Types of outlier detection:

- Statistical Distribution-Based Outlier Detection
- Distance-Based Outlier Detection
- Density-Based Local Outlier Detection
- Deviation-Based Outlier Detection

**Statistical Distribution-Based Outlier Detection:**

The statistical distribution-based approach to outlier detection assumes a distribution or probability model for the given data set (e.g., a normal or Poisson distribution) and then identifies outliers with respect to the model using a discordancy test. Application of the test requires knowledge of the data set parameters knowledge of distribution parameters such as the mean and variance and the expected number of outliers.

A statistical discordancy test examines two hypotheses:

- A working hypothesis
- An alternative hypothesis

A working hypothesis, H, is a statement that the entire data set of n objects comes from an initial distribution model, F, that is,

The hypothesis is retained if there is no statistically significant evidence supporting its rejection. A discordancy test verifies whether an object, oi, is significantly large (or small) in relation to the distribution F. Different test statistics have been proposed for use as a discordancy test, depending on the available knowledge of the data. Assuming that some statistic, T, has been chosen for discordancy testing, and the value of the statistic for object oi is vi, then the distribution of T is constructed. Significance probability, SP(vi)=Prob(T > vi), is evaluated. If SP(vi) is sufficiently small, then oi is discordant and the working hypothesis is rejected.

An alternative hypothesis, H, which states that oi comes from another distribution model, G, is adopted. The result is very much dependent on which model F is chosen because oi may be an outlier under one model and a perfectly valid value under another. The alternative distribution is very important in determining the power of the test, that is, the probability that the working hypothesis is rejected when oi is really an outlier. There are different kinds of alternative distributions.

**Inherent alternative distribution:**

In this case, the working hypothesis that all of the objects come from distribution F is rejected in favor of the alternative hypothesis that all of the objects arise from another distribution, G: H :oi € G, where i = 1, 2,…, n

F and G may be different distributions or differ only in parameters of the same distribution. There are constraints on the form of the G distribution in that it must have potential to produce outliers. For example, it may have a different mean or dispersion, or a longer tail.

**Mixture alternative distribution:**

The mixture alternative states that discordant values are not outliers in the F population, but contaminants from some other population, G.

**Slippage alternative distribution:**

This alternative states that all of the objects (apart from some prescribed small number) arise independently from the initial model, F, with its given parameters, whereas the remaining objects are independent observations from a modified version of F in which the parameters have been shifted.

There are two basic types of procedures for detecting outliers:**Block procedures:** In this case, either all of the suspect objects are treated as outliers or all of them are accepted as consistent.

**Consecutive procedures:**

An example of such a procedure is the insideout procedure. Its main idea is that the object that is least likely to be an outlier is tested first. If it is found to be an outlier, then all of the more extreme values are alsoconsidered outliers; otherwise, the next most extreme object is tested, and so on. This procedure tends to be more effective than block procedures.

**Distance-Based Outlier Detection:**

The notion of distance-based outliers was introduced to counter the main limitations imposed by statistical methods. An object, o, in a data set, D, is a distance-based (DB)outlier with parameters pct and dmin, that is, a DB(pct;dmin)-outlier, if at least a fraction, pct, of the objects in D lie at a distance greater than dmin from o. In other words, rather that relying on statistical tests, we can think of distance-based outliers as those objects that do not have enough neighbors, where neighbors are defined based on distance from the given object. In comparison with statistical-based methods, distance based outlier detection generalizes the ideas behind discordancy testing for various standarddistributions. Distance-based outlier detection avoids the excessive computation that can be associated with fitting the observed distribution into some standard distribution and in selecting discordancy tests.

For many discordancy tests, it can be shown that if an object, o, is an outlier according to the given test, then o is also a DB(pct, dmin)-outlier for some suitably defined pct and dmin. For example, if objects that lie three or more standard deviations from the mean are considered to be outliers, assuming a normal distribution, then this definition can be generalized by a DB(0.9988, 0.13s) outlier. Several efficient algorithms for mining distance-based outliers have been developed.

**Index-based algorithm:**

Given a data set, the index-based algorithm uses multidimensional indexing structures, such as R-trees or k-d trees, to search for neighbors of each object o within radius dmin around that object. Let M be the maximum number of objects within the dmin-neighborhood of an outlier. Therefore, onceM+1 neighbors of object o are found, it is clear that o is not an outlier. This algorithm has a worst-case complexity of O(n2k), where n is the number of objects in the data set and k is the dimensionality. The index-based algorithm scales well as k increases. However, this complexity evaluation takes only the search time into account, even though the task of building an index in itself can be computationally intensive.

**Nested-loop algorithm:**

The nested-loop algorithm has the same computational complexity as the index-based algorithm but avoids index structure construction and tries to minimize the number of I/Os. It divides the memory buffer space into two halves and the data set into several logical blocks. By carefully choosing the order in which blocks are loaded into each half, I/O efficiency can be achieved.

**Cell-based algorithm:**

To avoid O(n²) computational complexity, a cell-based algorithm was developed for memory-resident data sets. Its complexity is O(ck+n), where c is a constant depending on the number of cells and k is the dimensionality.

**Density-Based Local Outlier Detection:**

Statistical and distance-based outlier detection both depend on the overall or global distribution of the given set of data points, D. However, data are usually not uniformly distributed. These methods encounter difficulties when analyzing data with rather different density distributions.

**Deviation-Based Outlier Detection:**

Deviation-based outlier detection does not use statistical tests or distance-based measures to identify exceptional objects. Instead, it identifies outliers by examining the main characteristics of objects in a group. Objects that ―deviate‖ from this description are considered outliers. Hence, in this approach the term deviations is typically used to refer to outliers. In this section, we study two techniques for deviation-based outlier detection. The first sequentially compares objects in a set, while the second employs an OLAP data cube approach.