Robust Kernel Density Function Estimation

The classical kernel density estimation technique is the commonly used method to estimate the density function. It is now evident that the accuracy of such density function estimation technique is easily affected by outliers. To remedy this problem, Kim and Scott (2008) proposed an Iteratively Re-weighted Least Squares (IRWLS) algorithm for Robust Kernel Density Estimation (RKDE). However, the weakness of IRWLS based estimator is that its computation time is very long. The shortcoming of such RKDE has inspired us to propose new non-iterative and unsupervised based approaches which are faster, more accurate and more flexible. The proposed estimators are based on our newly developed Robust Kernel Weight Function (RKWF) and Robust Density Weight Function (RDWF). The basic idea of RKWF based method is to first define a function which measures the outlying distance of observation. The resultant distances are manipulated to obtain the robust weights. The statement of Chandola et al. (2009) that the normal (clean) data appear in high probability area of stochastic model, while the outliers appear in low probability area of stochastic model, has motivated us to develop RDWF. Based on this notion, we employ the pilot (preliminary) estimate of density function as initial similarity (or distance) measure of observations with the neighbours. The modified similarity measures produce the robust weights to estimate density function robustly. Subsequently, the robust weights are incorporated in the kernel function to formulate the robust density function estimation. An extensive simulation study has been carried out to assess the performance of the RKWF-based estimator and RDWF-based estimator. The RKDE based on RKWF and RDWF perform as good as the classical Kernel Density Estimator (KDE) in outlier free data sets. Nonetheless, their performances are faster, more accurate and more reliable than the IRWLS approach for contaminated data sets. The classical kernel density function estimation approach is widely used in various formula and methods. Unfortunately, many researchers are not aware that the KDE is easily affected by outliers. We have proposed the RKDE which is more efficient and consumes less time. Our work on RKDE or corresponding robust weights has motivated us to develop alternative location and scale estimators. A modification is made to the classical location and scale estimator by incorporating the robust weight and RKDE. To evaluate the efficiency of the proposed method, comprehensive contaminated models are designed and simulated. The accuracy of the proposed new method was compared with the location and scale estimators based on M. Minimum Covariance Determinant (MCD) and Minimum Volume Ellipsoid (MVE) estimator. The simulation study demonstrates that, on the whole, the accuracy of the proposed method is better than the competitor methods. The research also develops two new approaches for outlier and potential outlier detection in unimodal and multimodal distributions. The distance of observations from the center of data set is incorporated in the formulation of the first outlier detection method in unimodal distribution. The second method attempts to define an approach that is useable not only for unimodal distribution but also for multimodal distribution. This approach incorporates robust weights, whereby, high weights and low weights are assigned to normal (clean) and outlying observations, respectively. In this thesis, we also illustrate that the sensitivity of RKDE depends on the setting of the tuning constants of the employed loss function. The results of the study indicate that the proposed methods are capable of labelling normal observation and potential outliers in a data set. Additionally, they are able to assign anomaly scores to normal and outlying observations. Finally this thesis also addresses the estimation of Mutual Information (MI) for mixture distribution which prone to create two distant groups in the data. The formulation of MI involves estimation of density function. Mutual information estimate for bivariate random variables involves the bivariate density estimation. The bivariate density estimation employs the estimate of covariance matrix. The sensitivity of covariance matrix to the presence of outliers has motivated us to substitute it with robust estimate derived from MCD and MVE. The efficiency of the modified mutual information estimate is evaluated based on its accuracy. To do this evaluation, the mixtures of bivariate normal distribution with different percentage of contribution are simulated. Simulation results show that the new formulation of MI increases the accuracy of mutual information estimation

The performance of mutual information for mixture of bivariate normal disatributions based on robust kernel estimation.

Mutual Information (MI) measures the degree of association between variables in nonlinear model as well as linear models. It can also be used to measure the dependency between variables in mixture distribution. The MI is estimated based on the estimated values of the joint density function and the marginal density functions of X and Y. A variety of methods for the estimation of the density function have been recommended. In this paper, we only considered the kernel method to estimate the density function. However, the classical kernel density estimator is not reliable when dealing with mixture density functions which prone to create two distant groups in the data. In this situation a robust kernel density estimator is proposed to acquire a more efficient MI estimate in mixture distribution. The performance of the robust MI is investigated extensively by Monte Carlo simulations. The results of the study offer substantial improvement over the existing techniques

Molecular systematics of genus Bulbophyllum (Orchidaceae) in Peninsular Malaysia based on combined nuclear and plastid DNA sequences

Phylogenetic relationships were inferred for representative Bulbophyllum species of 13 sections from subtribe Bulbophyllinae (Epidendroideae, Orchidaceae) in Peninsular Malaysia. The combined data matrix consists of sequences from ITS nuclear gene region and trnL-F, matK, and rbcL plastid gene regions with 3114 characters. Molecular data were analysed using parsimony and Bayesian inference. The results show that several recognized sections are monophyletic. Section Hirtula with paraphyletic status must split up and section Desmosanthes contain misplaced elements. Furthermore, generic status of Cirrhopetalum and Epicrianthes cannot be supported, because they are deeply embedded within the genus Bulbophyllum. Section Desmosanthes is recognized as the closest group to section Cirrhopetalum; therefore, they can be merged in some aspects

Soil depth modelling using terrain analysis and satellite imagery: the case study of Qeshlaq mountainous watershed (Kurdistan, Iran)

Soil depth is a major soil characteristic, which is commonly used in distributed hydrological modelling in order to present watershed subsurface attributes. This study aims at developing a statistical model for predicting the spatial pattern of soil depth over the mountainous watershed from environmental variables derived from a digital elevation model (DEM) and remote sensing data. Among the explanatory variables used in the models, seven are derived from a 10 m resolution DEM, namely specific catchment area, wetness index, aspect, slope, plan curvature, elevation and sediment transport index. Three variables landuse, NDVI and pca1 are derived from Landsat8 imagery, and are used for predicting soil depth by the models. Soil attributes, soil moisture, topographic curvature, training samples for each landuse and major vegetation types are considered at 429 profiles within four subwatersheds. Random forests (RF), support vector machine (SVM) and artificial neural network (ANN) are used to predict soil depth using the explanatory variables. The models are run using 336 data points in the calibration dataset with all 31 explanatory variables, and soil depth as the response of the models. Mean decrease permutation accuracy is performed on Variable selection. Testing dataset is done with the model soil depth values at testing locations (93 points) using different efficiency criteria. Prediction error is computed for both the calibration and testing datasets. Results show that the variables landuse, specific surface area, slope, pca1, NDVI and aspect are the most important explanatory variables in predicting soil depth. RF and SVM models are appropriate for the mountainous watershed areas that have been limited in the depth of the soil and ANN model is more suitable for watershed with the fields of agricultural and deep soil depth

Ball Recovery Consistency as a Performance Indicator in Elite Soccer

In soccer, an attack begins with ball recovery. Therefore, the consistency of this performance indicator during a match and its balanced distribution in the field zones can be one of the distinct characteristics of successful soccer teams. This study aims to investigate the performance consistency of ball recovery during a match within several time periods (6 periods of 15 min) and zones (four zones). To this end, observational methodology and software Focus X2 were adopted to evaluate 28 matches of semi-final teams at FIFA 2014 including Germany, Argentina, Netherlands, and Brazil in terms of ball recovery frequency. In total, 3222 performances were recorded. All teams in each match and in whole competition had homogeneity of distribution of ball recovery during the time periods (χ2 3=1.597, p=0.66). The results of time-zone evaluation indicated that Netherlands and Brazil teams did not have performance consistency on all field zones (χ2 15=31.29, p=0.008 and χ2 15=37.53, p=0.001, respectively). Most ball recoveries were made in the defensive and middle-defensive zones in accordance with modern soccer. It was found that for a soccer team to be successful, it requires a space distribution of experienced players in the field, which leads to power balance for redesigning a team to be offensive in all zones

Spiritually-Oriented Cognitive Therapy in Reduction of Depression Symptoms in Mothers of Children with Cancer

Objectives: Some of the mothers of children with cancer suffer from reactive depression and confront existential crises, and benefit from their image of God in coping with it. The purpose of this study was to determine the effectiveness of spiritually-oriented cognitive therapy on reducing depression symptoms in mothers of children with cancer. Methods: A single case experimental design and an A-B form were used in this study. The participants were selected through purposeful sampling. We studied three of the mothers of children who had been admitted to the pediatric ward of ‘Mofid Pediatric Hospital’. These children were aged under 12years they suffered from any kind of cancer except brain tumor cancer had not metastasized to other parts of the body the mothers themselves had no history of psychiatric illness prior to their child’s illness, and had mild to moderate depression at the time of screening. These mothers were subjected to spirituallyoriented cognitive therapy for 10 individual sessions, 90 minutes per week. The depression grade and the changes were measured with Beck Depression Inventory (BDI-II). Results: Comparing the mothers’ scores through 8 times of completing the inventory (three at baseline, three during the therapy and two follow-ups), and calculating the percent of recovery showed a decrease in depression scores. Discussion: It seems that spiritually-oriented cognitive therapy can enhance the spiritual experience and reduce depression in cognitive and existential contexts