Adaptive Predictive Coding: Transmit signal AND prediction coefficients I'm not well versed in the literature so please forgive/inform me if my terminology is wrong:

I have a bounded real-valued, discrete-time signal $x_n \in \mathbb [0, 1]$, and I want to (lossily) transmit this signal via a bitstream $q_n \in \{0, 1\}$, so that it can be reconstructed into $\hat x_n$, and the average reconstruction error $\frac1N \sum_{n=1}^N (x_n - \hat x_n)^2$ is minimised.

$x_n$ will in general be temporally redundant, and I'd like to exploit this redundancy to make the most efficient use of my bits.  In my case, I expect $x_n$ to settle towards a fixed point for a long time and then suddenly jump towards a new fixed point (as you may expect from the trace of a single pixel in video).  



Example Signal: It goes though "settling" phases where it remains fairly static, and "dynamic" phases where it transitions quickly towards a new settling point.

From what I gather, Predictive Coding is the standard approach here - subtract the predictable component of the signal, encode the rest, and reconstruct the predictable component on the other end.  The trouble is that the optimal predictive coding parameters (at least for a Linear Predictive Coder) will change depending on the state of the signal: When $x$ is dynamic, I expect to encode it at a coarse resolution.  When $x$ is static, I expect to use my bits to communicate fine increments to the value of $x$.  So I need to communicate my predictive coding coefficients along the same bitstream $q_n$.

Is there a standard approach for this kind of problem?
This sounds like a variant of Differential Pulse-Code Modulation. In this set of techniques, the correlation of adjacent values and the effect of quantisation are exploited for "compression" (whether this is for transmission or storage). That is, instead of encoding the sample values, you encode the difference between them which, within limits, could be described by a much smaller number than the values themselves.

However, another technique that might be useful to you here is run length encoding. In run length encoding, the encoded word describes either the sample value itself or how many repetitions of the same value are to be decoded.

In either case, you would require higher bandwidth around the signal transients (many messages sent between the encoder and the decoder) and much lower during the longer times of constant values.

Hope this helps.