Recent developments in sensor technology, human centered computing and robotics have
brought to light new classes of multidimensional data which are naturally represented as
three- or four-dimensional vector-valued processes. Such signals are readily modeled as
real vectors in R3 and R4, however, it has become apparent that there are advantages in
processing such data in division algebras - the quaternion domain. The progress in the
statistics of quaternion variable, particularly augmented statistics and widely linear modeling,
has opened up a new front of research in vector sensor modeling, however, there are
several key problems that need to be addressed in order to exploit the full power of quaternions
in statistical signal processing. The principal problem lies in the lack of a mathematical
framework, such as the CR-calculus in the complex domain, for the differentiation of
non-holomorphic functions. Since most functions (including typical cost functions) in the
quaternion domain are non-holomorphic, as defined by the Cauchy-Riemann-Fueter (CRF)
condition, this presents a severe obstacle to solving optimisation problems and developing
adaptive filtering algorithms in the quaternion domain. To this end, we develop the HR-calculus,
an extension of the CR-calculus, allowing the differentiation of non-holomorphic
functions. This is followed by the introduction of the I-gradient, enabling for generic extensions
of complex valued algorithms to be derived. Using this unified framework we
introduce the quaternion least mean square (QLMS), quaternion recursive least squares
(QRLS), quaternion affine projection algorithm (QAPA) and quaternion Kalman filter.
These estimators are made optimal for the processing of noncircular data, by proposing
widely linear extensions of their standard versions. Convergence and steady state properties
of these adaptive estimators are analysed and validated experimentally via simulations
on both synthetic and real world signals.