Examples
The following example will construct a V-optimal histogram having a Sort Value of Value, a Source Value of Frequency, and a Partition Class of Serial. In practice, almost all histograms used in research or commercial products are of the Serial class, meaning that sequential sort values are placed in either the same bucket, or sequential buckets. For example, values 1, 2, 3 and 4 will be in buckets 1 and 2, or buckets 1, 2 and 3, but never in buckets 1 and 3. That will be taken as an assumption in any further discussion.
Take a simple set of data, for example, a list of integers:
1, 3, 4, 7, 2, 8, 3, 6, 3, 6, 8, 2, 1, 6, 3, 5, 3, 4, 7, 2, 6, 7, 2
Compute the value and frequency pairs (1, 2), (2, 4), (3, 5), (4, 2), (5, 1), (6, 4), (7, 3), (8, 2)
Our V-optimal histogram will have two buckets. Since one bucket must end at the data point for 8, we must decide where to put the other bucket boundary. The V-optimality rule states that the cumulative weighted variance of the buckets must be minimized. We will look at two options and compute the cumulative variance of those options.
Option 1: Bucket 1 contains values 1 through 4. Bucket 2 contains values 5 through 8.
Bucket 1:
Average frequency 3.25
Weighted variance 2.28
Bucket 2:
Average frequency 2.5
Weighted variance 2.19
Sum of Weighted Variance 4.47
Option 2: Bucket 1 contains values 1 through 2. Bucket 2 contains values 3 through 8.
Bucket 1:
Average frequency 3
Weighted variance 1.41
Bucket 2:
Average frequency 2.83
Weighted variance 3.29
Sum of Weighted Variance 4.70
The first choice is better, so the histogram that would wind up being stored is Bucket 1: Range(1 - 4), Average Frequency 3.25 Bucket 2: Range(5 - 8), Average Frequency 2.5
Read more about this topic: V-optimal Histograms
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