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<H2><A NAME="SECTION001722000000000000000">
Signal detection</A>
</H2>
<P>
Paradoxically, signal detection is concerned with fitting models to
supposedly random series similarly to mathematical proofs by <I>reductio ad absurdum</I> of the antithesis. That is, the hypothesis
(antithesis) <I>H</I><SUB><I>o</I></SUB> is made, that the observed series <I>X</I><SUP>(<I>o</I>)</SUP> has
properties of a pure noise series, <I>N</I><SUP>(<I>o</I>)</SUP>. Then, a model is fitted
and series <I>X</I><SUP>(<I>m</I>)</SUP> and <I>X</I><SUP>(<I>r</I>)</SUP> are obtained. If the quality of the
model fits to the observations <I>X</I><SUP>(<I>o</I>)</SUP> does not significantly differ
from the quality of a fit to pure noise <I>N</I><SUP>(<I>o</I>)</SUP>, then <I>H</I><SUB><I>o</I></SUB> is true
and we say that <I>X</I><SUP>(<I>o</I>)</SUP> contains no signal but noise. In the
opposite case of model fitting <I>X</I><SUP>(<I>o</I>)</SUP> significantly better than
<I>N</I><SUP>(<I>o</I>)</SUP>, we reject <I>H</I><SUB><I>o</I></SUB> and say that the model signal was detected
in <I>X</I><SUP>(<I>o</I>)</SUP>. The difference is significant (at some level) if it is
not likely (at this level) to occur between two different realizations
of the noise <I>N</I><SUP>(<I>o</I>)</SUP>.
<P>
The quality of the fit is evaluated using a function <I>S</I> of the series
<I>X</I><SUP>(<I>o</I>)</SUP>, <I>X</I><SUP>(<I>m</I>)</SUP>, and <I>X</I><SUP>(<I>r</I>)</SUP>. A function of random variables,
such as
<!-- MATH: $S(X^{(o)})$ -->
<I>S</I>(<I>X</I><SUP>(<I>o</I>)</SUP>), is a random variable itself and is called a <EM>statistic</EM>. A random variable S is characterized by its probability
distribution function. Following <I>H</I><SUB><I>o</I></SUB> we use the distribution of <I>S</I>for pure noise signal <I>N</I><SUP>(<I>o</I>)</SUP>, <I>N</I><SUP>(<I>m</I>)</SUP> and <I>N</I><SUP>(<I>r</I>)</SUP>, to be
denoted <I>p</I><SUB><I>N</I></SUB>(<I>S</I>) or simply <I>p</I>(<I>S</I>). Precisely, we shall use the <EM>cumulative probability distribution</EM> function which for a given
critical value of the statistic <I>S</I>=<I>S</I><SUB><I>o</I></SUB> supplies the probability
<I>p</I>(<I>S</I><SUB><I>o</I></SUB>) for the observed <I>S</I> to fall on one side of the <I>S</I><SUB><I>o</I></SUB>.
<P>
The observed value of the statistic and its probability distribution,
<!-- MATH: $S(X^{(o)})$ -->
<I>S</I>(<I>X</I><SUP>(<I>o</I>)</SUP>) and <I>p</I>(<I>S</I>) respectively, are used to obtain the
probability <I>p</I>(<I>S</I>(<I>X</I>)) of <I>H</I><SUB><I>o</I></SUB> being true. That is, if <I>p</I> turns out
small, <IMG
WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img441.gif"
ALT="$p<\alpha$">,
<I>H</I><SUB><I>o</I></SUB> is improbable and <I>X</I><SUP>(<I>o</I>)</SUP> has no properties
of <I>N</I><SUP>(<I>o</I>)</SUP>. Then we say that the model signal has been detected at
the confidence level <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img442.gif"
ALT="$\alpha$">.
The smaller <IMG
WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img443.gif"
ALT="$\alpha$">
is, the more
convincing (significant) is the detection. The special realization of
a random series which consists of independent variables of common
(gaussian) distribution is called (gaussian) white noise. We assume
here that the noise <I>N</I><SUP>(<I>o</I>)</SUP> is white noise. Note that in the signal
detection process, frequency <IMG
WIDTH="42" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img444.gif"
ALT="$\nu$">
and lag <I>l</I> are considered
independent variables and do not count as parameters.
<P>
Summarizing, the basis for the determination of the properties of a an
observed time series is a test statistic, <I>S</I>, with known probability
distribution for (white) noise, <I>p</I>(<I>S</I>).
<P>
Let <I>N</I><SUP>(<I>o</I>)</SUP> consist of <I>n</I><SUB><I>o</I></SUB> random variables and let a given model
have <I>n</I><SUB><I>m</I></SUB> parameters. Then the modeled series <I>N</I><SUP>(<I>m</I>)</SUP> corresponds
to a combination of <I>n</I><SUB><I>m</I></SUB> random variables and the residual series
<I>N</I><SUP>(<I>r</I>)</SUP> corresponds to a combination of
<!-- MATH: $n_r=n_o-n_m$ -->
<I>n</I><SUB><I>r</I></SUB>=<I>n</I><SUB><I>o</I></SUB>-<I>n</I><SUB><I>m</I></SUB> random
variables. The proof rests on the observation that orthogonal
transformations convert vectors of independent variables into vectors
of independent variables. Let us consider an approximately linear
model with matrix <IMG
WIDTH="56" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img445.gif"
ALT="${\cal M}$">
so that
<!-- MATH: $N^{(m)} = {\cal M} \circ P$ -->
<IMG
WIDTH="102" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img446.gif"
ALT="$N^{(m)} = {\cal M} \circ P$">,
where <I>P</I> is a vector of <I>n</I><SUB><I>m</I></SUB> parameters. Then <I>N</I><SUP>(<I>m</I>)</SUP> spans a
vector space with no more than <I>n</I><SUB><I>m</I></SUB> orthogonal vectors (dimensions).
The numbers <I>n</I><SUB><I>o</I></SUB>, <I>n</I><SUB><I>m</I></SUB> and <I>n</I><SUB><I>r</I></SUB> are called the numbers of degrees
of freedom of the observations, the model fit, and the residuals,
respectively.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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