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

Dimension-free Euler estimates of rough differential equations

We give a dimension-free Euler estimation of solution of rough differential equations in term of the driving rough path. In the meanwhile, we prove that, the solution of rough differential equation is close to the exponential of a Lie series, with a concrete error bound

Separation capacity of linear reservoirs with random connectivity matrix

We argue that the success of reservoir computing lies within the separation capacity of the reservoirs and show that the expected separation capacity of random linear reservoirs is fully characterised by the spectral decomposition of an associated generalised matrix of moments. Of particular interest are reservoirs with Gaussian matrices that are either symmetric or whose entries are all independent. In the symmetric case, we prove that the separation capacity always deteriorates with time; while for short inputs, separation with large reservoirs is best achieved when the entries of the matrix are scaled with a factor $\rho_T/\sqrt{N}$, where $N$ is the dimension of the reservoir and $\rho_T$ depends on the maximum length of the input time series. In the i.i.d. case, we establish that optimal separation with large reservoirs is consistently achieved when the entries of the reservoir matrix are scaled with the exact factor $1/\sqrt{N}$. We further give upper bounds on the quality of separation in function of the length of the time series. We complement this analysis with an investigation of the likelihood of this separation and the impact of the chosen architecture on separation consistency

Root Cause Analysis of Actuator Fault

This chapter develops a two-level fault diagnosis (FD) and root cause analysis (RCA) scheme for a class of interconnected invertible dynamic systems and aims at detecting and identifying actuator fault and the causes. By considering actuator as an individual dynamic subsystem connected with process dynamic subsystem in cascade, an interconnected system is then constituted. Invertibility of the interconnected system in faulty model is studied. An interconnected observer is introduced and aims at monitoring the performance of the interconnected system and providing information of actuator fault occurrence. A local fault filter algorithm is then triggered to identify the root causes of the detected actuator faults. According to real plant, outputs of the actuator subsystem are assumed inaccessible and are reconstructed by measurements of the global system, thus providing a means for monitoring and diagnosing the plant at both local and global level

Uncertainty-wise software anti-patterns detection: A possibilistic evolutionary machine learning approach

Context: Code smells (a.k.a. anti-patterns) are manifestations of poor design solutions that can deteriorate software maintainability and evolution. Research gap: Existing works did not take into account the issue of uncertain class labels, which is an important inherent characteristic of the smells detection problem. More precisely, two human experts may have different degrees of uncertainty about the smelliness of a particular software class not only for the smell detection task but also for the smell type identification one. Unluckily, existing approaches usually reject and/or ignore uncertain data that correspond to software classes (i.e. dataset instances) with uncertain labels. Throwing away and/or disregarding the uncertainty factor could considerably degrade the detection/identification process effectiveness. From a solution approach viewpoint, there is no work in the literature that proposed a method that is able to detect and/or identify code smells while preserving the uncertainty aspect. Objective: The main goal of our research work is to handle the uncertainty factor, issued from human experts, in detecting and/or identifying code smells by proposing an evolutionary approach that is able to deal with anti-patterns classification with uncertain labels. Method: We suggest Bi-ADIPOK, as an effective search-based tool that is capable to tackle the previously mentioned challenge for both detection and identification cases. The proposed method corresponds to an EA (Evolutionary Algorithm) that optimizes a set of detectors encoded as PK-NNs (Possibilistic K-nearest neighbors) based on a bi-level hierarchy, in which the upper level role consists on finding the optimal PK-NNs parameters, while the lower level one is to generate the PK-NNs. A newly fitness function has been proposed fitness function PomAURPC-OVA_dist (Possibilistic modified Area Under Recall Precision Curve One-Versus-All_distance, abbreviated PAURPC_d in this paper). Bi-ADIPOK is able to deal with label uncertainty using some concepts stemming from the Possibility Theory. Furthermore, the PomAURPC-OVA_dist is capable to process the uncertainty issue even with imbalanced data. We notice that Bi-ADIPOK is first built and then validated using a possibilistic base of smell examples that simulates and mimics the subjectivity of software engineers opinions. Results: The statistical analysis of the obtained results on a set of comparative experiments with respect to four relevant state-of-the-art methods shows the merits of our proposal. The obtained detection results demonstrate that, for the uncertain environment, the PomAURPC-OVA_dist of Bi-ADIPOK ranges between 0.902 and 0.932 and its IAC lies between 0.9108 and 0.9407, while for the certain environment, the PomAURPC-OVA_dist lies between 0.928 and 0.955 and the IAC ranges between 0.9477 and 0.9622. Similarly, the identification results, for the uncertain environment, indicate that the PomAURPC-OVA_dist of Bi-ADIPOK varies between 0.8576 and 0.9273 and its IAC is between 0.8693 and 0.9318. For the certain environment, the PomAURPC-OVA_dist lies between 0.8613 and 0.9351 and the IAC values are between 0.8672 and 0.9476. With uncertain data, Bi-ADIPOK can find 35% more code smells than the second best approach (i.e., BLOP). Furthermore, Bi-ADIPOK has succeeded to reduce the number of false alarms (i.e., misclassified smelly instances) by 12%. In addition, our proposed approach can identify 43% more smell types than BLOP and reduces the number of false alarms by 32%. The same results have been obtained for the certain environment, demonstrating Bi-ADIPOK's ability to deal with such environment

The Accessibility Problem for Geometric Rough Differential Equations

AbstractWe show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields [1] with the rough calculus on manifolds developed by Cass, Litterer and Lyons in [2].</jats:p

Multi-parameter fault isolation using trajectory-based envelope

International audienceParameter interval based fault isolation for single parameter fault has ideal isolation speed, the fault parameter value is estimated when fault is isolated. Analogous scheme can be built for multi-parameter fault isolation using envelope to replace scalar threshold for interval judgment. However wrapping effect should be considered when envelope is used. In this paper a multi-parameter fault isolation scheme is built using a trajectory-based envelope which is without of wrapping effect

Fault-tolerant control based on insufficient fault information

International audienc