The MLDS model (
Maloney & Yang, 2003) assumes that the perceptual scale is a scalar value denoted by
Display Formula\(\psi\)—in the ordinal embedding language, it always embeds in a one-dimensional space. In contrast to NMDS, the MLDS method uses a parametric model. For a quadruplet of stimulus levels
Display Formula\(S_i,S_j,S_k,S_l\), for simplicity denoted by
Display Formula\((i,j;k,l)\), a decision random variable is defined as
\begin{equation*}
Dec(i,j;k,l) = \vert \psi _i - \psi _j \vert - \vert \psi _k - \psi _l \vert + \epsilon,\end{equation*}
where
Display Formula\(\epsilon \sim \mathcal {N}(0,\sigma ^{2})\) is a zero-mean Gaussian noise with standard deviation
Display Formula\(\sigma \gt 0\). If
Display Formula\(Dec(i,j;k,l) \gt 0\), then the observer would respond that the pair
Display Formula\((i,j)\) has a larger difference than the pair
Display Formula\((k,l)\). In this case, the response to the quadruplet
Display Formula\(q=(i,j;k,l)\) is set to
Display Formula\(R_q=1\); otherwise, the response is
Display Formula\(R_q=0\). The goal of the MLDS is now to estimate the perception scale
Display Formula\(\psi\) that maximizes the likelihood of the observed quadruplet answers. We first set
Display Formula\(\psi _1 = 0, \psi _n = 1\) to remove degenerate solutions. Now, assuming that
Display Formula\(R_1,R_2, \ldots, R_m \in \lbrace 0,1\rbrace\) denote the independent responses to
Display Formula\(m\) quadruplet questions, the likelihood of the perceptual scales given the quadruplet answers is
\begin{eqnarray*}
&&\mathcal {L}\left(\psi _2,\ldots,\psi _{n-1},\sigma \vert R_1,\ldots,R_m\right) \\
&&= \prod _{q=1}^m{\Phi (\Delta _q)^{R_q}[1-\Phi (\Delta _q)]^{1-R_q}},\end{eqnarray*}
where
Display Formula\(\Phi (.)\) denotes the cumulative distribution function of
Display Formula\(\epsilon \sim \mathcal {N}(0,\sigma ^{2})\), and
Display Formula\(\Delta _q = \vert \psi _i - \psi _j \vert - \vert \psi _k - \psi _l \vert\) for the quadruplet
Display Formula\(q=(i,j;k,l)\).
2 As it is the case for the stress function of NMDS, the likelihood function of MLDS is not convex with respect to the perceptual scale values
Display Formula\(\psi _i\). Thus, the proposed numerical methods to maximize this likelihood might get stuck in a local maximum.