Data Mining: The Textbook

8.2. EXTREME VALUE ANALYSIS

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	0.4																			0.35
	0.35																			0.3
DENSITY																	DENSITY
	0.25																											BUT NOT
	0.3																			0.25
																										OUTLIERS
PROBABILITY																	PROBABILITY
																			0.2								EXTREME VALUES
	0.2
																				0.15
	0.15
	0.1	LOWER										UPPER								0.1
																						LOWER									UPPER
		TAIL										TAIL																				DENSITY
					DENSITY THRESHOLD															0.05
	0.05																															THRESHOLD
	0	−4	−3	−2	−1	0		1	2			3	4			5				0		−4		−3		−2		−1	0	1	2	3	4	5
	−5																			−5
						VALUE																							VALUE

(a) Symmetric distribution (b) Asymmetric distribution

Figure 8.2: Tails of a symmetric and asymmetric distribution

distributions. Some asymmetric distributions, such as an exponential distribution, may not even have a tail at one end of the distribution.

A model distribution is selected for quantifying the tail probability. The most commonly used model is the normal distribution. The density function f_X (x) of the normal distribution with mean μ and standard deviation σ is defined as follows:

fX (x) =		1			−(x−μ)^2			(8.1)
					e		.
						2·σ^2
		^√2·π		·
σ ·

A standard normal distribution is one in which the mean is 0, and the standard deviation σ is 1. In some application scenarios, the mean μ and standard deviation σ of the distribution may be known through prior domain knowledge. Alternatively, when a large number of data samples is available, the mean and standard deviation may be estimated very accurately. These can be used to compute the Z-value for a random variable. The Z-number z_i of an observed value x_i can be computed as follows:

zi = (xi − μ)/σ.

(8.2)

Large positive values of z_i correspond to the upper tail, whereas large negative values correspond to the lower tail. The normal distribution can be expressed directly in terms of the Z-number because it corresponds to a scaled and translated random variable with a mean 0 and standard deviation of 1. The normal distribution of Eq. 8.3 can be written directly in terms of the Z-number, with the use of a standard normal distribution as follows:

	1				2