For this book, we assume you’ve had an introductory statistics or experimental design class already! This chapter is a mini refresher of some critical concepts we’ll be using and lets you check you understand them correctly. The topics include understanding predictor and response variables, the common probability distributions that biologists encounter in their data, the common techniques, particularly ordinary least squares (OLS) and maximum likelihood (ML), for fitting models to data and estimating effects, including their uncertainty. You should be familiar with confidence intervals and understand what hypothesis tests and P-values do and don’t mean. You should recognize that we use data to decide, but these decisions can be wrong, so you need to understand the risk of missing important effects and the risk of falsely claiming an effect. Decisions about what constitutes an “important” effect are central.

In this chapter, we introduce the design of statistical anomaly detectors. We discuss types of data – continuous, discrete categorical, and discrete ordinal features – encountered in practice. We then discuss how to model such data, in particular to form a null model for statistical anomaly detection, with emphasis on mixture densities. The EM algorithm is developed for estimating the parameters of a mixture density. K-means is a specialization of EM for Gaussian mixtures. The Bayesian information criterion (BIC) is discussed and developed – widely used for estimating the number of components in a mixture density. We also discuss parsimonious mixtures, which economize on the number of model parameters in a mixture density (by sharing parameters across components). These models allow BIC to obtain accurate model order estimates even when the feature dimensionality is huge and the number of data samples is small (a case where BIC applied to traditional mixtures grossly underestimates the model order). Key performance measures are discussed, including true positive rate, false positive rate, and receiver operating characteristic (ROC) and associated area-under-the-curve (ROC AUC). The density models are used in attack detection defenses in Chapters 4 and 13. The detection performance measures are used throughout the book.

A one-sample t-test is an NHST procedure that is appropriate when a z-test cannot be performed because the population standard deviation is unknown. The one-sample t-test follows all of the eight steps of the z-test, but requires modifications to accommodate the unknown sample standard deviation. First, the formulas that used σy now use the estimated population standard deviation based on sample data instead. Second, degrees of freedom must be calculated. Finally, t-tests use a new probability distribution called a t-distribution.

This chapter also explains more about p-values. First, when p is lower than α, the null hypothesis is always rejected. Second, when p is higher than α, the null hypothesis is always retained. Therefore, we can determine whether p is smaller or larger than α by determining whether the null hypothesis was retained or rejected for α. This chapter also discusses confidence intervals (CIs), which are a range of plausible values for a population parameter. CIs can vary in width, which the researcher chooses. The 95% CI width is most common in social science research. Finally, one-sample t-tests can be used to test the hypothesis that the sample mean is equal to any value (not the population mean).

One of the most frequent research designs in the social sciences is to compare two groups’ scores. When there is no pairing between sets of scores, it is necessary to conduct an unpaired two-sample t-test. The steps of this NHST are the same eight steps of the previous statistical tests because all of these are members of the GLM. There are some modifications that make the unpaired two-sample t-test unique:

• The null hypothesis for an unpaired two-sample t-test is always.

• The equation for the number of degrees of freedom is different: df = (n1 – 1) + (n2 – 1).

• The formula to calculate the observed value is more complex:.

• The effect size for an unpaired two-sample t-test is still Cohen’s d, but the formula has been modified:.

The unpaired two-sample t-test shares many characteristics with other NHSTs: the sensitivity to sample size, the necessity of calculating an effect size, the arbitrary nature of selecting an α value, and the need to worry about Type I and Type II errors.

This chapter deals with

• The notion of exploratory spatial data analysis

• The presentation of descriptive statistics

• Spatial statistics and their importance in analyzing spatial data

• Analyzing univariate data

• Simple exploratory spatial data analysis tools such as histograms, boxplots and other visual methods to get a better insight of spatial datasets

• Bivariate analysis

• Correlation and pairwise correlation

• Normalization, rescaling and adjustments

• An introduction of basic notions of statistical significant tests

• The importance of hypothesis setting in a spatial context

• The importance of normal distribution in classic statistics and how it is integrated into spatial analysis

After a thorough study of the theory and lab sections, you will be able to

• Have a solid knowledge of descriptive statistics

• Use descriptive statistics for univariate analysis

• Understand and use exploratory spatial data analysis techniques to map and analyze variables attached to spatial objects

• Create several plots, link them to maps and identify interesting patterns of data

• Conduct bivariate analysis, identify whether two variables are linearly related and use plots to further examine their relation

• Rescale data to make comparisons between variables easier and also allow for better data handling

• Apply ESDA tools through ArcGIS and GeoDa