Compression-based methods for nonparametric density estimation, on-line prediction, regression and classification for time series

We address the problem of nonparametric estimation of characteristics for stationary and ergodic time series. We consider finite-alphabet time series and the real-valued ones and the following problems: estimation of the (limiting) probability P(u
<sub>0

hellip</sub>
u
_s
) for every s and each sequence u
₀
hellip u
_s
of letters from the process alphabet (or estimation of the density p(x
₀
,
_hellip
, x
_s
) for real-valued time series), so-called on-line prediction, where the conditional probability P(x
_t+1
/x
₁
x
<sub>2

hellip</sub>
x
_t
) (or the conditional density p(x
_t+1
/x
₁
x
<sub>2

hellip</sub>
x
_t
)) should be estimated (in the case where x
₁
x
₂
hellip x
_t
is known), regression and classification (or so-called problems with side information). We show that any universal code (or a universal data compressor) can be used as a basis for constructing asymptotically optimal methods for the above problems.