Streaming quantiles algorithms, or quantiles sketches, enable us to analyze the distributions of massive data very quickly using only a small amount of space.

They allow us to compute 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 the reader to some of the basic concepts of quantiles, ranks and their functions.

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 original set.The

is a number between 0.0 and 1.0 computed by dividing the**normalized rank***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.

*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 Probability Mass Function (PMF) offered by all the quantile sketches in the library.

*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 fourths. 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).

Let's examine the following table:

Quantile: | 10 | 20 | 30 | 40 | 50 |
---|---|---|---|---|---|

Natural Rank | 1 | 2 | 3 | 4 | 5 |

Normalized Rank | .2 | .4 | .6 | .8 | 1.0 |

The term “value” can be ambiguous because items that we stream into a sketch are values and numeric ranks are also values. To avoid this ambiguity, we will use the term “quantiles” to refer to values that are streamed into a sketch even before they have been associated with a rank.

Let's define the simple functions

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.

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

Quantile: | 10 | 20 | 20 | 20 | 50 |
---|---|---|---|---|---|

Natural Rank | 1 | 2 | 3 | 4 | 5 |

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

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

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

Quantile: | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
---|---|---|---|---|---|---|---|---|---|---|

Natural Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

The sketch might discard the even numbered quantiles producing something like this:

Quantile: | 10 | 30 | 50 | 70 | 90 |
---|---|---|---|---|---|

Natural Rank | 2 | 4 | 6 | 8 | 10 |

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

How do we resolve *q(3)* or *r(20)*?

The quantile sketch algorithms discussed in the literature primarily differ by how they choose which quantiles 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.

**Rule 1:**Every quantile that exists in the input stream or retained by the sketch has an associated rank.**Rule 2:**All of our quantile sketches only retain quantiles that exist in the actual input stream of quantiles.**Rule 3:**For the*getQuantile(rank)*queries, all of our quantile sketches only return quantiles that were retained by the sketch. (i.e, we do not interpolate between quantiles.)**Rule 4:**For the*getRank(quantile)*queries, all of our quantile sketches only return ranks that are associated with quantiles retained by the sketch. (i.e, we do not interpolate between ranks.)**Rule 5:**All of our quantile algorithms compensate for quantiles removed during the sketch quantile selection and compression process by increasing the weights of some of the quantiles not selected for removal, such that:- The sum of the natural weights of all quantiles retained by the sketch equals
**n**, the total count of all quantiles given to the sketch. - And by corollary, the largest quantile, when sorted by cumulative rank, has a cumulative natural rank of
**n**, or equivalently, a cumulative normalized rank of**1.0**.

- The sum of the natural weights of all quantiles retained by the sketch equals

Given

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 Exceptions:

**Boundary Rule 1:**If the given*q*is*>=*the quantile associated with the largest cumulative rank retained by the sketch, the function will return the largest cumulative rank,*1.0*.**Boundary Rule 2:**If the given*q*is*<*the quantile associated with the smallest cumulative rank retained by the sketch, the function will return a rank of*0.0*.

Examples using normalized ranks:

*r(30) = .786*Normal rule applies:*30 <= 30 < 40*, return*r(q1) = .786*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 |

Quantile input | 30 | |||||||

Qualifying pair | q1 | q2 | ||||||

Rank result | .786 |

*r(55) = 1.0*Use Boundary Rule 1:*50 <= 55*, return*1.0*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 | ? |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Quantile input | 55 | ||||||||

Qualifying pair | q1 | (q2) | |||||||

Rank result | 1.0 |

*r(5) = 0.0*Use Boundary Rule 2:*5 < 10*, return*0.0*.

Quantile[]: | ? | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Quantile input | 5 | ||||||||

Qualifying pair | (q1) | q2 | |||||||

Rank result | 0 |

Given

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 Exceptions:

**Boundary Rule 1:**If the given*q*is*>*the quantile associated with the largest cumulative rank retained by the sketch, the sketch will return the the largest cumulative rank,*1.0*.**Boundary Rule 2:**If the given*q*is*<=*the quantile associated with the smallest cumulative rank retained by the sketch, the sketch will return a rank of*0.0*.

Examples using normalized ranks:

*r(30) = .357*Normal rule applies:*20 < 30 <= 30*, return*r(q1) = .357*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.000 |

Quantile input | 30 | |||||||

Qualifying pair | q1 | q2 | ||||||

Rank result | .357 |

*r(55) = 1.0*Use Boundary Rule 1:*50 < 55*, return*1.0*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 | ? |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Quantile input | 55 | ||||||||

Qualifying pair | q1 | (q2) | |||||||

Rank result | 1.000 |

*r(5) = 0.0*Use Boundary Rule 2:*5 <= 10*, return*0*.

Quantile[]: | ? | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Quantile input | 5 | ||||||||

Qualifying pair | (q1) | q2 | |||||||

Rank result | 0 |

Given

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 Exceptions:

**Boundary Rule 2:**If the given normalized rank,*r*, is*<=*the smallest rank, the function will return the**quantile**associated with the smallest cumulative rank.

Examples using normalized ranks:

*q(.786) = 30*Normal rule applies:*.643 < .786 <= .786*, return*q(r2) = 30*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.000 |

Rank input | .786 | |||||||

Qualifying pair | r1 | r2 | ||||||

Quantile result | 30 |

*q(1.0) = 50*Normal rule applies:*.929 < 1.0 <= 1.0*, return*q(r2) = 50*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 |

Rank input | 1.0 | |||||||

Qualifying pair | r1 | r2 | ||||||

Quantile result | 50 |

*q(0.0 <= .071) = 10*Use Boundary Rule 2:*0.0 <= .071*, return*10*.

Quantile[]: | ? | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Rank input | 0.0 | ||||||||

Qualifying pair | (r1) | r2 | |||||||

Rank result | 10 |

Given

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 Exceptions:

**Boundary Rule 1:**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 quantile associated with the largest cumulative rank retained by the sketch.**Boundary Rule 2:**If the given normalized rank,*r*, is less than the smallest rank, the function will return the quantile associated with the smallest cumulative rank retained by the sketch.

Examples using normalized ranks:

*q(.357) = 30*Normal rule applies:*.357 <= .357 < .500*, return*q(r2) = 30*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.000 |

Rank input | .357 | |||||||

Qualifying pair | r1 | r2 | ||||||

Quantile result | 30 |

*q(1.0) = 50*Use Boundary Rule 1*1.0 <= 1.0 < ?*, return*50*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 | ? |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Rank input | 1.0 | ||||||||

Qualifying pair | r1 | (r2) | |||||||

Quantile result | 50 |

*q(0.99) = 50*Normal rule applies*.929 <= .99 < 1.0*, return*50*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 |

Rank input | .99 | |||||||

Qualifying pair | r1 | r2 | ||||||

Quantile result | 50 |

*q(0.0 <= .071) = 10*Use Boundary Rule 2:*0.0 < .071*, return*10*.

Quantile[]: | ? | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Rank input | 0.0 | ||||||||

Qualifying pair | (r1) | r2 | |||||||

Rank result | 10 |

Given

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 Exceptions:

**Boundary Rule 1:**If the given normalized rank,*r*, is equal to*1.0*, there is no quantile that satisfies this criterion. Return*NaN*or*null*.**Boundary Rule 2:**If the given normalized rank,*r*, is less than the smallest rank, the function will return the quantile associated with the smallest cumulative rank retained by the sketch..

Examples using normalized ranks:

*q(.357) = 30*Normal rule applies:*.357 <= .357 < .500*, return*q(r2) = 30*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.000 |

Rank input | .357 | |||||||

Qualifying pair | r1 | r2 | ||||||

Quantile result | 30 |

*q(1.0) = 50*Use Boundary Rule 1*1.0 <= 1.0 < ?*, return*NaN or null*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 | ? |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Rank input | 1.0 | ||||||||

Qualifying pair | r1 | (r2) | |||||||

Quantile result | NaN or null |

*q(0.99) = 50*Normal rule applies*.929 <= .99 < 1.0*, return*50*.

Quantile[]: | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 |

Rank input | .99 | |||||||

Qualifying pair | r1 | r2 | ||||||

Quantile result | 50 |

*q(0.0 <= .071) = 10*Use Boundary Rule 2:*0.0 < .071*, return*10*.

Quantile[]: | ? | 10 | 20 | 20 | 30 | 30 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|---|

Natural Rank[]: | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 14 | |

Normalized Rank[]: | .071 | .214 | .357 | .500 | .643 | .786 | .929 | 1.0 | |

Rank input | 0.0 | ||||||||

Qualifying pair | (r1) | r2 | |||||||

Rank result | 10 |

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.