Michael J. Johnson
Computer Software Apps Healthcare CS205
Grantham University
M8 Written Assignment: Insurance Claims Follow-up and Dispute
22 February 2013
1.
Explain why patients with dual insurance coverage may have a longer time frame for waiting
unt
algorithm towards working more at the long lengthscales (bigger, coarser blocks). Well, perhaps you can see
why the complexity of this algorithm has put people off
using it. The proof that the algorithm satisfies detailed
balance is quite involved, and, s
Metropolis algorithm has a slight edge in speed in this
regime because of its extreme simplicity. So, if the
Metropolis algorithm beats the Wolff algorithm (albeit
only 98 Chapter 4: Other algorithms for the Ising model
by a slim margin) at both high and
SwendsenWang algorithm. (A comparison with the
Wolff algorithm yields similar resultsthe performance
of the Wolff and SwendsenWang algorithms is
comparable.) So, how does the invaded cluster algorithm
work? Basically, it is just a variation of the Swendse
interaction energy is changed by a factor of two J 1 2 J
from Equation (3.1).) For higher values of q the Potts
model behaves similarly in some ways to the Ising model.
For J > 0 (the ferromagnetic case) it has q equivalent
ground states in which all the
making the clusters formed larger and larger. This gives
us a way of controlling the sizes of the clusters formed in
our algorithm, all the way up to clusters which
encompass (almost) every spin on the lattice at every
move. 2. If we choose E0 to be less
can use Equation (4.16) to calculate z without ever having
to measure the mean cluster size in the algorithm, which
eliminates one source of error in the measurement,
thereby making the value of z more accurate.6 The first
step in demonstrating Equation (
on average every two steps (rather than every step as in
the Wolff algorithmsee Figure 4.4), but otherwise the
two will behave almost identically. Thus, as with the
Wolff algorithm, we can expect 106 Chapter 4: Other
algorithms for the Ising model the per
temperature of the system is lower than the critical
temperature, the algorithm performs Monte Carlo steps
with T > Tc, and vice versa. Thus, it seems plausible that
the algorithm would drive itself towards the critical point
quicker than simply performin
a link between them with probability Padd = 1 e
2J . When we are done, we will have divided the whole
lattice into many different clusters of spins, as shown in
Figure 4.7, each of which will be a correct Wolff cluster
(since we have used the correct Wolf
have been frozen, deleted or marked as active. In this
case we leave those bonds as they are, and only go to
work on the others which have not yet been considered.
Notice also that if we come to a spin and it has already
been linked (frozen) to another sp
condition of detailed balance. The proof of this fact is
exactly the same as it was for the Wolff algorithm. If the
number of links broken and made in performing a move
are m and n respectively (and the reverse for the reverse
move), then the energy chang
large. To see this, let us consider an extreme case: the q =
100 Potts model on a square lattice in two dimensions. At
high temperatures, the acceptance ratio (4.39) is always
either 1 or close to it because is small, so the algorithm
is reasonably effici
methods are very general and can be applied to all sorts
of models, such as the glassy spin models that we will
study in Chapter 6. Here we will just consider their
application to the ordinary Ising model. Niedermayer
pointed out that it is not necessary
come as a great surprise that this algorithm also satisfies
the condition of detailed balance, though just to be
thorough lets prove it. The probability of making the
transition from a state in which sk = n to one in which sk
= n is just pn , and the prob
desirable states of the spins. The algorithm goes like this.
First we choose a spin k at random from the lattice. Then,
regardless of its current value, we choose a new value sk
for the spin in proportion to the Boltzmann weights for
the different valuest
than 1, which may be very many if q is large, and this
gives rise to many more low-lying excitation states than in
the Ising case. Monte Carlo simulation of Potts models is
quite similar to the simulation of the Ising model. The
simplest thing one can do
simpler than a direct measurement, it is also superior,
giving, as it turns out, smaller statistical errors. For this
reason the expression (4.24) is sometimes referred to as
an improved estimator for the susceptibility (Sweeny
1983, Wolff 1989). 4.3 Prop
Wolff algorithm really is. 100 Chapter 4: Other algorithms
for the Ising model 4.3.2 The dynamic exponent and the
susceptibility In most studies of the Wolff algorithm for
the Ising model one does not actually make use of
Equation (4.14) to calculate . If
the normal SwendsenWang algorithm. Then the whole
procedure is repeated from step 2 again. So whats the
point here? Why does the algorithm work? Well, consider
what happens if the system is below the critical
temperature. In that case, the spins will be m
questionhow we know when we have a percolating
cluster does not have such an elegant solution. Machta
et al. (1995), who invented the algorithm, suggest two
ways of resolving the problem. One is to measure the
dimensions of each of the clusters along the
in the relative complexity of the two algorithms. It is this
impressive performance on the part of the Wolff
algorithm which makes it a worthwhile algorithm to use if
we want to study the behaviour of the model close to Tc.
In Figure 4.6 we have plotted o
given by the average of the probability of a cluster being
chosen times the size of that cluster: hni = DX i pini E = 1
N DX i n 2 i E = Nhm2 i. (4.23) Now if we employ Equation
(1.36), and recall that hmi = 0 for T Tc, we get = hni
(4.24) as promised. Th
cluster, the time taken to do one Monte Carlo step should
scale with cluster size. 4.3 Properties of the Wolff
algorithm 99 10 100 lattice size L 1 2 3 correlation time
Figure 4.6 The correlation time for the 2D Ising model
simulated using the Wolff algo
the clusters first and then choose the seed spin
afterwards, rather than the other way around. This would
not be a very efficient way of implementing the Wolff
algorithm, since it requires us to create clusters all over
the lattice, almost all of which ne
configurations: they can be linked as in the Swendsen
Wang case, so that they must flip together, they can have
no connection between them at all, so that they can flip
however they like, or they can have a normal Ising
interaction between them of strengt
algorithm in this region, it clearly would not be fair to
measure it for both algorithms in terms of number of
Monte Carlo steps (or steps per lattice site). A single
Monte Carlo step in the Wolff algorithm is a very
complicated procedure, flipping maybe
the speed of our simulation. 4.4.4 The invaded cluster
algorithm Finally, in our round-up of Monte Carlo
algorithms for the Ising model, we come to an unusual
algorithm proposed by Jonathan Machta and coworkers
called the invaded cluster algorithm (Machta
temperature is beforehand in order to use the algorithm.
Starting with a system at any temperature (for example T
= 0 at which all the spins are pointing in the same
direction) the algorithm will adjust its simulation
temperature until it finds the critic
discussed in Chapter 14, the SwendsenWang algorithm
can be implemented more efficiently on a parallel
computer than can the Wolff algorithm. 4.4 Further
algorithms for the Ising model 107 dimension d
Metropolis Wolff SwendsenWang 2 2.167 0.001 0.25
0.01
measured in steps (i.e., clusters flipped) in the Wolff
algorithm. The conventional choice for the constant of
proportionality is 1. This makes the correlation times for
the Wolff and Metropolis algorithms equal in the limits of
low and high temperature,
the Wolff algorithm is a better choice than the
Metropolis algorithm; although it is more complex than
the Metropolis algorithm, the Wolff algorithm has a very
small dynamic exponent, which means that the time
taken to perform a simulation scales roughly
models Niedermayer considered it needs to be defined
elsewhere as well. Clearly, if for the Ising model we make
Padd(J) = 1e 2J and Padd(J) = 0, then we recover the
Wolff algorithm or the SwendsenWang algorithm,
depending on whether we flip only a single
is good because it minimizes the correlation between the
direction of a cluster before and after a move, the new
direction being chosen completely at random, regardless
of the old one. 3. The algorithm updates the entire lattice
on each move. In measuring