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- STATE-BASED MOBILITY MODEL (SMM)

By jaguar1637 2687 days ago Comments (16)

A mobility model, in the context of location management, understands the daily

movements of a user , and describes this understanding. Thus, various mobility models have been developed for mobile computing environments.

Mobility modeling in MOD is difficult because the higher location resolution compare to PCS. Moreover, a

matter of concern in MOD is not a logical/symbolic location, like cell-id, but the very

physical/geographical location of a moving object obtained by a location-sensing device,

such as GPS. As noted above, it is necessary to consider a complex movement containing

both random and linear movement patterns.

Basic Definitions

Man propose the state-based mobility model (SMM) that understands a complex mobility pattern as a set of simple movement components using a finite state Markov chain based on the classification scheme

Definition 1 A movement state si is a 3-tuple (vmin, vmax , φ), where vmin and vmax

are the minimum and maximum speeds of a moving object, respectively. φ is a function of

movement and is either probabilistic or non-probabilistic.

S is a finite set of movement states (called the state space).

Definition 2 The state-based mobility model (SMM) describes user mobility patterns

using a finite state Markov Chain {staten}, where staten denotes the movement state at

step n, staten ∈ S. Furthermore, the chain can be described completely by its transition probability as

pij ≡ Pr{staten+1 = j | staten = i} for all i, j ∈ S. (1)

These probabilities can be grouped together into a transition matrix as follows: P ≡ (pij )i,j∈S. (2)

An important question is why such a general mobility model is not as popular as the

restrictive models so abundant in the literature. The most important reason is that the

generalized model has nothing to be assumed to start the analysis.

In the SMM model, we also assume that a moving object has a tendency to remain in the

same state rather than to switch states.

This is generally called temporal locality.

The self-transition probability vector (STPV) is obtained by taking all of the

self-transition probabilities, which is equivalent to an 1 × |S| matrix.

## Comments

Determining the Transition Matrix

One of the most important tasks in the SMM model is to determine the transition

matrix P. The matrix can be determined using the user profile, the spatio-temporal data

mining process, or in an ad hoc manner. The simplest way to determine the transition

matrix P is to set pij S = 1 | | for all i, j ∈ S.

A more reasonable solution is to use statistical techniques to infer the values of the transition probabilities empirically based on past data .

For example, suppose the optimal state for each time unit is a Markov chain having state space S0. Furthermore, assuming that the optimal state for 36 time units has been PPPLLLLRRLPLLLRRRPPPRLLRLLLLRPPRRRRP.

Many tracking problems lie outside of the traditional situation of linear (or linearizable) measurements and dynamics addressed by the Kalman filter and its variants.

This is especially true in applications where target measurements are highly ambiguous and visibility is affected by unpredictable phenomena such as intermittent interference and low signal-to-noise ratios. Bayesian tracking provides the general solution to this more general class of problems.

Conceptually, Bayesian tracking is straightforward: given the target measurements, apply Bayes' rule to compute the probability density of the target location at any given time, all the while assuming a target motion model. Bayesian trackers are computationally expensive; there are two basic approaches to their implementation: sequential Monte Carlo methods such as particle filtering, and deterministic methods that compute the target density directly.

A new approach to the direct method that is theoretically and computationally novel in several ways.

First, recent results from the theory of adaptive moving meshes are modified and applied, an approach that is distinctly different from previously published direct methods that use fixed meshes to solve the Fokker-Planck equation.

Second, a straight-line motion model based on a Markov jump process for the velocity is assumed. Straight-line motion punctuated by jumps in target velocity may be a more suitable assumption for some target dynamics than the traditional random walk assumed in the Kalman filter and many Bayesian trackers.

The resulting linear partial differential equation that describes the target position density is relatively easy to solve numerically, especially compared to the Fokker- Planck equation that results from the random walk motion assumption.

The proposed Bayesian tracking algorithm is a promising alternative to competing methods. It is also shown that, like particle filters, the adaptive mesh approach necessarily suffers f- - rom the curse of dimensionality. Simulation results are shown using the example of bistatic radar.

How to compute a matrix position (based here on 36 previous bars)

The key in this way of thinking is how to compute the matrix transition. ( I was looking for an anwser since a long time ago, and finally I got it !)

I take this example for helping you =>

Counting the number of transitions Nij 2 from state i to state j gives

UPDownRanging (sleep)UP522Down194Ranging (Sleep)336¨Pij =UPDownRanging (sleep)UP5/(5+2+2)2/(5+2+2)2/(5+2+2)Down1/(1+9+4)3/(1+9+4)4/(/(1+9+4)Ranging (Sleep)3/(3+3+6)3/(3+3+6)6/(3+3+6)UPDownRanging (sleep)UP0.55560.22220.2222Down0.07140.64290.2857Ranging (Sleep)0.25.025.025At first glance, I can seeDown of Down = 0.6429 (the highest)Up of Up = 0.5556R of R = 0.25Solving for pi = p * TPi U = 0.257Pi D = 0.4Pi R= 0.343=> The result implies to take a Short positionI forgot to send the signals provided by the 36 last bars, regarding the previous explanation

UU DDDD RR DU DDD RRR UUU R DD R DDDD R UU RRRRR USo, the operation orders (Up, Down , and Range (nope)) can be put inside a mtraix

UPDownRanging (sleep)UP522Down194Ranging (Sleep)336Counting the number of transitions Nij 2 from state i to state j gives the matrix

I think this is a transition matrix ?

In mathematics, a

stochastic matrixis also named asprobability matrix,transition matrix,substitution matrix, orMarkov matrix !http://en.wikipedia.org/wiki/Stochastic_matrix

And much better : A stochastic matrix describes a Markov chain over a finite state space

S.This is connected to transition probabilities => the http://en.wikipedia.org/wiki/Transition_probabilities

Also, check this http://en.wikipedia.org/wiki/State-transition_matrix

For people interested into Markov Chains, and matrix calculations, there is this link

http://academic.uprm.edu/wrolke/esma6600/mark1.htm

the mQ4/R code could be this one,

The following R/mq4 code will do:

make.Markov_Chains = function (x)

{

N=max(x)

res=array(dim=c(N,N))

for(i in 1:max(x))

{

index=x[1:(length(x)-1)]==i

S=sum(index)

for(j in 1:N)

res[i,j]=sum( x[2:length(x)][index]==j)/S

}

return(res)

}

I check this formula

http://latex.codecogs.com/gif.latex?P1v(i,j)=\frac{{\sum_{u=1}^{^{S_{u}-1}}\sum_{v=1}^{S_{v}-1}\delta%20({F_{h}(u,v)=i,{F_{h}(u,v+1)=j}})}}{\sum_{u=1}^{{S_{u}-1}}\sum_{v=1}^{S_{v}-1}\delta%20(F_{h}(u,v)=i)}

So you really want to build something based on Markov transitional matrix? I begin slowly to understand it.

Maybe it would be a good idea to try to make something extremely simple first.

yes, this is exactly what I want, they used a transition matrix to fetch the next movment

Better, this transition Markov matrix will be reused in NN eas or w/ perceptrons

What is a markov chain ?

Suppose X1, X2, ... , Xn is a sequence of integer-valued independent random variables. Define

Y1=X1,

Yn + cY(n-1) = Xn,

where c is a positive integer. {Yn, n>=1} is a Markov Chain

First, the raw data are a time series of cathegories corresponding to the rows (or columns) in the transition matrix. For each cathegory, we fill in the corresponding row with the empirical distribution of the successor. Up Down Range (means nope)

I just asked a mathematics teacher to privide me the code in C

Nice!