Recent studies have shown that independent identical distributed Gaussian
random variables is not suitable for modelling extreme values observed during extremal
events. However, many real life data on extreme values are dependent and stationary
rather than the conventional independent identically distributed data. We propose a stationary
autoregressive (AR) process with Gumbel distributed innovation and characterise
the short-term dependence among maxima of an (AR) process over a range of sample
sizes with varying degrees of dependence. We estimate the maximum likelihood of the
parameters of the Gumbel AR process and its residuals, and evaluate the performance
of the parameter estimates. The AR process is fitted to the Gumbel-generalised Pareto
(GPD) distribution and we evaluate the performance of the parameter estimates fitted
to the cluster maxima and the original series. Ignoring the effect of dependence leads to
overestimation of the location parameter of the Gumbel-AR (1) process. The estimate
of the location parameter of the AR process using the residuals gives a better estimate.
Estimate of the scale parameter perform marginally better for the original series than the
residual estimate. The degree of clustering increases as dependence is enhance for the AR
process. The Gumbel-AR(1) fitted to the threshold exceedances shows that the estimates
of the scale and shape parameters fitted to the cluster maxima perform better as sample
size increases, however, ignoring the effect of dependence lead to an underestimation of
the parameter estimates of the scale parameter. The shape parameter of the original
series gives a superior estimate compare to the threshold excesses fitted to the Gumbel
distributed Generalised Pareto ditribution.
A boxplot is an exploratory data analysis (EDA) tool for a compact visual display of a distributional summary of a univariate data set. It is designed to capture all typical observations and displays the location, spread, skewness and the tail of the data. The precision of some of this functionality is considered to be more reliable for symmetric data type and thus less appropriate for skewed data such as the extreme data. Many observations from extreme data were mistakenly marked as outliers by the Tukey’s standard boxplot. A new boxplot implementation is presented which adopts a fence definition using the extent of skewness and enhances the plot with additional features such as a quantile region for the parameters of generalized extreme value (GEV) distribution in fitting an extreme data set. The advantage of the new superimposed region was illustrated in term of batch comparison of extreme samples and an EDA tool to determine search region or direction as contained in the optimisation routines of a maximum likelihood parameter estimation of GEV model. A simulated and real-life data were used to justify the advantages of the boxplot enhancement.
Phytase activity and growth of anaerobic rumen bacterium, Mitsuokella jalaludinii were investigated by semi-solid
state fermentation. Carbon source (rice bran, yam and cassava), nitrogen sources (soya bean, offal meal, fish meal and
feather meal) and growth factors (hemin, L-cysteine hydrochloride and minerals) were evaluated in a one-factor-at-atime
approach. Rice bran and fish meal produced better growth and phytase enzyme activity. The removal of L-cysteine
hydrochloride and minerals significantly decreased (p<0.05) phytase activity from 1178.72 U to 446.99 U and 902.54
U, respectively. The response surface methods (RSM) was conducted to optimize the phytase production and the results
showed the combination of 7.7% of rice bran and 3.7% of fish meal in semi-solid state fermentation gave the highest
phytase activity. Maximum phytase production and optimum growth of bacteria were detected at 12 h incubation in both
MF medium (control) and agro-medium. In this agro-medium, M. jalaludinii produced 2.5 fold higher phytase activity
compared to MF medium.