from functional neuroimaging experiments, at some computational expense. Introduced into the functional neuroimaging literature by Holmes et al. ([1996]: J Cereb Blood Flow Metab 16:7-22), the permutation approach readily accounts for the multiple comparisons problem implicit in the standard voxel-by-voxel hypothesis testing framework. When the appropriate assumptions hold, the nonparametric permutation approach gives results similar to those obtained from a comparable Statistical Parametric Mapping approach using a general linear model with multiple comparisons corrections derived from random field theory. For analyses with low degrees of freedom, such as single subject PET/SPECT experiments or multi-subject PET/SPECT or fMRI designs assessed for population effects, the nonparametric approach employing a locally pooled (smoothed) variance estimate can outperform the comparable Statistical Parametric Mapping approach. Thus, these nonparametric techniques can be used to verify the validity of less computationally expensive parametric approaches. Although the theory and relative advantages of permutation approaches have been discussed by various authors, there has been no accessible explication of the method, and no freely distributed software implementing it. Consequently, there have been few practical applications of the technique. This article, and the accompanying MATLAB software, attempts to address these issues. The standard nonparametric randomization and permutation testing ideas are developed at an accessible level, using practical examples from functional neuroimaging, and the extensions for multiple comparisons described. Three worked examples from PET and fMRI are presented, with discussion, and comparisons with standard parametric approaches made where appropriate. Practical considerations are given throughout, and relevant statistical concepts are expounded in appendices.
6,530 citations
TL;DR: This paper presents a generic framework for permutation inference for complex general linear models (glms) when the errors are exchangeable and/or have a symmetric distribution, and shows that, even in the presence of nuisance effects, these permutation inferences are powerful while providing excellent control of false positives in a wide range of common and relevant imaging research scenarios.
3,509 citations
Posted Content•
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TL;DR: The main theorem characterizes the permutation invariant objective functions and provides a family of functions to which any permutation covariant objective function must belong, which enables the design of a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks.
Abstract: We study the problem of designing models for machine learning tasks defined on \emph{sets}. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics \cite{poczos13aistats}, to anomaly detection in piezometer data of embankment dams \cite{Jung15Exploration}, to cosmology \cite{Ntampaka16Dynamical,Ravanbakhsh16ICML1}. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.
1,891 citations
TL;DR: Methods are shown how to adapt invertible two-dimensional chaotic maps on a torus or on a square to create new symmetric block encryption schemes to encrypt an N×N image.
Abstract: In this paper, methods are shown how to adapt invertible two-dimensional chaotic maps on a torus or on a square to create new symmetric block encryption schemes. A chaotic map is first generalized by introducing parameters and then discretized to a finite square lattice of points which represent pixels or some other data items. Although the discretized map is a permutation and thus cannot be chaotic, it shares certain properties with its continuous counterpart as long as the number of iterations remains small. The discretized map is further extended to three dimensions and composed with a simple diffusion mechanism. As a result, a symmetric block product encryption scheme is obtained. To encrypt an N×N image, the ciphering map is iteratively applied to the image. The construction of the cipher and its security is explained with the two-dimensional Baker map. It is shown that the permutations induced by the Baker map behave as typical random permutations. Computer simulations indicate that the cipher has g...
1,879 citations
Book•
1 Jan 1974
TL;DR: A vocabulary of combinatorial analysis can be found in this paper, where the authors define definitions of partitions of an integer [n]- 22 Generating Functions of p(n) and P(n, m)- 23 Conditional Partitions- 24 Ferrers Diagrams- 25 Special Identities 'Formal' and 'Combinatorial' Proofs- 26 Partitions with Forbidden Summands Denumerants- Supplement and Exercises- III Identities and Expansions- III Identity and Expansion of a Product of Sums Abel Identity- 31
Abstract: I Vocabulary of Combinatorial Analysis- 11 Subsets of a Set Operations- 12 Product Sets- 13 Maps- 14 Arrangements, Permutations- 15 Combinations (without repetitions) or Blocks- 16 Binomial Identity- 17 Combinations with Repetitions- 18 Subsets of [n], Random Walk- 19 Subsets of Z/nZ- 110 Divisions and Partitions of a Set Multinomial Identity- 111 Bound Variables- 112 Formal Series- 113 Generating Functions- 114 List of the Principal Generating Functions- 115 Bracketing Problems- 116 Relations- 117 Graphs- 118 Digraphs Functions from a Finite Set into Itself- Supplement and Exercises- II Partitions of Integers- 21 Definitions of Partitions of an Integer [n]- 22 Generating Functions of p(n) and P(n, m)- 23 Conditional Partitions- 24 Ferrers Diagrams- 25 Special Identities 'Formal' and 'Combinatorial' Proofs- 26 Partitions with Forbidden Summands Denumerants- Supplement and Exercises- III Identities and Expansions- 31 Expansion of a Product of Sums Abel Identity- 32 Product of Formal Series Leibniz Formula- 33 Bell Polynomials- 34 Substitution of One Formal Series into Another Formula of Faa di Bruno- 35 Logarithmic and Potential Polynomials- 36 Inversion Formulas and Matrix Calculus- 37 Fractionary Iterates of Formal Series- 38 Inversion Formula of Lagrange- 39 Finite Summation Formulas- Supplement and Exercises- IV Sieve Formulas- 41 Number of Elements of a Union or Intersection- 42 The 'probleme des rencontres'- 43 The 'probleme des menages'- 44 Boolean Algebra Generated by a System of Subsets- 45 The Method of Renyi for Linear Inequalities- 46 Poincare Formula- 47 Bonferroni Inequalities- 48 Formulas of Ch Jordan- 49 Permanents- Supplement and Exercises- V Stirling Numbers- 51 Stirling Numbers of the Second Kind S(n, k) and Partitions of Sets- 52 Generating Functions for S(n, k)- 53 Recurrence Relations between the S(n, k)- 54 The Number ?(n) of Partitions or Equivalence Relations of a Set with n Elements- 55 Stirling Numbers of the First Kind s(n, k) and their Generating Functions- 56 Recurrence Relations between the s(n, k)- 57 The Values of s(n, k)- 58 Congruence Problems- Supplement and Exercises- VI Permutations- 61 The Symmetric Group- 62 Counting Problems Related to Decomposition in Cycles Return to Stirling Numbers of the First Kind- 63 Multipermutations- 64 Inversions of a Permutation of [n]- 65 Permutations by Number of Rises Eulerian Numbers- 66 Groups of Permutations Cycle Indicator Polynomial Burnside Theorem- 67 Theorem of Polya- Supplement and Exercises- VII Examples of Inequalities and Estimates- 71 Convexity and Unimodality of Combinatorial Sequences- 72 Sperner Systems- 73 Asymptotic Study of the Number of Regular Graphs of Order Two on N- 74 Random Permutations- 75 Theorem of Ramsey- 76 Binary (Bicolour) Ramsey Numbers- 77 Squares in Relations- Supplement and Exercises- Fundamental Numerical Tables- Factorials with Their Prime Factor Decomposition- Binomial Coefficients- Partitions of Integers- Bell Polynomials- Logarithmic Polynomials- Partially Ordinary Bell polynomials- Multinomial Coefficients- Stirling Numbers of the First Kind- Stirling Numbers of the Second Kind and Exponential Numbers
1,488 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 18 |
| 2021 | 507 |
| 2020 | 546 |
| 2019 | 511 |
| 2018 | 456 |
| 2017 | 416 |