# Light Cone Reconstruction of States - Predictive State Estimation From Spatio-Temporal Data

### Description

A package for predictive state estimation from
spatio-temporal data. The main function is
`mixed_LICORS`

, which implements an EM
algorithm for predictive state recovery (see References).

This is an early release: some function names and arguments might/will (slightly) change in the future, so regularly check with new package updates.

### Details on Methodology - Predictive State Model for Spatio-temporal Processes

*For details and additional references please
consult Goerg and Shalizi (2012, 2013).*

Let *\mathcal{D} = \lbrace X(\mathbf{r}, t) \mid
\mathbf{r} \in \mathbf{S}, t = 1, …, T \rbrace =
(X_1, …, X_{\tilde{N}})* be a sample from a
spatio-temporal process, observed over an
*N*-dimensional spatial grid *\mathbf{S}* and for
*T* time steps. We want to find a model that is
optimal for forecasting a new *X(\mathbf{s}, u)*
given the data *\mathcal{D}*. To do this we need to
know

* P(X(\mathbf{s}, u) \mid \mathcal{D}) *

In general this is too complicated/time-intensive since
*\mathcal{D}* is very high-dimensional. But we know
that in any physical system, information can only
propagate at a finite speed, and thus we can restrict the
search for optimal predictors to a subset
*\ell^{-}(\mathbf{r}, t) \subset \mathcal{D}*; this
is the **past light cone (PLC)** at
*(\mathbf{r}, t)*.

There exists a mapping *ε: \ell^{-}
\rightarrow \mathcal{S}*, where *\mathcal{S} =
\lbrace s_1, …, s_K \rbrace* is the predictive state
space. This mapping is such that

* P(X_i \mid
\ell^{-}_i) = P(X_i \mid s_j), *

where *s_j =
ε(\ell^{-}_i)* is the predictive state of PLC
*i*. Furthermore, the future is independent of the
past given the predictive state:

* P(X_i \mid
\ell^{-}_i, s_j) = P(X_i \mid s_j) . *

The likelihood of the joint process factorizes as a product of predictive conditional distributions

*
P(X_1, …, X_N ) \propto ∏_{i=1}^{N} P(X_i \mid
\ell^{-}_i) = ∏_{i=1}^{N} P(X_i \mid
ε(\ell^{-}_i)). *

Since *s_j* is unknown this can be seen as the
complete data likelihood of a nonparametric finite
mixture model over predictive states:

* P(X_1,
…, X_N ) \propto ∏_{i=1}^{N} ∑_{j=1}^{K}
\mathbf{1}(ε(\ell^{-}_i) = s_j) \times P(X_i \mid
s_j). *

This predictive state model is a provably optimal finite
mixture model, where the “parameter” *ε* is
chosen to provide optimal forecasts.

The LICORS R package implements methods to estimate
this optimal mapping *ε*.

### Acronyms and common function arguments

The R package uses a lot of acronyms and terminology from the References, which are provided here for the sake of clarity/easier function navigation:

- LCs
light cones

- PLC
past light cone; notation:

*\ell^{-}*- FLC
future light cone; notation:

*\ell^{+}*- LICORS
LIght COne Reconstruction of States

Many functions use these acryonyms as part of their name. Function arguments that repeat over and over again are:

`weight.matrix`

an

*N \times K*matrix, where*N*are the samples and*K*are the states. That is, each row contains a vector of length*K*that adds up to one (the mixture weights).`states`

a vector of length

*N*with entry*i*being the label*k = 1, …, K*of PLC*i*

### Author(s)

Georg M. Goerg gmg@stat.cmu.edu

### References

Goerg and Shalizi (2013), JMLR W\&CP 31:289-297. Also available at arxiv.org/abs/1211.3760.

Goerg and Shalizi (2012). Available at arxiv.org/abs/1206.2398.

### See Also

The main function in this package:
`mixed_LICORS`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
## Not run:
# setup the light cone geometry
LC_geom <- setup_LC_geometry(speed = 1, horizon = list(PLC = 2, FLC = 0),
shape = "cone")
# load the field
data(contCA00)
# get LC configurations from field
contCA_LCs <- data2LCs(contCA00$observed, LC.coordinates = LC_geom$coordinates)
# run mixed LICORS
mod <- mixed_LICORS(contCA_LCs, num.states_start = 10, initialization = "KmeansPLC",
max_iter = 20)
plot(mod)
## End(Not run)
``` |