arXiv subjects

Terry Lyons

Publications and source records attributed to Terry Lyons.

At least 91 records · Page 5Linked to original sources

arXiv2019

Developing the Path Signature Methodology and its Application to Landmark-based Human Action Recognition

Landmark-based human action recognition in videos is a challenging task in computer vision. One key step is to design a generic approach that generates discriminative features for the spatial structure and temporal dynamics. To this end, we regard the evolving landmark data as a high-dimensional path and apply non-linear path signature techniques to provide an expressive, robust, non-linear, and interpretable representation for the sequential events. We do not extract signature features from the raw path, rather we propose path disintegrations and path transformations as preprocessing steps. Path disintegrations turn a high-dimensional path linearly into a collection of lower-dimensional paths; some of these paths are in pose space while others are defined over a multiscale collection of temporal intervals. Path transformations decorate the paths with additional coordinates in standard ways to allow the truncated signatures of transformed paths to expose additional features. For spatial representation, we apply the signature transform to vectorize the paths that arise out of pose disintegration, and for temporal representation, we apply it again to describe this evolving vectorization. Finally, all the features are collected together to constitute the input vector of a linear single-hidden-layer fully-connected network for classification. Experimental results on four datasets demonstrated that the proposed feature set with only a linear shallow network and Dropconnect is effective and achieves comparable state-of-the-art results to the advanced deep networks, and meanwhile, is capable of interpretation.

cs.CV↗

arXiv2019

A new definition of rough paths on manifolds

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in "Rough Paths on Manifolds" (Cass, T., Litterer, C. & Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.

math.CA↗

arXiv2019

Deep Signature Transforms

The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network. In this context it may be interpreted as a pooling operation. We present the results of empirical experiments to back up the theoretical justification. Code available at https://github.com/patrick-kidger/Deep-Signature-Transforms.

cs.LG↗

arXiv2019

Learning stochastic differential equations using RNN with log signature features

This paper contributes to the challenge of learning a function on streamed multimodal data through evaluation. The core of the result of our paper is the combination of two quite different approaches to this problem. One comes from the mathematically principled technology of signatures and log-signatures as representations for streamed data, while the other draws on the techniques of recurrent neural networks (RNN). The ability of the former to manage high sample rate streams and the latter to manage large scale nonlinear interactions allows hybrid algorithms that are easy to code, quicker to train, and of lower complexity for a given accuracy. We illustrate the approach by approximating the unknown functional as a controlled differential equation. Linear functionals on solutions of controlled differential equations are the natural universal class of functions on data streams. Following this approach, we propose a hybrid Logsig-RNN algorithm that learns functionals on streamed data. By testing on various datasets, i.e. synthetic data, NTU RGB+D 120 skeletal action data, and Chalearn2013 gesture data, our algorithm achieves the outstanding accuracy with superior efficiency and robustness.

cs.LG↗

arXiv2019

Insertion algorithm for inverting the signature of a path

In this article we introduce the insertion method for reconstructing the path from its signature, i.e. inverting the signature of a path. For this purpose, we prove that a converging upper bound exists for the difference between the inserted n-th term and the (n+1)-th term of the normalised signature of a smooth path, and we also show that there exists a constant lower bound for a subsequence of the terms in the normalised signature of a piecewise linear path. We demonstrate our results with numerical examples.

math.PR↗

arXiv2019

Nonparametric pricing and hedging of exotic derivatives

In the spirit of Arrow-Debreu, we introduce a family of financial derivatives that act as primitive securities in that exotic derivatives can be approximated by their linear combinations. We call these financial derivatives signature payoffs. We show that signature payoffs can be used to nonparametrically price and hedge exotic derivatives in the scenario where one has access to price data for other exotic payoffs. The methodology leads to a computationally tractable and accurate algorithm for pricing and hedging using market prices of a basket of exotic derivatives that has been tested on real and simulated market prices, obtaining good results.

q-fin.MF↗

arXiv2019

Optimal execution with rough path signatures

We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact function is a continuous function of the trading speed. Following an approximation of the optimisation problem, we are able to calculate an optimal solution for the trading speed in the space of linear functions on a truncation of the signature of the price process. We provide strong numerical evidence illustrating the accuracy and flexibility of the approach. Our numerical investigation both examines cases where exact solutions are known, demonstrating that the method accurately approximates these solutions, and models where exact solutions are not known. In the latter case, we obtain favourable comparisons with standard execution strategies.

q-fin.CP↗

arXiv2018

Random forest prediction of Alzheimer's disease using pairwise selection from time series data

Time-dependent data collected in studies of Alzheimer's disease usually has missing and irregularly sampled data points. For this reason time series methods which assume regular sampling cannot be applied directly to the data without a pre-processing step. In this paper we use a machine learning method to learn the relationship between pairs of data points at different time separations. The input vector comprises a summary of the time series history and includes both demographic and non-time varying variables such as genetic data. The dataset used is from the 2017 TADPOLE grand challenge which aims to predict the onset of Alzheimer's disease using including demographic, physical and cognitive data. The challenge is a three-fold diagnosis classification into AD, MCI and control groups, the prediction of ADAS-13 score and the normalised ventricle volume. While the competition proceeds, forecasting methods may be compared using a leaderboard dataset selected from the Alzheimer's Disease Neuroimaging Initiative (ADNI) and with standard metrics for measuring accuracy. For diagnosis, we find an mAUC of 0.82, and a classification accuracy of 0.73. The results show that the method is effective and comparable with other methods.

q-bio.QM↗

arXiv2018

Inverting the signature of a path

The aim of this article is to develop an explicit procedure that enables one to reconstruct any $C^1$ path (at natural parametrization) from its signature. We also explicitly quantify the distance between the reconstructed path and the original path in terms of the number of terms in the signature that are used for the construction and the modulus of continuity of the derivative of the path. A key ingredient in the construction is the use of a procedure of symmetrization that separates the behavior of the path at small and large scales.

math.CA↗

arXiv2018

Labelling as an unsupervised learning problem

Unravelling hidden patterns in datasets is a classical problem with many potential applications. In this paper, we present a challenge whose objective is to discover nonlinear relationships in noisy cloud of points. If a set of point satisfies a nonlinear relationship that is unlikely to be due to randomness, we will label the set with this relationship. Since points can satisfy one, many or no such nonlinear relationships, cloud of points will typically have one, multiple or no labels at all. This introduces the labelling problem that will be studied in this paper. The objective of this paper is to develop a framework for the labelling problem. We introduce a precise notion of a label, and we propose an algorithm to discover such labels in a given dataset, which is then tested in synthetic datasets. We also analyse, using tools from random matrix theory, the problem of discovering false labels in the dataset.

stat.ML↗

arXiv2017

A signature-based machine learning model for bipolar disorder and borderline personality disorder

Mobile technologies offer opportunities for higher resolution monitoring of health conditions. This opportunity seems of particular promise in psychiatry where diagnoses often rely on retrospective and subjective recall of mood states. However, getting actionable information from these rather complex time series is challenging, and at present the implications for clinical care are largely hypothetical. This research demonstrates that, with well chosen cohorts (of bipolar disorder, borderline personality disorder, and control) and modern methods, it is possible to objectively learn to identify distinctive behaviour over short periods (20 reports) that effectively separate the cohorts. Participants with bipolar disorder or borderline personality disorder and healthy volunteers completed daily mood ratings using a bespoke smartphone app for up to a year. A signature-based machine learning model was used to classify participants on the basis of the interrelationship between the different mood items assessed and to predict subsequent mood. The signature methodology was significantly superior to earlier statistical approaches applied to this data in distinguishing the participant three groups, clearly placing 75% into their original groups on the basis of their reports. Subsequent mood ratings were correctly predicted with greater than 70% accuracy in all groups. Prediction of mood was most accurate in healthy volunteers (89-98%) compared to bipolar disorder (82-90%) and borderline personality disorder (70-78%).

stat.ML↗

arXiv2017

Sketching the order of events

We introduce features for massive data streams. These stream features can be thought of as "ordered moments" and generalize stream sketches from "moments of order one" to "ordered moments of arbitrary order". In analogy to classic moments, they have theoretical guarantees such as universality that are important for learning algorithms.

stat.ML↗

arXiv2017

Detecting early signs of depressive and manic episodes in patients with bipolar disorder using the signature-based model

Recurrent major mood episodes and subsyndromal mood instability cause substantial disability in patients with bipolar disorder. Early identification of mood episodes enabling timely mood stabilisation is an important clinical goal. Recent technological advances allow the prospective reporting of mood in real time enabling more accurate, efficient data capture. The complex nature of these data streams in combination with challenge of deriving meaning from missing data mean pose a significant analytic challenge. The signature method is derived from stochastic analysis and has the ability to capture important properties of complex ordered time series data. To explore whether the onset of episodes of mania and depression can be identified using self-reported mood data.

stat.ML↗

arXiv2017

Hyperbolic development and inversion of signature

We develop a simple procedure that allows one to explicitly reconstruct any piecewise linear path from its signature. The construction is based on the development of the path onto the hyperbolic space.

math.CA↗

arXiv2017

Learning Spatial-Semantic Context with Fully Convolutional Recurrent Network for Online Handwritten Chinese Text Recognition

Online handwritten Chinese text recognition (OHCTR) is a challenging problem as it involves a large-scale character set, ambiguous segmentation, and variable-length input sequences. In this paper, we exploit the outstanding capability of path signature to translate online pen-tip trajectories into informative signature feature maps using a sliding window-based method, successfully capturing the analytic and geometric properties of pen strokes with strong local invariance and robustness. A multi-spatial-context fully convolutional recurrent network (MCFCRN) is proposed to exploit the multiple spatial contexts from the signature feature maps and generate a prediction sequence while completely avoiding the difficult segmentation problem. Furthermore, an implicit language model is developed to make predictions based on semantic context within a predicting feature sequence, providing a new perspective for incorporating lexicon constraints and prior knowledge about a certain language in the recognition procedure. Experiments on two standard benchmarks, Dataset-CASIA and Dataset-ICDAR, yielded outstanding results, with correct rates of 97.10% and 97.15%, respectively, which are significantly better than the best result reported thus far in the literature.

cs.CV↗

arXiv2017

Characteristic functions of measures on geometric rough paths

We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from Lévy, Gaussian and Markovian rough paths.

math.PR↗

arXiv2016

Discretely sampled signals and the rough Hoff process

We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead-lag increments. In particular, by sampling a $d$-dimensional continuous semimartingale $X:[0,1] \rightarrow \mathbb{R}^d$ at a set of times $D=(t_i)$, we construct a piecewise linear, axis-directed process $X^D: [0,1] \rightarrow\mathbb{R}^{2d}$ comprised of a past and future component. We call such an object the Hoff process associated with the discrete data $\{X_{t}\}_{t_i\in D}$. The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. We prove that the Itô integral can be recovered from a sequence of random ODEs driven by the components of $X^D$. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong-Zakai Theorem. Such random ODEs have a natural interpretation in the context of mathematical finance.

math.PR↗

arXiv2016

Learning from the past, predicting the statistics for the future, learning an evolving system

We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input variable is a stream of information, and the dependent response is also (potentially) a stream. A certain graded feature set of a stream, known in the rough path literature as the signature, has a universality that allows formally, linear regression to be used to characterise the functional relationship between independent explanatory variables and the conditional distribution of the dependent response. This approach, via linear regression on the signature of the stream, is almost totally general, and yet it still allows explicit computation. The grading allows truncation of the feature set and so leads to an efficient local description for streams (rough paths). In the statistical context this method offers potentially significant, even transformational dimension reduction. By way of illustration, our approach is applied to stationary time series including the familiar AR model and ARCH model. In the numerical examples we examined, our predictions achieve similar accuracy to the Gaussian Process (GP) approach with much lower computational cost especially when the sample size is large.

q-fin.ST↗