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commit ea484f7060d72220fb723599380c1ca20cb775da
Author: buildbot <[email protected]>
AuthorDate: Sat Aug 3 00:14:25 2024 +0000
Automatic Site Publish by Buildbot
---
output/docs/KLL/UnderstandingKLLBounds.html | 42 +++++++++++------------------
1 file changed, 16 insertions(+), 26 deletions(-)
diff --git a/output/docs/KLL/UnderstandingKLLBounds.html
b/output/docs/KLL/UnderstandingKLLBounds.html
index 333f2a07..b81e6f02 100644
--- a/output/docs/KLL/UnderstandingKLLBounds.html
+++ b/output/docs/KLL/UnderstandingKLLBounds.html
@@ -568,30 +568,20 @@
<span class="n">println</span><span class="o">(</span><span class="s">"r2:
"</span> <span class="o">+</span> <span class="n">r2</span><span
class="o">);</span>
<span class="n">println</span><span class="o">(</span><span
class="s">""</span><span class="o">);</span>
- <span class="kt">double</span> <span class="n">r1LB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getRankLowerBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"r1LB: "</span> <span class="o">+</span> <span
class="n">r1LB</span><span class="o">);</span>
- <span class="kt">double</span> <span class="n">r1UB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getRankUpperBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"r1UB: "</span> <span class="o">+</span> <span
class="n">r1UB</span><span class="o">);</span>
-
- <span class="kt">double</span> <span class="n">r2LB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getRankLowerBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"r2LB: "</span> <span class="o">+</span> <span
class="n">r2LB</span><span class="o">);</span>
- <span class="kt">double</span> <span class="n">r2UB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getRankUpperBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"r2UB: "</span> <span class="o">+</span> <span
class="n">r2UB</span><span class="o">);</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"r1LB(r1): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getRankLowerBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">));</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"r1UB(r1): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getRankUpperBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">));</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"r2LB(r2): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getRankLowerBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">));</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"r2UB(r2): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getRankUpperBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">));</span>
<span class="n">println</span><span class="o">(</span><span
class="s">""</span><span class="o">);</span>
- <span class="kt">double</span> <span class="n">q1LB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getQuantileLowerBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"q1LB(r1): "</span> <span class="o">+</span> <span
class="n">q1LB</span><span class="o">);</span>
- <span class="kt">double</span> <span class="n">q1UB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getQuantileUpperBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"q1UB(r1): "</span> <span class="o">+</span> <span
class="n">q1UB</span><span class="o">);</span>
-
- <span class="kt">double</span> <span class="n">q2LB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getQuantileLowerBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"q2LB(r2): "</span> <span class="o">+</span> <span
class="n">q2LB</span><span class="o">);</span>
- <span class="kt">double</span> <span class="n">q2UB</span> <span
class="o">=</span> <span class="n">sk</span><span class="o">.</span><span
class="na">getQuantileUpperBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">);</span>
- <span class="n">println</span><span class="o">(</span><span
class="s">"q2UB(r2): "</span> <span class="o">+</span> <span
class="n">q2UB</span><span class="o">);</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"q1LB(r1): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getQuantileLowerBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">));</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"q1UB(r1): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getQuantileUpperBound</span><span class="o">(</span><span
class="n">r1</span><span class="o">));</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"q2LB(r2): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getQuantileLowerBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">));</span>
+ <span class="n">println</span><span class="o">(</span><span
class="s">"q2UB(r2): "</span> <span class="o">+</span> <span
class="n">sk</span><span class="o">.</span><span
class="na">getQuantileUpperBound</span><span class="o">(</span><span
class="n">r2</span><span class="o">));</span>
<span class="n">println</span><span class="o">(</span><span
class="s">""</span><span class="o">);</span>
<span class="o">}</span>
- <span class="kd">static</span> <span class="kt">void</span> <span
class="nf">println</span><span class="o">(</span><span class="nc">Object</span>
<span class="n">o</span><span class="o">)</span> <span class="o">{</span> <span
class="nc">System</span><span class="o">.</span><span
class="na">out</span><span class="o">.</span><span
class="na">println</span><span class="o">(</span><span class="n">o</span><span
class="o">.</span><span class="na">toString</span><span class="o">());</span>
<s [...]
+ <span class="kd">private</span> <span class="kd">static</span> <span
class="kt">void</span> <span class="nf">println</span><span
class="o">(</span><span class="nc">Object</span> <span class="n">o</span><span
class="o">)</span> <span class="o">{</span> <span class="nc">System</span><span
class="o">.</span><span class="na">out</span><span class="o">.</span><span
class="na">println</span><span class="o">(</span><span class="n">o</span><span
class="o">.</span><span class="na">toString</spa [...]
<span class="o">}</span>
</code></pre></div></div>
@@ -605,10 +595,10 @@ q2: 620.0
r1: 0.5
r2: 0.52
-r1LB: 0.4932237572729862
-r1UB: 0.5067762427270138
-r2LB: 0.5132237572729862
-r2UB: 0.5267762427270138
+r1LB(r1): 0.4932237572729862
+r1UB(r1): 0.5067762427270138
+r2LB(r2): 0.5132237572729862
+r2UB(r2): 0.5267762427270138
q1LB(r1): 494.0
q1UB(r1): 608.0
@@ -627,15 +617,15 @@ PASSED: exampleQuantileBounds
<p>We choose two quantiles on either side of the discontinuity, 500 and 620,
and get their respective ranks of 0.5 and 0.52. Note that because of the
discontinuity the difference in the input quantiles is 120/1100 or ~10.9%,
while the difference in their respective ranks is only 2%.</p>
-<p>Next we compute the upper and lower rank bounds of the two resulting ranks
of 0.5 and 0.52, which are given above. Note that the UB - LB of each rank is
about .013 which is 2 X .0067. This means that the true rank of each quantile
is within the UB - LB range of ranks with a confidence of 99%, which is about
+/- 2.6 standard deviations from the mean.</p>
+<p>Next we compute the rank upper bound (UB) and rank lower bound (LB) of the
two resulting ranks of 0.5 and 0.52, which are given above. Note that the UB -
LB of each rank is about .013 which is 2 X .0067. This means that the true
rank of each quantile is within the UB - LB range of ranks with a confidence of
99%, which is about +/- 2.6 standard deviations from the estimate.</p>
-<p>Then we compute the upper and lower quantile bounds of the same two
resulting ranks of 0.5 and 0.52. Note that the UB - LB quantile range of
<em>r1</em> is 114/1100 or 10.4%, because in between the rank UB and LB is the
discontinuity. These points are shown in the next image.</p>
+<p>Then we compute the quantile UB and LB of the same two resulting ranks of
0.5 and 0.52. Note that the UB - LB quantile range of <em>r1</em> is 114/1100
or 10.4%, because in between the rank UB and LB is the discontinuity. These
points are shown in the next plot</p>
<p><img class="doc-img-half" src="/docs/img/kll/QuantileBounds2.png"
alt="QuantileBounds2.png" /></p>
<p>This graphically illustrates why the mathematical guarantee of error
applies only to the rank domain, because the input quantile domian could have
huge discontinuities. Nonetheless, we <strong>can</strong> say that the true
quantile does lie within that UB - LB quantile range with a confidence of 99%.
But we cannot guarantee anything about the UB - LB quantile difference and
relate that to a quantile accuracy compared to the total range of the input
values.</p>
-<p>Our Classic, KLL, and REQ quantiles sketches are input insensitive and do
not know or care what the input distribution looks like. It does not have to be
a smooth and well behaved function! This is not the case with other heuristic
quantile algorithms,</p>
+<p>Our Classic, KLL, and REQ quantiles sketches are input insensitive and do
not know or care what the input distribution looks like. It does not have to be
a smooth and well behaved function. This is not the case with other heuristic
quantile algorithms,</p>
</div> <!-- End content -->
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