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Categorical time series such the genomic DNA, the sleep state of a person $\cdots$ etc, are often treated and analyzed. The first question that came to my head was how can such type of series be a white noise? is there a definition for categorical white noise (I haven't found any definition).

Edit: $X_{t}$, for $t = 0, \pm 1, \pm 2 ,\cdots$ , is a categorical-valued time series with finite state-space $\mathbb{C}= \{c_{1}, c_{2},\cdots, c_{k}\}$ and $p_{j} = pr (X_{t} = c_{j}) >0$

enter image description here

As an example, here's the categorical time series that represent EEG sleep stat of an infant.

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    $\begingroup$ @StudentT, no, I'm talking about categorical time series, an example is the sleep state (per minute) of an infant :$$\{ah, qt, qt, tr, \cdots\}$$. $\endgroup$ – Toney Shields Jan 28 '17 at 7:25
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    $\begingroup$ What do you mean by categorical time-series not being Mark of Chains? You mean that you have N i.i.d categorical variables and there is no transition probabilities? I can't see how this makes any sense... $\endgroup$ – Tim♦ Jan 28 '17 at 8:32
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    $\begingroup$ There is no such thing as random white noise for categorical data. If you have categorical, each one must have some kind of probability. That's what Markov Chain is doing. MC applies to DNA data. I'd interested to learn if I'm proven wrong $\endgroup$ – SmallChess Jan 28 '17 at 9:38
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    $\begingroup$ @StudentT Although it wouldn't ordinarily be called "white noise," the term suggests models in which a continuous underlying variable is binned to create the categorical responses. Adding any kind of "noise" to that variable would model errors in the responses. (Logistic regression can be formulated in this manner. Probit regression is precisely a binned linear log-odds response with Gaussian white noise.) Thus, it might be a little too extreme to assert "there is no such thing." $\endgroup$ – whuber♦ Jan 28 '17 at 15:03
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    $\begingroup$ Nice question. I got confused trying to translate your small contingency table of infant sleep states, which are clearly not random, into a random signal. It might help to think in terms of massively categorical information. Note that there are many types of 'noise': white noise is just one. One key question is whether time is discrete or continuous, "In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance; a single realization of white noise is a random shock." (wiki/white noise) $\endgroup$ – Mike Hunter Jan 28 '17 at 20:27
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Great question!

A fairly simple but rather imprecise method

First, you need a theoretical prior about the distribution of the variable you are interested on (e.g. infant EEG sleep states). The example you give is that of a categorical distribution, with $p_j=pr(X=c_j)>0$, and $\sum_{j} p_j=1$, with $j= 1, 2, ... , k$

Second, I would test whether the sample distribution follows the a-priori theoretical distribution you are given. A simple Chi-square goodness-of-fit test would suffice for it. If the test indicates that the sample data could have been drawn from such theoretical distribution with a given confidence level, then the categorical white noise is given by the observed discrepancy between the predicted and observed distribution, at that confidence level.

For instance, in a sample of 100 minutes, your prior is that you should observe 30 $ah$, 40 $qt$, and 30 $tr$. Yet, you observe 27, 41, and 32 respectively. If the two are identical at 5%, then the discrepancy is due to white noise.

On the contrary, if you find that the theoretical and sample distributions are different at your desire confidence level, then you might conclude that (i) your theoretical distribution is wrong, and/or (ii) there is some "non-white (non-random) noise".

Spectral analysis

There seems to be a very recent literature on spectral analysis for time series of categorical variables (e.g. here). The only way to evaluate these series seems to be by means of "indicators" variables. For instance, you assume

$$Y_t = \begin{cases} 1 &\mbox{if } X_t=c_k \\ 0 & \mbox{if } X_t \neq c_k. \end{cases} $$

Applying this to your dataset gives a sequence like 1 0 1 0 0 1 0 1 0 1 1 0 0 1 ...

The idea is to analyze the frequency of all the possible cycles, with the length of the cycle being $k$. Then, a spectral analysis of these cycles will show you the strength of the signal at different frequencies. If these are fairly constant, there is indication of the series being white noise.

Here for instance in the spectral analysis of a white noise:

enter image description here

Image source: Wikipedia

The paper that I refer to have more examples.

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