Data Mining by Mehmed Kantardzic (good book recommendations TXT) 📕

n-dimensional space increases exponentially with dimensions. For example, if a one-dimensional (1-D) sample containing n data points has a satisfactory level of density, then to achieve the same density of points in k dimensions, we need nk data points. If integers 1 to 100 are values of 1-D samples, where the domain of the dimension is [0, 100], then to obtain the same density of samples in a 5-D space we will need 1005 = 1010 different samples. This is true even for the largest real-world data sets; because of their large dimensionality, the density of samples is still relatively low and, very often, unsatisfactory for data-mining purposes.

Figure 2.2. High-dimensional data look conceptually like a porcupine.

2. A larger radius is needed to enclose a fraction of the data points in a high-dimensional space. For a given fraction of samples, it is possible to determine the edge length e of the hypercube using the formula

where p is the prespecified fraction of samples, and d is the number of dimensions. For example, if one wishes to enclose 10% of the samples (p = 0.1), the corresponding edge for a 2-D space will be e2(0.1) = 0.32, for a 3-D space e3(0.1) = 0.46, and for a 10-D space e10(0.1) = 0.80. Graphical interpretation of these edges is given in Figure 2.3.

Figure 2.3. Regions enclose 10% of the samples for one-, two-, and three-dimensional spaces.

This shows that a very large neighborhood is required to capture even a small portion of the data in a high-dimensional space.

3. Almost every point is closer to an edge than to another sample point in a high-dimensional space. For a sample of size n, the expected distance D between data points in a d-dimensional space is

For example, for a 2-D space with 10,000 points the expected distance is D(2,10,000) = 0.005 and for a 10-D space with the same number of sample points D(10,10,000) = 0.4. Keep in mind that the maximum distance from any point to the edge occurs at the center of the distribution, and it is 0.5 for normalized values of all dimensions.

4. Almost every point is an outlier. As the dimension of the input space increases, the distance between the prediction point and the center of the classified points increases. For example, when d = 10, the expected value of the prediction point is 3.1 standard deviations away from the center of the data belonging to one class. When d = 20, the distance is 4.4 standard deviations. From this standpoint, the prediction of every new point looks like an outlier of the initially classified data. This is illustrated conceptually in Figure 2.2, where predicted points are mostly in the edges of the porcupine, far from the central part.

These rules of the “curse of dimensionality” most often have serious consequences when dealing with a finite number of samples in a high-dimensional space. From properties (1) and (2) we see the difficulty of making local estimates for high-dimensional samples; we need more and more samples to establish the required data density for performing planned mining activities. Properties (3) and (4) indicate the difficulty of predicting a response at a given point, since any new point will on average be closer to an edge than to the training examples in the central part.

One interesting experiment, performed recently by a group of students, shows the importance of understanding curse-of-dimensionality concepts for data-mining tasks. They generated randomly 500 points for different n-dimensional spaces. The number of dimensions was between 2 and 50. Then, they measured in each space all distances between any pair of points and calculated the parameter P:

where n is the number of dimensions, and MAX-DIST and MIN-DIST are maximum and minimum distances in the given space, respectively. The results are presented in Figure 2.4. What is interesting from the graph is that as the number of dimensions increases, the parameter Pn approaches the value of 0. That means maximum and minimum distances are becoming very close in these spaces; in other words, there are no differences in distances between any two points in these large-dimensional spaces. It is an experimental confirmation that traditional definitions of density and distance between points, which are critical for many data-mining tasks, change their meaning. When dimensionality of a data set increases, data become increasingly sparse, with mostly outliers in the space that they occupy. Therefore, we have to revisit and reevaluate traditional concepts from statistics: distance, similarity, data distribution, mean, standard deviation, and so on.

Figure 2.4. With large number of dimensions, the concept of a distance changes the meaning.

2.2 CHARACTERISTICS OF RAW DATA

All raw data sets initially prepared for data mining are often large; many are related to human beings and have the potential for being messy. A priori, one should expect to find missing values, distortions, misrecording, inadequate sampling, and so on in these initial data sets. Raw data that do not appear to show any of these problems should immediately arouse suspicion. The only real reason for the high quality of data could be that the presented data have been cleaned up and preprocessed before the analyst sees them, as in data of a correctly designed and prepared data warehouse.

Let us see what the sources and implications of messy data are. First, data may be missing for a huge variety of reasons. Sometimes there are mistakes in measurements or recordings, but in many cases, the value is unavailable. To cope with this in a data-mining process, one must be able to model with the data that are presented, even with their values missing. We will see later that some data-mining techniques are more or less sensitive to missing values. If the method is robust enough, then the missing values are not a problem. Otherwise, it is necessary to solve the problem of missing values before the application of a selected data-mining technique. The second cause of messy data is misrecording of data,