MA469 R-projects for 2015/6 from R.S.MacKay
1. Hubble’s law without Friedmann’s equations
Background: Hubble’s law (that redshift z is proportional to distance D) or its
refinements are nowadays based on assuming not only a Friedmann universe
ds^2 = -dt^2 + S^2(t)|dx|^2 where |dx| is Euclidean (or possibly standard
spherical or hyperbolic) metric on R^3 (or a 3-manifold) and S(t) is some
function of t, with matter moving roughly along x=constant, but also Friedmann’s
equations https://en.wikipedia.org/wiki/Friedmann_equations, which constrain
the form of the function S to satisfy two differential equations.
But one could attempt to fit the observations of redshift z and D (“corrected
luminosity distance” to be precise) directly to the equations 1+z = S(t_r)/S(t_e)
and D = S(t_r)\int dt/S(t) (which follow for a Friedmann universe in the
Euclidean case) to determine the function S. Namely, its inverse function is given
by S^{-1}(1/1+z) = t_r - \int_0^D dD/1+z.
Project: Find suitable data sources for (z,D) and plot the resulting function S.
Take care in interpreting the data: many sources provide luminosity distance
D_L instead of corrected luminosity distance; they are related by D_L = (1+z)D.
Or they may give -5 log_10{D_L}. A good reference on the variety of distance
measures is Perlick http://www.livingreviews.org/lrr-2004-9
Extrapolate your function S to determine t* such that S(t*)=0 (the time of the bigbang).
2. Hierarchical Aggregation of Markov processes
Background: To compute the stationary distribution and mean first passage time
for a Markov process (or semi-Markov process) one can eliminate parts of the
state space successively to make smaller processes with equivalent properties,
thereby speeding up the computation. This was developed by Wales in Int Rev
Phys Chem 25 (2006) 237 and J Chem Phys 130 (2009) 2014111. I proposed an
improvement on Wales’ methods, which works by aggregating groups of states
rather than eliminating them and I believe would be more efficient.
The idea is that for a continuous-time regular jump homogeneous Markov chain
with state space S, mean waiting times T_s, transition probabilities P_st from s to
t (equivalently transition rates q_st = P_st/T_s), one can aggregate groups of
states s in A_i for any chosen disjoint subsets A_i into “super-states” which we
call A_i again, but the process becomes one on the set of remaining (directed)
edges and is no longer Markov but mean times can be preserved. It may also
result in multiple edges from a state to a super-state or vice versa or between
super-states, but further aggregation produces nothing more complicated. Thus
we take a more general starting point where the process is on a directed graph G
that can have multiple edges between nodes (can allow self-edges if desired but
wlog eliminate them), each edge e has a mean waiting time T_e (in the standard
case can say this is the waiting time at the destination node for the edge [could
do origin node if preferred]) and each edge e has a probability distribution P_ef
for the next edge f, with the successive transitions being independent. The result
of aggregating edges a in A is to update the mean waiting times on incoming
edges e to A to
T’_e = T_e + P_eA (I-P_AA)^{-1} T_A
and transition probabilities form incoming edges e to outgoing ones f to
P’_ef = P_ef + P_eA (I-P_AA)^{-1} P_Af,
where P_eA is the vector of transition probabilities P_ea for edges a in A, P_AA is
the matrix of probabilities for transitions between edges within A, T_A is the
vector of mean waiting times T_a on edges within A, and P_Af is the vector of
probabilities P_af for transition from edges a in A to f. The resulting process
(incompletely defined because I didn’t specify the distribution of waiting times,
but it is only mean times that I aim to extract) has the induced mean first passage
times and stationary distribution. Once small enough one could solve the
standard system of equations
T_eB = T_e + \sum_f P_ef T_fB
for the mean first passage time from edge e to set B of edges (B can be a
singleton).
Project: Check the above theory to make sure you understand it. Add things like
how to compute the stationary distribution and how to compute quantities for
edges that have at some stage been aggregated from the hierarchical aggregation
tree. Implement in software. This will require heuristics for which groups of
edges to aggregate. Test on some realistic examples (David Wales can probably
provide some from physical chemistry).
3. Metrics on probability distributions for probabilistic cellular automata
Background: There are many metrics on spaces of probability distributions, but
most are not suitable for probability distributions on large or infinite products,
such as for the state of a probabilistic cellular automaton. In my paper
Robustness of Markov processes on large networks, J Difference Eqns & Applns
17 (2011) 1155–67, I rejected a lot and proposed a good one, which I named
after Dobrushin. Project: The project has three related parts:
1. Decide whether Prokhorov metric is good in my sense or not (e.g.
Parthasarathy, Probability measures on metric spaces).
2. Compare Dobrushin metric with Föllmer & Horst, Stoch Process Appl 96
(2001) 99-121
3. Compare Dobrushin metric with Steif, Convergence to equilibrium and
space—time bernoullicity for spin systems in the M < ε case. Ergodic
Theory and Dynamical Systems, 11 (1991) 547-575.
4. Dynamic dependency graphs for evaluation of policy
Background: In evaluating proposed public or commercial policy it is important
to estimate the interdependency between the dynamics of the relevant agents.
For a simple class of stochastic systems (probabilistic cellular automata), which
can include time-inhomogeneity, if the interdependence is weak then the
statistical outcome is unique and varies smoothly with parameters. In contrast, if
it is strong then the outcome may depend on the initial conditions and the
realisation, and may exhibit jumps as parameters are varied. See my notes,
Dependency graphs for stochastic dynamics of complex systems.
Project: Extend the above theory to continuous-time processes (known as
interacting particle systems: many results already exist so it is more a question
of formulating and treating them in the same way as I did the discrete-time case)
and to cases where the independent updates are to pairs of units rather than
single units.