Read-noise characterization of focal plane array detectors via mean-variance analysis

Chemical Measurement and Fluctuation Scaling

Fluctuation scaling reports on all processes producing a data set. Some fluctuation scaling relationships, such as the Horwitz curve, follow exponential dispersion models which have useful properties. The mean-variance method applied to Poisson distributed data is a special case of these properties allowing the gain of a system to be measured. Here, a general method is described for investigating gain (G), dispersion (β), and process (α) in any system whose fluctuation scaling follows a simple exponential dispersion model, a segmented exponential dispersion model, or complex scaling following such a model locally. When gain and dispersion cannot be obtained directly, relative parameters, G_R and β_R, may be used. The method was demonstrated on data sets conforming to simple, segmented, and complex scaling. These included mass, fluorescence intensity, and absorbance measurements and specifications for classes of calibration weights. Changes in gain, dispersion, and process were observed in the scaling of these data sets in response to instrument parameters, photon fluxes, mathematical processing, and calibration weight class. The process parameter which limits the type of statistical process that can be invoked to explain a data set typically exhibited 0 < α < 1, with α > 4 possible. With two exceptions, calibration weight class definitions only affected β. Adjusting photomultiplier voltage while measuring fluorescence intensity changed all three parameters (0 < α < 0.8; 0 < β_R < 3; 0 < G_R < 4.1). The method provides a framework for calibrating and interpreting uncertainty in chemical measurement allowing robust comparison of specific instruments, conditions, and methods.

Fluctuation scaling, Taylor's law, and crime

Fluctuation scaling relationships have been observed in a wide range of processes ranging from internet router traffic to measles cases. Taylor’s law is one such scaling relationship and has been widely applied in ecology to understand communities including trees, birds, human populations, and insects. We show that monthly crime reports in the UK show complex fluctuation scaling which can be approximated by Taylor’s law relationships corresponding to local policing neighborhoods and larger regional and countrywide scales. Regression models applied to local scale data from Derbyshire and Nottinghamshire found that different categories of crime exhibited different scaling exponents with no significant difference between the two regions. On this scale, violence reports were close to a Poisson distribution (α = 1.057±0.026) while burglary exhibited a greater exponent (α = 1.292±0.029) indicative of temporal clustering. These two regions exhibited significantly different pre-exponential factors for the categories of anti-social behavior and burglary indicating that local variations in crime reports can be assessed using fluctuation scaling methods. At regional and countrywide scales, all categories exhibited scaling behavior indicative of temporal clustering evidenced by Taylor’s law exponents from 1.43±0.12 (Drugs) to 2.094±0081 (Other Crimes). Investigating crime behavior via fluctuation scaling gives insight beyond that of raw numbers and is unique in reporting on all processes contributing to the observed variance and is either robust to or exhibits signs of many types of data manipulation.