Data Mining_ Concepts and Techniques - Jiawei Han [43]
Quantile–quantile plot
Figure 2.5 shows a quantile–quantile plot for unit price data of items sold at two branches of AllElectronics during a given time period. Each point corresponds to the same quantile for each data set and shows the unit price of items sold at branch 1 versus branch 2 for that quantile. (To aid in comparison, the straight line represents the case where, for each given quantile, the unit price at each branch is the same. The darker points correspond to the data for Q1, the median, and Q3, respectively.)
We see, for example, that at Q1, the unit price of items sold at branch 1 was slightly less than that at branch 2. In other words, 25% of items sold at branch 1 were less than or equal to $60, while 25% of items sold at branch 2 were less than or equal to $64. At the 50th percentile (marked by the median, which is also Q2), we see that 50% of items sold at branch 1 were less than $78, while 50% of items at branch 2 were less than $85. In general, we note that there is a shift in the distribution of branch 1 with respect to branch 2 in that the unit prices of items sold at branch 1 tend to be lower than those at branch 2.
Figure 2.5 A q-q plot for unit price data from two AllElectronics branches.
Histograms
Histograms (or frequency histograms) are at least a century old and are widely used. “Histos” means pole or mast, and “gram” means chart, so a histogram is a chart of poles. Plotting histograms is a graphical method for summarizing the distribution of a given attribute, X. If X is nominal, such as automobile_model or item_type, then a pole or vertical bar is drawn for each known value of X. The height of the bar indicates the frequency (i.e., count) of that X value. The resulting graph is more commonly known as a bar chart.
If X is numeric, the term histogram is preferred. The range of values for X is partitioned into disjoint consecutive subranges. The subranges, referred to as buckets or bins, are disjoint subsets of the data distribution for X. The range of a bucket is known as the width. Typically, the buckets are of equal width. For example, a price attribute with a value range of $1 to $200 (rounded up to the nearest dollar) can be partitioned into subranges 1 to 20, 21 to 40, 41 to 60, and so on. For each subrange, a bar is drawn with a height that represents the total count of items observed within the subrange. Histograms and partitioning rules are further discussed in Chapter 3 on data reduction.
Histogram
Figure 2.6 shows a histogram for the data set of Table 2.1, where buckets (or bins) are defined by equal-width ranges representing $20 increments and the frequency is the count of items sold.
Figure 2.6 A histogram for the Table 2.1 data set.
Although histograms are widely used, they may not be as effective as the quantile plot, q-q plot, and boxplot methods in comparing groups of univariate observations.
Scatter Plots and Data Correlation
A scatter plot is one of the most effective graphical methods for determining if there appears to be a relationship, pattern, or trend between two numeric attributes. To construct a scatter plot, each pair of values is treated as a pair of coordinates in an algebraic sense and plotted as points in the plane. Figure 2.7 shows a scatter plot for the set of data in Table 2.1.
Figure 2.7 A scatter plot for the Table 2.1 data set.
The scatter plot is a useful method for providing a first look at bivariate data to see clusters of points and outliers, or to explore the possibility of correlation relationships. Two attributes, X, and Y, are correlated if one attribute implies the other. Correlations can be positive, negative, or null (uncorrelated). Figure 2.8 shows examples of positive and negative correlations between two attributes. If the plotted points pattern slopes from lower left to upper right, this means that the values of X increase as the values of Y increase, suggesting a positive correlation (Figure 2.8a). If the pattern