532 research outputs found
Time-warping invariants of multidimensional time series
In data science, one is often confronted with a time series representing
measurements of some quantity of interest. Usually, as a first step, features
of the time series need to be extracted. These are numerical quantities that
aim to succinctly describe the data and to dampen the influence of noise. In
some applications, these features are also required to satisfy some invariance
properties. In this paper, we concentrate on time-warping invariants. We show
that these correspond to a certain family of iterated sums of the increments of
the time series, known as quasisymmetric functions in the mathematics
literature. We present these invariant features in an algebraic framework, and
we develop some of their basic properties.Comment: 18 pages, 1 figur
Recommended from our members
Time-warping invariants of multidimensional time series
In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants.We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties
Time-warping invariants of multidimensional time series
In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants.We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties
Tropical time series, iterated-sum signatures and quasisymmetric functions
Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects
The adaptive advantage of symbolic theft over sensorimotor toil: Grounding language in perceptual categories
Using neural nets to simulate learning and the genetic algorithm to simulate evolution in a toy world of mushrooms and mushroom-foragers, we place two ways of acquiring categories into direct competition with one another: In (1) "sensorimotor toil,” new categories are acquired through real-time, feedback-corrected, trial and error experience in sorting them. In (2) "symbolic theft,” new categories are acquired by hearsay from propositions – boolean combinations of symbols describing them. In competition, symbolic theft always beats sensorimotor toil. We hypothesize that this is the basis of the adaptive advantage of language. Entry-level categories must still be learned by toil, however, to avoid an infinite regress (the “symbol grounding problem”). Changes in the internal representations of categories must take place during the course of learning by toil. These changes can be analyzed in terms of the compression of within-category similarities and the expansion of between-category differences. These allow regions of similarity space to be separated, bounded and named, and then the names can be combined and recombined to describe new categories, grounded recursively in the old ones. Such compression/expansion effects, called "categorical perception" (CP), have previously been reported with categories acquired by sensorimotor toil; we show that they can also arise from symbolic theft alone. The picture of natural language and its origins that emerges from this analysis is that of a powerful hybrid symbolic/sensorimotor capacity, infinitely superior to its purely sensorimotor precursors, but still grounded in and dependent on them. It can spare us from untold time and effort learning things the hard way, through direct experience, but it remain anchored in and translatable into the language of experience
Recommended from our members
Iterated-sums signature, quasi-symmetric functions and time series analysis
We survey and extend results on a recently defined character on the quasi-shuffle algebra. This character, termed iterated-sums signature, appears in the context of time series analysis and originates from a problem in dynamic time warping. Algebraically, it relates to (multidimensional) quasisymmetric functions as well as (deformed) quasi-shuffle algebras
Signal Classification in Quotient Spaces via Globally Optimal Variational Calculus
A ubiquitous problem in pattern recognition is that of matching an observed
time-evolving pattern (or signal) to a gold standard in order to recognize or
characterize the meaning of a dynamic phenomenon. Examples include matching
sequences of images in two videos, matching audio signals in speech
recognition, or matching framed trajectories in robot action recognition. This
paper shows that all of these problems can be aided by reparameterizing the
temporal dependence of each signal individually to a universal standard
timescale that allows pointwise comparison at each instance of time. Given two
sequences, each with timesteps, the complexity of the algorithm has a cost
of , which is an improvement on the most common method for matching two
signals, i.e., dynamic time warping. The core of the approach presented here is
that the universal standard timescale is found by solving a variational
calculus problem in which the cost functional reflects the amount of change
that takes place as measured in the original temporal variable, and then
produces a mapping to a new temporal variable in which the amount of change is
globally minimized. The result builds on known facts in differential geometry
Simultaneous inference for misaligned multivariate functional data
We consider inference for misaligned multivariate functional data that
represents the same underlying curve, but where the functional samples have
systematic differences in shape. In this paper we introduce a new class of
generally applicable models where warping effects are modeled through nonlinear
transformation of latent Gaussian variables and systematic shape differences
are modeled by Gaussian processes. To model cross-covariance between sample
coordinates we introduce a class of low-dimensional cross-covariance structures
suitable for modeling multivariate functional data. We present a method for
doing maximum-likelihood estimation in the models and apply the method to three
data sets. The first data set is from a motion tracking system where the
spatial positions of a large number of body-markers are tracked in
three-dimensions over time. The second data set consists of height and weight
measurements for Danish boys. The third data set consists of three-dimensional
spatial hand paths from a controlled obstacle-avoidance experiment. We use the
developed method to estimate the cross-covariance structure, and use a
classification setup to demonstrate that the method outperforms
state-of-the-art methods for handling misaligned curve data.Comment: 44 pages in total including tables and figures. Additional 9 pages of
supplementary material and reference
Invariance transformations for processing NDE signals
The ultimate objective in nondestructive evaluation (NDE) is the characterization of materials, on the basis of information in the response from energy/material interactions. This is commonly referred to as the inverse problem. Inverse problems are in general ill-posed and full analytical solutions to these problems are seldom tractable. Pragmatic approaches for solving them employ a constrained search technique by limiting the space of all possible solutions. A more modest goal is therefore to use the received signal for characterizing defects in objects in terms of the location, size and shape. However, the NDE signal received by the sensors is influenced not only by the defect, but also by the operational parameters associated with the experiment. This dissertation deals with the subject of invariant pattern recognition techniques that render NDE signals insensitive to operational variables, while at the same time, preserve or enhance defect related information. Such techniques are comprised of invariance transformations that operate on the raw signals prior to interpretation using subsequent defect characterization schemes. Invariance transformations are studied in the context of the magnetostatic flux leakage (MFL) inspection technique, which is the method of choice for inspecting natural gas transmission pipelines buried underground;The magnetic flux leakage signal received by the scanning device is very sensitive to a number of operational parameters. Factors that have a major impact on the signal include those caused by variations in the permeability of the pipe-wall material and the velocity of the inspection tool. This study describes novel approaches to compensate for the effects of these variables;Two types of invariance schemes, feature selection and signal compensation, are studied. In the feature selection approach, the invariance transformation is recast as a problem in interpolation of scattered, multi-dimensional data. A variety of interpolation techniques are explored, the most powerful among them being feed-forward neural networks. The second parametric variation is compensated by using restoration filters. The filter kernels are derived using a constrained, stochastic least square optimization technique or by adaptive methods. Both linear and non-linear filters are studied as tools for signal compensation;Results showing the successful application of these invariance transformations to real and simulated MFL data are presented
- …