‘Which road do I take?’ Alice asked. ‘Where do you want to go?’ responded the Cheshire Cat. ‘I don't know,’ Alice answered. ‘Then,’ said the Cat, ‘it doesn't matter’ (Lewis Carroll, Alice's Adventures in Wonderland, 1865).
Its name, Projection Pursuit, highlights a key aspect of the method: the search for projections worth pursuing. Projection Pursuit can be regarded as embracing the classical multivariate methods while at the same time striving to find something ‘interesting’. This invites the question of what we call interesting. For scores in mathematics, language and literature, and comprehensive tests that psychologists, for example, use to find a person's hidden indicators of intelligence, we could attempt to find as many indicators as possible, or one could try to find the most interesting or most informative indicator. In Independent Component Analysis, one attempts to find all indicators, whereas Projection Pursuit typically searches for the most interesting one.
In Principal Component Analysis, the directions or projections of interest are those which capture the variability in the data. The stress and strain criteria in Multidimensional Scaling variously broaden this set of directions. Of a different nature are the directions of interest in Canonical Correlation Analysis: they focus on the strength of the correlation between different parts of the data. Projection Pursuit covers a rich set of directions and includes those of the classical methods. The directions of interest in Principal Component Analysis, the eigen vectors of the covariance matrix, are obtained by solving linear algebraic equations.