Introduction Apert, Crouzon, and Pfeiffer syndromes are common genetic syndromes related to syndromic craniosynostosis (SC), whereby it is a congenital defect that occurs when the cranial growth is distorted. Identifying cranial angles associated with these 3 syndromes may assist the surgical team to focus on a specific cranial part during the intervention planning, thus optimizing surgical outcomes and reducing potential morbidity. Objective The aim of this study is to identify the cranial angles, which are associated with Apert, Crouzon, and Pfeiffer syndromes. Methods The cranial computed tomography scan images of 17 patients with SC and 22 control groups aged 0 to 12 years who were treated in the University Malaya Medical Centre were obtained, while 12 angular measurements were attained using the Mimics software. The angular data were then divided into 2 groups (patients aged 0 to 24 months and >24 months). This work proposes a 95% confidence interval (CI) for angular mean to detect the abnormality in patient's cranial growth for the SC syndromes. Results The 95% CI of angular mean for the control group was calculated and used as an indicator to confirm the abnormality in patient's cranial growth that is associated with the 3 syndromes. The results showed that there are different cranial angles associated with these 3 syndromes. Conclusions All cranial angles of the patients with these syndromes lie outside the 95% CI of angular mean of control group, indicating the reliability of the proposed CI in the identification of abnormality in the patient's cranial growth.

Descriptive statistics are commonly used in data analysis to describe the basic features of raw data. Descriptive summaries enable us to present the data in a more simple and meaningful way so that the interpretation will be easier to understand. The descriptive analysis of circular data with outliers is discussed in this study. Circular data is different from linear data in many aspects such as statistical modeling, descriptive statistics and etc. Hence, unlike linear data, the availability of statistical software specialises in analysing circular data is very limited. Python is a programming language which frequently used by data analysts nowadays. However, the package for circular statistics is not fully developed and it is not ready to use like in Splus or R programming language. In this study, the descriptive analysis of circular data is performed using the in-demand programming language, Python. Descriptive statistics of the circular data especially with the existence of outliers are discussed and the proposed Python code is available to use.

Synthetic data is artificial data that is created based on the statistical properties of the original data. The aim of this study is to generate a synthetic or simulated data for univariate circular data that follow von Mises (VM) distribution with various outliers scenario using Python programming language. The procedure of formulation a synthetic data generation is proposed in this study. The synthetic data is generated from various combinations of seven sample size, n and five concentration parameters, K. Moreover, a synthetic data will be generated by formulating a data generation procedure with different condition of outliers scenarios. Three outliers scenarios are proposed in this study to introduce the outliers in synthetic dataset by placing them away from inliers at a specific distance. The number of outliers planted in the dataset are fixed with three outliers. The synthetic data is randomly generated by using Python library and package which are ‘numpy’, ‘random’ and von Mises’. In conclusion, the synthetic data of univariate circular data from von Mises distribution is generated and the outliers are successfully introduced in the dataset with three outliers scenarios using Python. This study will be valuable for those who are interested to study univariate circular data with outliers and choose Python as an analysis tool.

The procedure of outliers detection in univariate circular data can be developed using clustering algorithm. In clustering, it is necessary to calculate the similarity measure in order to cluster the observations into their own group. The similarity measure in circular data can be determined by calculating circular distance between each point of angular observation. In this paper, clustering-based procedure for outlier detection in univariate circular biological data with different similarity distance measures will be developed and the performance will be investigated. Three different circular similarity distance measures are used for the outliers detection procedure using single-linkage clustering algorithm. However, there are two similarity measures namely Satari distance and Di distance that are found to have similarity in formula for univariate circular data. The aim of this study is to develop and demonstrate the effectiveness of proposed clustering-based procedure with different similarity distance measure in detecting outliers. Therefore, in this study the circular similarity distance of SL-Satari/Di and another similarity measure namely SL-Chang will be compared at certain cutting rule. It is found that clustering-based procedure using single-linkage algorithm with different similarity distances are applicable and promising approach for outlier detection in univariate circular data, particularly for biological data. The result also found that at a certain condition of data, the SL-Satari/Di distance seems to overperform the performance of SL-Chang distance.

This study focuses on detecting a single outlier in circular data generated from a wrapped normal (WN) distribution. The discordancy tests of M, A and G 1 statistics are used to detect single outlier in simulated data generated from wrapped normal distribution. The purpose of this study is to make a comparison on the performance of these statistics via Monte Carlo simulation by obtaining the proportion of correct outlier detection for each statistic. In this study, Splus-language and R-language programming are used to carry out the simulation study. The power performance of these statistics have been investigated and the result revealed that these statistics performed better as the increment in contamination value, λ and the value of concentration parameter, ρ gets larger and close to 1 in the case of small and large sample size, n. In general, the A statistic is found to be outperformed the M and G 1statistics in all cases. As an illustration, a practical example is included in this study by using the Kuantan wind direction dataset.