The measure also corrects for correlations and anti-correlations between components of the data . The following figure illustrates this advantage over a Euclidean distance measure for an unstandardised data set. The Mahalanobis threshold correctly excludes a point from a non-related cluster, whereas the simpler Euclidean distance does not. When the variances of the data set components are equal, and there is no correlation, the Mahalanobis measure degenerates to the Euclidean distance. .
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The Euclidean and Mahalanobis Distances Compared.
The high computational cost of calculating the covariance matrix means that other standardisation methods, discussed earlier, are used in practical clustering algorithms. .
Limitations of Linear Discriminate Functions.
1. The features may be inadequate to distinguish the different classes .
2. The features may be highly correlated .
3. The decision boundary may have to be curved .
4. There may be distinct subclasses in the data .
5. The feature space may simply be too complex .
Mahalanobis Advantages .
1. Scale invariance --it doesn't matter what units the features are measured in .
2. Determines probability of membership if population features are jointly Gaussian .
3. Can represent curved boundaries between classes .
4. Works well on a wide class of problems even when populations aren't Gaussian .
Now we check seven distance metric for validity.
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USE OF DISTANCE METRICS IN INSTANCE BASED LEARNING AND VALUE ADDED METRIC (VDM).
Instance-Based Learning (IBL) is a paradigm of learning in which algorithms typically store some or all of the n available training examples (instances) from a training set, T, during learning. Each instance has an input vector x, and an output class c. During generalization, these systems use a distance function to determine how close a new input vector y is to each stored instance, and use the nearest instance or instances to predict the output class of y (i.
