Clasificación de las Competencias Gerenciales

The Beal model is a variation of the Linear Dynamical System or Kalman filter,

which in turn is a subclass of Dynamic Bayesian Networks (DBN) used to model

times series data. DBNs use probabilities given a sequence of observed variables to

calculate the relationships between them. For the purposes of explanation, in this

example the observed variables can represent a sequence of expression

measurements for a single gene at a set of given time points on a microarray. At each

time point there are factors that can affect the measured expression levels of the gene

on the microarray such as poor RNA extraction or low mRNA levels. These factors

cannot be quantified directly and are therefore “hidden” from the user and can be

referred to as the hidden state. At each time point the hidden state variables impact

upon the observed expression values, therefore when modelling regulatory

relationships between genes, the hidden variables must be taken into account. The

Kalman filter captures this process of change in the hidden state from time point to

time point, which in turn impacts upon the observed expression values at each time

point (Kalman 1960). The basic Kalman filter model is as follows:

,

w. ~Gaussian (0, Q)

v. ~Gaussian (0, R)

Where x represents a k-vector of hidden state variables that cannot be observed directly but impact upon y, a p-vector of observed variables that can be measured. A

represents a (k x k) transition matrix, which captures the process of change in the

state of hidden variables over time and C is a (p x k) observation matrix that captures

the change in observed variables over time. w and v are variables that represent the

state and observed variable noise respectively and Q and R represent the covariance

matrices associated with them. The noise represents imperfections in the data (in this

case a microarray) caused by a random set of variables such as temperature,

vibrations or even dust specks on the laser. These noise variables are thought to

occur randomly irrespective of any time index hence why w and v are followed by a “.” to emphasise their independence from any time step. It is important to take this noise into account when estimating the hidden state and observed variables as they

invariably have an impact. The state noise is also considered Gaussian; that is to say

normally distributed.Because the model captures the process of change in the hidden

state (and its subsequent impact on the observed values) over a single time step it is

defined as being a 1st order Markov model. This is because it has a memory of 1;

therefore the probabilities of the possible values of the next hidden state depend on

the values of the previous state.

In practical terms the model answers the following question:

“Given a set of observed variables and parameters what can be said of the hidden

state at time point t? “ (Roweis and Ghahramani 1999).

In the context of this chapter, determining the regulatory relationships between genes

over time is the main aim of this study, therefore it is important that the effect of both

the hidden and observed gene expression levels at each time point is not only

captured but that its impact is also incorporated into the values at the next time point.

The Beal model incorporates the principles of the Kalman filter whilst extending it to

include matrix B, which captures the influence of the observed variables from a

previous time point on the current hidden state and matrix D, which captures the

influence of the observed variables from a previous time point on the current

observed variables. By including these matrices Beal is able to model the influence

of previous observed measurements back on to the current hidden state and observed

measurements. The Beal model is thus defined below:

,

w. ~Gaussian (0, Q)

,

v. ~Gaussian (0, R)

Where x represents a set of hidden state variables, y represents a set of observed

variables. Matrix A represents a transition matrix capturing the change in the hidden

state variables over time t. Matrix B represents the effect of observed values from a

previous time point on the current hidden state. C symbolises an observation matrix

capturing the change in observed variables over time and D, a matrix containing the

effect of the previous observed values on the current observed values. w and v

symbolise the state and observed variable noise respectively. By incorporating the

influence of observed variables from a previous time point into the values of the

current time point, the model acts as a feedback loop whereby the outputted observed

expression levels and hidden state variables at time t-1 are used as input values to

help determine the hidden state and explain the gene expression values at t. Whilst

the Beal model is still a 1st order Markov model the inclusion of matrices B and D

has allowed the capture of gene relationships which are higher than 1st order.

Until now the example provided centred upon modelling the expression values of a

single gene over a given set of time points, however the model is designed to

estimate the influences of a set of genes on one another over a given set of time

points. In this case all matrices (A, B, C and D) would contain values for a set of

genes at time t. The model can then be used to characterise both direct gene-gene

regulatory relationships and those that occur through the hidden state. For example,

to observe the direct effect of gene a at t-1 on gene b at t, one must look at matrix

element [D]ba. Thus to capture all the effects of the hidden and observed states of

one gene on another over a single time step, the matrices must be combined. A

function of the model to do this is shown below:

( )

Where yt represents the observed expression level at time t and CB + D represents

the influences of the hidden state on gene expression values, the effect of gene levels

from a previous time point on the current hidden state and the effect of gene levels

Standard score or Z score values from this matrix (CBDZ) will be used to determine

which genes are showing “significant” regulatory relationships. This is defined as those genes whose relationship scores differ from a default normal distribution of

zero. Values at or close to zero indicate the genes involved do not have any

regulatory influence on one another. For genes to be considered as having regulatory

influences on one another, either directly or indirectly, their CBDZ values must be at

least 1.69 standard deviations away from the mean of zero, which equates to around

a 90% confidence that the values are significantly different.