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Hypothesis Testing

Module by: Clayton Scott, Robert Nowak. E-mail the authors

Suppose you measure a collection of scalars x 1 , , x N x 1 , , x N . You believe the data is distributed in one of two ways. Your first model, call it H 0 H 0 , postulates the data to be governed by the density f 0 x f 0 x (some fixed density). Your second model, H 1 H 1 , postulates a different density f 1 x f 1 x . These models, termed hypotheses, are denoted as follows: H 0 : x n f 0 x , n = 1 N H 0 : x n f 0 x , n = 1 N H 1 : x n f 1 x , n = 1 N H 1 : x n f 1 x , n = 1 N A hypothesis test is a rule that, given a measurement xx, makes a decision as to which hypothesis best "explains" the data.

Example 1

Suppose you are confident that your data is normally distributed with variance 1, but you are uncertain about the sign of the mean. You might postulate H 0 : x n 𝒩-11 H 0 : x n -1 1 H 1 : x n 𝒩11 H 1 : x n 1 1 These densities are depicted in Figure 1.

Figure 1
Figure 1 (GaussOppMeanUnitVar.png)
Assuming each hypothesis is a priori equally likely, an intuitively appealing hypothesis test is to compute the sample mean x-=1N n =1N x n x 1 N n 1 N x n , and choose H 0 H 0 if x-0 x 0 , and H 1 H 1 if x->0 x 0 . As we will see later, this test is in fact optimal under certain assumptions.

Generalizations and Nomenclature

The concepts introduced above can be extended in several ways. In what follows we provide more rigorous definitions, describe different kinds of hypothesis testing, and introduce terminology.

Data

In the most general setup, the observation is a collection x 1 , , x N x 1 , , x N of random vectors. A common assumption, which facilitates analysis, is that the data are independent and identically distributed (IID). The random vectors may be continuous, discrete, or in some cases mixed. It is generally assumed that all of the data is available at once, although for some applications, such as Sequential Hypothesis Testing, the data is a never ending stream.

Binary Versus M-ary Tests

When there are two competing hypotheses, we refer to a binary hypothesis test. When the number of hypotheses is M2 M 2 , we refer to an M-ary hypothesis test. Clearly, binary is a special case of MM-ary, but binary tests are accorded a special status for certain reasons. These include their simplicity, their prevalence in applications, and theoretical results that do not carry over to the MM-ary case.

Example 2

Phase-Shift Keying

Suppose we wish to transmit a binary string of length rr over a noisy communication channel. We assign each of the M=2r M 2 r possible bit sequences to a signal s k s k , k=1M k 1 M where s n k =cos2π f 0 n+2π(k1)M s n k 2 f 0 n 2 k 1 M This symboling scheme is known as phase-shift keying (PSK). After transmitting a signal across the noisy channel, the receiver faces an MM-ary hypothesis testing problem: H 0 : x= s 1 +w H 0 : x s 1 w H M : x= s M +w H M : x s M w where w𝒩0σ2I w 0 σ 2 I .

In many binary hypothesis tests, one hypothesis represents the absence of a ceratin feature. In such cases, the hypothesis is usually labelled H 0 H 0 and called the null hypothesis. The other hypothesis is labelled H 1 H 1 and called the alternative hypothesis.

Example 3

Waveform Detection

Consider the problem of detecting a known signal s= s 1 s N T s s 1 s N in additive white Gaussian noise (AWGN). This scenario is common in sonar and radar systems. Denoting the data as x= x 1 x N T x x 1 x N , our hypothesis testing problem is H 0 : x=w H 0 : x w H 1 : x=s+w H 1 : x s w where w𝒩0σ2I w 0 σ 2 I . H 0 H 0 is the null hypothesis, corresponding to the absence of a signal.

Tests and Decision Regions

Consider the general hypothesis testing problem where we have NN dd-dimensional observations x 1 , , x N x 1 , , x N and MM hypotheses. If the data are real-valued, for example, then a hypothesis test is a mapping φ : RdN 1M φ : d N 1 M For every possible realization of the input, the test outputs a hypothesis. The test φφ partitions the input space into a disjoint collection R 1 , , R M R 1 , , R M , where R k = ( x 1 , , x N ) | φ x 1 x N =k R k ( x 1 , , x N ) | φ x 1 x N k The sets R k R k are called decision regions. The boundary between two decision regions is a decision boundary. Figure 2 depicts these concepts when d=2 d 2 , N=1 N 1 , and M=3 M 3 .

Figure 2
Figure 2 (decisionRegions.png)

Simple Versus Composite Hypotheses

If the distribution of the data under a certain hypothesis is fully known, we call it a simple hypothesis. All of the hypotheses in the examples above are simple. In many cases, however, we only know the distribution up to certain unknown parameters. For example, in a Gaussian noise model we may not know the variance of the noise. In this case, a hypothesis is said to be composite.

Example 4

Consider the problem of detecting the signal s n =cos2π f 0 (nk) n :n=1N s n 2 f 0 n k n n 1 N where kk is an unknown delay parameter. Then H 0 : x=w H 0 : x w H 1 : x=s+w H 1 : x s w is a binary test of a simple hypothesis ( H 0 H 0 ) versus a composite alternative. Here we are assuming w n 𝒩0σ2 w n 0 σ 2 , with σ2 σ 2 known.

Often a test involving a composite hypothesis has the form H 0 : θ= θ 0 H 0 : θ θ 0 H 1 : θ θ 0 H 1 : θ θ 0 where θ 0 θ 0 is fixed. Such problems are called two-sided because the composite alternative "lies on both sides of H 0 H 0 ." When θθ is a scalar, the test H 0 : θ θ 0 H 0 : θ θ 0 H 1 : θ> θ 0 H 1 : θ θ 0 is called one-sided. Here, both hypotheses are composite.

Example 5

Suppose a coin turns up heads with probability pp. We want to assess whether the coin is fair ( p=12 p 1 2 ). We toss the coin NN times and record x 1 , , x N x 1 , , x N ( x n =1 x n 1 means heads and x n =0 x n 0 means tails). Then H 0 : p=12 H 0 : p 1 2 H 1 : p12 H 1 : p 1 2 is a binary test of a simple hypothesis ( H 0 H 0 ) versus a composite alternative. This is also a two-sided test.

Errors and Probabilities

In binary hypothesis testing, assuming at least one of the two models does indeed correspond to reality, there are four possible scenarios:

  • Case 1: H 0 H 0 is true, and we declare H 0 H 0 to be true
  • Case 2: H 0 H 0 is true, but we declare H 1 H 1 to be true
  • Case 3: H 1 H 1 is true, and we declare H 1 H 1 to be true
  • Case 4: H 1 H 1 is true, but we declare H 0 H 0 to be true
In cases 2 and 4, errors occur. The names given to these errors depend on the area of application. In statistics, they are called type I and type II errors respectively, while in signal processing they are known as a false alarm or a miss.

Consider the general binary hypothesis testing problem H 0 : x f θ x , θ Θ 0 H 0 : x f θ x , θ Θ 0 H 1 : x f θ x , θ Θ 1 H 1 : x f θ x , θ Θ 1 If H 0 H 0 is simple, that is, Θ 0 = θ 0 Θ 0 θ 0 , then the size (denoted αα), also called the false-alarm probability ( P F P F ), is defined to be α= P F =Pr θ 0 declare H 1 α P F θ 0 declare H 1 When Θ 0 Θ 0 is composite, we define α= P F = sup θ Θ 0 Pr θ declare H 1 α P F sup θ Θ 0 θ declare H 1 For θ Θ 1 θ Θ 1 , the power (denoted ββ), or detection probability ( P D P D ), is defined to be β= P D =Pr θ declare H 1 β P D θ declare H 1 The probability of a type II error, also called the miss probability, is P M =1 P D P M 1 P D If H 1 H 1 is composite, then β=βθ β β θ is viewed as a function of θθ.

Criteria in Hypothesis Testing

The design of a hypothesis test/detector often involves constructing the solution to an optimization problem. The optimality criteria used fall into two classes: Bayesian and frequent.

Representing the former approach is the Bayes Risk Criterion. Representing the latter is the Neyman-Pearson Criterion. These two approaches are developed at length in separate modules.

Statistics Versus Engineering Lingo

The following table, adapted from Kay, p.65, summarizes the different terminology for hypothesis testing from statistics and signal processing:

Table 1
Statistics Signal Processing
Hypothesis Test Detector
Null Hypothesis Noise Only Hypothesis
Alternate Hypothesis Signal + Noise Hypothesis
Critical Region Signal Present Decision Region
Type I Error False Alarm
Type II Error Miss
Size of Test (αα) Probability of False Alarm ( P F P F )
Power of Test (ββ) Probability of Detection ( P D P D )

References

  1. S. Kay. (1998). Detection Theory. Prentice Hall.

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