Estimates of the Distribution Properties | While these methods are precise for a full power law distribution, they do not converge to the proper value when the distribution is truncated either from above or from below (Fig 1A). |
Estimates of the Distribution Properties | This truncated power law estimate does converge to the y >|< — . |
Haplotype Distribution Formalism | Also, as noted above, the power law eXponent of yH(x) is 1 unit greater than the power law eXponent of VH(x)since these two quantities share the properties linking CDFs to PDFs, namely that a PDF is proportional to the first derivative of its CDF. |
Haplotype Distribution Formalism | Formally, the expected number of unique haplotypes (U(R)) in a sample size (R) can be estimated, assuming a truncated power law formalism as defined above, to be (see Sl—S3 Texts): |
Haplotype Numbers in US Sub-populations | For most populations, the haplotype frequency distribution follows a power law with eXponent or of the PDF between 1.4 and 1.9. |
Haplotype Numbers in US Sub-populations | Further, estimates of the power law eXponent converged before the full sample size was analyzed (Table 3 and 83—88 Figs). |
Introduction | Occupancy distributions [7]offer a more natural representation of our data sampling process and use the mathematical concept of regular variation to extrapolate A and H using a power law function that describes the distribution of allele and haplotype frequencies in the population [7,8]. |
Introduction | The power law relationship was constructed under the assumption that an infinite number of categories (or kinds) exist in the population; so for our purposes we apply a boundary constraint to estimate a finite number of categories under this general framework. |
Methodology Validation | We generated H haplotypes with relative frequencies (pj ’ 5) taken from a pure power law distribution with exponent of a. |
Methodology Validation | A similar analysis with a truncated power law gave similar results. |
Methodology Validation | We then compared the convergence rate of our method with other frameworks for estimating H. We used a simulated population from a truncated power law , as described above and formed sub samples of incremental size. |
Abstract | The equilibrium constants of the binding follow a log-normal distribution around the mean and a power law distribution in the tail. |
Abstract | Furthermore, the kinetics of binding follows a log-normal distribution near the mean and a power law distribution at the tail. |
Analytical Models of Distribution of Affinity, Equilibrium Constants, Specificity and Kinetics | On the other hand, since partition function Z has an asymptotic form and a power law distribution at the tail, the associated free energy is therefore exponentially distributed at the tail (x—>oo)(See details in SI Text). |
Analytical Models of Distribution of Affinity, Equilibrium Constants, Specificity and Kinetics | So, the distribution function f of the equilibrium constants K or the affinity can be shown to be: which is a log-normal distribution near the mean above TC While which shows a power law decay near the low K value tail of the distribution (near or below TC). |
Analytical Models of Distribution of Affinity, Equilibrium Constants, Specificity and Kinetics | This power law behavior is familiar in the physics and chemistry community as a signature of the long range order and self similarity often appeared in the critical phenomena of phase transition, fractals, turbulence and earthquakes etc. |
The Warwick model, control actions and ensembles | The parameters, ps, 3, pc, 5, p5, T and p6, T, are power law parameters that account for a nonlinear increase in susceptibility and transmissibility as animal numbers on a farm increase. |
The Warwick model, control actions and ensembles | Previous work has indicated that a model with power laws provides a closer fit to the 2001 data than when these powers are set to unity [43,51,52]. |
The Warwick model, control actions and ensembles | Region-specific transmissibility and susceptibility parameters (and associated power laws ) capture specific epidemiological characteristics and policy measures used in the main hot spots of Cumbria, Devon and the Welsh and Scottish borders. |