Abstract
A way to compare two or more measurements for the same random variable can be achieved by using a negligible error reference measurement, which is called the gold standard, obtained by consolidated measurement methods. This paper presents a new methodology for comparing measurements in the presence of a gold standard with random variables from the multivariate three-parameter (shape, scale, and location) gamma distribution. The errors between gold standard measures and approximate measures have a gamma difference distribution with the same three parameters of the gamma distribution. The concordance measurements were obtained by mean of a coefficient, which measures the degree of agreement as a ratio between the variances of the gold standard and the errors. The developed methodology is illustrated with climatic data which is divided into four ranges. The measurements analyzed are rainfall forecasts of the following four national centers: Canadian Meteorological Center (CMC), European Center for Medium-Range Weather Forecasts (ECMWF), National Centers for Environmental Prediction (NCEP), and Center for Weather Forecasting and Climate Studies (CPTEC). The forecast range was 240 hours for the West mesoregion of Paraná – Brazil, and in the October 1–March 31 period of the 2010/2011 –2015/2016 harvest years. The period was selected because it is related to soybean crop development in the region and because several crop estimation models use rainfall forecast data in this period. The methodology applied spatially indicated the center to be selected in each geographical location according to each rainfall range interval. The gamma model fit well with the data and is an alternative to the normal one for modelling rainfall, in particular to estimate concordances between rainfall forecasts and the gold standard, which are used to improve the selection of rainfall forecast centers.
Highlights
The three-parameter gamma probability distribution according to Johnson et al (1994) has several applications in stochastic modeling and hydrology
The period was selected because it is related to soybean crop development in the region and because several crop estimation models use rainfall forecast data in this period
The qq-plots are presented in Figure 3 for the three-parameter gamma distribution, using the gold standard data grouped in ten-day periods and the corresponding 240 h range of the centers Canadian Meteorological Center (CMC), European Center for Medium-Range Weather Forecasts (ECMWF), National Centers for Environmental Prediction (NCEP), and CPTEC
Summary
The three-parameter (shape, scale, and location) gamma probability distribution according to Johnson et al (1994) has several applications in stochastic modeling and hydrology. The three-parameter (shape), (scale), and (location) gamma probability density function is defined by Mathal and Moschopoulos (1992) as:. For evaluation of the degree of agreement (concordance) for measurements of a random variable with gamma distribution which were obtained by approximation methods, one can use the standard model (Lord & Novick, 1968; Donner, 1986; Fleiss, 1999; Galea, 2013) of the reproducibility for measures (agreement) with respect to a reference measure, called gold standard,. Of the measurements performed via gold standard and the approximation methods on the i-th unity
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