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| <div class="collapse" id="collapse_background"> |
| <li><a href="/docs/Background/TheChallenge.html">•The Challenge</a></li> |
| <li><a href="/docs/Background/SketchOrigins.html">•Sketch Origins</a></li> |
| <li><a href="/docs/Background/SketchElements.html">•Sketch Elements</a></li> |
| <li><a href="/docs/Background/Presentations.html">•Presentations</a></li> |
| <li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/DataSketches_deck.pdf">•Overview Slide Deck</a></li> |
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| <li><a href="/docs/Architecture/MajorSketchFamilies.html">•The Major Sketch Families</a></li> |
| <li><a href="/docs/Architecture/LargeScale.html">•Large Scale Computing</a></li> |
| <li><a href="/docs/Architecture/KeyFeatures.html">•Key Features</a></li> |
| <li><a href="/docs/Architecture/SketchFeaturesMatrix.html">•Sketch Features Matrix</a></li> |
| <li><a href="/docs/Architecture/Components.html">•Components</a></li> |
| <li><a href="/docs/Architecture/SketchesByComponent.html">•Sketches by Component</a></li> |
| <li><a href="/docs/Architecture/SketchCriteria.html">•Sketch Criteria</a></li> |
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| <p id="memory-component"> |
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| <li><a href="/docs/Memory/MemoryComponent.html">•Memory Component</a></li> |
| <li><a href="/docs/Memory/MemoryPerformance.html">•Memory Component Performance</a></li> |
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| <li><a href="/docs/Architecture/OrderSensitivity.html">•Notes on Order Sensitivity</a></li> |
| <li><a href="/docs/Architecture/Concurrency.html">•Notes on Concurrency</a></li> |
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| <li><a href="/docs/DistinctCountFeaturesMatrix.html">•Features Matrix</a></li> |
| <li><a href="/docs/DistinctCountMeritComparisons.html">•Figures-of-Merit Comparison</a></li> |
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| <p id="cpc-sketches"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_cpc_sketches">CPC Sketches</a> |
| </p> |
| <div class="collapse" id="collapse_cpc_sketches"> |
| <li><a href="/docs/CPC/CPC.html">•CPC Sketch</a></li> |
| <li><a href="/docs/CPC/CpcPerformance.html">•CPC Sketch Performance</a></li> |
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| <a data-toggle="collapse" class="menu collapsed" href="#collapse_cpc_examples">CPC Examples</a> |
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| <li><a href="/docs/CPC/CpcJavaExample.html">•CPC Sketch Java Example</a></li> |
| <li><a href="/docs/CPC/CpcCppExample.html">•CPC Sketch C++ Example</a></li> |
| <li><a href="/docs/CPC/CpcPigExample.html">•CPC Sketch Pig UDFs</a></li> |
| <li><a href="/docs/CPC/CpcHiveExample.html">•CPC Sketch Hive UDFs</a></li> |
| </div> |
| </div> |
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| <p id="hyperloglog-sketches"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_hyperloglog_sketches">HyperLogLog Sketches</a> |
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| <li><a href="/docs/HLL/HLL.html">•HLL Sketch</a></li> |
| <li><a href="/docs/HLL/HllMap.html">•HLL Map Sketch</a></li> |
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| <p id="hll-examples"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_hll_examples">HLL Examples</a> |
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| <li><a href="/docs/HLL/HllJavaExample.html">•HLL Sketch Java Example</a></li> |
| <li><a href="/docs/HLL/HllCppExample.html">•HLL Sketch C++ Example</a></li> |
| <li><a href="/docs/HLL/HllPigUDFs.html">•HLL Sketch Pig UDFs</a></li> |
| <li><a href="/docs/HLL/HllHiveUDFs.html">•HLL Sketch Hive UDFs</a></li> |
| </div> |
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| <p id="hll-studies"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_hll_studies">HLL Studies</a> |
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| <div class="collapse" id="collapse_hll_studies"> |
| <li><a href="/docs/HLL/HllPerformance.html">•HLL Sketch Performance</a></li> |
| <li><a href="/docs/HLL/Hll_vs_CS_Hllpp.html">•HLL vs Clearspring HLL++</a></li> |
| <li><a href="/docs/HLL/HllSketchVsDruidHyperLogLogCollector.html">•HLL Sketch vs Druid HyperLogLogCollector</a></li> |
| </div> |
| </div> |
| |
| <p id="theta-sketches"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_theta_sketches">Theta Sketches</a> |
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| <div class="collapse" id="collapse_theta_sketches"> |
| <li><a href="/docs/Theta/ThetaSketchFramework.html">•Theta Sketch Framework</a></li> |
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| <p id="theta-examples"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_theta_examples">Theta Examples</a> |
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| <div class="collapse" id="collapse_theta_examples"> |
| <li><a href="/docs/Theta/ConcurrentThetaSketch.html">•Concurrent Theta Sketch</a></li> |
| <li><a href="/docs/Theta/ThetaJavaExample.html">•Theta Sketch Java Example</a></li> |
| <li><a href="/docs/Theta/ThetaSparkExample.html">•Theta Sketch Spark Example</a></li> |
| <li><a href="/docs/Theta/ThetaPigUDFs.html">•Theta Sketch Pig UDFs</a></li> |
| <li><a href="/docs/Theta/ThetaHiveUDFs.html">•Theta Sketch Hive UDFs</a></li> |
| </div> |
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| <p id="kmv-tutorial"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_kmv_tutorial">KMV Tutorial</a> |
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| <div class="collapse" id="collapse_kmv_tutorial"> |
| <li><a href="/docs/Theta/InverseEstimate.html">•The Inverse Estimate</a></li> |
| <li><a href="/docs/Theta/KMVempty.html">•Empty Sketch</a></li> |
| <li><a href="/docs/Theta/KMVfirstEst.html">•First Estimator</a></li> |
| <li><a href="/docs/Theta/KMVbetterEst.html">•Better Estimator</a></li> |
| <li><a href="/docs/Theta/KMVrejection.html">•Rejection Rules</a></li> |
| <li><a href="/docs/Theta/KMVupdateVkth.html">•Update V(kth) Rule</a></li> |
| </div> |
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| <p id="set-operations-and-p-sampling"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_set_operations_and_p-sampling">Set Operations and P-sampling</a> |
| </p> |
| <div class="collapse" id="collapse_set_operations_and_p-sampling"> |
| <li><a href="/docs/Theta/ThetaSketchSetOps.html">•Set Operations</a></li> |
| <li><a href="/docs/Theta/ThetaSetOpsCornerCases.html">•Model & Test Set Operations</a></li> |
| <li><a href="/docs/Theta/ThetaPSampling.html">•<i>p</i>-Sampling</a></li> |
| </div> |
| |
| <p id="accuracy"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_accuracy">Accuracy</a> |
| </p> |
| <div class="collapse" id="collapse_accuracy"> |
| <li><a href="/docs/Theta/ThetaAccuracy.html">•Basic Accuracy</a></li> |
| <li><a href="/docs/Theta/ThetaAccuracyPlots.html">•Accuracy Plots</a></li> |
| <li><a href="/docs/Theta/ThetaErrorTable.html">•Relative Error Table</a></li> |
| <li><a href="/docs/Theta/ThetaSketchSetOpsAccuracy.html">•SetOp Accuracy</a></li> |
| <li><a href="/docs/Theta/AccuracyOfDifferentKUnions.html">•Unions With Different k</a></li> |
| </div> |
| |
| <p id="size"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_size">Size</a> |
| </p> |
| <div class="collapse" id="collapse_size"> |
| <li><a href="/docs/Theta/ThetaSize.html">•Theta Sketch Size</a></li> |
| </div> |
| |
| <p id="speed"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_speed">Speed</a> |
| </p> |
| <div class="collapse" id="collapse_speed"> |
| <li><a href="/docs/Theta/ThetaUpdateSpeed.html">•Update Speed</a></li> |
| <li><a href="/docs/Theta/ThetaMergeSpeed.html">•Merge Speed</a></li> |
| </div> |
| |
| <p id="theta-sketch-theory"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_theta_sketch_theory">Theta Sketch Theory</a> |
| </p> |
| <div class="collapse" id="collapse_theta_sketch_theory"> |
| <li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/ThetaSketchFramework.pdf">•Theta Sketch Framework (PDF)</a></li> |
| <li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/ThetaSketchEquations.pdf">•Theta Sketch Equations (PDF)</a></li> |
| <li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/DataSketches.pdf">•DataSketches (PDF)</a></li> |
| <li><a href="/docs/Theta/ThetaConfidenceIntervals.html">•Confidence Intervals Notes</a></li> |
| <li><a href="/docs/Theta/ThetaMergingAlgorithm.html">•Merging Algorithm Notes</a></li> |
| <li><a href="/docs/Theta/ThetaReferences.html">•Theta References</a></li> |
| </div> |
| </div> |
| |
| <p id="tuple-sketches"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_tuple_sketches">Tuple Sketches</a> |
| </p> |
| <div class="collapse" id="collapse_tuple_sketches"> |
| <li><a href="/docs/Tuple/TupleOverview.html">•Tuple Overview</a></li> |
| |
| <p id="tuple-examples"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_tuple_examples">Tuple Examples</a> |
| </p> |
| <div class="collapse" id="collapse_tuple_examples"> |
| <li><a href="/docs/Tuple/TupleJavaExample.html">•Tuple Java Example</a></li> |
| <li><a href="/docs/Tuple/TupleEngagementExample.html">•Tuple Engagement Example</a></li> |
| <li><a href="/docs/Tuple/TuplePigUDFs.html">•Tuple Pig UDFs</a></li> |
| <li><a href="/docs/Tuple/TupleHiveUDFs.html">•Tuple Hive UDFs</a></li> |
| </div> |
| </div> |
| </div> |
| |
| <p id="most-frequent"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_most_frequent">Most Frequent</a> |
| </p> |
| <div class="collapse" id="collapse_most_frequent"> |
| <li><a href="/docs/Frequency/FrequencySketchesOverview.html">•Frequency Sketches Overview</a></li> |
| |
| <p id="frequent-item-sketches"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_frequent_item_sketches">Frequent Item Sketches</a> |
| </p> |
| <div class="collapse" id="collapse_frequent_item_sketches"> |
| <li><a href="/docs/Frequency/FrequentItemsOverview.html">•Frequent Items Overview</a></li> |
| <li><a href="/docs/Frequency/FrequentItemsErrorTable.html">•Frequent Items Error Table</a></li> |
| <li><a href="/docs/Frequency/FrequentItemsReferences.html">•Frequent Items References</a></li> |
| <li><a href="/docs/Frequency/FrequentItemsPerformance.html">•Frequent Items Performance</a></li> |
| |
| <p id="most-frequent-examples"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_most_frequent_examples">Most Frequent Examples</a> |
| </p> |
| <div class="collapse" id="collapse_most_frequent_examples"> |
| <li><a href="/docs/Frequency/FrequentItemsJavaExample.html">•Frequent Items Java Example</a></li> |
| <li><a href="/docs/Frequency/FrequentItemsCppExample.html">•Frequent Items C++ Example</a></li> |
| <li><a href="/docs/Frequency/FrequentItemsPigUDFs.html">•Frequent Items Pig UDFs</a></li> |
| <li><a href="/docs/Frequency/FrequentItemsHiveUDFs.html">•Frequent Items Hive UDFs</a></li> |
| </div> |
| </div> |
| |
| <p id="frequent-distinct-sketches"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_frequent_distinct_sketches">Frequent Distinct Sketches</a> |
| </p> |
| <div class="collapse" id="collapse_frequent_distinct_sketches"> |
| <li><a href="/docs/Frequency/FrequentDistinctTuplesSketch.html">•Frequent Distinct Tuples Sketch</a></li> |
| </div> |
| </div> |
| |
| <p id="quantiles-and-histograms"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_and_histograms">Quantiles And Histograms</a> |
| </p> |
| <div class="collapse" id="collapse_quantiles_and_histograms"> |
| <li><a href="/docs/Quantiles/SketchingQuantilesAndRanksTutorial.html">•Quantiles and Ranks Tutorial</a></li> |
| <li><a href="/docs/Quantiles/QuantilesOverview.html">•Quantiles Overview</a></li> |
| <li><a href="/docs/KLL/KLLSketch.html">•KLL Floats sketch</a></li> |
| <li><a href="/docs/KLL/KLLAccuracyAndSize.html">•KLL Sketch Accuracy and Size</a></li> |
| <li><a href="/docs/REQ/ReqSketch.html">•REQ Floats sketch</a></li> |
| <li><a href="/docs/Quantiles/OrigQuantilesSketch.html">•Original QuantilesSketch</a></li> |
| |
| <p id="quantiles-examples"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_examples">Quantiles Examples</a> |
| </p> |
| <div class="collapse" id="collapse_quantiles_examples"> |
| <li><a href="/docs/Quantiles/QuantilesJavaExample.html">•Quantiles Sketch Java Example</a></li> |
| <li><a href="/docs/KLL/KLLCppExample.html">•KLL Quantiles Sketch C++ Example</a></li> |
| <li><a href="/docs/Quantiles/QuantilesPigUDFs.html">•Quantiles Sketch Pig UDFs</a></li> |
| <li><a href="/docs/Quantiles/QuantilesHiveUDFs.html">•Quantiles Sketch Hive UDFs</a></li> |
| </div> |
| |
| <p id="quantiles-studies"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_studies">Quantiles Studies</a> |
| </p> |
| <div class="collapse" id="collapse_quantiles_studies"> |
| <li><a href="/docs/QuantilesStudies/DruidApproxHistogramStudy.html">•Druid Approximate Histogram</a></li> |
| <li><a href="/docs/QuantilesStudies/MomentsSketchStudy.html">•Moments Sketch Study</a></li> |
| <li><a href="/docs/QuantilesStudies/QuantilesStreamAStudy.html">•Quantiles StreamA Study</a></li> |
| <li><a href="/docs/QuantilesStudies/ExactQuantiles.html">•Exact Quantiles for Studies</a></li> |
| </div> |
| |
| <p id="quantiles-sketch-theory"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_quantiles_sketch_theory">Quantiles Sketch Theory</a> |
| </p> |
| <div class="collapse" id="collapse_quantiles_sketch_theory"> |
| <li><a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/Quantiles_KLL.pdf">•Optimal Quantile Approximation in Streams</a></li> |
| <li><a href="/docs/Quantiles/QuantilesReferences.html">•Quantiles References</a></li> |
| </div> |
| </div> |
| |
| <p id="sampling"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_sampling">Sampling</a> |
| </p> |
| <div class="collapse" id="collapse_sampling"> |
| <li><a href="/docs/Sampling/ReservoirSampling.html">•Reservoir Sampling</a></li> |
| <li><a href="/docs/Sampling/ReservoirSamplingPerformance.html">•Reservoir Sampling Performance</a></li> |
| <li><a href="/docs/Sampling/VarOptSampling.html">•VarOpt Sampling</a></li> |
| |
| <p id="sampling-examples"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_sampling_examples">Sampling Examples</a> |
| </p> |
| <div class="collapse" id="collapse_sampling_examples"> |
| <li><a href="/docs/Sampling/ReservoirSamplingJava.html">•Reservoir Sampling Java Example</a></li> |
| <li><a href="/docs/Sampling/ReservoirSamplingPigUDFs.html">•Reservoir Sampling Pig UDFs</a></li> |
| <li><a href="/docs/Sampling/VarOptSamplingJava.html">•VarOpt Sampling Java Example</a></li> |
| <li><a href="/docs/Sampling/VarOptPigUDFs.html">•VarOpt Sampling Pig UDFs</a></li> |
| </div> |
| </div> |
| </div> |
| |
| <p id="system-integrations"> |
| <a data-toggle="collapse" class="menu collapsed" href="#collapse_system_integrations">System Integrations</a> |
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| <h2 id="theta-sketch-set-operations-accuracy">Theta Sketch Set Operations Accuracy</h2> |
| |
| <h3 id="the-theta-rules">The Theta Rules</h3> |
| |
| <p>All set operations (Union, Intersection, Difference) between two sketches <i>A</i> and <i>B</i> |
| must obey the following two rules:</p> |
| |
| <ul> |
| <li><i>θ<sub>result</sub> = min( θ<sub>A</sub> , θ<sub>B</sub> )</i>.</li> |
| <li>All entries retained in the result sketch must be less than <i>θ<sub>result</sub></i>.</li> |
| </ul> |
| |
| <p>These rules can be extended to arbitrary set expressions as:</p> |
| |
| <ul> |
| <li><i>θ<sub>result</sub> = min{ θ<sub>i</sub> }</i>.</li> |
| <li>All entries retained in the result sketch must be less than <i>θ<sub>result</sub></i>.</li> |
| </ul> |
| |
| <h3 id="union-set-expressions">Union Set Expressions</h3> |
| |
| <h4 id="source-sketches-and-target-with-the-same-nominal-entries-or-k">Source sketches and target with the same <i>Nominal Entries</i> or <i>k</i></h4> |
| |
| <p>As long as all source sketches and target have been configured with the same <i>Nominial Entries</i> |
| or <i>k</i>, and there has been no other intervening <i>Intersection</i> or <i>AnotB</i> operations, |
| the accuracy of the resulting <i>Union</i> sketch will have the same Relative Standard Error (RSE) |
| as determined by the <i>Nominal Entries</i> or <i>k</i> value. In other words,</p> |
| |
| <ul> |
| <li><i>Union Estimate = k/θ<sub>result</sub></i>.</li> |
| <li>RSE<i><sub>union</sub> = 1 / sqrt(k - 1)</i>.</li> |
| </ul> |
| |
| <p>This remains true no matter how many sketches are unioned together.</p> |
| |
| <h4 id="source-sketches-and-target-with-different-nominal-entries-or-k">Source sketches and target with different <i>Nominal Entries</i> or <i>k</i></h4> |
| |
| <p>The Relative Standard Error (RSE) of the resulting <i>Union</i> sketch will determined by |
| the <i>Theta Rules</i> above. Ultimately, the source sketch with the smallest theta will |
| dominate the overall resulting error of the result. Given two sketches with equal cardinalities and |
| different values of <i>k</i>, the sketch with the smaller value of <i>k</i> will have the smallest |
| value of theta and will largely determine the error distribution of the result.</p> |
| |
| <h3 id="mixed-set-expressions-union-intersection-anotb">Mixed Set Expressions (Union, Intersection, AnotB)</h3> |
| |
| <p>Conceptually, the Intersection and AnotB functions operate by first performing a Union of all the |
| values from both source sketches and then identifying the appropriate proper subset of that |
| Union set creating a result sketch with the Union theta (which is the minimum theta of the source |
| sketches) and the qualifying subset of values.</p> |
| |
| <p>This means, of course, that depending on the operations and the data, the result set could have |
| zero, all, or some number in between of the retained values of the Union sketch. |
| Mixed set expressions can produce an error distribution that is larger that of a standard sketch |
| of a given <i>Nominal Entries</i> or <i>k</i> and is mathematically described in |
| <a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/SketchEquations.pdf">Sketch Equations / Subsets of Fixed <i>k</i> Sampling</a>.</p> |
| |
| <h4 id="source-sketches-and-target-with-the-same-nominal-entries-or-k-1">Source sketches and target with the same <i>Nominal Entries</i> or <i>k</i></h4> |
| |
| <p>When the source and target sketches have the same value of <i>k</i>, |
| the accuracy of the result sketch has a relatively straightforward mathematical solution and intuition:</p> |
| |
| <center>RSE = <i>sqrt( est(Union(A,B)) / est(SetOperation(A,B)) ) * sqrt( 1/ (k-1) )</i></center> |
| |
| <h5 id="example">Example</h5> |
| |
| <p>The intuition for this can be explained using this simple example of intersection |
| (set difference would behave similarly):</p> |
| |
| <p>Suppose we have as inputs two segments of unique identifiers, <i>A</i> and <i>B</i>.</p> |
| |
| <p>Assumptions:</p> |
| |
| <ul> |
| <li>We will use a sketch size of <i>k = 4096</i> entries with an RSE ~ 1/sqrt(k) = 1.6%.</li> |
| <li>Segment <i>A</i> has 4 million (4M) unique identifiers.</li> |
| <li>Segment <i>B</i> has 4 thousand (4K) unique identifiers and is a proper subset of <i>A</i>.</li> |
| </ul> |
| |
| <p>Steps:</p> |
| |
| <ul> |
| <li>Build the Sketches. Sketch <i>S<sub>A</sub></i> will be fed segment <i>A</i> and Sketch <i>S<sub>B</sub></i> |
| will be fed segment <i>B</i>. |
| <ul> |
| <li>Segment <i>B</i> fits exactly in <i>S<sub>B</sub></i>, so <i>θ<sub>B</sub></i> = 4K/4K = 1.0.</li> |
| <li>Sketch <i>S<sub>A</sub></i> will end up with <i>θ<sub>A</sub></i> = 4K/4M = .001.</li> |
| </ul> |
| </li> |
| </ul> |
| |
| <p>The hash values retained in <i>S<sub>A</sub></i> represent a uniform sampling of all of the unique identifiers |
| in segment <i>A</i> and the resulting value of <i>θ<sub>A</sub></i> represents the effective sampling rate |
| required to end up with <i>k</i> samples, which is ~ 4K/4M = .001.</p> |
| |
| <p>Even though in the raw data all the values of segment <i>B</i> are in segment <i>A</i>, the probability |
| that all the 4K samples of <i>S<sub>B</sub></i> appear <i>S<sub>A</sub></i> is extremely low since |
| only one in one thousand can be randomly chosen.</p> |
| |
| <p>Applying the <i>Theta Rules</i>:</p> |
| |
| <ul> |
| <li><i>θ<sub>result</sub> = min(0.001, 1.0) = .001</i></li> |
| <li>All the entries from <i>S<sub>A</sub></i> already qualify.</li> |
| <li>Since all 4K values in <i>S<sub>B</sub></i> are uniformly distributed between 0 and 1.0, only .001 of them |
| or approximately 4 of the bottom values will remain.</li> |
| <li>The resulting intersection sketch will have, on average, only 4 values and a theta of .001.</li> |
| </ul> |
| |
| <p>The mean estimate from the intersection sketch will be 4/.001 = 4K. |
| This happens to be correct using this hand-wavy analysis but in general is a random result with a variance. |
| The proof that the estimate will be unbiased is in the attached |
| <a href="https://github.com/apache/datasketches-website/tree/master/docs/pdf/SketchEquations.pdf">Sketch Equations</a>.</p> |
| |
| <p>The RSE of a sketch with only 4 values is ~ 1/sqrt(4) = .5 or 50% error. |
| This is considerably larger than the RSE of either <i>S<sub>A</sub></i> or <i>S<sub>B</sub></i>, |
| which is ~ 1.6% as stated above.</p> |
| |
| <p>The insight to be gained from this example is that it was the theta (sampling rate) of the sketch of the |
| larger population that caused the increase in error. |
| And, for this example, increasing the sketch size of <i>S<sub>A</sub></i> would improve the resulting error. |
| The general case may be more complex.</p> |
| |
| <p>More formally, if we define a factor <i>F</i> to be the ratio:</p> |
| |
| <center><i>F</i> = (size of Union(A,B) ) / (size of Intersection(A,B).</center> |
| |
| <p>Then a simple way to compute the resulting RSE of an intersection (or difference) is</p> |
| |
| <center>RSE<sub><i>intersection</i></sub> = <i>sqrt(F) * (RSE of input Sketches)</i></center> |
| |
| <p>And in our example:</p> |
| |
| <center>RSE<sub><i>intersection</i></sub> = sqrt(4M/4K) * 1/sqrt(4K) = 31.63 * 0.016 = 0.5 = 50%.</center> |
| |
| <h4 id="source-sketches-and-target-with-different-nominal-entries-or-k-1">Source sketches and target with different <i>Nominal Entries</i> or <i>k</i></h4> |
| |
| <p>In the general case this scenario is even more complex and difficult to predict mathematically, but a conservative |
| estimate would be to use the above equations substituting <i>k</i> with the smallest of the participating |
| <i>k</i> values.</p> |
| |
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