Self-modeling for two-dimensional response curves

Two-dimensional response curves are an important experimental outcome in speech kinematics and other areas of research. These parameterized curves are usually obtained by recording the two-dimensional location of an object over time. In this setting, time is the independent variable and the x and y locations on specified coordinate axes define the multivariate response. Collections of such parameterized curves can be obtained either from one subject or from a number of different subjects, each producing one or several realizations of the response curve. When only one dependent variable is observed over time and no parametric model is specified, self-modeling regression (SEMOR) is an attractive modeling approach. SEMOR assumes that each of a collection of curves differs from a smooth, average shape function by some simple parametric transformation of the coordinate axes (usually linear). We will describe the extension of SEMOR to two-dimensional parameterized curves using affine transformations of a two-dimensional, time-parameterized shape function.