  The problem of universally predicting an individual continuous sequence using
a deterministic finite-state machine (FSM) is considered. The empirical mean is
used as a reference as it is the constant that fits a given sequence within a
minimal square error. With this reference, a reasonable prediction performance
is the regret, namely the excess square-error over the reference loss, the
empirical variance. The paper analyzes the tradeoff between the number of
states of the universal FSM and the attainable regret. It first studies the
case of a small number of states. A class of machines, denoted Degenerated
Tracking Memory (DTM), is defined and the optimal machine in this class is
shown to be the optimal among all machines for small enough number of states.
Unfortunately, DTM machines become suboptimal as the number of available states
increases. Next, the Exponential Decaying Memory (EDM) machine, previously used
for predicting binary sequences, is considered. While this machine has poorer
performance for small number of states, it achieves a vanishing regret for
large number of states. Following that, an asymptotic lower bound of
O(k^{-2/3}) on the achievable regret of any k-state machine is derived. This
bound is attained asymptotically by the EDM machine. Furthermore, a new
machine, denoted the Enhanced Exponential Decaying Memory machine, is shown to
outperform the EDM machine for any number of states.
