# Data transformation

I was writing with a question regarding a time-varying state space model of the form: \begin{align} y(t) &= \mu_1(t) + A(t)x(t) + v(t); &v(t) &\sim (0, R(t)) \\ x(t) &= \mu_2(t) + \Phi(t)\!\times\!x(t-1) + s(t); &s(t) &\sim (0, Q(t)) \end{align} $x(t)$ is of the form: $x(t) = \log[1-X(t)]$

obviously, $$|X(t)| \leq 1 \\ \Rightarrow x(t) \text{ lies in } (-\infty, \log(2) ]$$ Could anyone suggest a transformation that I could use, so that $x(t)$ becomes unbounded?

• I took the liberty of formatting your post using the $\LaTeX$ markup our site affords. Please ensure it still says what you want it to. – gung - Reinstate Monica Sep 3 '14 at 15:00

If $-1 < X(t) < 1$, then $T(X(t)) = sign(X(t)) \exp(1/\log(|X(t)|))$ would be unbounded, i.e. $(-\infty, +\infty)$.