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<title>MADlib: Prediction Metrics</title>
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<div class="header">
  <div class="headertitle">
<div class="title">Prediction Metrics<div class="ingroups"><a class="el" href="group__grp__mdl.html">Model Selection</a></div></div>  </div>
</div><!--header-->
<div class="contents">
<div class="toc"><b>Contents</b> <ul>
<li>
<a href="#list">List of Prediction Metric Functions</a> </li>
<li>
<a href="#specs">Function Specific Details</a> </li>
<li>
<a href="#examples">Examples</a> </li>
<li>
<a href="#literature">Literature</a> </li>
<li>
<a href="#related">Related Topics</a> </li>
</ul>
</div><p>This module provides a set of metrics to evaluate the quality of predictions of a model. A typical function will take a set of "prediction" and "observation" values and use them to calculate the desired metric, unless noted otherwise. Grouping is supported for all functions (except confusion matrix).</p>
<p><a class="anchor" id="list"></a></p><dl class="section user"><dt>Prediction Metrics Functions</dt><dd><table class="output">
<tr>
<th>mean_abs_error(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>Mean absolute error  </td></tr>
<tr>
<th>mean_abs_perc_error(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>Mean absolute percentage error  </td></tr>
<tr>
<th>mean_perc_error(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>Mean percentage error  </td></tr>
<tr>
<th>mean_squared_error(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>Mean squared error </td></tr>
<tr>
<th>r2_score(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>R-squared  </td></tr>
<tr>
<th>adjusted_r2_score(table_in, table_out, prediction_col, observed_col, num_predictors, training_size, grouping_cols)</th><td>Adjusted R-squared  </td></tr>
<tr>
<th>binary_classifier(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>Collection of prediction metrics related to binary classification </td></tr>
<tr>
<th>area_under_roc(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>Area under the ROC curve (in binary classification)  </td></tr>
<tr>
<th>confusion_matrix(table_in, table_out, prediction_col, observed_col, grouping_cols)</th><td>Confusion matrix for a multi-class classifier  </td></tr>
</table>
</dd></dl>
<p><b>Arguments</b> </p><dl class="arglist">
<dt>table_in </dt>
<dd>TEXT. Name of the input table. </dd>
<dt>table_out </dt>
<dd>TEXT. Name of the output table. For consistency, a table is created for all metric outputs even when grouping is not used, which may mean there is only a single value in the output table in some cases.  </dd>
<dt>prediction_col </dt>
<dd>TEXT. Name of the column of predicted values from input table. </dd>
<dt>observed_col </dt>
<dd>TEXT. Name of the column of observed values from input table. </dd>
<dt>num_predictors (for adjusted R-squared score only) </dt>
<dd>INTEGER. The number of parameters in the predicting model, not counting the constant term. </dd>
<dt>training_size (for adjusted R-squared score only) </dt>
<dd>INTEGER. The number of rows used for training, excluding any NULL rows. </dd>
<dt>grouping_cols (optional) </dt>
<dd>TEXT, default: NULL. Name of the column of grouping values from input table. </dd>
</dl>
<p><a class="anchor" id="specs"></a></p><dl class="section user"><dt>Function Specific Details</dt><dd></dd></dl>
<p><b>R-squared Score</b></p>
<p>This function returns the coefficient of determination (R2) between the predicted and observed values. An R2 of 1 indicates that the regression line perfectly fits the data, while an R2 of 0 indicates that the line does not fit the data at all. Negative values of R2 may occur when fitting non-linear functions to data. Please refer to reference <a href="#r2">[1]</a> for more details.</p>
<p><b>Adjusted R-squared Score</b></p>
<p>This function returns the adjusted R2 score in addition to the R-squared score described above. Adjusted R2 score is used to counter the problem of the R2 automatically increasing when extra explanatory variables are added to the model. It takes two additional parameters describing the degrees of freedom of the model (num_predictors) and the size of the training set over which it was developed (training_size):</p>
<ul>
<li>num_predictors: Indicates the number of parameters the model has other than the constant term. For example, if it is set to '3' the model may take the following form as an example: 7 + 5x + 39y + 0.91z.</li>
<li>training_size: Indicates the number of rows in the training set (excluding any NULL rows).</li>
</ul>
<p>Neither of these arguments can be deduced from the predicted values and the test data alone which is why they are explicit inputs. Please refer to reference <a href="#r2">[1]</a> for more details.</p>
<p><a class="anchor" id="bc"></a><b>Binary Classification</b></p>
<p>This function returns an output table with a number of metrics commonly used in binary classification.</p>
<p>The definitions of the various metrics are as follows:</p>
<ul>
<li>\(\textit{tp}\) is the count of correctly-classified positives.</li>
<li>\(\textit{tn}\) is the count of correctly-classified negatives.</li>
<li>\(\textit{fp}\) is the count of misclassified negatives.</li>
<li>\(\textit{fn}\) is the count of misclassified positives.</li>
<li>\(\textit{tpr}=\textit{tp}/(\textit{tp}+\textit{fn})\).</li>
<li>\(\textit{tnr}=\textit{tn}/(\textit{fp}+\textit{tn})\).</li>
<li>\(\textit{ppv}=\textit{tp}/(\textit{tp}+\textit{fp})\).</li>
<li>\(\textit{npv}=\textit{tn}/(\textit{tn}+\textit{fn})\).</li>
<li>\(\textit{fpr}=\textit{fp}/(\textit{fp}+\textit{tn})\).</li>
<li>\(\textit{fdr}=1-\textit{ppv}\).</li>
<li>\(\textit{fnr}=\textit{fn}/(\textit{fn}+\textit{tp})\).</li>
<li>\(\textit{acc}=(\textit{tp}+\textit{tn})/(\textit{tp}+\textit{tn}+\textit{fp} +\textit{fn})\).</li>
<li>\(\textit{f1}=2*\textit{tp}/(2*\textit{tp}+\textit{fp}+\textit{fn})\).</li>
</ul>
<p><b>Area Under ROC Curve</b></p>
<p>This function returns the area under the Receiver Operating Characteristic curve for binary classification (the AUC). The ROC curve is the curve relating the classifier's TPR and FPR metrics. (See <a href="#bc">Binary Classification</a> above for a definition of these metrics). Please refer to reference <a href="#aoc">[2]</a> for more details. Note that the binary classification function can be used to obtain the data (TPR and FPR values) required for drawing the ROC curve.</p>
<dl class="section note"><dt>Note</dt><dd>For 'binary_classifier' and 'area_under_roc' functions:<ul>
<li>The 'observed_col' column is assumed to be a numeric column with two values: 0 and 1, or a Boolean column. For the purposes of the metric calculation, 0 is considered to be negative and 1 to be positive.</li>
<li>The 'pred_col' column is expected to contain numeric values corresponding to likelihood/probability. A larger value corresponds to greater certainty that the observed value will be '1', and a lower value corresponds to a greater certainty that it will be '0'.</li>
</ul>
</dd></dl>
<p><b>Confusion Matrix</b></p>
<p>This function returns the confusion matrix of a multi-class classification. Each column of the matrix represents the instances in a predicted class while each row represents the instances in an actual class. This allows more detailed analysis than mere proportion of correct guesses (accuracy). Please refer to the reference <a href="#cm">[3]</a> for more details. Please note that grouping is not supported for the confusion matrix.</p>
<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
<ol type="1">
<li>Create the sample data: <pre class="example">
DROP TABLE IF EXISTS test_set;
CREATE TABLE test_set(
                  pred FLOAT8,
                  obs FLOAT8
                );
INSERT INTO test_set VALUES
  (37.5,53.1), (12.3,34.2), (74.2,65.4), (91.1,82.1);
</pre></li>
<li>Run the Mean Absolute Error function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.mean_abs_error( 'test_set', 'table_out', 'pred', 'obs');
SELECT * FROM table_out;
</pre> Result <pre class="result">
 mean_abs_error
&#160;----------------
         13.825
</pre></li>
<li>Run the Mean Absolute Percentage Error function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.mean_abs_perc_error( 'test_set', 'table_out', 'pred', 'obs');
SELECT * FROM table_out;
</pre> Result <pre class="result">
 mean_abs_perc_error
&#160;---------------------
   0.294578793636013
</pre></li>
<li>Run the Mean Percentage Error function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.mean_perc_error( 'test_set', 'table_out', 'pred', 'obs');
SELECT * FROM table_out;
</pre> Result <pre class="result">
 mean_perc_error
&#160;-------------------
   -0.17248930032771
</pre></li>
<li>Run the Mean Squared Error function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.mean_squared_error( 'test_set', 'table_out', 'pred', 'obs');
SELECT * FROM table_out;
</pre> Result <pre class="result">
 mean_squared_error
&#160;--------------------
   220.3525
</pre></li>
<li>Run the R2 Score function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.r2_score( 'test_set', 'table_out', 'pred', 'obs');
SELECT * FROM table_out;
</pre> Result <pre class="result">
 r2_score
&#160;------------------------
   0.27992908844337695865
</pre></li>
<li>Run the Adjusted R2 Score function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.adjusted_r2_score( 'test_set', 'table_out', 'pred', 'obs', 3, 100);
SELECT * FROM table_out;
</pre> Result <pre class="result">
       r2_score      | adjusted_r2_score
&#160;--------------------+------------------
   0.279929088443375 | 0.257426872457231
</pre></li>
<li>Create the sample data for binary classifier metrics: <pre class="example">
DROP TABLE IF EXISTS test_set;
CREATE TABLE test_set AS
    SELECT ((a*8)::integer)/8.0 pred,
        ((a*0.5+random()*0.5)&gt;0.5) obs
    FROM (select random() as a from generate_series(1,100)) x;
</pre></li>
<li>Run the Binary Classifier metrics function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.binary_classifier( 'test_set', 'table_out', 'pred', 'obs');
</pre></li>
<li>View the True Positive Rate and the False Positive Rate: <pre class="example">
SELECT threshold, tpr, fpr FROM table_out ORDER BY threshold;
</pre> Result (your results for this and other functions below will look different due to the presence of the random function in sample data generator): <pre class="result">
       threshold        |          tpr           |          fpr
------------------------+------------------------+------------------------
 0.00000000000000000000 | 1.00000000000000000000 | 1.00000000000000000000
 0.12500000000000000000 | 1.00000000000000000000 | 0.94915254237288135593
 0.25000000000000000000 | 0.92682926829268292683 | 0.64406779661016949153
 0.37500000000000000000 | 0.80487804878048780488 | 0.47457627118644067797
 0.50000000000000000000 | 0.70731707317073170732 | 0.35593220338983050847
 0.62500000000000000000 | 0.63414634146341463415 | 0.25423728813559322034
 0.75000000000000000000 | 0.48780487804878048780 | 0.06779661016949152542
 0.87500000000000000000 | 0.29268292682926829268 | 0.03389830508474576271
 1.00000000000000000000 | 0.12195121951219512195 | 0.00000000000000000000
</pre></li>
<li>View all metrics at a given threshold value: <pre class="example">
-- Set extended display on for easier reading of output
\x on
SELECT * FROM table_out WHERE threshold=0.5;
</pre> Result <pre class="result">
-[ RECORD 1 ]---------------------
threshold | 0.50000000000000000000
tp        | 29
fp        | 21
fn        | 12
tn        | 38
tpr       | 0.70731707317073170732
tnr       | 0.64406779661016949153
ppv       | 0.58000000000000000000
npv       | 0.76000000000000000000
fpr       | 0.35593220338983050847
fdr       | 0.42000000000000000000
fnr       | 0.29268292682926829268
acc       | 0.67000000000000000000
f1        | 0.63736263736263736264
</pre></li>
<li>Run the Area Under ROC curve function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.area_under_roc( 'test_set', 'table_out', 'pred', 'obs');
SELECT * FROM table_out;
</pre> Result <pre class="result">
 area_under_roc
&#160;---------------------------------------------
0.77428689541132699462698842496899545266640
</pre></li>
<li>Create the sample data for confusion matrix. <pre class="example">
DROP TABLE IF EXISTS test_set;
CREATE TABLE test_set AS
    SELECT (x+y)%5+1 AS pred,
        (x*y)%5 AS obs
    FROM generate_series(1,5) x,
        generate_series(1,5) y;
</pre></li>
<li>Run the confusion matrix function: <pre class="example">
DROP TABLE IF EXISTS table_out;
SELECT madlib.confusion_matrix( 'test_set', 'table_out', 'pred', 'obs');
SELECT * FROM table_out ORDER BY class;
</pre> Result <pre class="result">
 class | confusion_arr
-------+---------------
     0 | {0,1,2,2,2,2}
     1 | {0,2,0,1,1,0}
     2 | {0,0,0,2,2,0}
     3 | {0,0,2,0,0,2}
     4 | {0,2,1,0,0,1}
     5 | {0,0,0,0,0,0}
</pre></li>
</ol>
<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
<p><a class="anchor" id="r2"></a> [1] <a href="https://en.wikipedia.org/wiki/Coefficient_of_determination">https://en.wikipedia.org/wiki/Coefficient_of_determination</a></p>
<p><a class="anchor" id="aoc"></a> [2] <a href="https://en.wikipedia.org/wiki/Receiver_operating_characteristic">https://en.wikipedia.org/wiki/Receiver_operating_characteristic</a></p>
<p><a class="anchor" id="cm"></a> [3] <a href="https://en.wikipedia.org/wiki/Confusion_matrix">https://en.wikipedia.org/wiki/Confusion_matrix</a></p>
<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
<p>File <a class="el" href="pred__metrics_8sql__in.html" title="A collection of summary statistics to gauge model accuracy based on predicted values vs...">pred_metrics.sql_in</a> for list of functions and usage. </p>
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