Mixed Gamma Distribution Research Articles

Statistical analysis of the crack sensitivity of fiber networks

Fracturing two-dimensional random fiber networks of different densities (porosities) were statistically analyzed using both high-resolution finite element models and image analysis algorithms. Under small strains, the finite element fracture models revealed that networks with high relative densities were able to localize evolving fractures to small cracks while surprisingly larger cracks were required to localize fractures in networks of lower density. Further, it is indicated that the pore size distribution in fiber networks is rather diverse and can be captured using two mixed Gamma distributions; one part describing the background size distribution containing the vast majority of pores, and a second part describing the size distribution of larger pores and open regions. The second part covers less than 5% of the total network area but seems to be of paramount importance for the network’s global fracture behavior. It seems as a fiber network’s crack sensitivity is related to the average pore size in the second part of the mixed Gamma distribution. The analysis is remarkably consistent with reported experiments.

Using Mixed Probability Distribution Functions for Modelling Non-Zero Sub-Daily Rainfall in Australia

Probabilistic models for sub-daily rainfall predictions are important tools for understanding catchment hydrology and estimating essential rainfall inputs for agricultural and ecological studies. This research aimed at achieving theoretical probability distribution to non-zero, sub-daily rainfall using data from 1467 rain gauges across the Australian continent. A framework was developed for estimating rainfall data at ungauged locations using the fitted model parameters from neighbouring gauges. The Lognormal, Gamma and Weibull distributions, as well as their mixed distributions were fitted to non-zero six-minutes rainfall data. The root mean square error was used to evaluate the goodness of fit for each of these distributions. To generate data at ungauged locations, parameters of well-fit models were interpolated from the four closest neighbours using inverse weighting distance method. Results show that the Gamma and Weibull distributions underestimate and lognormal distributions overestimate the high rainfall events. In general, a mixed model of two distributions was found better compared to the results of an individual model. Among the five models studied, the mixed Gamma and Lognormal (G-L) distribution produced the minimum root mean square error. The G-L model produced the best match to observed data for high rainfall events (e.g., 90th, 95th, 99th, 99.9th and 99.99th percentiles).

Efficiency of Modified Mixed Gamma Distribution in Estimating Annual Maximum and Minimum Flows at Moniya Gauging Station, Nigeria

Efficiency of Modified Mixed Gamma Distribution in Estimating Annual Maximum and Minimum Flows at Moniya Gauging Station, Nigeria

Climatology and interannual variability in statistical characteristics of cloud cover over the North Atlantic during 1950-2017

A three-parameter mixed gamma distribution is applied for characterizing the statistical properties of the fractional cloud cover in the North Atlantic using visual cloud cover reports from voluntary observing ships (VOS) for the period from 1950 to 2017. Seasonal climatologies of the distribution parameters and the cloud cover are analyzed. The PDF-based analysis allows for the identification of the areas with different probability distributions even under the same mean total cloud cover. Analysis of interannual variability shows that in different regions of the North Atlantic, there are noticeable changes in the form of the distribution over time. In some regions, these changes are coincident with significant trends in the total cloud cover and result in changes in the distribution shape.

A Bayesian Plan to Reliability Demonstration Based on Mixed Gamma Distribution in Exponential Case

A Bayesian Plan to Reliability Demonstration Based on Mixed Gamma Distribution in Exponential Case

Exploring the physical controls of regional patterns of flow duration curves – Part 1: Insights from statistical analyses

Abstract. The flow duration curve (FDC) is a classical method used to graphically represent the relationship between the frequency and magnitude of streamflow. In this sense it represents a compact signature of temporal runoff variability that can also be used to diagnose catchment rainfall-runoff responses, including similarity and differences between catchments. This paper is aimed at extracting regional patterns of the FDCs from observed daily flow data and elucidating the physical controls underlying these patterns, as a way to aid towards their regionalization and predictions in ungauged basins. The FDCs of total runoff (TFDC) using multi-decadal streamflow records for 197 catchments across the continental United States are separated into the FDCs of two runoff components, i.e., fast flow (FFDC) and slow flow (SFDC). In order to compactly display these regional patterns, the 3-parameter mixed gamma distribution is employed to characterize the shapes of the normalized FDCs (i.e., TFDC, FFDC and SFDC) over the entire data record. This is repeated to also characterize the between-year variability of "annual" FDCs for 8 representative catchments chosen across a climate gradient. Results show that the mixed gamma distribution can adequately capture the shapes of the FDCs and their variation between catchments and also between years. Comparison between the between-catchment and between-year variability of the FDCs revealed significant space-time symmetry. Possible relationships between the parameters of the fitted mixed gamma distribution and catchment climatic and physiographic characteristics are explored in order to decipher and point to the underlying physical controls. The baseflow index (a surrogate for the collective impact of geology, soils, topography and vegetation, as well as climate) is found to be the dominant control on the shapes of the normalized TFDC and SFDC, whereas the product of maximum daily precipitation and the fraction of non-rainy days was found to control the shape of the FFDC. These relationships, arising from the separation of total runoff into its two components, provide a potential physical basis for regionalization of FDCs, as well as providing a conceptual framework for developing deeper process-based understanding of the FDCs.

Exploring the physical controls of regional patterns of flow duration curves – Part 4: A synthesis of empirical analysis, process modeling and catchment classification

Abstract. The paper reports on a four-pronged study of the physical controls on regional patterns of the flow duration curve (FDC). This involved a comparative analysis of long-term continuous data from nearly 200 catchments around the US, encompassing a wide range of climates, geology, and ecology. The analysis was done from three different perspectives – statistical analysis, process-based modeling, and data-based classification – followed by a synthesis, which is the focus of this paper. Streamflow data were separated into fast and slow flow responses, and associated signatures, and both total flow and its components were analyzed to generate patterns. Regional patterns emerged in all aspects of the study. The mixed gamma distribution described well the shape of the FDC; regression analysis indicated that certain climate and catchment properties were first-order controls on the shape of the FDC. In order to understand the spatial patterns revealed by the statistical study, and guided by the hypothesis that the middle portion of the FDC is a function of the regime curve (RC, mean within-year variation of flow), we set out to classify these catchments, both empirically and through process-based modeling, in terms of their regime behavior. The classification analysis showed that climate seasonality and aridity, either directly (empirical classes) or through phenology (vegetation processes), were the dominant controls on the RC. Quantitative synthesis of these results determined that these classes were indeed related to the FDC through its slope and related statistical parameters. Qualitative synthesis revealed much diversity in the shapes of the FDCs even within each climate-based homogeneous class, especially in the low-flow tails, suggesting that catchment properties may have become the dominant controls. Thus, while the middle portion of the FDC contains the average response of the catchment, and is mainly controlled by climate, the tails of the FDC, notably the low-flow tails, are mainly controlled by catchment properties such as geology and soils. The regime behavior explains only part of the FDC; to gain a deeper understanding of the physical controls on the FDC, these extremes must be analyzed as well. Thus, to completely separate the climate controls from the catchment controls, the roles of catchment properties such as soils, geology, topography etc. must be explored in detail.

Bias correction of monthly precipitation and temperature fields from Intergovernmental Panel on Climate Change AR4 models using equidistant quantile matching

A new quantile‐based mapping method is developed for the bias correction of monthly global circulation model outputs. Compared to the widely used quantile‐based matching method that assumes stationarity and only uses the cumulative distribution functions (CDFs) of the model and observations for the baseline period, the proposed method incorporates and adjusts the model CDF for the projection period on the basis of the difference between the model and observation CDFs for the training (baseline) period. Thus, the method explicitly accounts for distribution changes for a given model between the projection and baseline periods. We demonstrate the use of the new method over northern Eurasia. We fit a four‐parameter beta distribution to monthly temperature fields and discuss the sensitivity of the results to the choice of distribution range parameters. For monthly precipitation data, a mixed gamma distribution is used that accounts for the intermittent nature of rainfall. To test the fidelity of the proposed method, we choose 1970–1999 as the baseline training period and then randomly select 30 years from 1901–1999 as the projection test period. The bootstrapping is repeated 30 times to mimic different climate conditions that may occur, and the results suggest that both methods are comparable when applied to the 20th century for both temperature and precipitation for the examined quartiles. We also discuss the dependence of the bias correction results on the choice of time period for training. This indicates that the remaining biases in the bias‐corrected time series are directly tied to the model's performance during the training period, and therefore care should be taken when using a particular training time period. When applied to the Intergovernmental Panel on Climate Change fourth assessment report (AR4) A2 climate scenario projection, the data time series after bias correction from both methods exhibit similar spatial patterns. However, over regions where the climate model shows large changes in projected variability, there are discernable differences between the methods. The proposed method is more sensitive to a reduction in variability, exemplified by wintertime temperature. Further synthetic experiments using the lower 33% and upper 33% of the full data set as the validation data suggest that the proposed equidistance quantile‐matching method is more efficient in reducing biases than the traditional CDF mapping method for changing climates, especially for the tails of the distribution. This has important consequences for the occurrence and intensity of future projected extreme events such as heat waves, floods, and droughts. As the new method is simple to implement and does not require substantial computational time, it can be used to produce auxiliary ensemble scenarios for various climate impact‐oriented applications.

Rainfall pattern and moisture availability index in relation to rice (Oryza sativa L.) crop planning in eastern plateau region of India

Studies were undertaken to identify the quantum and distribution of rainfall with the frequency of dry spell occurrence during monsoon at Giridih, Jharkhand state, India. Rainfall at different confidence levels was calculated using mixed gamma distribution. The normal onset of monsoon at Giridih was 24th standard meteorological week (SMW) and the mean monsoon rainfall was 1112 mm. If monsoon onsets two weeks earlier (22nd SMW) than the normal (24th SMW) the total monsoonal rainfall was more than the normal with increased number of dry spells. The co-efficient of variation of June and September rainfall was very high. Moisture availability index (MAI) indicated possibility of rice cultivation from 25th SMW and the flowering stage of rice should be completed within 39th SMW (normal withdraw of monsoon). Thus to minimize crop failure, conventional cultivation of 135 ± 10 days rice could be replaced by 95 ± 10 days one, particularly in upland (without bund; direct seeded rice) and medium land (low bund land) situation.

Estimation of areal reduction factors using a mixed gamma distribution

Summary This study proposes a methodology to estimate the areal reduction factor (ARF) using mixed probability density functions. In practice, ARFs are estimated by using contemporaneous rainfall data from a high density rain gauge network over the basin, which are rarely available. The methodology proposed in this study uses all rainfall data available, whether or not they are concurrent with others. A mixed probability density function is introduced to explain both the rainfall intermittency and variability, and is applied to both point rainfall and areal average rainfall, which are compared to estimate the ARFs. In this study, a mixed gamma distribution was applied for the Geum river basin, Korea, to estimate the ARFs of a one-day (or, 24 h) rainfall as an example. The new method was easier to apply and provided similar results with those of other studies.

Rainfall frequency analysis using a mixed Gamma distribution: evaluation of the global warming effect on daily rainfall. Chulsang Yoo, Kwang‐Sik Jung and Tae‐Woong Kim, hyp.5985

AbstractThe original article to which this Erratum refers was published in Hydrological Processes, hyp.5985. Copyright © 2006 John Wiley & Sons, Ltd.

Rainfall frequency analysis using a mixed Gamma distribution: evaluation of the global warming effect on daily rainfall

AbstractThe effect of global warming (represented by general circulation model monthly rainfall predictions) on the daily rainfall distribution is investigated using a mixed Gamma distribution to estimate the change of rainfall quantiles. A mixed distribution is used to overcome the limitation of conventional frequency analysis, which uses a continuous distribution, as this is not applicable for the assessment of the effects of global warming. To summarize the results: (1) Even though the variation of daily rainfall distribution is high due to the variation of monthly rainfall amounts, the scale parameter and the wet probability of a mixed Gamma distribution are found to be closely related to the monthly rainfall amounts. On the other hand, the shape factor remains almost the same regardless of the monthly rainfall amount. (2) The rainfall quantiles estimated using the daily rainfall data from June to September were found to be the most similar to those using the annual maximum data. (3) Regardless of the increasing uncertainty as the return period becomes longer, flood risk is found to be increasing as a result of global warming. Copyright © 2005 John Wiley & Sons, Ltd.

혼합분포를 이용한 면적감소계수의 산정

본 연구에서는 혼합 확률밀도함수를 이용한 면적감소계수의 추정법을 제안한다. 기존 면적감소계수의 추정에는 동시간 강우자료가 필요하나 그런 자료를 충분히 구하기는 쉽지 않다. 본 연구에서 제안하는 방법은 보다 가용한 일 강우자료를 이용하는 방법으로 강우의 간헐성을 고려하기 위해 연속분포가 아닌 혼합분포를 이용한다. 본 연구에서는 혼합감마분포를 이용하여 금강유역의 면적감소계수를 추정하였으며, 그 결과 보다 쉽게 아울러 기존의 방법에 의 한 결과와 잘 대비되는 결과를 얻을 수 있었다. This study proposes a new methodology to derive the areal reduction factor (ARF) using mixed probability density functions. Estimation of ARFs requires using the simultaneous rainfall data over the basin, which is rarely available in general. The new methodology Proposed in this study uses more available daily rainfall data during a given period, so the mixed probability density functions should be introduced to explain both the rainfall intermittency and variability. This study applied the mixed gamma distribution for the derivation of ARFs for the Keum river basin, and found that the new method is easier for application as well as it provides very comparable results.

A climatic index for soil erosion potential (CSEP) including seasonal and vegetation factors

The concept of Cumulative Erosion Potential (CEP) proposed by De Ploey, Kirkby and Ahnert in 1991 ( Earth Surf. Process. Landforms, 16: 399–409) provides a simple climatic index of erosion potential based on annual rainfalls and their distribution, which is assumed to be exponential (or gamma with shape parameter 1). A modified CEP (CSEP) is presented here, extending the original in three ways. First, the index is disaggregated by month to include the effects of seasonality. Second, a mixed gamma(1) distribution is used to provide a better fit to the whole of the daily rainfall distribution. Third, the index is calculated both for a bare soil and for a natural vegetation cover, giving an additional index to the potential for managing erosion through landuse change. For the natural vegetation, the index is based on an estimate of equilibrium soil organic matter which is associated with differences in soil moisture storage capacities and derived from monthly rainfall and temperature values. The index thus remains a measure of climatic potential only, based on widely available monthly data on temperature, rainfall and number of rain days. It may be compared with the patterns shown by the European Soil Erosion Map and by the CORINE classification, which are dominated by a combination of climate and lithology.

The mixed gamma ageing model in life data analysis

Parametric and non-parametric lifetime data analyses in practical applications require sensitive tools if non-monotonic ageing properties (trend changes) are to be examined. The well known bathtub-shaped hazard rate is a special model for describing a trend change in ageing properties over time. The identification of trend changes in the hazard rate can be supported by graphical tools. This paper discusses the combined application of graphical tools and parametric estimation in the flexible mixed gamma distribution family to identify trend changes and model bathtub-shaped hazard rates.

Short term probabilities of rainfall in a semi-arid monsoonal climate

Application of Binomial distribution to the long term rainfall probabilities offers a method of estimating short term probabilities for periods like 4 out of 5 years, 2, 3 or 5 successive years etc. In view of their importance in irrigation scheduling, short term hydrological and agricultural planning, short term probabilities for the south west monsoon rainfall distribution over weekly periods in the Delhi region with a semi-arid monsoon climate based on data for the period 1941–1970 have been estimated. These are compared with the long term probability values derived using mixed gamma distribution. Probability values have been derived for occurrence of rainfall > Potential Evapotranspiration (PET) representing humid conditions and for rainfall < PET/2 representing drought conditions.

Fluctuations in the deficiency of the summer monsoon over India, and their effect on economy

To assess the deficiency in the activity of the monsoon over India during the season, an index for the country based on the percentage area with a specified percentage seasonal rainfall deficiency and termed the Monsoon Deficiency Index (MDI), has been utilized. The statistical properties of the MDI series for the period 1871–1978 have been examined. The series which can be taken to be homogeneous and random has a high variability. MDI is generally not observed to persist at a high level; a high value is invariably followed by a low value. The deficiency over the country is considered as largescale and is termed as monsoon failure when the MDI value equals or exceeds the nineth decile viz. 40, of the mixed gamma distribution fitted to the MDI series. Using this criterion, the years of monsoon failure have been identified. The monsoon failures are found to occur randomly. The effect of monsoon deficiency on the Indian economy has been assessed.

Double label estimation of the mean duration of the S-phase

The theoretical values of the double label index and the estimator of the mean duration of the DNA-synthesis phase are obtained for the situation where a steady-state renewal cell population is subjected to two instantaneous labels. The stochastic model used to describe this situation allows for a correlation structure among the durations of the phases of the cell cycle. An explicit form for the double label index is presented for the case where the durations of the phases of the cell cycle are jointly distributed in accordance with a multivariate mixed gamma distribution. The effect of the length sampling bias and the time between label points on the theoretical value of the estimator is demonstrated.