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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}=\{({x}_{1},,{x}_{N})|({x}_{1}, , {x}_{N})=k\}$$ The sets ${R}_{k}$ are called decision regions . The boundary between two decision regions is a decision boundary . depicts these concepts when $d=2$ , $N=1$ , and $M=3$ .
If the distribution of the data under a certain hypothesis is fully known, we call it a simple hypothesis. All of the hypotheses in the examples above are simple. In many cases, however, we onlyknow the distribution up to certain unknown parameters. For example, in a Gaussian noise model we may not know thevariance of the noise. In this case, a hypothesis is said to be composite .
Consider the problem of detecting the signal $${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.
In binary hypothesis testing, assuming at least one of the two models does indeed correspond to reality, thereare four possible scenarios:
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 $$ .
The design of a hypothesis test/detector often involves constructing the solution to an optimizationproblem. The optimality criteria used fall into two classes: Bayesian and frequent.
Representing the former approach is the Bayes Risk Criterion . Representing the latter is the Neyman-Pearson Criterion . These two approaches are developed at length in separate modules.
The following table, adapted from Kay, p.65 , summarizes the different terminology for hypothesis testing from statistics and signal processing:
Statistics | Signal Processing |
---|---|
Hypothesis Test | Detector |
Null Hypothesis | Noise Only Hypothesis |
Alternate Hypothesis | Signal + Noise Hypothesis |
Critical Region | Signal Present Decision Region |
Type I Error | False Alarm |
Type II Error | Miss |
Size of Test ( $$ ) | Probability of False Alarm ( ${P}_{F}$ ) |
Power of Test ( $$ ) | Probability of Detection ( ${P}_{D}$ ) |
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