# Sketching Quantiles and Ranks, the Basics

Streaming quantiles algorithms, or quantiles sketches, enable us to analyze the distributions of massive data very quickly using only a small amout of space.
They allow us to compute a quantile values given a desired rank, or compute a rank given a quantile value. Quantile sketches enable us to plot the CDF, PMF or histograms of a distribution.

The goal of this short tutorial it to introduce to the reader some of the basic concepts of quantiles, ranks and their functions.

## What is a rank?

### A rank identifies the numeric position of a specific value in an enumerated, ordered set if values.

The actual enumeration can be done in several ways, but for our use here we will define the two common ways that rank can be specified and that we will use.

• The natural rank is a natural number from the set of one-based, natural numbers, ℕ1, and is derived by enumerating an ordered set of values, starting with the value 1, up to n, the number of values in the set.

• The normalized rank is a number between 0.0 and 1.0 computed by dividing the natural rank by the total number of values in the set, n. Thus, for finite sets, any normalized rank is in the range (0, 1]. Normalized ranks are often written as a percent. But don't confuse percent with percentile! This will be explained below.

• A rank of 0, whether natural or normalized, represents the empty set.

In our sketch library and documentation, when we refer to rank, we imply normalized rank. However, in this tutorial, we will sometimes use natural ranks to simplify the examples.

### Rank and Mass

Normalized rank is closely associated with the concept of mass. The value associated with the rank 0.5 represents the median value, or the center of mass of the entire set, where half of the values are below the median and half are above. The concept of mass is important to understanding the Prabability Mass Function (PMF) offered by all the quantile sketches in the library.

## What is a quantile?

### A quantile is a value that is associated with a particular rank.

Quantile is the general term that includes other terms that are also quantiles. To wit:

• A percentile is a quantile where the rank domain is divided into hundredths. For example, “An SAT Math score of 740 is at the 95th percentile”. The score of 740 is the quantile and .95 is the normalized rank.
• A decile is a quantile where the rank domain is divided into tenths. For example, "An SAT Math score of 690 is at the 9th decile (rank = 0.9).
• A quartile is a quantile where the rank domain is divided into forths. For example, "An SAT Math score of 600 is at the third quartile (rank = 0.75).
• The median is a quantile that splits the rank domain in half. For example, "An SAT Math score of 520 is at the median (rank = 0.5).

## The simple quantile and rank functions

Let's examine the following table:

Quantile:1020304050
Natural Rank12345
Normalized Rank.2.4.6.81.0

Let's define the simple functions

### rank(quantile) or r(q) := return the rank r associated with a given quantile, q.

Using an example from the table:

• Using natural ranks:
• q(3) = 30
• r(30) = 3
• Using normalized ranks:
• q(.6) = 30
• r(30) = .6

Because of the close, two-way relationship of quantiles and ranks,
r(q) and q(r) form a 1:1 functional pair if, and only if

• q = q(r(q))
• r = r(q(r))

And this is certainly true of the table above.

## The challenge of duplicates

With real data we often encounter duplicate values in the stream. Let's examine this next table.

Quantile:1020202050
Natural Rank12345

As you can see q(r) is straightforward. But how about r(q)? Which of the rank values 2, 3, or 4 should the function return given the value 20? Given this data, and our definitions so far, the function r(q) is ambiguous. We will see how to resolve this shortly.

## The challenge of approximation

By definiton, sketching algorithms are approximate, and they achieve their high performance by discarding data. Suppose you feed n items into a sketch that retains only m < n items. This means n-m values were discarded. The sketch must track the value n used for computing the rank and quantile functions. When the sketch reconstructs the relationship between ranks and values n-m rank values are missing creating holes in the sequence of ranks. For example, examine the following tables.

The raw data might look like this, with its associated natural ranks.

Quantile:102030405060708090100
Natural Rank12345678910

The sketch might discard the even values producing something like this:

Quantile:1030507090
Natural Rank246810

When the sketch deletes values it adjusts the associated ranks by effectively increasing the “weight” of adjacent items so that they are positionally approximately correct and the top rank corresponds to n.

How do we resove q(3) or r(20)?

## The need for inequality search

The quantile sketch algorithms discussed in the literature primarily differ by how they choose which values in the stream should be discarded. After the elimination process, all of the quantiles sketch implementations are left with the challenge of how to reconstruct the actual distribution, approximately and with good accuracy.

Given the presence of duplicates and absence of values from the stream we must redefine the above quantile and rank functions as inequalities while retaining the properties of 1:1 functions.

One can find examples of the following definitions in the research literature. All of our library quantile sketches allow the user to choose the searching criteria.

These next examples use a small data set that mimics what could be the result of both duplication and sketch data deletion.

## The rank functions with inequalities

### rank(quantile, NON_INCLUSIVE) or r(q, LT) :=Given q, return the rank, r, of the largest quantile that is strictly Less Thanq.

Implementation: Given q, search the quantile array until we find the adjacent pair {q1, q2} where q1 < q <= q2. Return the rank, r, associated with q1, the first of the pair.

Boundary Notes:

• If the given q is larger than the largest quantile retained by the sketch, the sketch will return the rank of the largest retained quantile.
• If the given q is smaller than the smallest quantile retained by the sketch, the sketch will return a rank of zero.

Examples using normalized ranks:

• r(55) = 1.0
• r(5) = 0.0
• r(30) = .357 (Illustrated in table)
Quantile[]:1020203030304050
Natural Rank[]:13579111314
Normalized Rank[]:.071.214.357.500.643.786.9291.000
Quantile input303030
Qualifying pairq1q2
Rank result.786

### rank(quantile, INCLUSIVE) or r(q, LE) :=Given q, return the rank, r, of the largest quantile that is less than or equal to q.

Implementation: Given q, search the quantile array until we find the adjacent pair {q1, q2} where q1 <= q < q2. Return the rank, r, associated with q1, the first of the pair.

Boundary Notes:

• If the given q is larger than the largest quantile retained by the sketch, the function will return the rank of the largest retained quantile.
• If the given q is smaller than the smallest quantile retained by the sketch, the function will return a rank of zero.

Examples using normalized ranks:

• r(55) = 1.0
• r(5) = 0.0
• r(30) = .786 (Illustrated in table)
Quantile[]:1020203030304050
Natural Rank[]:13579111314
Normalized Rank[]:.071.214.357.500.643.786.9291.000
Quantile input303030
Qualifying pairq1q2
Rank result.786

## The quantile functions with inequalities

### quantile(rank, NON_INCLUSIVE) or q(r, GT) :=Given r, return the quantile, q, of the smallest rank that is strictly Greater Than r.

Implementation: Given r, search the rank array until we find the adjacent pair {r1, r2} where r1 <= r < r2. Return the quantile associated with r2, the second of the pair.

Boundary Notes:

• If the given normalized rank, r, is equal to 1.0, there is no quantile that satisfies this criterion. However, for convenience, the function will return the largest quantile retained by the sketch.
• If the given normalized rank, r, is less than the smallest rank, the function will return the smallest quantile.

Examples using normalized ranks:

• q(1.0) = 50
• q(0.0) = 10
• q(.357) = 30 (Illustrated in table)
Quantile[]:1020203030304050
Natural Rank[]:13579111314
Normalized Rank[]:.071.214.357.500.643.786.9291.000
Rank input.357
Qualifying pairr1r2
Quantile result30

### quantile(rank, NON_INCLUSIVE_STRICT) or q(r, GT_STRICT) :=Given r, return the quantile, q, of the smallest rank that is strictly Greater Than r.

In STRICT mode, the only difference is the following:

Boundary Notes:

• If the given normalized rank, r, is equal to 1.0, there is no quantile that satisfies this criterion. The function will return NaN.

### quantile(rank, INCLUSIVE) or q(r, GE) :=Given r, return the quantile, q, of the smallest rank that is strictly Greater than or Equal to r.

Implementation: Given r, search the rank array until we find the adjacent pair {r1, r2} where r1 < r <= r2. Return the quantile, q, associated with r2, the second of the pair.

Boundary Notes:

• If the given normalized rank, r, is equal to 1.0, the function will return the largest quantile retained by the sketch.
• If the given normalized rank, r, is less than the smallest rank, the function will return the smallest quantile.

Examples using normalized ranks:

For example q(.786) = 30

Quantile[]:1020203030304050
Natural Rank[]:13579111314
Normalized Rank[]:.071.214.357.500.643.786.9291.000
Rank input.786
Qualifying pairr1r2
Quantile result30

## Summary

The power of these inequality search algorithms is that they produce repeatable and accurate results, are insensitive to duplicates and sketch deletions, and maintain the property of 1:1 functions.