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of plotted points slopes from upper left to lower right, the values of X increase as the values of Y decrease, suggesting a negative correlation (Figure 2.8b). A line of best fit can be drawn to study the correlation between the variables. Statistical tests for correlation are given in Chapter 3 on data integration (Eq. (3.3)). Figure 2.9 shows three cases for which there is no correlation relationship between the two attributes in each of the given data sets. Section 2.3.2 shows how scatter plots can be extended to n attributes, resulting in a scatter-plot matrix.

Figure 2.8 Scatter plots can be used to find (a) positive or (b) negative correlations between attributes.

Figure 2.9 Three cases where there is no observed correlation between the two plotted attributes in each of the data sets.

In conclusion, basic data descriptions (e.g., measures of central tendency and measures of dispersion) and graphic statistical displays (e.g., quantile plots, histograms, and scatter plots) provide valuable insight into the overall behavior of your data. By helping to identify noise and outliers, they are especially useful for data cleaning.

2.3. Data Visualization

How can we convey data to users effectively? Data visualization aims to communicate data clearly and effectively through graphical representation. Data visualization has been used extensively in many applications—for example, at work for reporting, managing business operations, and tracking progress of tasks. More popularly, we can take advantage of visualization techniques to discover that are otherwise not easily observable by looking at the raw data. Nowadays, people also use data visualization to create fun and interesting graphics.

In this section, we briefly introduce the basic concepts of data visualization. We start with multidimensional data such as those stored in relational databases. We discuss several representative approaches, including pixel-oriented techniques, geometric projection techniques, icon-based techniques, and hierarchical and graph-based techniques. We then discuss the visualization of complex data and relations.

2.3.1. Pixel-Oriented Visualization Techniques

A simple way to visualize the value of a dimension is to use a pixel where the color of the pixel reflects the dimension's value. For a data set of m dimensions, pixel-oriented techniques create m windows on the screen, one for each dimension. The m dimension values of a record are mapped to m pixels at the corresponding positions in the windows. The colors of the pixels reflect the corresponding values.

Inside a window, the data values are arranged in some global order shared by all windows. The global order may be obtained by sorting all data records in a way that's meaningful for the task at hand.

Pixel-oriented visualization

AllElectronics maintains a customer information table, which consists of four dimensions: income, credit_limit, transaction_volume, and age. Can we analyze the correlation between income and the other attributes by visualization?

We can sort all customers in income-ascending order, and use this order to lay out the customer data in the four visualization windows, as shown in Figure 2.10. The pixel colors are chosen so that the smaller the value, the lighter the shading. Using pixel-based visualization, we can easily observe the following: credit_limit increases as income increases; customers whose income is in the middle range are more likely to purchase more from AllElectronics; there is no clear correlation between income and age.

Figure 2.10 Pixel-oriented visualization of four attributes by sorting all customers in income ascending order.

In pixel-oriented techniques, data records can also be ordered in a query-dependent way. For example, given a, we can sort all records in descending order of similarity to the.

Filling a window by laying out the data records in a linear way may not work well for a wide window. The first pixel in a row is far away from the last pixel in the previous row, though they are next to each other in