Hypothesis testing  (Page 2/2)

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Tests and decision regions

Consider the general hypothesis testing problem where we have $N$ $d$ -dimensional observations ${x}_{1},,{x}_{N}$ and $M$ hypotheses. If the data are real-valued, for example, then a hypothesistest is a mapping $(:\mathbb{R}^{d}^{N}, \{1, , M\})$ For every possible realization of the input, the test outputs a hypothesis. The test  partitions the input space into a disjoint collection ${R}_{1},,{R}_{M}$ , where ${R}_{k}=\{\left({x}_{1},,{x}_{N}\right)|({x}_{1}, , {x}_{N})=k\}$ The sets ${R}_{k}$ are called decision regions . The boundary between two decision regions is a decision boundary . depicts these concepts when $d=2$ , $N=1$ , and $M=3$ .

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 onlyknow the distribution up to certain unknown parameters. For example, in a Gaussian noise model we may not know thevariance of the noise. In this case, a hypothesis is said to be composite .

Consider the problem of detecting the signal ${s}_{n}=\cos (2\pi {f}_{0}(n-k))\forall n\colon n=\{1, , N\}$ where $k$ is an unknown delay parameter. Then ${H}_{0}:x=w$ ${H}_{1}:x=s+w$ is a binary test of a simple hypothesis ( ${H}_{0}$ ) versus a composite alternative. Here we are assuming $({w}_{n}, (0, ^{2}))$ , with $^{2}$ known.

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

Suppose a coin turns up heads with probability $p$ . We want to assess whether the coin is fair( $p=\frac{1}{2}$ ). We toss the coin $N$ times and record ${x}_{1},,{x}_{N}$ ( ${x}_{n}=1$ means heads and ${x}_{n}=0$ means tails). Then ${H}_{0}:p=\frac{1}{2}$ ${H}_{1}:p\neq \frac{1}{2}$ is a binary test of a simple hypothesis ( ${H}_{0}$ ) 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, thereare four possible scenarios:

• Case 1

${H}_{0}$ is true, and we declare ${H}_{0}$ to be true
• Case 2

${H}_{0}$ is true, but we declare ${H}_{1}$ to be true
• Case 3

${H}_{1}$ is true, and we declare ${H}_{1}$ to be true
• Case 4

${H}_{1}$ is true, but we declare ${H}_{0}$ to be true
In cases 2 and 4, errors occur. The names given to these errors depend on the area of application. In statistics, theyare 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 ${H}_{0}:(x, {f}_{}(x)),\in {}_{0}$ ${H}_{1}:(x, {f}_{}(x)),\in {}_{1}$ If ${H}_{0}$ is simple, that is, ${}_{0}=\{{}_{0}\}$ , then the size (denoted  ), also called the false-alarm probability ( ${P}_{F}$ ), is defined to be $={P}_{F}=({}_{0}, \text{declare}{H}_{1})$ When ${}_{0}$ is composite, we define $={P}_{F}={\mathrm{sup}}_{{}_{0}}((, \text{declare}{H}_{1}))$ For $\in {}_{1}$ , the power (denoted  ), or detection probability ( ${P}_{D}$ ), is defined to be $={P}_{D}=(, \text{declare}{H}_{1})$ The probability of a type II error, also called the miss probability , is ${P}_{M}=1-{P}_{D}$ If ${H}_{1}$ is composite, then $=()$ is viewed as a function of  .

Criteria in hypothesis testing

The design of a hypothesis test/detector often involves constructing the solution to an optimizationproblem. 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:

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 (  ) Probability of False Alarm ( ${P}_{F}$ )
Power of Test (  ) Probability of Detection ( ${P}_{D}$ )

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