Neurocomputing in Complex Domain - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 69

3.3.3.5 Sine-Hyperbolic Error Function
The sinh EF is symmetric about the origin. The computation of slope should be
directed toward the origin in the first and the third quadrants separately, for which
the update rule must be defined in these quadrants accordingly. The function is
smooth and maintains convexity through out the x -axis. The slope computed from
this EF varies according to cosh(x). The complex Sinh EF extends the hyperbolic
sine function to the complex domain. A complex conjugation was employed in the
argument to the function that makes the extended complex function an even function.
This is rotationally symmetric about the z -axis. The surface maintains convexity with
respect to xy -plane. The steepness of the slope increases as the index rises.
3.3.3.6 Cauchy Error Function
The Cauchy EF has one minimum point at the origin. The function is symmetric
about the y -axis. The training steers the weights so as to reach the minimum of the
function. The function is defined through out the real line, it is continuous everywhere
and differentiable all through. The function changes convexity as x increases. As a
ramification of this fact, the update from the slope function based on this EF at
larger values of x would be smaller in comparison with that of the quadratic EF. The
complex Cauchy EF was defined to perform the Cauchy function for the complex
variables. The surface plot reveals a unique point of global minimum. The surface
is differentiable through out the real plane. The surface is also characterized by
changing convexity as the radius vector increases. It is rotationally symmetric about
the z -axis.
3.3.3.7 Huber Error Function
The definition of Huber EF has both features of the quadratic error and the absolute
error. The function enables one to optimally choose EFs. If the data were prone
to outliers and if their scatter is biased to one side, an obvious choice would be to
suppress the influence of these spurious points by assigning an absolute EF to the side
and set quadratic function to operate on the other side. It was found that in statistical
analysis such choice indeed bettered the results as a judicious assignment was proven
to be effective. The complex Huber function was defined to generalize the real Huber
function to the complex variables retaining the form the function. The principle of
operation of the function remains the same as the choice of the parameter c assigns
the domains of operation of the EF. The quadratic and the absolute functions exist
in the definition and the assignment will accordingly suppress the ill-effects of the
outliers and other spurious data by evaluating the assigned function accordingly. It
must be noted that the extended function is a paraboloid of revolution for the part
of the definition that was quadratic and for the part that was absolute function, the
extended version is a cone.