The main advantage of the structural approach is that in addition to classification, it provides a description of the pattern. It specifies how the given pattern can be constructed from primitives. This paradigm has been used in the situations where the patterns have a definite structure which can be captured in terms of a set of rules Here there is a correlation between the structure of the pattern and the syntax of a language. The patterns belonging to a category are viewed as sentences belonging to a language. A large set of complex patterns can be described by using a small number of primitives and grammatical rules. The grammar for each pattern class must be inferred from training samples.
Syntactic patterns have a great deal of intuitive appeal. However, implementation of this approach leads to many difficulties, such as segmenting noisy patterns to detect the primitives, and inferring grammars.
We talked about distributions. Lets look into this a little more closely. We mentioned that Gaussian distribution leads to optimal rules. This is because the mathematical properties of this distribution is well understood i.e they are justified by a central limit theorem. Second, many applications find it easier to summarize data in the form of a mean vector and a covariance matrix. If the density
function is unimodal, then this is simple and easy to apply. If X is a vector with d features, the density function f(x1, x2, ...., xd) is described in terms of the d * d covariance matrix, its determinant, mu- the d-vector of expected values and x which is a d vector containing variables x1, x2, ... xd. The mean vector mu and the covariance matrix define a Gaussian distribution. The parameters for a multivariate Gaussian density are estimated by sample co-variances and sample means to fit a Gaussian distribution to a set of patterns The mean is found with some objective or expectation function i.e mu-i = E(Xi). The population covariance between Xi and Xj is sigma-ij = E[(Xi-mui)(xj - muj)] and this works for diagonal elements. This can also be written as sigma-ij = ro-ij.sigma-i.sigma-j where
ro-ij is the correlation co-efficient between sigma-i and sigma-j. If ro-ij turns out to be zero, then Xi and Xj are both linearly and statistically independent. Features are pictured as random variables and each pattern is a realization of these random variables.
Syntactic patterns have a great deal of intuitive appeal. However, implementation of this approach leads to many difficulties, such as segmenting noisy patterns to detect the primitives, and inferring grammars.
We talked about distributions. Lets look into this a little more closely. We mentioned that Gaussian distribution leads to optimal rules. This is because the mathematical properties of this distribution is well understood i.e they are justified by a central limit theorem. Second, many applications find it easier to summarize data in the form of a mean vector and a covariance matrix. If the density
function is unimodal, then this is simple and easy to apply. If X is a vector with d features, the density function f(x1, x2, ...., xd) is described in terms of the d * d covariance matrix, its determinant, mu- the d-vector of expected values and x which is a d vector containing variables x1, x2, ... xd. The mean vector mu and the covariance matrix define a Gaussian distribution. The parameters for a multivariate Gaussian density are estimated by sample co-variances and sample means to fit a Gaussian distribution to a set of patterns The mean is found with some objective or expectation function i.e mu-i = E(Xi). The population covariance between Xi and Xj is sigma-ij = E[(Xi-mui)(xj - muj)] and this works for diagonal elements. This can also be written as sigma-ij = ro-ij.sigma-i.sigma-j where
ro-ij is the correlation co-efficient between sigma-i and sigma-j. If ro-ij turns out to be zero, then Xi and Xj are both linearly and statistically independent. Features are pictured as random variables and each pattern is a realization of these random variables.
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