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---
.../SketchingQuantilesAndRanksTutorial.html | 1275 ++++++++++++++++++--
1 file changed, 1187 insertions(+), 88 deletions(-)
diff --git a/output/docs/Quantiles/SketchingQuantilesAndRanksTutorial.html
b/output/docs/Quantiles/SketchingQuantilesAndRanksTutorial.html
index 53b67257..a9647ac3 100644
--- a/output/docs/Quantiles/SketchingQuantilesAndRanksTutorial.html
+++ b/output/docs/Quantiles/SketchingQuantilesAndRanksTutorial.html
@@ -723,26 +723,53 @@ the function <em>r(q)</em> is ambiguous. We will see how
to resolve this shortly
<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="the-rules-for-returned-quantiles-or-ranks">The rules for returned
quantiles or ranks</h2>
+
+<ul>
+ <li>
+ <p><strong>Rule 1:</strong> Every Quantile that exists in the input stream
or retained by the sketch has an associated Rank.</p>
+ </li>
+ <li>
+ <p><strong>Rule 2:</strong> All of our quantile sketches only retain
quantiles that exist in the actual input stream of quantiles.</p>
+ </li>
+ <li>
+ <p><strong>Rule 3:</strong> For the <em>getQuantile(rank)</em> queries,
all of our quantile sketches only return quantiles that were retained by the
sketch. (i.e, we do not interpolate between quantiles.)</p>
+ </li>
+ <li>
+ <p><strong>Rule 4:</strong> For the <em>getRank(quantile)</em> 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.)</p>
+ </li>
+ <li>
+ <p><strong>Rule 5:</strong> 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:</p>
+
+ <ul>
+ <li>The sum of the natural weights of all quantiles retained by the
sketch equals <strong>N</strong>, the total count of all quantiles given to the
sketch.</li>
+ <li>And by corollary, the largest quantile, when sorted by cumulative
rank, has a cumulative natural rank of <strong>N</strong>, or equivalently, a
cumulative normalized rank of <strong>1.0</strong>.</li>
+ </ul>
+ </li>
+</ul>
+
<h2 id="the-rank-functions-with-inequalities">The rank functions with
inequalities</h2>
-<h3
id="rankquantile-exclusive-or-rq-lt-given-q-return-the-rank-r-of-the-largest-quantile-that-is-strictly-less-than-q"><strong><em>rank(quantile,
EXCLUSIVE)</em></strong> or <strong><em>r(q, LT)</em></strong> :=<br />Given
<em>q</em>, return the rank, <em>r</em>, of the largest quantile that is
strictly <em>Less Than</em> <em>q</em>.</h3>
+<h3
id="rankquantile-inclusive-or-rq-le-given-q-return-the-rank-r-of-the-largest-quantile-that-is-less-than-or-equal-to-q"><strong><em>rank(quantile,
INCLUSIVE)</em></strong> or <strong><em>r(q, LE)</em></strong> :=<br />Given
<em>q</em>, return the rank, <em>r</em>, of the largest quantile that is less
than or equal to <em>q</em>.</h3>
+
+<p><b>Implementation:</b></p>
-<p><b>Implementation:</b>
-Given <em>q</em>, search the quantile array until we find the adjacent pair
<em>{q1, q2}</em> where <em>q1 < q <= q2</em>. Return the rank,
<em>r</em>, associated with <em>q1</em>, the first of the pair.</p>
+<ul>
+ <li>Given <em>q</em>, search the quantile array until we find the adjacent
pair <em>{q1, q2}</em> where <em>q1 <= q < q2</em>.</li>
+ <li>Return the rank, <em>r</em>, associated with <em>q1</em>, the first of
the pair.</li>
+</ul>
-<p><b>Boundary Notes:</b></p>
+<p><b>Boundary Exceptions:</b></p>
<ul>
- <li>If the given <em>q</em> is larger than the largest quantile retained by
the sketch, the sketch will return the rank of the largest retained
quantile.</li>
- <li>If the given <em>q</em> is smaller than the smallest quantile retained
by the sketch, the sketch will return a rank of zero.</li>
+ <li><strong>Boundary Rule 1:</strong> If the given <em>q</em> is
<em>>=</em> the quantile associated with the largest cumulative rank
retained by the sketch, the function will return the largest cumulative rank,
<em>1.0</em>.</li>
+ <li><strong>Boundary Rule 2:</strong> If the given <em>q</em> is
<em><</em> the quantile associated with the smallest cumulative rank
retained by the sketch, the function will return a rank of <em>0.0</em>.</li>
</ul>
-<p><b>Examples using normalized ranks:</b></p>
+<h4 id="examples-using-normalized-ranksb">Examples using normalized
ranks:</b></h4>
<ul>
- <li><em>r(55) = 1.0</em></li>
- <li><em>r(5) = 0.0</em></li>
- <li><em>r(30) = .357</em> (Illustrated in table)</li>
+ <li><em>r(30) = .786</em> Normal rule applies: <em>30 <= 30 < 40</em>,
return <em>r(q1) = .786</em>.</li>
</ul>
<table>
@@ -780,15 +807,15 @@ Given <em>q</em>, search the quantile array until we find
the adjacent pair <em>
<td>.643</td>
<td>.786</td>
<td>.929</td>
- <td>1.000</td>
+ <td>1.0</td>
</tr>
<tr>
<td>Quantile input</td>
<td> </td>
<td> </td>
<td> </td>
- <td>30</td>
- <td>30</td>
+ <td> </td>
+ <td> </td>
<td>30</td>
<td> </td>
<td> </td>
@@ -797,53 +824,119 @@ Given <em>q</em>, search the quantile array until we
find the adjacent pair <em>
<td>Qualifying pair</td>
<td> </td>
<td> </td>
- <td>q1</td>
- <td>q2</td>
<td> </td>
<td> </td>
<td> </td>
+ <td>q1</td>
+ <td>q2</td>
<td> </td>
</tr>
<tr>
<td>Rank result</td>
<td> </td>
<td> </td>
- <td>.357</td>
<td> </td>
<td> </td>
<td> </td>
+ <td>.786</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
-<hr />
-
-<h3
id="rankquantile-inclusive-or-rq-le-given-q-return-the-rank-r-of-the-largest-quantile-that-is-less-than-or-equal-to-q"><strong><em>rank(quantile,
INCLUSIVE)</em></strong> or <strong><em>r(q, LE)</em></strong> :=<br />Given
<em>q</em>, return the rank, <em>r</em>, of the largest quantile that is less
than or equal to <em>q</em>.</h3>
-
-<p><b>Implementation:</b>
-Given <em>q</em>, search the quantile array until we find the adjacent pair
<em>{q1, q2}</em> where <em>q1 <= q < q2</em>. Return the rank,
<em>r</em>, associated with <em>q1</em>, the first of the pair.</p>
-
-<p><b>Boundary Notes:</b></p>
-
<ul>
- <li>If the given <em>q</em> is larger than the largest quantile retained by
the sketch, the function will return the rank of the largest retained
quantile.</li>
- <li>If the given <em>q</em> is smaller than the smallest quantile retained
by the sketch, the function will return a rank of zero.</li>
+ <li><em>r(55) = 1.0</em> Use Boundary Rule 1: <em>50 <= 55</em>, return
<em>1.0</em>.</li>
</ul>
-<p><b>Examples using normalized ranks:</b></p>
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ <th>?</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Quantile input</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>55</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>q1</td>
+ <td>(q2)</td>
+ </tr>
+ <tr>
+ <td>Rank result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>1.0</td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
<ul>
- <li><em>r(55) = 1.0</em></li>
- <li><em>r(5) = 0.0</em></li>
- <li><em>r(30) = .786</em> (Illustrated in table)</li>
+ <li><em>r(5) = 0.0</em> Use Boundary Rule 2: <em>5 < 10</em>, return
<em>0.0</em>.</li>
</ul>
<table>
<thead>
<tr>
<th>Quantile[]:</th>
+ <th>?</th>
<th>10</th>
<th>20</th>
<th>20</th>
@@ -857,6 +950,7 @@ Given <em>q</em>, search the quantile array until we find
the adjacent pair <em>
<tbody>
<tr>
<td>Natural Rank[]:</td>
+ <td> </td>
<td>1</td>
<td>3</td>
<td>5</td>
@@ -868,6 +962,7 @@ Given <em>q</em>, search the quantile array until we find
the adjacent pair <em>
</tr>
<tr>
<td>Normalized Rank[]:</td>
+ <td> </td>
<td>.071</td>
<td>.214</td>
<td>.357</td>
@@ -875,64 +970,69 @@ Given <em>q</em>, search the quantile array until we find
the adjacent pair <em>
<td>.643</td>
<td>.786</td>
<td>.929</td>
- <td>1.000</td>
+ <td>1.0</td>
</tr>
<tr>
<td>Quantile input</td>
+ <td>5</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
<td> </td>
<td> </td>
<td> </td>
- <td>30</td>
- <td>30</td>
- <td>30</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>Qualifying pair</td>
+ <td>(q1)</td>
+ <td>q2</td>
+ <td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
- <td>q1</td>
- <td>q2</td>
<td> </td>
</tr>
<tr>
<td>Rank result</td>
+ <td>0</td>
+ <td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
- <td>.786</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
-<h2 id="the-quantile-functions-with-inequalities">The quantile functions with
inequalities</h2>
+<hr />
-<h3
id="quantilerank-exclusive-or-qr-gt-given-r-return-the-quantile-q-of-the-smallest-rank-that-is-strictly-greater-than-r"><strong><em>quantile(rank,
EXCLUSIVE)</em></strong> or <strong><em>q(r, GT)</em></strong> :=<br />Given
<em>r</em>, return the quantile, <em>q</em>, of the smallest rank that is
strictly Greater Than <em>r</em>.</h3>
+<h3
id="rankquantile-exclusive-or-rq-lt-given-q-return-the-rank-r-of-the-largest-quantile-that-is-strictly-less-than-q"><strong><em>rank(quantile,
EXCLUSIVE)</em></strong> or <strong><em>r(q, LT)</em></strong> :=<br />Given
<em>q</em>, return the rank, <em>r</em>, of the largest quantile that is
strictly <em>Less Than</em> <em>q</em>.</h3>
-<p><b>Implementation:</b>
-Given <em>r</em>, 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.</p>
+<p><b>Implementation:</b></p>
+
+<ul>
+ <li>Given <em>q</em>, search the quantile array until we find the adjacent
pair <em>{q1, q2}</em> where <em>q1 < q <= q2</em>.</li>
+ <li>Return the rank, <em>r</em>, associated with <em>q1</em>, the first of
the pair.</li>
+</ul>
-<p><b>Boundary Notes:</b></p>
+<p><b>Boundary Exceptions:</b></p>
<ul>
- <li>If the given normalized rank, <em>r</em>, 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.</li>
- <li>If the given normalized rank, <em>r</em>, is less than the smallest
rank, the function will return the smallest quantile.</li>
+ <li><strong>Boundary Rule 1:</strong> If the given <em>q</em> is
<em>></em> the quantile associated with the largest cumulative rank retained
by the sketch, the sketch will return the the largest cumulative rank,
<em>1.0</em>.</li>
+ <li><strong>Boundary Rule 2:</strong> If the given <em>q</em> is
<em><=</em> the quantile associated with the smallest cumulative rank
retained by the sketch, the sketch will return a rank of <em>0.0</em>.</li>
</ul>
<p><b>Examples using normalized ranks:</b></p>
<ul>
- <li><em>q(1.0) = 50</em></li>
- <li><em>q(0.0) = 10</em></li>
- <li><em>q(.357) = 30</em> (Illustrated in table)</li>
+ <li><em>r(30) = .357</em> Normal rule applies: <em>20 < 30 <= 30</em>,
return <em>r(q1) = .357</em>.</li>
</ul>
<table>
@@ -973,11 +1073,11 @@ Given <em>r</em>, search the rank array until we find
the adjacent pair <em>{r1,
<td>1.000</td>
</tr>
<tr>
- <td>Rank input</td>
+ <td>Quantile input</td>
<td> </td>
<td> </td>
- <td>.357</td>
<td> </td>
+ <td>30</td>
<td> </td>
<td> </td>
<td> </td>
@@ -987,19 +1087,19 @@ Given <em>r</em>, search the rank array until we find
the adjacent pair <em>{r1,
<td>Qualifying pair</td>
<td> </td>
<td> </td>
- <td>r1</td>
- <td>r2</td>
+ <td>q1</td>
+ <td>q2</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
- <td>Quantile result</td>
+ <td>Rank result</td>
<td> </td>
<td> </td>
+ <td>.357</td>
<td> </td>
- <td>30</td>
<td> </td>
<td> </td>
<td> </td>
@@ -1008,36 +1108,10 @@ Given <em>r</em>, search the rank array until we find
the adjacent pair <em>{r1,
</tbody>
</table>
-<hr />
-
-<h3
id="quantilerank-exclusive_strict-or-qr-gt_strict-given-r-return-the-quantile-q-of-the-smallest-rank-that-is-strictly-greater-than-r"><strong><em>quantile(rank,
EXCLUSIVE_STRICT)</em></strong> or <strong><em>q(r, GT_STRICT)</em></strong>
:=<br />Given <em>r</em>, return the quantile, <em>q</em>, of the smallest rank
that is strictly Greater Than <em>r</em>.</h3>
-
-<p>In <b>STRICT</b> mode, the only difference is the following:</p>
-
-<p><b>Boundary Notes:</b></p>
-
-<ul>
- <li>If the given normalized rank, <em>r</em>, is equal to 1.0, there is no
quantile that satisfies this criterion. The function will return
<em>NaN</em>.</li>
-</ul>
-
-<hr />
-
-<h3
id="quantilerank-inclusive-or-qr-ge-given-r-return-the-quantile-q-of-the-smallest-rank-that-is-strictly-greater-than-or-equal-to-r"><strong><em>quantile(rank,
INCLUSIVE)</em></strong> or <strong><em>q(r, GE)</em></strong> :=<br />Given
<em>r</em>, return the quantile, <em>q</em>, of the smallest rank that is
strictly Greater than or Equal to <em>r</em>.</h3>
-
-<p><b>Implementation:</b>
-Given <em>r</em>, search the rank array until we find the adjacent pair
<em>{r1, r2}</em> where <em>r1 < r <= r2</em>. Return the quantile,
<em>q</em>, associated with <em>r2</em>, the second of the pair.</p>
-
-<p><b>Boundary Notes:</b></p>
-
<ul>
- <li>If the given normalized rank, <em>r</em>, is equal to 1.0, the function
will return the largest quantile retained by the sketch.</li>
- <li>If the given normalized rank, <em>r</em>, is less than the smallest
rank, the function will return the smallest quantile.</li>
+ <li><em>r(55) = 1.0</em> Use Boundary Rule 1: <em>50 < 55</em>, return
<em>1.0</em>.</li>
</ul>
-<p><b>Examples using normalized ranks:</b></p>
-
-<p>For example <em>q(.786) = 30</em></p>
-
<table>
<thead>
<tr>
@@ -1050,6 +1124,7 @@ Given <em>r</em>, search the rank array until we find the
adjacent pair <em>{r1,
<th>30</th>
<th>40</th>
<th>50</th>
+ <th>?</th>
</tr>
</thead>
<tbody>
@@ -1063,6 +1138,7 @@ Given <em>r</em>, search the rank array until we find the
adjacent pair <em>{r1,
<td>11</td>
<td>13</td>
<td>14</td>
+ <td> </td>
</tr>
<tr>
<td>Normalized Rank[]:</td>
@@ -1073,18 +1149,20 @@ Given <em>r</em>, search the rank array until we find
the adjacent pair <em>{r1,
<td>.643</td>
<td>.786</td>
<td>.929</td>
- <td>1.000</td>
+ <td>1.0</td>
+ <td> </td>
</tr>
<tr>
- <td>Rank input</td>
+ <td>Quantile input</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
- <td>.786</td>
<td> </td>
<td> </td>
+ <td> </td>
+ <td>55</td>
</tr>
<tr>
<td>Qualifying pair</td>
@@ -1092,19 +1170,1040 @@ Given <em>r</em>, search the rank array until we find
the adjacent pair <em>{r1,
<td> </td>
<td> </td>
<td> </td>
- <td>r1</td>
- <td>r2</td>
<td> </td>
<td> </td>
+ <td> </td>
+ <td>q1</td>
+ <td>(q2)</td>
</tr>
<tr>
- <td>Quantile result</td>
+ <td>Rank result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>1.000</td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>r(5) = 0.0</em> Use Boundary Rule 2: <em>5 <= 10</em>, return
<em>0</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>?</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td> </td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td> </td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Quantile input</td>
+ <td>5</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td>(q1)</td>
+ <td>q2</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Rank result</td>
+ <td>0</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
+
+<h2 id="the-quantile-functions-with-inequalities">The quantile functions with
inequalities</h2>
+
+<h3
id="quantilerank-inclusive-or-qr-ge-given-r-return-the-quantile-q-of-the-smallest-rank-that-is-strictly-greater-than-or-equal-to-r"><strong><em>quantile(rank,
INCLUSIVE)</em></strong> or <strong><em>q(r, GE)</em></strong> :=<br />Given
<em>r</em>, return the quantile, <em>q</em>, of the smallest rank that is
strictly Greater than or Equal to <em>r</em>.</h3>
+
+<p><b>Implementation:</b></p>
+
+<ul>
+ <li>Given <em>r</em>, search the rank array until we find the adjacent pair
<em>{r1, r2}</em> where <em>r1 < r <= r2</em>.</li>
+ <li>Return the quantile, <em>q</em>, associated with <em>r2</em>, the second
of the pair.</li>
+</ul>
+
+<p><b>Boundary Exceptions:</b></p>
+
+<ul>
+ <li><strong>Boundary Rule 2:</strong> If the given normalized rank,
<em>r</em>, is <em><=</em> the smallest rank, the function will return the
<strong>quantile</strong> associated with the smallest cumulative rank.</li>
+</ul>
+
+<p><b>Examples using normalized ranks:</b></p>
+
+<ul>
+ <li><em>q(.786) = 30</em> Normal rule applies: <em>.643 < .786 <=
.786</em>, return <em>q(r2) = 30</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.000</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>.786</td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>r2</td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>30</td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(1.0) = 50</em> Normal rule applies: <em>.929 < 1.0 <=
1.0</em>, return <em>q(r2) = 50</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>r2</td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>50</td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(0.0 <= .071) = 10</em> Use Boundary Rule 2: <em>0.0 <=
.071</em>, return <em>10</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>?</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td> </td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td> </td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td>0.0</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td>(r1)</td>
+ <td>r2</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Rank result</td>
+ <td> </td>
+ <td>10</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
+
+<hr />
+
+<h3
id="quantilerank-exclusive-or-qr-gt-given-r-return-the-quantile-q-of-the-smallest-rank-that-is-strictly-greater-than-r"><strong><em>quantile(rank,
EXCLUSIVE)</em></strong> or <strong><em>q(r, GT)</em></strong> :=<br />Given
<em>r</em>, return the quantile, <em>q</em>, of the smallest rank that is
strictly Greater Than <em>r</em>.</h3>
+
+<p><b>Implementation:</b></p>
+
+<ul>
+ <li>Given <em>r</em>, search the rank array until we find the adjacent pair
<em>{r1, r2}</em> where <em>r1 <= r < r2</em>.</li>
+ <li>Return the quantile, <em>q</em>, associated with <em>r2</em>, the second
of the pair.</li>
+</ul>
+
+<p><b>Boundary Exceptions:</b></p>
+
+<ul>
+ <li><strong>Boundary Rule 1:</strong> If the given normalized rank,
<em>r</em>, 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.</li>
+ <li><strong>Boundary Rule 2:</strong> If the given normalized rank,
<em>r</em>, is less than the smallest rank, the function will return the
quantile associated with the smallest cumulative rank retained by the
sketch.</li>
+</ul>
+
+<p><b>Examples using normalized ranks:</b></p>
+
+<ul>
+ <li><em>q(.357) = 30</em> Normal rule applies: <em>.357 <= .357 <
.500</em>, return <em>q(r2) = 30</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.000</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td>.357</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>r2</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>30</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(1.0) = 50</em> Use Boundary Rule 1 <em>1.0 <= 1.0 < ?</em>,
return <em>50</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ <th>?</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>1.0</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>(r2)</td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>50</td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(0.99) = 50</em> Normal rule applies <em>.929 <= .99 <
1.0</em>, return <em>50</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>.99</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>r2</td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>50</td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(0.0 <= .071) = 10</em> Use Boundary Rule 2: <em>0.0 <
.071</em>, return <em>10</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>?</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td> </td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td> </td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td>0.0</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td>(r1)</td>
+ <td>r2</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Rank result</td>
+ <td> </td>
+ <td>10</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
+
+<hr />
+
+<h3
id="quantilerank-exclusive_strict-or-qr-gt_strict-given-r-return-the-quantile-q-of-the-smallest-rank-that-is-strictly-greater-than-r"><strong><em>quantile(rank,
EXCLUSIVE_STRICT)</em></strong> or <strong><em>q(r, GT_STRICT)</em></strong>
:=<br />Given <em>r</em>, return the quantile, <em>q</em>, of the smallest rank
that is strictly Greater Than <em>r</em>.</h3>
+
+<p><b>Implementation:</b></p>
+
+<ul>
+ <li>Given <em>r</em>, search the rank array until we find the adjacent pair
<em>{r1, r2}</em> where <em>r1 <= r < r2</em>.</li>
+ <li>Return the quantile, <em>q</em>, associated with <em>r2</em>, the second
of the pair.</li>
+</ul>
+
+<p><b>Boundary Exceptions:</b></p>
+
+<ul>
+ <li><strong>Boundary Rule 1:</strong> If the given normalized rank,
<em>r</em>, is equal to <em>1.0</em>, there is no quantile that satisfies this
criterion. Return <em>NaN</em> or <em>null</em>.</li>
+ <li><strong>Boundary Rule 2:</strong> If the given normalized rank,
<em>r</em>, is less than the smallest rank, the function will return the
quantile associated with the smallest cumulative rank retained by the
sketch..</li>
+</ul>
+
+<p><b>Examples using normalized ranks:</b></p>
+
+<ul>
+ <li><em>q(.357) = 30</em> Normal rule applies: <em>.357 <= .357 <
.500</em>, return <em>q(r2) = 30</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.000</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td>.357</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>r2</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>30</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(1.0) = 50</em> Use Boundary Rule 1 <em>1.0 <= 1.0 < ?</em>,
return <em>NaN or null</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ <th>?</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>1.0</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>(r2)</td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>NaN or null</td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(0.99) = 50</em> Normal rule applies <em>.929 <= .99 <
1.0</em>, return <em>50</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>.99</td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>r1</td>
+ <td>r2</td>
+ </tr>
+ <tr>
+ <td>Quantile result</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td>50</td>
+ </tr>
+ </tbody>
+</table>
+
+<ul>
+ <li><em>q(0.0 <= .071) = 10</em> Use Boundary Rule 2: <em>0.0 <
.071</em>, return <em>10</em>.</li>
+</ul>
+
+<table>
+ <thead>
+ <tr>
+ <th>Quantile[]:</th>
+ <th>?</th>
+ <th>10</th>
+ <th>20</th>
+ <th>20</th>
+ <th>30</th>
+ <th>30</th>
+ <th>30</th>
+ <th>40</th>
+ <th>50</th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <td>Natural Rank[]:</td>
+ <td> </td>
+ <td>1</td>
+ <td>3</td>
+ <td>5</td>
+ <td>7</td>
+ <td>9</td>
+ <td>11</td>
+ <td>13</td>
+ <td>14</td>
+ </tr>
+ <tr>
+ <td>Normalized Rank[]:</td>
+ <td> </td>
+ <td>.071</td>
+ <td>.214</td>
+ <td>.357</td>
+ <td>.500</td>
+ <td>.643</td>
+ <td>.786</td>
+ <td>.929</td>
+ <td>1.0</td>
+ </tr>
+ <tr>
+ <td>Rank input</td>
+ <td>0.0</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Qualifying pair</td>
+ <td>(r1)</td>
+ <td>r2</td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ </tr>
+ <tr>
+ <td>Rank result</td>
+ <td> </td>
+ <td>10</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
- <td>30</td>
<td> </td>
<td> </td>
</tr>
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