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.Comment: 12 pages, 3 tables, 10 figure

A new approach to SLE phase transition

It is well know that SLEκ curves exhibit a phase transition at κ=4. For κ≤4 they are simple curves with probability one, for κ>4 they are not. The standard proof is based on the analysis of the Bessel SDE of dimension d=1+4/κ. We propose a different approach which is based on the analysis of the Bessel SDE with d=1−4/κ. This not only gives a new perspective, but also allows to describe the formation of the SLE `bubbles' for κ>4

Deriving information from missing data: implications for mood prediction

The availability of mobile technologies has enabled the efficient collection prospective longitudinal, ecologically valid self-reported mood data from psychiatric patients. These data streams have potential for improving the efficiency and accuracy of psychiatric diagnosis as well predicting future mood states enabling earlier intervention. However, missing responses are common in such datasets and there is little consensus as to how this should be dealt with in practice. A signature-based method was used to capture different elements of self-reported mood alongside missing data to both classify diagnostic group and predict future mood in patients with bipolar disorder, borderline personality disorder and healthy controls. The missing-response-incorporated signature-based method achieves roughly 66\% correct diagnosis, with f1 scores for three different clinic groups 59\% (bipolar disorder), 75\% (healthy control) and 61\% (borderline personality disorder) respectively. This was significantly more efficient than the naive model which excluded missing data. Accuracies of predicting subsequent mood states and scores were also improved by inclusion of missing responses. The signature method provided an effective approach to the analysis of prospectively collected mood data where missing data was common and should be considered as an approach in other similar datasets

On the rough Gronwall lemma and its applications

We present a rough path analog of the classical Gronwall Lemma introduced recently by A. Deya, M. Gubinelli, M. Hofmanov\'a, S. Tindel in [arXiv:1604.00437] and discuss two of its applications. First, it is applied in the framework of rough path driven PDEs in order to establish energy estimates for weak solutions. Second, it is used in order to prove uniqueness for reflected rough differential equations