In the standard classification image method with additive white Gaussian noise, an unbiased estimate
Display Formula\({\bf{\widehat w}}\) of the observer's inner template
w can be computed as
\begin{equation}\tag{13}{\bf{\widehat w}} = ({{\bf{\overline n}}_{01}} + {{\bf{\overline n}}_{11}}) - ({{\bf{\overline n}}_{00}} + {{\bf{\overline n}}_{10}}),\!\end{equation}
where
Display Formula\({{\bf{\overline n}}_{ij}}\) is the mean of the added noise over all trials in which the stimulus contained signal
i and the observer indicated signal
j (Ahumada,
2002; Murray et al.,
2002).
What if the noise is Gaussian but not white? Abbey and Eckstein (
2002) considered the discrimination of two distinct real signals embedded in additive nonwhite (correlated) Gaussian noise, within a 2AFC experimental paradigm. In particular, they showed that, for a linear observer with additive Gaussian internal noise, generalizing from white to nonwhite noise involves normalization of the estimated template by the covariance of the noise. Here we show that this result generalizes to a yes/no task. We first prove the claim for real signals and then show that it easily generalizes to complex-valued stimuli.
Let
Display Formula\({\bf{\tilde s}} = {\bf{s}} + {\bf{n}}\) be an
m-dimensional real-valued random vector representing a visual stimulus, where
s is a binary signal variable taking one of two values
s0 or
s1 and
Display Formula\({\bf{n}}\sim {\cal N}\left( {0,\Sigma } \right)\) is added zero-mean multivariate normal stimulus noise with covariance matrix Σ. Following Ahumada (
2002), we model the internal source of variability by assuming that the criterion
β is also a random variable. We let
Display Formula\(R \in \{ 0,1\} \) represent the two possible responses of the observer.
Let
Display Formula\({{\bf{\overline n}}_{ij}} = {\mathbb{E}}\left[ {{\bf{n}}|{\bf{s}} = {{\bf{s}}_i},R = j} \right]{}\). Then
\begin{equation}\tag{14}{{\bf{\overline n}}_{i0}} = {{\mathbb{E} }_{{\bf{n}}\beta }}\left[ {{\bf{n}}|{{\bf{w}}^ \top }\left( {{{\bf{s}}_i} + {\bf{n}}} \right) \lt \beta } \right],\!\end{equation}
and
\begin{equation}\tag{15}{{\bf{\overline n}}_{i1}} = {{\mathbb{E} }_{{\bf{n}}\beta }}\left[ {{\bf{n}}|{{\bf{w}}^ \top }\left( {{{\bf{s}}_i} + {\bf{n}}} \right) \gt \beta } \right],\!\end{equation}
where
w is an
m-vector representing the observer template. Without loss of generality, we assume that
Display Formula\(||{\bf{w}}||\; = 1\).
Let
U be an orthonormal rotation matrix with first column
w, so that
Display Formula\({{\bf{U}}^ \top }{\bf{w}} = {{\bf{e}}_1} \buildrel \Delta \over = {\left[ {1,0, \ldots ,0} \right]^ \top }\), and let
Display Formula\({\bf{n^{\prime} }} \buildrel \Delta \over = {{\bf{U}}^ \top }{\bf{n}}\) and
Display Formula\({\bf{s}}_i^\prime \buildrel \Delta \over = {{\bf{U}}^ \top }{{\bf{s}}_i}\). Note that
Display Formula\({\bf{n^{\prime} }}\sim {\cal N}\left( {0,\Sigma ^{\prime} } \right)\) is also zero-mean multivariate normal with covariance matrix
Display Formula\(\Sigma ^{\prime} = {{\bf{U}}^ \top }\Sigma {\bf{U}}\). Note also that
Display Formula\({{\bf{w}}^ \top }\left( {{{\bf{s}}_i} + {\bf{n}}} \right) = {\left( {{{\bf{U}}^ \top }{\bf{w}}} \right)^ \top }{{\bf{U}}^ \top }\left( {{{\bf{s}}_i} + {\bf{n}}} \right) = {\bf{e}}_1^ \top \left( {{\bf{s}}_i^\prime + {\bf{n^{\prime} }}} \right) = {s^{\prime} _{i1}} + {n^{\prime} _1}\), where
Display Formula\({s^{\prime} _{i1}}\) and
Display Formula\({n^{\prime} _1}\) are the first elements of
Display Formula\({\bf{s}}_i^\prime\) and
n′, respectively.
In this new coordinate frame,
Display Formula\({\bf \overline n_{\it i\rm 1}}\) can be expressed as
\begin{equation}\tag{16}{{\bf{\overline n}}_{i1}} = {{\mathbb{E} }_{{\bf{n}}\beta }}\left[ {{\bf{n}}|{{\bf{w}}^ \top }\left( {{{\bf{s}}_i} + {\bf{n}}} \right) \gt \beta } \right] = {\bf{U}}{{\mathbb{E} }_{{\bf{n^{\prime} }}\beta }}\left[ {{\bf{n^{\prime} }}|s_{i1}^\prime + {n_i^\prime} \gt \beta } \right].\end{equation}
Let us now consider the conditional expectation of each element
Display Formula\({n^{\prime}_k}\) of the noise
Display Formula\({\bf n^{\prime}}\) in this new coordinate frame. Consider first the conditional expectation of the first element
Display Formula\({n^{\prime}_1}\):
\begin{equation}\tag{17}{\mathbb{E} }_{{\bf n}^{\prime}}\left[ {n_1^\prime|{n_1^\prime} \gt \beta - s_{i1}^\prime} \right] = \int_{\beta - {s_{i1}^\prime}}^\infty {{n_1^\prime}} p\left( {{n_1^\prime}} \right)d{n^{\prime} _1} = {1 \over {\sqrt {2\pi } {{\sigma_1 ^{\prime} }}}}\int_{\beta - {s_{i1}^\prime}}^\infty {{n_1^\prime}} \exp \left( { - {{n^{\prime2} _1} \over {2\sigma ^{\prime2} _1}}} \right)d{n^{\prime}_1} = {{ - {{\sigma_1^{\prime} }}} \over {\sqrt {2\pi } }}\left. {\exp \left( { - {{n^{\prime2} _1} \over {2\sigma ^{\prime2} _1}}} \right)} \right|_{\beta - {{s_{i1}^\prime}}}^\infty = {{{{\sigma_1^{\prime} }}} \over {\sqrt {2\pi } }}\exp \left( { - {{{{\left( {\beta - {s_{i1}^\prime}} \right)}^2}} \over {2\sigma ^{\prime2} _1}}} \right),\!\end{equation}
where
Display Formula\({\sigma^{\prime}_1}\) is the standard deviation of
Display Formula\({n^{\prime}_1}\).
Now consider the conditional expectations of the remaining elements
Display Formula\({{n^{\prime}_k}, k \ne 1}\):
\begin{equation}\tag{18}{\mathbb{E} }_{\bf{n^{\prime}}}\left[ {n^{\prime} _k}|{s_{i1}^\prime} + {n^{\prime} _1} \gt \beta \right] = {\mathbb{E} }_{\bf{n^{\prime} }}\left[ {n^{\prime} _k}|{n^{\prime} _1} \gt \beta - {s_{i1}^\prime} \right] = \int_{\beta - {s_{i1}^\prime}}^\infty p \left( {n^{\prime} _1} \right)\int_{ - \infty }^\infty {n^{\prime} _k} p\left( {n^{\prime} _k}|{n^{\prime} _1} \right)d{n^{\prime} _k}d{n^{\prime} _1}.\end{equation}
Because
Display Formula\({n^{\prime} _1}\) and
Display Formula\({n^{\prime} _k}\) are jointly normal, the conditional random variable
Display Formula\({n^{\prime} _k}|{n^{\prime} _1}\sim {\cal N}\left( {{\mu ^{\prime}_{k|1}},\sigma ^{\prime2} _{k|1}} \right)\) is univariate normal with mean and variance given by
\begin{equation}\tag{19}{\mu ^{\prime} _{k|1}} = {\left( {{\sigma ^{\prime}_{1k}}/{\sigma ^{\prime}_1}} \right)^2}{n^{\prime} _1},\!\end{equation}
\begin{equation}\tag{20}\sigma ^{\prime2} _{k|1} = \sigma ^{\prime2} _k - \sigma ^{\prime4} _{1k}/\sigma ^{\prime2} _1,\!\end{equation}
where
Display Formula\(\sigma ^{\prime2} _k\) is the variance of
Display Formula\({n^{\prime} _k}\) and
Display Formula\(\sigma ^{\prime2} _{1k}\) is the covariance of
Display Formula\({n^{\prime} _1}\) and
Display Formula\({n^{\prime} _k}\). (See, for example, Bishop,
2006, p. 87, equations 2.81–2.82.)
As a result, we can write
\begin{equation}\tag{22}{{\bf{\overline n}}_{i1}} = {\bf{U}}{{\mathbb{E}}_{{\bf{n^{\prime} }}\beta }}\left[ {{\bf{n^{\prime} }}|{s_{i1}^\prime} + {n^{\prime} _1} \gt \beta } \right] = {1 \over {\sqrt {2\pi } {\sigma ^{\prime} _1}}}{{\mathbb{E}}_\beta }\left[ {\exp \left( { - {{{{\left( {\beta - {s_{i1}^\prime}} \right)}^2}} \over {2\sigma ^{\prime2} _1}}} \right)} \right]\times {\bf{U}}{\left[ {\sigma ^{\prime2} _1,\sigma ^{\prime2} _{12}, \ldots ,\sigma ^{\prime2} _{1m}} \right]^ \top } = {c_{i1}}{\bf{U}}{{\mathbb{E}}_{{\bf{n^{\prime} }}}}\left[ {{\bf{n^{\prime} }}{n^{\prime} _1}} \right],\!\end{equation}
where
Display Formula\({c_{i1}} = {1 \over {\sqrt {2\pi } {\sigma ^\prime _1}}}{{\mathbb{E}}_\beta }\left[ {\exp \left( { - {{{{\left( {\beta - {s^{\prime} _{t1}}} \right)}^2}} \over {2\sigma ^{\prime2} _1}}} \right)} \right]{}\) is a positive proportionality constant.
This result can be transformed back to the original pixel coordinates by applying the inverse rotation
U and making the substitution
Display Formula\({n^{\prime} _1} = {{\bf{w}}^ \top }{\bf{n}} = {{\bf{n}}^ \top }{\bf{w}}\) before taking the expectation:
\begin{equation}\tag{23}{{\bf{\overline n}}_{i1}} = {c_{i1}}{{\mathbb{E}}_{{\bf{n^{\prime} }}}}\left[ {{\bf{Un^{\prime} }}{n^{\prime} _1}} \right] = {c_{i1}}{{\mathbb{E}}_{\bf{n}}}\left[ {{\bf{n}}{{\bf{n}}^ \top }} \right]{\bf{w}} = {c_{i1}}\Sigma {\bf{w}}.\end{equation}
Thus, we have that an unbiased estimate of the observer template
w can be obtained by premultiplying
Display Formula\({{\bf{\overline n}}_{i1}}\) by the inverse covariance of the stimulus noise:
\begin{equation}\tag{24}{\bf{w}} = {\left( {{c_{i1}}\Sigma } \right)^{ - 1}}{{\bf{\overline n}}_{i1}},\quad {\rm{with}}\quad {c_{i1}} = {1 \over {\sqrt {2\pi } {\sigma ^{\prime} _1}}}{{\mathbb{E}}_\beta }\left[ {\exp \left( { - {{{{\left( {\beta - {s_{i1}^\prime}} \right)}^2}} \over {2\sigma ^{\prime2} _1}}} \right)} \right].\end{equation}
It is straightforward to show that for
Display Formula\({{\bf{\overline n}}_{i0}}\) an analogous equation holds but with a negative proportionality constant. Specifically,
\begin{equation}\tag{25}{\bf{w}} = {\left( {{c_{i0}}\Sigma } \right)^{ - 1}}{{\bf{\overline n}}_{i0}},\quad {\rm{with}}\quad {c_{i0}} = - {1 \over {\sqrt {2\pi } {\sigma ^{\prime} _1}}}{{\mathbb{E}}_\beta }\left[ {\exp \left( { - {{{{\left( {\beta - {s^{\prime} _{i0}}} \right)}^2}} \over {2\sigma ^{\prime2} _1}}} \right)} \right].\end{equation}
We now generalize this result to complex-valued signals. In our complex-valued linear template model, discrimination is based on a real-valued scalar decision variable
r given by
Display Formula\(r = \rm{Re}( {{{\bf{w}}^{\it{H}}}{\bf{\tilde s}}}\ )\), where
w is the complex-valued observer template and
Display Formula\({\bf{\tilde s}}\) is the complex-valued noisy stimulus. This can be re-expressed as a sum of two real-valued inner products:
\begin{equation}\tag{26}r = {\bf{w}}_x^ \top {{\bf{\tilde s}}_x} + {\bf{w}}_y^ \top {{\bf{\tilde s}}_y},\!\end{equation}
where
wx and
wy are the real and imaginary components of
w and
Display Formula\({{\bf{\tilde s}}_x}\) and
Display Formula\({{\bf{\tilde s}}_y}\) are the real and imaginary components of
Display Formula\({\bf{\tilde s}}\), respectively. This can be reduced to a single real-valued inner product if we stack the real and imaginary components of the template and stimulus:
\begin{equation}\tag{27}{{\bf{w}}_{xy}} = {\left[ {{\bf{w}}_x^ \top ,{\bf{w}}_y^ \top } \right]^ \top },{{\bf{\tilde s}}_{xy}} = {\left[ {{\bf{\tilde s}}_x^ \top ,{\bf{\tilde s}}_y^ \top } \right]^ \top } \to r = {\bf{w}}_{xy}^ \top {\bf{\tilde s}}_{xy}.\end{equation}
From the proof above, we know that an unbiased estimate
Display Formula\({{\bf{\widehat w^{\prime} }}_{xy}}\) of the real-valued template
wxy can be obtained by normalizing the biased estimate (
Equation 13) by the covariance of
Display Formula\({{\bf{\tilde s}}_{xy}}\):
\begin{equation}\tag{28}{\bf\widehat w}^\prime_{xy} = \Sigma _{xy}^{ - 1}{{\bf{\widehat w}}_{xy}}.\end{equation}
Because there is a 1:1 identification of the real-valued coefficients of the template
wxy with the real and imaginary coefficients of the complex-valued template
w,
Equation 28 also yields an unbiased estimate
Display Formula\({\bf{\widehat w^{\prime} }}\) of the latter. In our particular case, because the same real-valued independent and identically distributed noise process
Display Formula\(\sim {\cal N}\left( {0,\Sigma } \right)\) is used to generate both real and imaginary components of the stimulus, Σ
xy is block-diagonal and can be written as
\begin{equation}\tag{29}{\Sigma _{xy}} = \left[ {\matrix{ \Sigma&{{{\bf{0}}_m}} \cr {{{\bf{0}}_m}}&\Sigma \cr } } \right],\!\end{equation}
and so
\begin{equation}\tag{30}{\bf{\widehat w^{\prime} }} = {\Sigma ^{ - 1}}{\bf{\widehat w}}.\end{equation}