Advancing Shannon entropy for measuring diversity in systems

From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy H has been used to re-think diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon's approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a re-normalization of probability distributions based on the notion of case-based entropy Cc as a function of the cumulative probability c. Given a probability density p(x), Cc measures the diversity of the distribution up to a cumulative probability of c, by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of ^pc(x) up to cumulative probability c. We illustrate the utility of our approach by re-normalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions for energy of sub-atomic particles. The comparison shows that Cc is a vast improvement over H as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions

Atmospheric noise on the bispectrum in optical speckle interferometry

Based on a simple picture of speckle phenomena in optical interferometry it is shown that the recent signal-to-noise ratio estimate for the so called bispectrum, due to Wirnitzer (1985), does not possess the right limit when photon statistics is unimportant. In this wave-limit, which is true for bright sources, his calculations over-estimate the signal-to-noise ratio for the bispectrum by a factor of the order of the square root of the number of speckles

Mathematical diversity of parts for a continuous distribution

The current paper is part of a series exploring how to link diversity measures (e.g., Gini-Simpson index, Shannon entropy, Hill numbers) to a distribution’s original shape and to compare parts of a distribution, in terms of diversity, with the whole. This linkage is crucial to understanding the exact relationship between the density of an original probability distribution, denoted by p(x), and the diversity D in non-uniform distributions, both within parts of a distribution and the whole. Empirically, our results are an important advance since we can compare various parts of a distribution, noting that systems found in contemporary data often have unequal distributions that possess multiple diversity types and have unknown and changing frequencies at different scales (e.g. income, economic complexity ratings, rankings, etc.). To date, we have proven our results for discrete distributions. Our focus here is continuous distributions. In both instances, we do so by linking case-based entropy, a diversity approach we developed, to a probability distribution’s shape for continuous distributions. This allows us to demonstrate that the original probability distribution g 1, the case-based entropy curve g 2, and the slope of diversity g 3 (c (a, x) versus the c(a, x)*lnA(a, x) curve) are one-to-one (or injective). Put simply, a change in the probability distribution, g 1, leads to variations in the curves for g 2 and g 3. Consequently, any alteration in the permutation of the initial probability distribution, which results in a different form, will distinctly define the graphs g 2 and g3 . By demonstrating the injective property of our method for continuous distributions, we introduce a unique technique to gauge the level of uniformity as indicated by D/c. Furthermore, we present a distinct method to calculate D/c for different forms of the original continuous distribution, enabling comparison of various distributions and their components

Transportation of lymph node biopsy specimens in selective Kirchner’s liquid medium for culture of tubercle bacilli

Lymph node biopsy specimens, obtained from 297 paediatric and adult patients with tuberculous lymphadenitis at Madurai, were transported in selective Kirchner’s liquid medium (KL-T) to the Tuberculosis Research Centre, Madras and processed for culture. Mycobucterium tuberculosis was isolated from 201 (68%) specimens. Of the 192 specimens received within 4 days of resection, 134 (69.8%) yielded M. tuberculosis on culture and of the 105 specimens received after 5 days, 67 (63.8%) were culture positive; the difference was not statistically significant. By incubating KL-T alone further, after removing the gland for processing, it was found that mere contact with the excised node during transportation was enough to retrieve 77 (38.3%) of the total of 201 positive isolates obtained, the delay did not affect the culture positivity rate. Thus, lymph node specimens for culture of tubercle bacilli can be stored in the refrigerator for up to 15 days and transported in KL-T at ambient temperature for 18-20 h without any loss in culture positivity

A causal investigation of soy isoflavone intake for primary prevention of post-menopausal breast cancer among Asian women

The incidence of breast cancer is increasing at an alarming rate across Asia, by up to 6% annually, compared to near stable incidence rates in many Western countries. While selective oestrogen receptor modulators and aromatase inhibitors are actively being studied as chemoprevention among high-risk Caucasian women, the risks may outweigh the benefits among Asian women with lower population risk of breast cancer. Modifiable lifestyle targets for primary prevention have long been identified, such as post-menopausal obesity, alcohol intake, and hormone replacement therapy use, but these risk factors are less prevalent among women in Asian countries. There remains an urgent need to find primary prevention strategies that are low risk, acceptable, and effective for Asian women. Epidemiological evidence in Asian women suggests that high soy intake is associated with lower risk of breast cancer, but these findings were not observed in epidemiological studies of Caucasian women nor in clinical trials of soy isoflavone supplements. To date, there are no clinical trials that examine the effect of soy isoflavone intake from diet nor supplement on breast cancer risk among Asian women. In this thesis, I present the research studies undertaken to investigate if soy isoflavone intake is causally and inversely associated with post-menopausal breast cancer risk among Asian women. The objective of the first research study was to identify mammographic density measures that are suitable biomarkers of breast cancer risk in the target population (Chapter 3). In this study, volume-based mammographic density measures and breast cancer risk factors were compared between 1,501 Malaysian women and 4,501 age- and BMI-matched Swedish women with no personal history of cancer. The analysis demonstrated that absolute dense volume, rather than percent density, may be a better biomarker of breast cancer risk among post-menopausal Asian women. Based on the above findings, the second research study sought to determine if mammographic density mediates the association between soy intake and breast cancer risk in the target population (Chapter 4). A cross-sectional analysis of 3,277 healthy Malaysian women showed that mammographic density was lower among women with frequent soy intake compared to non-consumers, by up to 2.5cm3 dense volume or 2.0cm2 dense area, but this was not statistically significant. Intriguingly, regular soy intake was associated with lower mammographic density among overweight or obese women, but for leaner women, regular soy intake was associated with higher mammographic density. This interaction was statistically significant among pre-menopausal women (pinteraction = 0.029). Prior to designing a robust clinical trial to test the causal association between soy intake and mammographic density as a biomarker of breast cancer risk, the feasibility of a dietary soy intervention was assessed in a small sample of the target population (n=10, Chapter 5). Overall, women in the study were able to maintain a diet of 70-90mg/day of soy isoflavones for 2 months, but the target of 100mg/day was not feasible and may have led to some adverse events. Thematic analysis of semi-structured interviews revealed that women participated in the study for altruistic reasons and due to emotional attachments to the cause, and that adherence was largely influenced by the practicability of the new diet or routine. Building from the results of the previous three chapters, the primary objective of the last research study was to test the effect of daily soy isoflavone intake for 1 year on breast cancer risk among Asian women, using mammographic density as a biomarker of risk (Chapter 6). In this clinical trial, 57 healthy post-menopausal Malaysian women were randomized into the Supplement arm (100mg/day isoflavones, with >90% daidzein), the Dietary Soy arm (50mg/day isoflavones), or the Control arm. After 1 year of intervention, women in the Supplement arm experienced 4.1cm2 lower dense area and 2.4% lower area-based percent density compared to women in the Control arm, but these associations were not statistically significant. The associations were weaker for women in the Dietary Soy arm and for volume-based mammographic density measures. Interestingly, stronger associations were observed when the analysis was limited to women with high BMI or low dietary fat intake, but the sample size was too small for robust analyses. In conclusion, the data presented in this thesis suggest a causal association between soy isoflavone intake and lower post-menopausal breast cancer risk among Asian women. However, due to the small sample size, the analysis was underpowered to show statistically significant effects and will require confirmation in a larger trial. Nonetheless, the research undertaken here adds to existing evidence that the soy isoflavone daidzein may be responsible for the protective effect of soy. Furthermore, it proposes new hypotheses in understanding the association between soy intake and breast cancer risk across populations, including possible effect modification by BMI or dietary fat intake

Translational diffusion of fluorescent probes on a sphere: monte carlo simulations, theory, and fluorescence anisotropy experiment

Translational diffusion of fluorescent molecules on curved surfaces (micelles, vesicles, and proteins) depolarizes the fluorescence. A Monte Carlo simulation method was developed to obtain the fluorescence anisotropy decays for the general case of molecular dipoles tilted at an angle a to the surface normal. The method is used to obtain fluorescence anisotropy decay due to diffusion of tilted dipoles on a spherical surface, which matched well with the exact solution for the sphere. The anisotropy decay is a single exponential for α = 0° , a double exponential for α = 90° , and three exponentials for intermediate angles. The slower decay component(s) for α ≠ 0 arise due to the geometric phase factor. Although the anisotropy decay equation contains three exponentials, there are only two parameters, namely a and the rate constant, Dtr/R2, where Dtr is the translational diffusion coefficient and R is the radius of the sphere. It is therefore possible to determine the orientation angle and translational diffusion coefficient from the experimental fluorescence anisotropy data. This method was applied in interpreting the fluorescence anisotropy decay of Nile red in SDS micelles. It is necessary, however, to include two other independent mechanisms of fluorescence depolarization for molecules intercalated in micelles. These are the wobbling dynamics of the molecule about the molecular long axis, and the rotation of the spherical micelle as a whole. The fitting of the fluorescence anisotropy decay to the full equation gave the tilt angle of the molecular dipoles to be 1± 2° and the translational diffusion coefficient to be 1.3± 0.1×10-10 m2/s

A causal investigation of soy isoflavone intake for primary prevention of post-menopausal breast cancer among Asian women

The incidence of breast cancer is increasing at an alarming rate across Asia, by up to 6% annually, compared to near stable incidence rates in many Western countries. While selective oestrogen receptor modulators and aromatase inhibitors are actively being studied as chemoprevention among high-risk Caucasian women, the risks may outweigh the benefits among Asian women with lower population risk of breast cancer. Modifiable lifestyle targets for primary prevention have long been identified, such as post-menopausal obesity, alcohol intake, and hormone replacement therapy use, but these risk factors are less prevalent among women in Asian countries. There remains an urgent need to find primary prevention strategies that are low risk, acceptable, and effective for Asian women. Epidemiological evidence in Asian women suggests that high soy intake is associated with lower risk of breast cancer, but these findings were not observed in epidemiological studies of Caucasian women nor in clinical trials of soy isoflavone supplements. To date, there are no clinical trials that examine the effect of soy isoflavone intake from diet nor supplement on breast cancer risk among Asian women. In this thesis, I present the research studies undertaken to investigate if soy isoflavone intake is causally and inversely associated with post-menopausal breast cancer risk among Asian women. The objective of the first research study was to identify mammographic density measures that are suitable biomarkers of breast cancer risk in the target population (Chapter 3). In this study, volume-based mammographic density measures and breast cancer risk factors were compared between 1,501 Malaysian women and 4,501 age- and BMI-matched Swedish women with no personal history of cancer. The analysis demonstrated that absolute dense volume, rather than percent density, may be a better biomarker of breast cancer risk among post-menopausal Asian women. Based on the above findings, the second research study sought to determine if mammographic density mediates the association between soy intake and breast cancer risk in the target population (Chapter 4). A cross-sectional analysis of 3,277 healthy Malaysian women showed that mammographic density was lower among women with frequent soy intake compared to non-consumers, by up to 2.5cm3 dense volume or 2.0cm2 dense area, but this was not statistically significant. Intriguingly, regular soy intake was associated with lower mammographic density among overweight or obese women, but for leaner women, regular soy intake was associated with higher mammographic density. This interaction was statistically significant among pre-menopausal women (pinteraction = 0.029). Prior to designing a robust clinical trial to test the causal association between soy intake and mammographic density as a biomarker of breast cancer risk, the feasibility of a dietary soy intervention was assessed in a small sample of the target population (n=10, Chapter 5). Overall, women in the study were able to maintain a diet of 70-90mg/day of soy isoflavones for 2 months, but the target of 100mg/day was not feasible and may have led to some adverse events. Thematic analysis of semi-structured interviews revealed that women participated in the study for altruistic reasons and due to emotional attachments to the cause, and that adherence was largely influenced by the practicability of the new diet or routine. Building from the results of the previous three chapters, the primary objective of the last research study was to test the effect of daily soy isoflavone intake for 1 year on breast cancer risk among Asian women, using mammographic density as a biomarker of risk (Chapter 6). In this clinical trial, 57 healthy post-menopausal Malaysian women were randomized into the Supplement arm (100mg/day isoflavones, with >90% daidzein), the Dietary Soy arm (50mg/day isoflavones), or the Control arm. After 1 year of intervention, women in the Supplement arm experienced 4.1cm2 lower dense area and 2.4% lower area-based percent density compared to women in the Control arm, but these associations were not statistically significant. The associations were weaker for women in the Dietary Soy arm and for volume-based mammographic density measures. Interestingly, stronger associations were observed when the analysis was limited to women with high BMI or low dietary fat intake, but the sample size was too small for robust analyses. In conclusion, the data presented in this thesis suggest a causal association between soy isoflavone intake and lower post-menopausal breast cancer risk among Asian women. However, due to the small sample size, the analysis was underpowered to show statistically significant effects and will require confirmation in a larger trial. Nonetheless, the research undertaken here adds to existing evidence that the soy isoflavone daidzein may be responsible for the protective effect of soy. Furthermore, it proposes new hypotheses in understanding the association between soy intake and breast cancer risk across populations, including possible effect modification by BMI or dietary fat intake

A direct rifampicin sensitivity test for tubercle bacilli

A direct sensitivity test for rifampicin has been standardised for early detection of resistance for the mycobacterium tuberculosis smear positive sputum samples. Indirect sensitivity tests set up from primary cultures of the same samples served as controls. The direct test showed 95 per cent agreement with the standard indirect test and as such 74 per cent and 90 per cent of the resistant strains were detected by the fourth week and fifth week, respectively, with an overall gain of 4-5 wks time. Resistance could be detected earlier for multibacillary specimens. This direct sensitivity test on Lowenstein Jensen (LJ) medium offers a feasible alternative for laboratories which lack facilities to perform drug susceptibility tests by the rapid but sophisticated and costly BACTEC method. The method is simple to perform, economic, reliable and amenable to confirmation by the indirect test, if needed

Tuberculosis awareness among educated public in two cities in Tamil Nadu

A questionnaire on source of information regarding tuberculosis, signs and symptoms, diagnostic methods, treatment duration and personal and community hygiene relating to tuberculosis, was administered to 446 students and employees with an educational status of high school certificate and above. The main source of information were books and magazines and 86% were aware that the Tuberculosis germ was the causative agent. Symptoms of Tuberculosis such as cough (85%) and loss of weight (74%) were well known. Other symptoms suchchest pain (29%), fever (27%) were less known. Sputum examination as a diagnostic tool was known to 68% while 80% knew about radiograph being used to diagnose the disease. Cough as a method of spread was known to 91%. In this questionnaire the duration of treatment was the least known fact. 28% felt that treatment could be stopped if symptoms disappeared. 16% were aware that the method of sputum disposal was by incineration. The implications are discussed

On the mathematical quantification of inequality in probability distributions

A fundamental challenge in the study of probability distributions is the quantification of inequality that is inherently present in them. Some parts of the distribution are more probable and some others are not, and we are interested in the quantification of this inequality through the lens of mathematical diversity, which is a new approach to studying inequality. We offer a theoretical advance, based on case-based entropy and slope of diversity, which addresses inequality for arbitrary probability distributions through the concept of mathematical diversity. Our approach is useful in three important ways: (1) it offers a universal way to measure inequality in arbitrary probability distributions based purely on the entropic uncertainty that is inherent in them and nothing else; (2) it allows us to compare the degree of inequality of arbitrary parts of any distribution (not just tails) and entire distributions alike; and (3) it can glean out empirical rules similar to the 80/20 rule, not just for the power law but for any given distribution or its parts thereof. The techniques shown in this paper demonstrate a more general machinery to quantify inequality, compare the degree of inequality of parts or whole of general distributions, and prove or glean out empirical rules for general distributions based on mathematical diversity. We demonstrate the utility of this new machinery by applying it to the power law, the exponential and the geometric distributions. The 60 − 40 rule of restricted diversity states that 60 percent or more of cases following a power law (or more generally a right skewed distribution) reside within 40 percent or less of the lower bound of Shannon equivalent equi-probable (SEE) types as measured by case-based entropy. In this paper, we prove the 60 − 40 rule for power law distributions analytically. We also show that in all power law distributions, the second half of the distribution is at least 4 times more uniformly distributed as the first. Lastly, we also show a scale-free way of comparing probability distributions based on the idea of mathematical diversity of parts of a distribution. We use this comparison technique to compare the exponential and power law distribution, and obtain the exponential distribution as an entropic limit of the power law distribution. We also demonstrate that the machinery is applicable to discrete distributions by proving a general result regarding the comparison of parts of the geometric distribution