Official repository - An Algorithm for Building Markov Models from Time Series
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CSSR tries to infer the minimal Markovian model capable of generating a time-series, or set of time-series from the same source. The program implements the algorithm proposed in the paper, hereafter BC, Blind Construction of Optimal Nonlinear Recursive Predictors for Discrete Sequences 1. We won't describe the algorithm in any detail here (see the paper for that), but the next two paragraphs say a little about what it produces and how it works at a high level.

The output of the algorithm is a set of states which form a Markov chain. Each state has a certain probability of emitting any of the symbols in the original time series. The current state and the symbol it emits fix the next state. (The states are "future-resolving", if you're from nonlinear dynamics, or "deterministic", if you're from automata theory.) Each state, moreover, corresponds to a distinct set of strings, in the following sense. If the state A contains a string w, and at time t the time-series ends with w, then at time t the Markov chain is in state A. The set of states, their transition probabilities and connections, is called the state machine.

The algorithm uses a recursive inference procedure to find the simplest set of states with the above properties that can reproduce the statistical properties of the data. If we could give the algorithm an infinitely long time series, and let it consider infinitely long sub-strings, it would produce the causal states of the process, which are its ideal predictors (see BC for a formal definition). Since we have only finite data, there is always some probability that the inferred or estimated states are not the true causal states. Nonetheless, for the rest of this file, when we say "causal states", we mean the estimated causal states.

1 Blind Construction of Optimal Nonlinear Recursive Predictors for Discrete Sequences, Cosma Rohilla Shalizi and Kristina Lisa Shalizi, pp. 504--511 of Max Chickering and Joseph Halpern (eds.), Uncertainty in Artificial Intelligence: Proceedings of the Twentieth Conference. link

Installing the Original Program

There are several versions of CSSR in this repository. The original code, written in C++, can be found in the cpp folder. This is not actively maintained, however it is stable and is the tried-and-true implementation. Deployment to different package managers will happen further down the line, for now:

# the recursive flag is optional
git clone --recursive
cd CSSR/cpp
make                   # warnings can be safely ignored
cp CSSR ~/.local/bin   # whereever you keep your binaries

Details of how to use this program can be found in cpp/

Preview v0.2.0

Cosma, Kristi, and I are currently working on large improvements for CSSR. The source for this can be found in the scala folder. If you would like to preview this implementation, a release candidate can be [found at github][v0.2.0-RC1].

From the command line:

curl -LO
java -jar cssr-assembly-0.1.0.jar --alphabet test-machines/perl/alphabets/binary --data tmp/Misiurewicz_timeseq
# where tmp/Misiurewicz_timeseq is generated from test-machines/perl/


The program will create the following four files after running, where <dataname> is the name of the file the data is in:

  1. <dataname>_results - contains the information on the inferred states and their properties. For each state, it gives:
    • the histories of length L-1 and L in the state
    • the probability that the state will emit different symbols (e.g. P(a) = x) and the states transitioned to when those symbols are emitted (e.g. T(a) = s)
    • the observed probability of the state in the data-stream
  2. <dataname>_state_series - is the series of causal states in the data. That is, the program scans through the data, looks up which state the history-to-date is in, and writes the corresponding symbol to this file. What you see is then the trajectory through estimated causal state space of the data/process. Multiline data results in a multiline state-series file.
  3. <dataname> - represents the state machine as a labeled directed graph, where each state has its own node, and each transition between states its own edge. The file is for use with the program dot, available from
  4. <dataname>_info - is the file containing the command-line settings and the metrics run on the causal state machine. These are:
    • the name of the alphabet file
    • the alphabet size
    • the name of the data file
    • the history length
    • the significance level
    • whether multi-line mode was used
    • whether the chi-squared test was used
    • the number of states
    • the statistical complexity (entropy of the states)
    • the entropy rate
    • three measures of the difference between the empirical distribution of symbol sequences, and that generated by the inferred causal state machine. These are:
      • the divergence or relative entropy between the inferred and empirical distribution*
      • the relative entropy rate, or increase per symbol in the divergence*
      • the total variational distance ("variation") between the two distributions.

*Note that the relative entropy and the relative entropy rate can be infinite; this indicates that the inferred model gives a probability of zero to a sequence in the data.

*Note that sometimes, when the relative entropy should be very small (order of 1e-6 bits or less), numerical rounding errors result in a negative number being calculated. In these cases, the program outputs zero. Similarly, the complexity of one-state machines is sometimes reported as -0.

Some Suggestions About Parameters

It is always good to use as much data as you can. While it is generally good practice to hold back some data for testing or cross-validation, we recommend that this be minimized. High-entropy processes are especially data-hungry. (See BC.) For reference, let us call the number of data-points N.

The two key parameters of the program are the maximum history length, L, and the significance level used in the test, s. For any given process, there is a minimum history length M, such that the true states cannot be found if L < M. The number of states returned may be less than the correct number or higher. If L >= M, and there is enough data, there will generally be a "plateau" of values of L where the correct number of states is returned. For fixed N, if we keep increasing L, then past a certain point there are not enough examples of each string in the data. This tends to erroneously create new states, which spawn others through determinization. Thus there is generally a "blow-up" when L is too large (relative to N and s). A rough guide-line is to limit L to no more than log(N)/log(k), where k is the alphabet size (see BC for details).

In general, one should use as small an L as possible, since under-sampling, even before the blow-up, will reduce the accuracy of many probability estimates. Blow-up can be delayed by reducing s --- that is, reducing the probability of mistakenly splitting a state --- but this carries the risk of failing to create valid new states. We suggest exploring the data at low L and high s initially, and then increasing L and lowering s. If a stable architecture is found, it should be recorded at the lowest possible L.

Known Issues

There are a few known issues with CSSR's behavior in the C++ code. These arise from certain unavoidable approximations in the determinization procedure. For details, see cpp/ While v0.2.0 will not have the same limitations as the original C++, these issues still need to be reviewed in the context of the new algorithm.

Bug Reports, Fixes, Modifications

We welcome bug reports or reports of strange behavior. These reports are welcomed with more enthusiasm when accompanied by successful modifications to the code! (See the accompanying file on the Gnu Public License for information about modifying the code.) Even if you can't fix it, however, please do tell us about it; at the least it will be documented for other users.

If you modify CSSR, and want to make the resulting program available, please let us know. We are happy to provide a link, and have a (limited) capability to host alternate versions and descendants. Also, if you use CSSR successfully in some application, we'd love to hear about it.

Please check for up-to-date contact information.