The entropy rate of Linear Additive Markov Processes.

This work derives a theoretical value for the entropy of a Linear Additive Markov Process (LAMP), an expressive but simple model able to generate sequences with a given autocorrelation structure. Our research establishes that the theoretical entropy rate of a LAMP model is equivalent to the theoretical entropy rate of the underlying first-order Markov Chain. The LAMP model captures complex relationships and long-range dependencies in data with similar expressibility to a higher-order Markov process. While a higher-order Markov process has a polynomial parameter space, a LAMP model is characterised only by a probability distribution and the transition matrix of an underlying first-order Markov Chain. This surprising result can be explained by the information balance between the additional structure imposed by the next state distribution of the LAMP model, and the additional randomness of each new transition. Understanding the entropy of the LAMP model provides a tool to model complex dependencies in data while retaining useful theoretical results. To emphasise the practical applications, we use the LAMP model to estimate the entropy rate of the LastFM, BrightKite, Wikispeedia and Reuters-21578 datasets. We compare estimates calculated using frequency probability estimates, a first-order Markov model and the LAMP model, also considering two approaches to ensure the transition matrix is irreducible. In most cases the LAMP entropy rates are lower than those of the alternatives, suggesting that LAMP model is better at accommodating structural dependencies in the processes, achieving a more accurate estimate of the true entropy.

Patient-reported distress and survival among patients receiving definitive radiation therapy.

OBJECTIVE: Patient-reported distress (PRD) has not been well assessed in association with survival after radiation therapy (RT). The aims of this study were to evaluate the association between PRD level and survival after definitive RT and to identify the main causes of distress in definitive RT patients. METHODS AND MATERIALS: A total of 678 consecutive patients receiving definitive RT at our institution from April 2012 through May 2015 were included. All patients answered a PRD questionnaire that contained 30 items related to possible causes of distress, which could be rated from 1 (no distress) to 5 (high distress). Additionally, patients were asked to rate their overall distress level from 0 (no distress) to 10 (extreme distress). This overall distress level was our primary patient-reported distress measure and was examined as a continuous variable and as a categorical variable with 3 PRD levels (low, 0-3 [n = 295]; moderate, 4-6 [n = 222]; and high, 7-10 [n = 161]). RESULTS: As a continuous variable in multivariable Cox regression analysis, a higher overall PRD level was associated with poorer survival after RT (hazard ratio [HR], 1.39; P = .004). As a categorical variable, compared with patients with low distress, survival was poorer for patients with moderate distress (HR, 1.62; P = .038) or high distress (HR, 1.49; P = .12), but the latter difference was not significant. When the moderate and high distress levels were combined, survival was significantly poorer compared with the low distress level (HR, 1.57; P = .034). The top 5 specific causes of distress that patients mentioned were "How I feel during treatment," "Fatigue," "Out-of-pocket medical costs," "Pain that affects my daily functioning," and "Sleep difficulties." CONCLUSIONS: PRD before or during RT is a prognostic factor associated with decreased survival. Distress screening guidelines and interventions should be implemented for patients receiving definitive RT.