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for analysis, dimensionality reduction is required in order for any reliable model to be mined or to be of any practical use. On the other hand, data overload, because of high dimensionality, can make some data-mining algorithms non-applicable, and the only solution is again a reduction of data dimensions. For example, a typical classification task is to separate healthy patients from cancer patients, based on their gene expression “profile.” Usually fewer than 100 samples (patients’ records) are available altogether for training and testing. But the number of features in the raw data ranges from 6000 to 60,000. Some initial filtering usually brings the number of features to a few thousand; still it is a huge number and additional reduction is necessary. The three main dimensions of preprocessed data sets, usually represented in the form of flat files, are columns (features), rows (cases or samples), and values of the features.

Therefore, the three basic operations in a data-reduction process are delete a column, delete a row, and reduce the number of values in a column (smooth a feature). These operations attempt to preserve the character of the original data by deleting data that are nonessential. There are other operations that reduce dimensions, but the new data are unrecognizable when compared with the original data set, and these operations are mentioned here just briefly because they are highly application-dependent. One approach is the replacement of a set of initial features with a new composite feature. For example, if samples in a data set have two features, person height and person weight, it is possible for some applications in the medical domain to replace these two features with only one, body mass index, which is proportional to the quotient of the initial two features. Final reduction of data does not reduce the quality of results; in some applications, the results of data mining are even improved.

In performing standard data-reduction operations (deleting rows, columns, or values) as a preparation for data mining, we need to know what we gain and/or lose with these activities. The overall comparison involves the following parameters for analysis:

1. Computing Time. Simpler data, a result of the data-reduction process, can hopefully lead to a reduction in the time taken for data mining. In most cases, we cannot afford to spend too much time on the data-preprocessing phases, including a reduction of data dimensions, although the more time we spend in preparation the better the outcome.

2. Predictive/Descriptive Accuracy. This is the dominant measure for most data-mining models since it measures how well the data are summarized and generalized into the model. We generally expect that by using only relevant features, a data-mining algorithm can learn not only faster but also with higher accuracy. Irrelevant data may mislead a learning process and a final model, while redundant data may complicate the task of learning and cause unexpected data-mining results.

3. Representation of the Data-Mining Model. The simplicity of representation, obtained usually with data reduction, often implies that a model can be better understood. The simplicity of the induced model and other results depends on its representation. Therefore, if the simplicity of representation improves, a relatively small decrease in accuracy may be tolerable. The need for a balanced view between accuracy and simplicity is necessary, and dimensionality reduction is one of the mechanisms for obtaining this balance.

It would be ideal if we could achieve reduced time, improved accuracy, and simplified representation at the same time, using dimensionality reduction. More often than not, however, we gain in some and lose in others, and balance between them according to the application at hand. It is well known that no single data-reduction method can be best suited for all applications. A decision about method selection is based on available knowledge about an application (relevant data, noise data, meta-data, correlated features), and required time