##### A Sequential Third Order Rotatable Design of Eighty Points in Four Dimensions with an Hypothetical Case Study

Nyakundi Omwando Cornelious, Matunde Nambilo Cruyff

Asian Journal of Probability and Statistics, Page 1-9
DOI: 10.9734/ajpas/2019/v4i430119

In research, experiments must be performed at pre determined levels of the controllable factors, meaning that an experimental design must be selected before the experiment takes place. Once an experimenter has chosen a polynomial model of suitable order, the problem arises on how best to choose the settings for the independent variables over which he has control. A particular selection of settings or factor levels at which observations are to be taken is called a design. A design may become inappropriate under special circumstances requiring an increase in factors or levels to make it more desirable. In agriculture for instance, continuous cultivation of crops may exhaust the previously available mineral elements necessitating a sequential appendage of the mineral elements which become deficient in the soil over time.

In current study, an eighty  points four  dimensional  third order rotatable design is constructed by combining two, four dimensional second order rotatable  designs and a practical hypothetical case study is given by converting coded levels to natural levels. We present an illustration on how to obtain the mathematical parameters of the coded values and its corresponding natural levels for a third order rotatable design in four dimensions by utilizing response surface methodology to approximate the functional relationship between the performance characteristics and the design variables.  This design permits a response surface to be fitted easily and provides spherical information contours besides the economic use of scarce resources in relevant production processes.

##### Almost Periodic Sequence in a Discrete Logistic Equation

Tianwei Zhang

Asian Journal of Probability and Statistics, Page 1-15
DOI: 10.9734/ajpas/2019/v4i430120

This paper is concerned with an almost periodic discrete logistic equation. By using thecontinuation theorem of Mawhin’s coincidence degree theory, this paper investigates theexistence and stability of a unique positive almost periodic sequence solution of the equation.These results generalize and improve the previous works, and they are easy to check. An examplewith a numerical simulation is also given to demonstrate the effectiveness of the results in this paper.

##### A New Smoothing Method for Time Series Data in the Presence of Autocorrelated Error

Asian Journal of Probability and Statistics, Page 1-19
DOI: 10.9734/ajpas/2019/v4i430121

Spline Smoothing is used to filter out noise or disturbance in an observation, its performance depends on the choice of smoothing parameters. There are many methods of estimating smoothing parameters; most popular among them are; Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR), this methods tend to overfit smoothing parameters in the presence of autocorrelation error. A new Spline Smoothing estimation method is proposed and compare with three existing methods in order to eliminate the problem of over fitting associated with the presence of Autocorrelation in the error term. It is demonstrated through a simulation study performed by using a program written in R based on the predictive Mean Score Error criteria. The result indicated that the predictive mean square error (PMSE) of the four smoothing methods decreases as the smoothing parameters increases and decreases as the sample sizes increases. This study discovered that the proposed smoothing method is the best for time series observations with Autocorrelated error because it doesn’t over fit and works well for large sample sizes. This study will help researchers overcome the problem of over fitting associated with applying Smoothing spline method time series observation.

##### Numerical Study on Micropolar Nanofluid Flow over an Inclined Surface by Means of Keller-Box

Khuram Rafique, Muhammad Imran Anwar, Masnita Misiran

Asian Journal of Probability and Statistics, Page 1-21
DOI: 10.9734/ajpas/2019/v4i430122

In this paper, micropolar nanofluid boundary layer flow over a linear inclined stretching surface with the magnetic effect is investigated. Buongiorno’s model utilized in this study for the thermal efficiencies of the fluid flow in the presence of Brownian motion and thermophoresis properties. The nonlinear problem for micropolar nanofluid flow over an inclined sheet is established to study the heat and mass exchange phenomenon by considering portent flow parameters to strengthen the boundary layers. The governing nonlinear partial differential equations are changed to nonlinear ordinary differential equations by using suitable similarity transformations and then solved numerically by applying the Keller-Box method. A comparison of the setup results in the absence of the incorporated impacts is performed with the accessible results and perceived in a decent settlement. Numerical and graphical outcomes are additionally presented in tables and diagrams.

##### Application of Seasonal Autoregressive Moving Average Models to Analysis and Forecasting of Time Series Monthly Rainfall Patterns in Embu County, Kenya

Tartisio Njoki Filder, Moses Mahugu Muraya, Robert Mathenge Mutwiri

Asian Journal of Probability and Statistics, Page 1-15
DOI: 10.9734/ajpas/2019/v4i430123

Rainfall is of critical importance for many people, particularly those whose livelihoods depend on rain-fed agriculture. Predicting the trend of rainfall is a difficult task, and statistical approaches such as time series analysis provide a means for predicting the patterns of rainfall. The models also offer the potential to improve areas such as increased food production, profitability, and improved food security policing. However, these forecasts and information systems may, in some instances, not be suitable for direct use by stakeholders in their decision-making. The objective of this study was to investigate rainfall variability and develop a Seasonal Auto-Regressive Integrated Moving Average (SARIMA) model for fitting the monthly rainfall using time series data. Secondary monthly data from 1998 to 2017 for Embu County was collected from the Kenya Meteorological Department, Embu and recorded into an excel sheet. R-software was utilized to analyse data for descriptive statistics, rainfall variability, and model fitting. The coefficient of variation for annual and seasonal rainfall was calculated. The Box Jenkin's ARIMA modelling procedure (model identification, model estimation, model validation) was used to determine the best models for the data. The main study findings indicated the existence of annual variability of 34%, March-April-May rainfall variability of 44%, and October-November-December variability of 44%. A first-order differenced SARIMA (1, 1, 1) (0, 1, 2)12 model with an AIC score of 9.99356 was found suitable for predicting rainfall pattern in Embu, County. The study outcome revealed that Embu County experiences high seasonal and rainfall variation of rainfall, thus requires a reliable model for better prediction.