Although it is not necessary to understand recurrence analysis, Takens' theorem, or the rest of nonlinear theory in order to use VRA, it is instructive to see the mathematics. First, the definition of a time series. A time series:
(x(t1), ... , x(tn))
is generated by a dynamical system on an d-dimensional manifold M:
s(t) = pt(s(0))
The original d-dimensional states of the system cannot be observed directly. Instead, the observations consist of the (possibly) noisy values x(t) of a 1-dimensional measurement function h which are related to the original states by:
x(t) = h(s(t)) + n(t)
where n(t) represents a noise process which corrupts the observations (for a noise-free observation n(t) = 0). To model a system described by the equations above it is necessary to reconstruct the original state-space (or its equivalent), namely to find the structure of the manifold M. Two spaces are considered to be topologically equivalent if there exists a continuous mapping with a continuous inverse between them. In this case it is sufficient that the equivalent structure is a part of a larger one. This leads to the concept of embedding, defined as follows. A function: f : X Y is an embedding if it is a continuous mapping with a continuous inverse f -1 : f(X) X from its range to its domain. Any d-dimensional manifold can be embedded into the space R2d+1. The theorem can be extended to show that with a proper definition of almost all for an infinite-dimensional function space, almost all smooth mappings from a given d-dimensional manifold to R2d+1 are embeddings. Having only a single time series, how does one get 2d+1 different coordinates? This can usually be solved by introducing delay coordinates.
Let p be a flow on a manifold M with t > 0, and h : M R be a smooth measurement function. The delay coordinate map F(h,p,t) : M Rn (with embedding delay t) is defined by:
s (h(s), h(p -t(s)), h(p -2t(s)), ... , h(p -(n-1)t(s)))
Takens proved in 1980 that such mappings can indeed be used to reconstruct the state-space of the original dynamic system. This results is known as Takens' embedding theorem.
(Takens): Let M be a compact manifold of dimension d. For triples (f, h, t) where f is a vector field on M with a flow p, and h : M R a measurement function, and the embedding delay t > 0, it is a generic property that the delay coordinate map F(h, p, t) : M R2d+1 is an embedding.
Takens' theorem states that in the general case the dynamics of the system recovered by delay coordinate embedding are the same as the dynamics of the original system.
In recurrence analysis, a one-dimensional time series is expanded into a higher-dimensional phase space, in which the dynamic of the underlying generator takes place (phase space contains all the possible states of a system). This is done using a technique called delayed coordinate embedding, which recreates a phase space portrait of the dynamical system under study from a single (scalar) time series. To expand a one-dimensional signal into an M-dimensional phase space, one substitutes each observation in the original signal:
X(t)
with vector:
yi = {xi, xi-d, xi-2d, ... , xi-(m-1)d}
where i is the time index, m is the embedding dimension, and d is the time delay. As a result, we have a series of vectors:
Y = {y1, y2, y3, ... , yN-(m-1)d}
where N is the length of the original series. The idea of such reconstruction is to capture the original system states at each time we have an observation of that system output. Each unknown state St at time t is approximated by a vector of delayed coordinates:
Yt = { xt, xt - d, xt - 2d, ... , xt - (m-1)d}
Two of the most important parameters in a recurrence model are dimension and delay. They work together as a mathematical 'comb'. Dimension is the number of teeth on the comb, and delay the separation between them. The comb is moved along the data, one value at a time. Data are read off from the 'teeth', and form a coordinate in state space. It is this state space reconstruction of the initial phase space data from which predictions are made:
Taken's embedding theorem states that if the underlying system has an attractor of dimension d, reconstruction using delay coordinates can be unfolded in dimension e < 2d+1 (unfolding means that a given trajectory does not cross itself).
The method of false nearest neighbours (FNN) can be used to estimate e. False neighbours are points that are close together because they are projected into a lower dimension than points in the the original system. These false neighbours diverge in successively higher dimensions. FNN looks for a reduction in the number of false neighbours as dimension increases. The embedding dimension e is found when the number of false neighbours reaches its minimum. Once this occurs the value d can then be found.
Regarding lottery numbers, how are the combinations calculated? For a set of n objects how many subsets (combinations) of size r can be taken from the set n? The formula is:
where n! is the factorial of n and r! the factorial of r (e.g. 6! = 1 x 2 x 3 x 4 x 5 x 6 = 720). For a 6-from-49 lottery there are:
13,983,816 distinct 6-number combinations. Therefore there is a 1 in 13,983,816 chance of choosing the winning 6 numbers. Compare this to the odds of being struck by a meteorite, 1 in 10,000,000. Other combinations are summarised below:
Y is an embedding if it is a continuous mapping with a continuous inverse f -1 : f(X) 
