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@@ -508,7 +508,8 @@
under the License.
-->
<h1 id="quantiles-and-ranks-definitions">Quantiles and Ranks Definitions</h1>
-<p>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 extract values given a desired rank, or the reverse.
Quantiles sketches enable us to plot the CDF, PMF or histograms of a
distribution.</p>
+<p>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.</p>
<p>The goal of this short tutorial it to introduce to the reader some of the
basic concepts of quantiles, ranks and their functions.</p>
@@ -521,16 +522,14 @@
<li>
<p>The <strong>natural rank</strong> is a <strong>natural number</strong>
from the set of one-based, natural numbers, ℕ<sub>1</sub>, and is derived by
enumerating an ordered set of values, starting with the value 1, up to
<em>n</em>, the number of values in the set.</p>
</li>
- <li>
- <p>The <strong><em>normalized rank</em></strong> is a number between 0 and
1 computed by dividing the <em>natural rank</em> by the total number of values
in the set, <em>n</em>. Thus, for finite sets, any <em>normalized rank</em> 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.</p>
- </li>
+ <li>The <strong><em>normalized rank</em></strong> is a number between 0.0
and 1.0 computed by dividing the <em>natural rank</em> by the total number of
values in the set, <em>n</em>. Thus, for finite sets, any <em>normalized
rank</em> 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.</li>
+ <li>A rank of 0, whether natural or normalized, represents the empty
set.</li>
</ul>
<p>In our sketch library and documentation, when we refer to <em>rank</em>, we
imply <em>normalized rank</em>. However, in this tutorial, we will sometimes
use <em>natural ranks</em> to simplify the examples.</p>
<h3 id="rank-and-mass">Rank and Mass</h3>
-<p><em>Normalized rank</em> is closely associated with the concept of
<em>mass</em>. The value associated with the rank 0.5 represents the median
value, or the center of <em>mass</em> 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 the
quantile sketches in the library.
-A rank of <em>0</em> means a mass of <em>0</em> or an empty set.</p>
+<p><em>Normalized rank</em> is closely associated with the concept of
<em>mass</em>. The value associated with the rank 0.5 represents the median
value, or the center of <em>mass</em> 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.</p>
<h2 id="what-is-a-quantile">What is a quantile?</h2>
@@ -538,7 +537,7 @@ A rank of <em>0</em> means a mass of <em>0</em> or an empty
set.</p>
<p>A <strong><em>quantile</em></strong> is a <em>value</em> that achieves a
particular <strong><em>rank</em></strong>.</p>
</blockquote>
-<p><em>Quantile</em> is the general term that describes other terms that are
also quantiles.
+<p><em>Quantile</em> is the general term that includes other terms that are
also quantiles.
To wit:</p>
<ul>
@@ -549,234 +548,341 @@ To wit:</p>
</ul>
<h2 id="the-quantile-and-rank-functions">The quantile and rank functions</h2>
-<p>Because of the relationship of quantiles and ranks, we can define</p>
+<p>Let’s examine the following table:</p>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank</td>
+ <td>1</td>
+ <td>2</td>
+ <td>3</td>
+ <td>4</td>
+ <td>5</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank</td>
+ <td>.2</td>
+ <td>.4</td>
+ <td>.6</td>
+ <td>.8</td>
+ <td>1.0</td>
+ </tr>
+ </tbody>
+</table>
+
+<p>Let’s define the functions</p>
<ul>
- <li>The <strong><em>r-quantile</em></strong> is a value
<strong><em>q</em></strong> such that <strong><em>rank(q) = r</em></strong>,
and <strong><em>quantile(r) = q</em></strong>, assuming no duplicates. In this
tutorial, we shorten these two functions to <em>r(q)</em> and
<em>q(r)</em>.</li>
+ <li>
+ <p><em>quantile(rank)</em> or <em>q(r)</em> := given a <em>rank, r</em>,
return the quantile value <em>q</em> associated with the given <em>r</em>.</p>
+ </li>
+ <li>
+ <p><em>rank(quantile)</em> or <em>r(q)</em> := given a quantile value
<em>q</em>, return the rank <em>r</em> associated with the given <em>q</em>.</p>
+ </li>
</ul>
-<h2 id="the-challenge-of-duplicates">The challenge of duplicates</h2>
-<p>The functions <em>q(r)</em> and <em>r(q)</em> would form a 1:1 functional
pair if <em>q = q(r(q))</em> and <em>r = r(q(r))</em>.
-However, duplicate values are quite common in real data so exact 1:1
functionality is not possible. As a result it is often the case that <em>q !=
q(r(q))</em> and <em>r != r(q(r))</em>. Duplicate values also could make the
rank function, <em>r(q)</em>, ambiguous. If there are multiple adjacent ranks
with the same value, which rank should the rank function return?</p>
-
-<h2 id="the-challenge-of-approximation">The challenge of approximation</h2>
-<p>By definiton, sketching algorithms are approximate, and they achieve their
high performance by discarding a vast amount of the data. Suppose you feed
<em>n</em> items into a sketch that retains only <em>m</em> items. This means
<em>n-m</em> values were discarded. The sketch must track the value <em>n</em>
used for computing the rank and quantile functions. When the sketch
reconstructs the relationship between ranks and values <em>n-m</em> rank values
are missing creating holes in th [...]
+<p>Using an example from the table:</p>
-<h2 id="the-need-for-inequality-search">The need for inequality search</h2>
-<p>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.</p>
+<ul>
+ <li>Using natural ranks:
+ <ul>
+ <li><em>q(3) = 30</em></li>
+ <li><em>r(30) = 3</em></li>
+ </ul>
+ </li>
+ <li>Using normalized ranks:
+ <ul>
+ <li><em>q(.6) = 30</em></li>
+ <li><em>r(30) = .6</em></li>
+ </ul>
+ </li>
+</ul>
-<p>Given the presence of duplicates and absence of values from the stream we
must redefine the above quantle and rank functions as inequalities. Let’s start
with a simple example.</p>
+<p>Because of the close, two-way relationship of quantiles and ranks,<br />
+<em>r(q)</em> and <em>q(r)</em> form a <em>1:1 functional pair</em> if, and
only if</p>
-<h2 id="two-conventions-used-for-searching-for-ranks">Two conventions used for
searching for ranks</h2>
<ul>
- <li>The first convention, called the <em>Less-Than</em> (<em>LT</em>)
criterion, finds the mass of a distribution, denoted by a rank, that is
strictly less-than the given rank.</li>
- <li>The second convention, called the <em>Less-Than-or-Equal</em>
(<em>LE</em>) criterion, finds the mass of a distribution, denoted by a rank,
that is strictly less-than-or-equal to the given rank.</li>
+ <li><em>q = q(r(q))</em></li>
+ <li><em>r = r(q(r))</em></li>
</ul>
-<p>You will find both of these in the literature. Our older
<em>quantiles/DoublesSketch</em> and our <em>KLL</em> quantiles sketch use the
<em>LT</em> criterion. Our newest <em>REQ</em> sketch allows the user to
choose.</p>
+<p>And this is certainly true of the table above.</p>
-<h2 id="two-complementing-conventions-used-for-searching-for-quantiles">Two
complementing conventions used for searching for quantiles</h2>
-<p>When searching for quantiles, we require that search to return a quantile,
such that our given <em>rank ~ r(q(r))</em> as close a possible.</p>
+<h2 id="the-challenge-of-duplicates">The challenge of duplicates</h2>
+<p>With real data we often encounter duplicate values in the stream. Let’s
examine this next table.</p>
-<p>In order to do that we use two complementing criteria.</p>
+<table>
+ <thead>
+ <tr>
+ <th>Quantile:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>20</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank</td>
+ <td>1</td>
+ <td>2</td>
+ <td>3</td>
+ <td>4</td>
+ <td>5</td>
+ </tr>
+ </tbody>
+</table>
-<ul>
- <li>To match the <em>LT</em> criterion for rank, we use the greater-than,
<em>GT</em>, criterion for quantiles</li>
- <li>To match the <em>LE</em> criterion for rank, we use the
greater-than-or-equal, <em>GE</em>, criterion for quantiles.</li>
-</ul>
+<p>As you can see <em>q(r)</em> is straightforward. But how about
<em>r(q)</em>? 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 <em>r(q)</em> is ambiguous. We will see how to resolve this
shortly.</p>
+
+<h2 id="the-challenge-of-approximation">The challenge of approximation</h2>
+<p>By definiton, sketching algorithms are approximate, and they achieve their
high performance by discarding data. Suppose you feed <em>n</em> items into a
sketch that retains only <em>m < n</em> items. This means <em>n-m</em>
values were discarded. The sketch must track the value <em>n</em> used for
computing the rank and quantile functions. When the sketch reconstructs the
relationship between ranks and values <em>n-m</em> rank values are missing
creating holes in the sequence of [...]
-<h2 id="example">Example</h2>
-<p>Given the ordered values <em>{10,20,20,20,30}</em>, we can construct the
following table of raw ranks and values. For simplicity we will use natural
ranks.</p>
+<p>The raw data might look like this, with its associated natural ranks.</p>
<table>
<thead>
<tr>
- <th style="text-align: center">Ranks, <em>r</em></th>
- <th style="text-align: center">1</th>
- <th style="text-align: center">2</th>
- <th style="text-align: center">3</th>
- <th style="text-align: center">4</th>
- <th style="text-align: center">5</th>
+ <th>Quantile:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ <th>60</th>
+ <th>70</th>
+ <th>80</th>
+ <th>90</th>
+ <th>100</th>
</tr>
</thead>
<tbody>
<tr>
- <td style="text-align: center">Values, <em>q</em></td>
- <td style="text-align: center">10</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">30</td>
+ <td>Natural Rank</td>
+ <td>1</td>
+ <td>2</td>
+ <td>3</td>
+ <td>4</td>
+ <td>5</td>
+ <td>6</td>
+ <td>7</td>
+ <td>8</td>
+ <td>9</td>
+ <td>10</td>
</tr>
</tbody>
</table>
-<p>Table 1: Raw data mapping of ranks to values</p>
-
-<p>After processing the stream the actual representation inside the sketch
might look like the following. This compresses out duplicate values and
effectively skips over missing values. Note that the top rank will always be
<em>n</em>.</p>
+<p>The sketch might discard the even values producing something like this:</p>
<table>
<thead>
<tr>
- <th style="text-align: center">Ranks, <em>r</em></th>
- <th style="text-align: center">1</th>
- <th style="text-align: center">4</th>
- <th style="text-align: center">5</th>
+ <th>Quantile:</th>
+ <th>10</th>
+ <th>30</th>
+ <th>50</th>
+ <th>70</th>
+ <th>90</th>
</tr>
</thead>
<tbody>
<tr>
- <td style="text-align: center">Values, <em>q</em></td>
- <td style="text-align: center">10</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">30</td>
+ <td>Natural Rank</td>
+ <td>2</td>
+ <td>4</td>
+ <td>6</td>
+ <td>8</td>
+ <td>10</td>
</tr>
</tbody>
</table>
-<p>Table 1B: Raw data mapping compressed</p>
+<p>So how do we resove <em>q(3)</em> or <em>r(20)</em>?</p>
-<p>We will use Table 1B for the following.</p>
+<h2 id="the-need-for-inequality-search">The need for inequality search</h2>
+<p>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.</p>
-<h3 id="convention-lt">Convention <em>LT</em></h3>
+<p>Given the presence of duplicates and absence of values from the stream we
must redefine the above quantile and rank functions as inequalities. <strong>We
also want the quantile and rank functions to retain the properties of 1:1
functions.</strong></p>
-<h4 id="the-lt-less-than-criterion-for-finding-ranks">The <em>LT</em>
(less-than) criterion for finding ranks</h4>
-<p>Given a value, <em>V</em>, find an adjacent pair of values, <em>q1,q2</em>,
where <em>q1 < V <= q2</em>. Return the rank of <em>q1</em>.</p>
+<p>You will find examples of both of the following definitions in the research
literature. All of our library quantile sketches allow the user to choose
between the two searching criteria.</p>
+<p>These next examples use a small data set that mimics what could be the
result of both duplication and sketch data deletion.</p>
+
+<h2 id="two-search-conventions-used-when-finding-ranks-rq">Two search
conventions used when finding ranks, r(q)</h2>
<ul>
- <li>Given <em>V=10</em>, <em>? < V <= 10</em>. Return 0. There is no
value in the set < <em>10</em>.</li>
- <li>Given <em>V=20</em>, <em>10 < V <= 20</em>. Return 1.</li>
- <li>Given <em>V=30</em>, <em>20 < V <= 30</em>. Return 4.</li>
+ <li>The <em>non inclusive</em> criterion for <em>r(q)</em>: (a.k.a. the
<em>LT</em> criterion)
+ <ul>
+ <li>Search the quantile array until we find the adjacent pair <em>{q1,
q2}</em> where <em>q1 < q <= q2</em>. Return the rank associated with
<em>q1</em>, the first of the pair.</li>
+ </ul>
+ </li>
</ul>
-<p>Table 2 represents this mapping.</p>
+<p>For example <em>r(30) = 5</em></p>
<table>
<thead>
<tr>
- <th style="text-align: center">Given <em>q</em></th>
- <th style="text-align: center">10</th>
- <th style="text-align: center">20</th>
- <th style="text-align: center">30</th>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>q1=20</th>
+ <th>q2=30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
</tr>
</thead>
<tbody>
<tr>
- <td style="text-align: center">Find <em>r</em> (LT)</td>
- <td style="text-align: center">0</td>
- <td style="text-align: center">1</td>
- <td style="text-align: center">4</td>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>r=5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
</tr>
</tbody>
</table>
-<p>Table 2: Using the <em>LT</em> criterion for finding ranks.</p>
-
-<p>Obtaining the quantile value given the rank is going the opposite
direction, so we use the <em>GT</em> (greater-than) criterion.</p>
-
-<h4 id="the-gt-greater-than-criterion-for-finding-quantiles">The <em>GT</em>
(greater-than) criterion for finding quantiles.</h4>
-<p>Given a rank, <em>R</em>, find an adjacent pair of ranks, <em>r1,r2</em>,
where <em>r1 <= R < r2</em>. Return <em>q(r2)</em>.</p>
-
<ul>
- <li>Given <em>R=1, 2 or 3</em>, <em>1 <= R < 4</em>. Return
<em>20</em>.</li>
- <li>Given <em>R=4</em>, <em>4 <= R < 5</em>. Return <em>30</em></li>
- <li>Given <em>R=5</em>, <em>5 <= R < ?</em>. Return <em>30</em>. There
is no rank > 5, but because it is at the top of the range we can safely
return the top value.</li>
+ <li>The <em>inclusive</em> criterion for <em>r(q)</em>: (a.k.a. the
<em>LE</em> criterion)
+ <ul>
+ <li>Search the quantile array until we find the adjacent pair <em>{q1,
q2}</em> where <em>q1 <= q < q2</em>. Return the rank associated with
<em>q1</em>, the first of the pair.</li>
+ </ul>
+ </li>
</ul>
+<p>For example <em>r(30) = 11</em></p>
+
<table>
<thead>
<tr>
- <th style="text-align: center">Given <em>r</em></th>
- <th style="text-align: center">0</th>
- <th style="text-align: center">1</th>
- <th style="text-align: center">2</th>
- <th style="text-align: center">3</th>
- <th style="text-align: center">4</th>
- <th style="text-align: center">5</th>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>q1=30</th>
+ <th>q2=40</th>
+ <th>50</th>
</tr>
</thead>
<tbody>
<tr>
- <td style="text-align: center">Find <em>q</em> (GT)</td>
- <td style="text-align: center">10</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">30</td>
- <td style="text-align: center">30</td>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>r=11</td>
+ <td>13</td>
+ <td>14</td>
</tr>
</tbody>
</table>
-<p>Table 3: Using the <em>GT</em> criterion for finding quantiles</p>
-
-<h3 id="convention-le">Convention <em>LE</em></h3>
-
-<h4 id="the-le-less-than-or-equals-criterion-for-finding-ranks">The
<em>LE</em> (less-than or equals) criterion for finding ranks</h4>
-<p>Given a value, <em>V</em>, find an adjacent pair of values, <em>q1,q2</em>,
where <em>q1 <= V < q2</em>. Return the rank of <em>q1</em>.</p>
-
+<h2 id="two-search-conventions-when-finding-quantiles-qr">Two search
conventions when finding quantiles, q(r)</h2>
<ul>
- <li>Given <em>V=10</em>, <em>10 <= V < 20</em>. Return 1.</li>
- <li>Given <em>V=20</em>, <em>20 <= V < 30</em>. Return 4.</li>
- <li>Given <em>V=30</em>, <em>30 <= V < ?</em>. Return 5.</li>
+ <li>The <em>non inclusive</em> criterion for <em>q(r)</em> : (a.k.a. the
<em>GT</em> criterion)
+ <ul>
+ <li>Search the rank array until we find the adjacent pair <em>{r1,
r2}</em> where <em>r1 <= r < r2</em>. Return the quantile associated with
<em>r2</em>, the second of the pair.</li>
+ </ul>
+ </li>
</ul>
+<p>For example <em>q(5) = 30</em></p>
+
<table>
<thead>
<tr>
- <th style="text-align: center">Given <em>q</em></th>
- <th style="text-align: center">10</th>
- <th style="text-align: center">20</th>
- <th style="text-align: center">30</th>
+ <th>Natural Rank[]:</th>
+ <th>1</th>
+ <th>3</th>
+ <th>r1=5</th>
+ <th>r2=7</th>
+ <th>9</th>
+ <th>11</th>
+ <th>13</th>
+ <th>14</th>
</tr>
</thead>
<tbody>
<tr>
- <td style="text-align: center">Find <em>r</em> (LE)</td>
- <td style="text-align: center">1</td>
- <td style="text-align: center">4</td>
- <td style="text-align: center">5</td>
+ <td>Quantile[]:</td>
+ <td>10</td>
+ <td>20</td>
+ <td>20</td>
+ <td>q=30</td>
+ <td>30</td>
+ <td>30</td>
+ <td>40</td>
+ <td>50</td>
</tr>
</tbody>
</table>
-<p>Table 4: The <em>LE</em> criterion for finding ranks.</p>
-
-<p>Obtaining the quantile value given the rank is going the opposite
direction, so we use the <em>GE</em> (greater-than-or-equals) criterion.</p>
-
-<h4 id="the-ge-greater-than-or-equals-criterion-for-finding-quantiles">The
<em>GE</em> (greater-than or equals) criterion for finding quantiles</h4>
-<p>Given a rank, <em>R</em>, find an adjacent pair of ranks, <em>r1,r2</em>,
where <em>r1 < R <= r2</em>. Return <em>q(r2)</em>.</p>
-
<ul>
- <li>Given <em>R=1</em>, <em>? < R <= 1</em>. Return <em>10</em>.</li>
- <li>Given <em>R=2, 3 or 4</em>, <em>1 < R <= 4</em>. Return
<em>20</em>.</li>
- <li>Given <em>R=5</em>, <em>4 < R <= 5</em>. Return <em>30</em>.</li>
+ <li>The <em>inclusive</em> criterion for <em>q(r)</em>: (a.k.a. the
<em>GE</em> criterion)
+ <ul>
+ <li>Search the rank array until we find the adjacent pair <em>{r1,
r2}</em> where <em>r1 < r <= r2</em>. Return the quantile associated with
<em>r2</em>, the second of the pair.</li>
+ </ul>
+ </li>
</ul>
+<p>For example <em>q(11) = 30</em></p>
+
<table>
<thead>
<tr>
- <th style="text-align: center">Given <em>r</em></th>
- <th style="text-align: center">1</th>
- <th style="text-align: center">2</th>
- <th style="text-align: center">3</th>
- <th style="text-align: center">4</th>
- <th style="text-align: center">5</th>
+ <th>Natural Rank[]:</th>
+ <th>1</th>
+ <th>3</th>
+ <th>5</th>
+ <th>7</th>
+ <th>r1=9</th>
+ <th>r2=11</th>
+ <th>13</th>
+ <th>14</th>
</tr>
</thead>
<tbody>
<tr>
- <td style="text-align: center">Find <em>q</em> (GE)</td>
- <td style="text-align: center">10</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">20</td>
- <td style="text-align: center">30</td>
+ <td>Quantile[]:</td>
+ <td>10</td>
+ <td>20</td>
+ <td>20</td>
+ <td>30</td>
+ <td>30</td>
+ <td>q=30</td>
+ <td>40</td>
+ <td>50</td>
</tr>
</tbody>
</table>
-<p>Table 5: The <em>GE</em> criterion for finding quantiles.</p>
+<p>The power of these inequality search algorithms is that the will produce
repeatable and accurate results and insensitive to duplicates and sketch
deletions.
+In addition, the way these algorithms are designed, the property of 1:1
functions are maintained.</p>
+
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