Context trees

Variable Length Markov Chain (VLMC) models provide parsimonious high order Markov chains which can have a finite but long memory without suffering from the computational and estimation problems associated to dense high order Markov chain. This is achieved using the notion of context.

Contexts

Definition

We consider a time series \(x=(x_i)_{i\geq 1}\) with values in a finite set \(S\), its state space. A context \(c\) is a finite sequence of elements of \(S\), \(c=(c_k, \ldots, c_1)\) observed in \(x\). \(c\) is observed in \(x\) if there is \(t\) such that \[ x_{t-1}=c_1, x_{t-2}=c_2, \ldots, x_{t-k}=c_k. \] Notice that \(c\) is numbered in a reverse order, in the sense that \(c_1\) is the most recent value in \(x\) while \(c_k\) is the oldest one. Thus the sub-sequence observed in \(x\). In numerous papers, for instance in the works of Bühlmann and Wyner, the convention is to reverse the temporal order and to write \(c=(c_1, \ldots, c_k)\), keeping \(c_1\) as the most recent value and \(c_k\) as the oldest one. In mixvlmc we use the more natural convention of writing contexts in temporal order, but many functions offer a reverse parameter than can be used to switch to Bühlmann and Wyner convention.

Back to examples, if \(S=\{0, 1\}\) and \(x=(0, 0, 0, 1, 1, 1)\)

\((0, 0)\) is a context of \(x\) with \(t=3\) and \(t=4\);
\((0, 1, 0)\) is not a context of \(x\).

Context trees

The contexts of a time series can be represented by a tree. The root of the tree stands for the empty context. The children of the root represent the contexts of length 1. In general, if a node represents the context \(c=(c_k, \ldots, c_1)\), then contexts of the form \(c'=(c_{k+1}, c_k, \ldots, c_1)\) are represented by the children of the node. Descending in the context tree corresponds to adding to the past of the context.

Let us consider again \(x=(0, 0, 0, 1, 1, 1)\) and all contexts that appear at least twice in \(x\) (i.e. which are observed for at least two different values of \(t\)). An ASCII art representation of the corresponding context tree is:

*
+-- 0
|   '-- 0
'-- 1

The tree represents 2 size one contexts (\((0)\) and \((1)\)), the direct children of the root (shown as a star *). It represents in addition 1 size 2 contexts, \((0, 0)\). Notice that for instance, \((1, 0)\) is not a context in the tree as the node of context \((0)\) has only one child labelled by \(0\).

Extracting contexts from a time series in mixvlmc

Mixvlmc can be used to compute all the contexts of a time series using the ctx_tree() function/constructor as follows:

x <- c(0, 0, 0, 1, 1, 1)
library(mixvlmc)
x_ctx <- ctx_tree(x)
x_ctx
#> Context tree on 0, 1 
#>  Number of contexts: 3 
#>  Maximum context length: 2

The result of ctx_tree() is a ctx_tree object. It can be drawn using ascii art

draw(x_ctx)
#> *
#> +-- 0
#> |   '-- 0
#> '-- 1

The default extraction is done with min_size=2 and max_depth=10 which means that

contexts are included only if they appear at least twice in the time series;
the maximum length of a context is 10 (the term depth is used in reference to the tree representation).

Notice that the number of potential contexts grows exponentially with the length of the time series and it is therefore advisable to keep max_depth to a reasonable value. Let us consider a simple example.

set.seed(0)
y <- sample(c("a", "b", "c"), 100, replace = TRUE)
y_ctx_def <- ctx_tree(y)
y_ctx_def
#> Context tree on a, b, c 
#>  Number of contexts: 77 
#>  Maximum context length: 6

With the default parameters, we end up with already 77 contexts. Setting min_size=1 gives an unreasonable number of contexts:

y_ctx_min_1 <- ctx_tree(y, min_size = 1)
y_ctx_min_1
#> Context tree on a, b, c 
#>  Number of contexts: 4607 
#>  Maximum context length: 99

Even if we decrease the depth limit the number of contexts remains very large:

y_ctx_min_1_d_15 <- ctx_tree(y, min_size = 1, max_depth = 15)
y_ctx_min_1_d_15
#> Context tree on a, b, c 
#>  Number of contexts: 1037 
#>  Maximum context length: 15

Contexts can be extracted from a context tree using the contexts() function as follows:

contexts(x_ctx)
#> Contexts:
#>  0, 0
#>  0
#>  1

In general, the raw list of contexts is not directly useful and one should use the node manipulation functions to leverage it (see below). A simple approach consists in asking to contexts() a data.frame output that will contain additional information about the contexts. This is done implicitly when additional parameters are given to contexts(). In the simple case of ctx_tree, setting the frequency parameter to "total" or "detailed" gives access to the distribution of \(x_t\) for all the \(t\) at which a context appears.

contexts(x_ctx, frequency = "total")

context	freq
0, 0	2
0	3
1	2

With frequency = "total", we obtain a data frame with a column freq that contains the number of occurrences of each context.

contexts(x_ctx, frequency = "detailed")

context	freq	0	1
0, 0	2	1	1
0	3	2	1
1	2	0	2

With frequency = "detailed", we obtain in addition a column for each value in the state space \(S\) which contains the distribution of \(x_t\) for the occurrences of each context. For instance in the table above, the context \((0, 0)\) appears twice in \(x\) and is followed once by \(0\) and once by \(1\).

Direct manipulation of nodes

Another way to extract information from a context tree, especially large ones, it to operate at the node level, using the find_sequence() function (or the contexts() function). For instance exploring the 4607 contexts obtained above with min_size=1 is not convenient, but we may be interested by e.g. the sequence c("a", "a", "a"). We look for a corresponding node in the tree with

node_aaa <- find_sequence(y_ctx_min_1, c("a", "a", "a"))
node_aaa
#> Sequence [T]: a, a, a 
#>  followed by a (3), b (2), c (2)

As the result is not NULL, we know that the sequence appears in the original time series. It is not a context, as shown by

is_context(node_aaa)
#> [1] FALSE

In this particular case, it is likely to be caused by longer contexts for which c("a", "a", "a") is a suffix. This can be verified by looking at the children of node_aaa:

children(node_aaa)
#> $a
#> Context [T]: a, a, a, a 
#>  followed by a (0), b (1), c (2)
#> 
#> $b
#> Context [T]: b, a, a, a 
#>  followed by a (1), b (0), c (0)
#> 
#> $c
#> Context [T]: c, a, a, a 
#>  followed by a (2), b (1), c (0)

This shows that we have indeed three contexts that all end by c("a", "a", "a").

Nodes carry information about the sequences or contexts they represent. The number of occurrences of a sequence in the original time series is obtained with the counts() function as follows:

counts(node_aaa, frequency = "total")
#> [1] 7

Notice that as with contexts() those occurrences cover only positions where the sequence is followed by at least one value. The distribution of those values is given by another call to counts():

counts(node_aaa, frequency = "detailed")

total	a	b	c
7	3	2	2

If sequence positions were saved during the construction of the context tree, the ctx_node can report them using the positions() function:

positions(node_aaa)
#> [1] 15 25 26 73 74 83 84

See the documentation of the function for the definition of a position.