Univariate median

Box plots, also known as box and whisker plots, are commonly used to display univariate statistics for a given variable across another variable. The statistics typically displayed in a box plot are the minimum, first quartile, median, third quartile, and maximum values. Mean values are often included in box plots as well. The following is a sample box plot of a clinical response measure showing how three different drug therapies compare to one another. 

Input mapping methods can be divided into univariate, multivariate, and probabalistic methods. Univariate methods analyze the inputs by extracting the relationship between the measurements. These methods include various types of single-scale and multiscale filtering such as exponential smoothing, wavelet thresholding, and median filtering. Multivariate methods analyze... 

In addition to the usual statistical methods based on univariate descriptors (mean, median, and standard deviation) and analysis of variance, multivariate techniques of statistics and chemometrics are increasingly being used in data evaluation. Whereas the former are more rigorous in theoretical background and assumptions, the latter are useful in the presentation of the data, pattern recognition, and multivariate calibrations. Several good monographs on chemometrics are available (see for example [58-61]). 

However, if the possible outlier xi = 15) is omitted, the median is barely changed to 5.00 while the mean x is shifted to 4.857 (Figure 8.3). Typically the median value is a more robust indicator of a typical value of a set of univariate data than is the mean, e.g., in comparisons of family incomes in two different countries if a small fraction of families can have very high incomes well removed from the vast majority this can lead to misleading conclusions based on the mean values. In analytical chemistry, the main value of comparisons exemplified by the fictional data in Figure 8.3 lies in their ability to highlight suspicious values x that should be examined as possible outliers whose exclusion can be justified by appropriate statistical tests (Section 8.2.7). 

In a case-control study in 543 lung transplant recipients, 17 (3.1%) developed squamous cell carcinomas Mer a median follow-up of 36 months [62 ]. The median time to development was 19 months after transplantation. Risk factors by univariate analysis included older age, residence in locations with high levels of sun exposure, single-lung transplantation, and duration and cumulative dose of voriconazole. The duration of voriconazole therapy and residence in locations with high sun exposure were independent risk factors by multivariate analysis. The lesions were located on the head and neck in 94% of cases and 53% had multiple lesions. After surgery at least one further independent lesion developed in 47% of patients. There was local spread and distant metastases in 7% of cases. There were no deaths. 

The Ll-median, also known as the spatial median, is a generalization of the standard median. It is a highly robust multivariate estimator of data location with a breakdown point of 50%. The Ll-median estimator is equivalent to the classic median for a univariate case. Specifically, the Ll-median centre of the data is the point p-Li in the multivariate data space that minimizes the sum of Euclidean distances between this point and all of the data points [3] (see Figure 2). 

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