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Posted to commits@madlib.apache.org by ok...@apache.org on 2017/12/28 22:52:08 UTC
[34/51] [abbrv] [partial] madlib-site git commit: Additional updates
for 1.13 release
http://git-wip-us.apache.org/repos/asf/madlib-site/blob/6c103d3e/docs/v1.13/graph_legend.html
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diff --git a/docs/v1.13/graph_legend.html b/docs/v1.13/graph_legend.html
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+ <div class="headertitle">
+<div class="title">All Pairs Shortest Path<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#apsp">APSP</a> </li>
+<li>
+<a href="#notes">Notes</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#literature">Literature</a> </li>
+</ul>
+</div><p>The all pairs shortest paths (APSP) algorithm finds the length (summed weights) of the shortest paths between all pairs of vertices, such that the sum of the weights of the path edges is minimized.</p>
+<dl class="section warning"><dt>Warning</dt><dd>APSP is an expensive algorithm for run-time because it finds the shortest path between all nodes in the graph. It is recommended that you start with a small graph to get a sense of run-time for your use case, then increase size carefully from there. The worst case run-time for this implementation is O(V^2 * E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, but it depends on the graph.</dd></dl>
+<p><a class="anchor" id="apsp"></a></p><dl class="section user"><dt>APSP</dt><dd><pre class="syntax">
+graph_apsp( vertex_table,
+ vertex_id,
+ edge_table,
+ edge_args,
+ out_table,
+ grouping_cols
+ )
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>vertex_table </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.</p>
+<p class="enddd"></p>
+</dd>
+<dt>vertex_id </dt>
+<dd><p class="startdd">TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids are of type INTEGER with no duplicates. They do not need to be contiguous.</p>
+<p class="enddd"></p>
+</dd>
+<dt>edge_table </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex, destination vertex and edge weight. Column naming convention is described below in the 'edge_args' parameter.</p>
+<p class="enddd"></p>
+</dd>
+<dt>edge_args </dt>
+<dd><p class="startdd">TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:</p><ul>
+<li>src (INTEGER): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.</li>
+<li>dest (INTEGER): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.</li>
+<li>weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.</li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to store the result of APSP. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):</p><ul>
+<li>source_vertex: The id for the source vertex. Will use the input edge column 'src' for column naming.</li>
+<li>dest_vertex: The id for the destination vertex. Will use the input edge column 'dest' for column naming.</li>
+<li>weight: The total weight of the shortest path from the source vertex to the destination vertex. Will use the input parameter 'weight' for column naming.</li>
+<li>parent: The parent of the destination vertex in the shortest path from source. Parent will equal dest_vertex if there are no intermediate vertices. Will use 'parent' for column naming.</li>
+</ul>
+<p>A summary table named <out_table>_summary is also created. This is an internal table that keeps a record of the input parameters and is used by the path retrieval function described below. </p>
+<p class="enddd"></p>
+</dd>
+<dt>grouping_cols (optional) </dt>
+<dd>TEXT, default = NULL. List of columns used to group the input into discrete subgraphs. These columns must exist in the edge table. When this value is null, no grouping is used and a single APSP result is generated. </dd>
+</dl>
+<dl class="section user"><dt>Path Retrieval</dt><dd></dd></dl>
+<p>The path retrieval function returns the shortest path from the source vertex to a specified desination vertex.</p>
+<pre class="syntax">
+graph_apsp_get_path( apsp_table,
+ source_vertex,
+ dest_vertex,
+ path_table
+ )
+</pre><p><b>Arguments</b> </p><dl class="arglist">
+<dt>apsp_table </dt>
+<dd><p class="startdd">TEXT. Name of the table that contains the APSP output.</p>
+<p class="enddd"></p>
+</dd>
+<dt>source_vertex </dt>
+<dd><p class="startdd">INTEGER. The vertex that will be the source of the desired path.</p>
+<p class="enddd"></p>
+</dd>
+<dt>dest_vertex </dt>
+<dd><p class="startdd">INTEGER. The vertex that will be the destination of the desired path.</p>
+<p class="enddd"></p>
+</dd>
+<dt>path_table </dt>
+<dd><p class="startdd">TEXT. Name of the output table that contains the path. It contains a row for every group and has the following columns:</p><ul>
+<li>grouping_cols: The grouping columns given in the creation of the APSP table. If there are no grouping columns, these columns will not exist and the table will have a single row.</li>
+<li>path (ARRAY): The shortest path from the source vertex to the destination vertex. </li>
+</ul>
+<p class="enddd"></p>
+</dd>
+</dl>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<p>Graphs with negative edges are supported but graphs with negative cycles are not.</p>
+<p>The implementation is analogous to a matrix multiplication procedure. Please refer to the MADlib design document and references [1] and [2] for more details.</p>
+<p>Also see the Grail project [3] for more background on graph analytics processing in relational databases.</p>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Create vertex and edge tables to represent the graph: <pre class="syntax">
+DROP TABLE IF EXISTS vertex, edge;
+CREATE TABLE vertex(
+ id INTEGER
+ );
+CREATE TABLE edge(
+ src INTEGER,
+ dest INTEGER,
+ weight FLOAT8
+ );
+INSERT INTO vertex VALUES
+(0),
+(1),
+(2),
+(3),
+(4),
+(5),
+(6),
+(7);
+INSERT INTO edge VALUES
+(0, 1, 1.0),
+(0, 2, 1.0),
+(0, 4, 10.0),
+(1, 2, 2.0),
+(1, 3, 10.0),
+(2, 3, 1.0),
+(2, 5, 1.0),
+(2, 6, 3.0),
+(3, 0, 1.0),
+(4, 0, -2.0),
+(5, 6, 1.0),
+(6, 7, 1.0);
+</pre></li>
+<li>Calculate the shortest paths: <pre class="syntax">
+DROP TABLE IF EXISTS out, out_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ NULL, -- Edge arguments (NULL means use default naming)
+ 'out'); -- Output table of shortest paths
+SELECT * FROM out ORDER BY src,dest;
+</pre> <pre class="result">
+ src | dest | weight | parent
+-----+------+----------+--------
+ 0 | 0 | 0 | 0
+ 0 | 1 | 1 | 1
+ 0 | 2 | 1 | 2
+ 0 | 3 | 2 | 2
+ 0 | 4 | 10 | 4
+ 0 | 5 | 2 | 2
+ 0 | 6 | 3 | 5
+ 0 | 7 | 4 | 6
+ 1 | 0 | 4 | 3
+ 1 | 1 | 0 | 1
+ 1 | 2 | 2 | 2
+ 1 | 3 | 3 | 2
+ 1 | 4 | 14 | 0
+ 1 | 5 | 3 | 2
+ 1 | 6 | 4 | 5
+ 1 | 7 | 5 | 6
+(showing only 16 of 64 rows)
+</pre></li>
+<li>Get the shortest path from vertex 0 to vertex 5: <pre class="syntax">
+DROP TABLE IF EXISTS out_path;
+SELECT madlib.graph_apsp_get_path('out',0,5,'out_path');
+SELECT * FROM out_path;
+</pre> <pre class="result">
+ path
+---------
+ {0,2,5}
+</pre></li>
+<li>Now let's do a similar example except using different column names in the tables (i.e., not the defaults). Create the vertex and edge tables: <pre class="syntax">
+DROP TABLE IF EXISTS vertex_alt, edge_alt;
+CREATE TABLE vertex_alt AS SELECT id AS v_id FROM vertex;
+CREATE TABLE edge_alt AS SELECT src AS e_src, dest, weight AS e_weight FROM edge;
+</pre></li>
+<li>Calculate the shortest paths: <pre class="syntax">
+DROP TABLE IF EXISTS out_alt, out_alt_summary;
+SELECT madlib.graph_apsp(
+ 'vertex_alt', -- Vertex table
+ 'v_id', -- Vertex id column
+ 'edge_alt', -- Edge table
+ 'src=e_src, weight=e_weight', -- Edge arguments
+ 'out_alt'); -- Output table of shortest paths
+SELECT * FROM out_alt ORDER BY e_src, dest;
+</pre> <pre class="result">
+ e_src | dest | e_weight | parent
+-------+------+----------+--------
+ 0 | 0 | 0 | 0
+ 0 | 1 | 1 | 1
+ 0 | 2 | 1 | 2
+ 0 | 3 | 2 | 2
+ 0 | 4 | 10 | 4
+ 0 | 5 | 2 | 2
+ 0 | 6 | 3 | 5
+ 0 | 7 | 4 | 6
+ 1 | 0 | 4 | 3
+ 1 | 1 | 0 | 1
+ 1 | 2 | 2 | 2
+ 1 | 3 | 3 | 2
+ 1 | 4 | 14 | 0
+ 1 | 5 | 3 | 2
+ 1 | 6 | 4 | 5
+ 1 | 7 | 5 | 6
+(showing only 16 of 64 rows)
+</pre></li>
+<li>Create a graph with 2 groups and find APSP for each group: <pre class="syntax">
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src < 6 AND dest < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+</pre></li>
+<li>Find APSP for all groups: <pre class="syntax">
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ NULL, -- Edge arguments (NULL means use default naming)
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+SELECT * FROM out_gr WHERE src < 2 ORDER BY grp,src,dest;
+</pre> <pre class="result">
+ grp | src | dest | weight | parent
+-----+-----+------+--------+--------
+ 0 | 0 | 0 | 0 | 0
+ 0 | 0 | 1 | 1 | 1
+ 0 | 0 | 2 | 1 | 2
+ 0 | 0 | 3 | 2 | 2
+ 0 | 0 | 4 | 10 | 4
+ 0 | 0 | 5 | 2 | 2
+ 0 | 0 | 6 | 4 | 2
+ 0 | 0 | 7 | 5 | 6
+ 0 | 1 | 0 | 4 | 3
+ 0 | 1 | 1 | 0 | 1
+ 0 | 1 | 2 | 2 | 2
+ 0 | 1 | 3 | 3 | 2
+ 0 | 1 | 4 | 14 | 0
+ 0 | 1 | 5 | 3 | 2
+ 0 | 1 | 6 | 4 | 5
+ 0 | 1 | 7 | 5 | 6
+ 1 | 0 | 0 | 0 | 0
+ 1 | 0 | 1 | 1 | 1
+ 1 | 0 | 2 | 1 | 2
+ 1 | 0 | 3 | 2 | 2
+ 1 | 0 | 4 | 10 | 4
+ 1 | 0 | 5 | -10 | 4
+ 1 | 1 | 0 | 4 | 3
+ 1 | 1 | 1 | 0 | 1
+ 1 | 1 | 2 | 2 | 2
+ 1 | 1 | 3 | 3 | 2
+ 1 | 1 | 4 | 14 | 0
+ 1 | 1 | 5 | -6 | 4
+(28 rows)
+</pre></li>
+<li>Find the path from vertex 0 to vertex 5 in every group <pre class="syntax">
+DROP TABLE IF EXISTS out_gr_path;
+SELECT madlib.graph_apsp_get_path('out_gr',0,5,'out_gr_path');
+SELECT * FROM out_gr_path ORDER BY grp;
+</pre> <pre class="result">
+ grp | path
+-----+---------
+ 0 | {0,2,5}
+ 1 | {0,4,5}
+</pre></li>
+</ol>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] <a href="http://www.columbia.edu/~cs2035/courses/ieor6614.S11/apsp.pdf">http://www.columbia.edu/~cs2035/courses/ieor6614.S11/apsp.pdf</a></p>
+<p>[2] <a href="http://users.cecs.anu.edu.au/~Alistair.Rendell/Teaching/apac_comp3600/module4/all_pairs_shortest_paths.xhtml">http://users.cecs.anu.edu.au/~Alistair.Rendell/Teaching/apac_comp3600/module4/all_pairs_shortest_paths.xhtml</a></p>
+<p>[3] The case against specialized graph analytics engines, J. Fan, G. Soosai Raj, and J. M. Patel. CIDR 2015. <a href="http://cidrdb.org/cidr2015/Papers/CIDR15_Paper20.pdf">http://cidrdb.org/cidr2015/Papers/CIDR15_Paper20.pdf</a> </p>
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+ <div id="projectname">
+ <span id="projectnumber">1.13</span>
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+ <div id="projectbrief">User Documentation for MADlib</div>
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+ <div class="headertitle">
+<div class="title">ARIMA<div class="ingroups"><a class="el" href="group__grp__tsa.html">Time Series Analysis</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li class="level1">
+<a href="#train">Training Function</a> </li>
+<li class="level1">
+<a href="#forecast">Forecasting Function</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#background">Technical Background</a> </li>
+<li class="level1">
+<a href="#literature">Literature</a> </li>
+<li class="level1">
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>Given a time series of data X, the Autoregressive Integrated Moving Average (ARIMA) model is a tool for understanding and, perhaps, predicting future values in the series. The model consists of three parts, an autoregressive (AR) part, a moving average (MA) part, and an integrated (I) part where an initial differencing step can be applied to remove any non-stationarity in the signal. The model is generally referred to as an ARIMA(p, d, q) model where parameters p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively.</p>
+<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function</dt><dd></dd></dl>
+<p>The ARIMA training function has the following syntax. </p><pre class="syntax">
+arima_train( input_table,
+ output_table,
+ timestamp_column,
+ timeseries_column,
+ grouping_columns,
+ include_mean,
+ non_seasonal_orders,
+ optimizer_params
+ )
+</pre><p><b>Arguments</b> </p><dl class="arglist">
+<dt>input_table </dt>
+<dd><p class="startdd">TEXT. The name of the table containing time series data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_table </dt>
+<dd><p class="startdd">TEXT. The name of the table to store the ARIMA model. Three tables are created, with names based on the value of the <em>output_table</em> argument in the training function:</p>
+<ol type="1">
+<li><em>output_table</em>: Table containing the ARIMA model. Contains the following columns: <table class="output">
+<tr>
+<th>mean </th><td>Model mean (only if 'include_mean' is TRUE) </td></tr>
+<tr>
+<th>mean_std_error </th><td>Standard errors for mean </td></tr>
+<tr>
+<th>ar_params </th><td>Auto-regressions parameters of the ARIMA model </td></tr>
+<tr>
+<th>ar_std_errors </th><td>Standard errors for AR parameters </td></tr>
+<tr>
+<th>ma_params </th><td>Moving average parameters of the ARIMA model </td></tr>
+<tr>
+<th>ma_std_errors </th><td>Standard errors for MA parameters </td></tr>
+</table>
+</li>
+<li><em>output_table</em>_summary: Table containing descriptive statistics of the ARIMA model. Contains the following columns: <table class="output">
+<tr>
+<th>input_table </th><td>Table name with the source data </td></tr>
+<tr>
+<th>timestamp_col </th><td>Column name in the source table that contains the timestamp index to data </td></tr>
+<tr>
+<th>timeseries_col </th><td>Column name in the source table that contains the data values </td></tr>
+<tr>
+<th>non_seasonal_orders </th><td>Orders of the non-seasonal ARIMA model </td></tr>
+<tr>
+<th>include_mean </th><td>TRUE if intercept was included in ARIMA model </td></tr>
+<tr>
+<th>residual_variance </th><td>Variance of the residuals </td></tr>
+<tr>
+<th>log_likelihood </th><td>Log likelihood value (when using MLE) </td></tr>
+<tr>
+<th>iter_num </th><td>The number of iterations executed </td></tr>
+<tr>
+<th>exec_time </th><td>Total time taken to train the model </td></tr>
+</table>
+</li>
+<li><em>output_table</em>_residual: Table containing the residuals for each data point in 'input_table'. Contains the following columns: <table class="output">
+<tr>
+<th>timestamp_col </th><td>Same as the 'timestamp_col' parameter (all indices from source table included except the first <em>d</em> elements, where <em>d</em> is the differencing order value from 'non_seasonal_orders') </td></tr>
+<tr>
+<th>residual </th><td>Residual value for each data point </td></tr>
+</table>
+</li>
+</ol>
+<p></p>
+<p class="enddd"></p>
+</dd>
+<dt>timestamp_column </dt>
+<dd><p class="startdd">TEXT. The name of the column containing the timestamp (index) data. This could be a serial index (INTEGER) or date/time value (TIMESTAMP).</p>
+<p class="enddd"></p>
+</dd>
+<dt>timeseries_column </dt>
+<dd><p class="startdd">TEXT. The name of the column containing the time series data. This data is currently restricted to DOUBLE PRECISION.</p>
+<p class="enddd"></p>
+</dd>
+<dt>grouping_columns (optional) </dt>
+<dd><p class="startdd">TEXT, default: NULL. <em>Not currently implemented. Any non-NULL value is ignored.</em></p>
+<p>A comma-separated list of column names used to group the input dataset into discrete groups, training one ARIMA model per group. It is similar to the SQL <code>GROUP BY</code> clause. When this value is null, no grouping is used and a single result model is generated.</p>
+<p class="enddd"></p>
+</dd>
+<dt>include_mean (optional) </dt>
+<dd><p class="startdd">BOOLEAN, default: FALSE. Mean value of the data series is added in the ARIMA model if this variable is True. </p>
+<p class="enddd"></p>
+</dd>
+<dt>non_seasonal_orders (optional) </dt>
+<dd><p class="startdd">INTEGER[], default: 'ARRAY[1,1,1]'. Orders of the ARIMA model. The orders are [p, d, q], where parameters p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively. </p>
+<p class="enddd"></p>
+</dd>
+<dt>optimizer_params (optional) </dt>
+<dd>TEXT. Comma-separated list of optimizer-specific parameters of the form ‘name=value'. The order of the parameters does not matter. The following parameters are recognized:<ul>
+<li><b>max_iter:</b> Maximum number of iterations to run learning algorithm (Default = 100)</li>
+<li><b>tau:</b> Computes the initial step size for gradient algorithm (Default = 0.001)</li>
+<li><b>e1:</b> Algorithm-specific threshold for convergence (Default = 1e-15)</li>
+<li><b>e2:</b> Algorithm-specific threshold for convergence (Default = 1e-15)</li>
+<li><b>e3:</b> Algorithm-specific threshold for convergence (Default = 1e-15)</li>
+<li><b>hessian_delta:</b> Delta parameter to compute a numerical approximation of the Hessian matrix (Default = 1e-6) </li>
+</ul>
+</dd>
+</dl>
+<p><a class="anchor" id="forecast"></a></p><dl class="section user"><dt>Forecasting Function</dt><dd></dd></dl>
+<p>The ARIMA forecast function has the following syntax. </p><pre class="syntax">
+arima_forecast( model_table,
+ output_table,
+ steps_ahead
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>model_table </dt>
+<dd><p class="startdd">TEXT. The name of the table containing the ARIMA model trained on the time series dataset.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_table </dt>
+<dd><p class="startdd">TEXT. The name of the table to store the forecasted values. The output table produced by the forecast function contains the following columns. </p><table class="output">
+<tr>
+<th>group_by_cols </th><td>Grouping column values (if grouping parameter is provided) </td></tr>
+<tr>
+<th>step_ahead </th><td>Time step for the forecast </td></tr>
+<tr>
+<th>forecast_value </th><td>Forecast of the current time step </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>steps_ahead </dt>
+<dd>INTEGER. The number of steps to forecast at the end of the time series. </dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd><ol type="1">
+<li>View online help for the ARIMA training function. <pre class="example">
+SELECT madlib.arima_train();
+</pre></li>
+<li>Create an input data set. <pre class="example">
+DROP TABLE IF EXISTS arima_beer;
+CREATE TABLE arima_beer (time_id integer NOT NULL, value double precision NOT NULL );
+COPY arima_beer (time_id, value) FROM stdin WITH DELIMITER '|';
+1 | 93.2
+2 | 96.0
+3 | 95.2
+4 | 77.0
+5 | 70.9
+6 | 64.7
+7 | 70.0
+8 | 77.2
+9 | 79.5
+10 | 100.5
+11 | 100.7
+12 | 107.0
+13 | 95.9
+14 | 82.7
+15 | 83.2
+16 | 80.0
+17 | 80.4
+18 | 67.5
+19 | 75.7
+20 | 71.0
+21 | 89.2
+22 | 101.0
+23 | 105.2
+24 | 114.0
+25 | 96.2
+26 | 84.4
+27 | 91.2
+28 | 81.9
+29 | 80.5
+30 | 70.4
+31 | 74.7
+32 | 75.9
+33 | 86.2
+34 | 98.7
+35 | 100.9
+36 | 113.7
+37 | 89.7
+38 | 84.4
+39 | 87.2
+40 | 85.5
+\.
+</pre></li>
+<li>Train an ARIMA model. <pre class="example">
+-- Train ARIMA model with 'grouping_columns'=NULL, 'include_mean'=TRUE,
+-- and 'non_seasonal_orders'=[1,1,1]
+SELECT madlib.arima_train( 'arima_beer',
+ 'arima_beer_output',
+ 'time_id',
+ 'value',
+ NULL,
+ FALSE,
+ ARRAY[1, 1, 1]
+ );
+</pre></li>
+<li>Examine the ARIMA model. <pre class="example">
+\x ON
+SELECT * FROM arima_beer_output;
+</pre> Result: <pre class="result">
+-[ RECORD 1 ]-+------------------
+ar_params | {0.221954769696}
+ar_std_errors | {0.575367782602}
+ma_params | {-0.140623564576}
+ma_std_errors | {0.533445214346}
+</pre></li>
+<li>View the summary statistics table. <pre class="example">
+SELECT * FROM arima_beer_output_summary;
+</pre> Result: <pre class="result">
+-[ RECORD 1 ]-------+---------------
+input_table | arima_beer
+timestamp_col | time_id
+timeseries_col | value
+non_seasonal_orders | {1,1,1}
+include_mean | f
+residual_variance | 100.989970539
+log_likelihood | -145.331516396
+iter_num | 28
+exec_time (s) | 2.75
+</pre></li>
+<li>View the residuals. <pre class="example">
+\x OFF
+SELECT * FROM arima_beer_output_residual;
+</pre> Result: <pre class="result">
+ time_id | residual
+---------+--------------------
+ 2 | 0
+ 4 | -18.222328834394
+ 6 | -5.49616627282665
+...
+ 35 | 1.06298837051437
+ 37 | -25.0886854003757
+ 39 | 3.48401666299571
+(40 rows)
+</pre></li>
+<li>Use the ARIMA forecast function to forecast 10 future values. <pre class="example">
+SELECT madlib.arima_forecast( 'arima_beer_output',
+ 'arima_beer_forecast_output',
+ 10
+ );
+SELECT * FROM arima_beer_forecast_output;
+</pre> Result: <pre class="result">
+ steps_ahead | forecast_value
+-------------+----------------
+ 1 | 85.3802343659
+ 3 | 85.3477516875
+ 5 | 85.3461514635
+ 7 | 85.3460726302
+ 9 | 85.3460687465
+ 2 | 85.3536518121
+ 4 | 85.3464421267
+ 6 | 85.3460869494
+ 8 | 85.3460694519
+ 10 | 85.34606859
+(10 rows)
+</pre></li>
+</ol>
+</dd></dl>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd>An ARIMA model is an <em>a</em>uto-<em>r</em>egressive <em>i</em>ntegrated <em>m</em>oving <em>a</em>verage model. An ARIMA model is typically expressed in the form <p class="formulaDsp">
+\[ (1 - \phi(B)) Y_t = (1 + \theta(B)) Z_t, \]
+</p>
+</dd></dl>
+<p>where \(B\) is the backshift operator. The time \( t \) is from \( 1 \) to \( N \).</p>
+<p>ARIMA models involve the following variables:</p><ul>
+<li>The values of the time series: \( X_t \).</li>
+<li>Parameters of the model: \( p \), \( q \), and \( d \); \( d \) is the differencing order, \( p \) is the order of the AR operator, and \( q \) is the order of the MA operator.</li>
+<li>The AR operator: \( \phi(B) \).</li>
+<li>The MA operator: \( \theta(B) \).</li>
+<li>The lag difference: \( Y_{t} \), where \( Y_{t} = (1-B)^{d}(X_{t} - \mu) \).</li>
+<li>The mean value: \( \mu \), which is set to be zero for \( d>0 \) and estimated from the data when d=0.</li>
+<li>The error terms: \( Z_t \).</li>
+</ul>
+<p>The auto regression operator models the prediction for the next observation as some linear combination of the previous observations. More formally, an AR operator of order \( p \) is defined as</p>
+<p class="formulaDsp">
+\[ \phi(B) Y_t= \phi_1 Y_{t-1} + \dots + \phi_{p} Y_{t-p} \]
+</p>
+<p>The moving average operator is similar, and it models the prediction for the next observation as a linear combination of the errors in the previous prediction errors. More formally, the MA operator of order \( q \) is defined as</p>
+<p class="formulaDsp">
+\[ \theta(B) Z_t = \theta_{1} Z_{t-1} + \dots + \theta_{q} Z_{t-q}. \]
+</p>
+<p>We estimate the parameters using the Levenberg-Marquardt Algorithm. In mathematics and computing, the Levenberg-Marquardt algorithm (LMA), also known as the damped least-squares (DLS) method, provides a numerical solution to the problem of minimizing a function, generally nonlinear, over a space of parameters of the function.</p>
+<p>Like other numeric minimization algorithms, LMA is an iterative procedure. To start a minimization, the user has to provide an initial guess for the parameter vector, $p$, as well as some tuning parameters \(\tau, \epsilon_1, \epsilon_2, \epsilon_3,\).</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] Rob J Hyndman and George Athanasopoulos: Forecasting: principles and practice, <a href="http://otexts.com/fpp/">http://otexts.com/fpp/</a></p>
+<p>[2] Robert H. Shumway, David S. Stoffer: Time Series Analysis and Its Applications With R Examples, Third edition Springer Texts in Statistics, 2010</p>
+<p>[3] Henri Gavin: The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems, 2011</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="arima_8sql__in.html" title="Arima function for forecasting of timeseries data. ">arima.sql_in</a> documenting the ARIMA functions </p>
+</div><!-- contents -->
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+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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+ <div id="projectname">
+ <span id="projectnumber">1.13</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+ <img id="MSearchSelect" src="search/mag_sel.png"
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+<div class="header">
+ <div class="headertitle">
+<div class="title">Array Operations<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transformations</a> » <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#notes">Implementation Notes</a> </li>
+<li>
+<a href="#list">List of Array Operations</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>This module provides a set of basic array operations implemented in C and SQL. It is a support module for several machine learning algorithms that require fast array operations.</p>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Implementation Notes</dt><dd></dd></dl>
+<p>All functions (except <a class="el" href="array__ops_8sql__in.html#acb57ea4521dcb717f9e3148e0acccc74" title="This function normalizes an array as sum of squares to be 1. ">normalize()</a> and <a class="el" href="array__ops_8sql__in.html#acc295a568878940ffc3e2c9a75990efb" title="This function takes an array as the input and keep only elements that satisfy the operator on specifi...">array_filter()</a>) described in this module work with 2-D arrays.</p>
+<p>These functions support several numeric types:</p><ul>
+<li>SMALLINT</li>
+<li>INTEGER</li>
+<li>BIGINT</li>
+<li>REAL</li>
+<li>DOUBLE PRECISION (FLOAT8)</li>
+<li>NUMERIC (internally casted into FLOAT8, loss of precisions can happen)</li>
+</ul>
+<p>Additionally, <a class="el" href="array__ops_8sql__in.html#af057b589f2a2cb1095caa99feaeb3d70" title="This function takes a 2-D array as the input and unnests it by one level. It returns a set of 1-D arr...">array_unnest_2d_to_1d()</a> supports other data types such as TEXT or VARCHAR.</p>
+<p>Several of the function require NO NULL VALUES, while others omit NULLs and return results. See details in description of individual functions.</p>
+<p><a class="anchor" id="list"></a></p><dl class="section user"><dt>Array Operations</dt><dd><table class="output">
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a91c8d3715142927b3967f05a4fbf1575" title="Adds two arrays. It requires that all the values are NON-NULL. Return type is the same as the input t...">array_add()</a></th><td><p class="starttd">Adds two arrays. It requires that all the values are NON-NULL. Return type is the same as the input type.</p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a26e8508a2bae10a6574cec697a270eea" title="Aggregate, element-wise sum of arrays. It requires that all the values are NON-NULL. Return type is the same as the input type. ">sum()</a></th><td><p class="starttd">Aggregate, sums vector element-wisely. It requires that all the values are NON-NULL. Return type is the same as the input type.</p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a2875a161a01c7dcdea9a4997b074eefc" title="Subtracts two arrays. It requires that all the values are NON-NULL. Return type is the same as the in...">array_sub()</a></th><td><p class="starttd">Subtracts two arrays. It requires that all the values are NON-NULL. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a652d70c480d484c4a1a92ded384b0dd7" title="Element-wise product of two arrays. It requires that all the values are NON-NULL. Return type is the ...">array_mult()</a></th><td><p class="starttd">Element-wise product of two arrays. It requires that all the values are NON-NULL. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a6cc05e7052495f8b64692faf40219576" title="Element-wise division of two arrays. It requires that all the values are NON-NULL. Return type is the same as the input type. ">array_div()</a></th><td><p class="starttd">Element-wise division of two arrays. It requires that all the values are NON-NULL. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#acde10964ed23b7c8da515fb84cb8d5e0" title="Dot-product of two arrays. It requires that all the values are NON-NULL. Return type is the same as t...">array_dot()</a></th><td><p class="starttd">Dot-product of two arrays. It requires that all the values are NON-NULL. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#aedf6cb13eb4803bcc12dc4d95ea8ff4e" title="Checks whether one array contains the other. This function returns TRUE if each non-zero element in t...">array_contains()</a></th><td><p class="starttd">Checks whether one array contains the other. This function returns TRUE if each non-zero element in the right array equals to the element with the same index in the left array. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#ae891429cc50705c530f3e5ca15541849" title="This function finds the maximum value in the array. NULLs are ignored. Return type is the same as the...">array_max()</a></th><td><p class="starttd">This function finds the maximum value in the array. NULLs are ignored. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#aa415256a9064aecc600dfb5e377fb7b1" title="This function finds the maximum value and corresponding index in the array. NULLs are ignored...">array_max_index()</a></th><td><p class="starttd">This function finds the maximum value and corresponding index in the array. NULLs are ignored. Return type is array in format [max, index], and its element type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a6659bf9d9363eb179fab34f81f8ac59b" title="This function finds the minimum value in the array. NULLs are ignored. Return type is the same as the...">array_min()</a></th><td><p class="starttd">This function finds the minimum value in the array. NULLs are ignored. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a813a4d9ffc1c18b1b3e18f6ecdb2051f" title="This function finds the minimum value and corresponding index in the array. NULLs are ignored...">array_min_index()</a></th><td><p class="starttd">This function finds the minimum value and corresponding index in the array. NULLs are ignored. Return type is array in format [min, index], and its element type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a4c98f20e6a737358806f63318daea5ec" title="This function finds the sum of the values in the array. NULLs are ignored. Return type is the same as...">array_sum()</a></th><td><p class="starttd">This function finds the sum of the values in the array. NULLs are ignored. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a418de59800833aa95f9b7cbd6b12901c" title="This function finds the sum of the values in the array. NULLs are ignored. Return type is always FLOA...">array_sum_big()</a></th><td><p class="starttd">This function finds the sum of the values in the array. NULLs are ignored. Return type is always FLOAT8 regardless of input. This function is meant to replace <a class="el" href="array__ops_8sql__in.html#a4c98f20e6a737358806f63318daea5ec" title="This function finds the sum of the values in the array. NULLs are ignored. Return type is the same as...">array_sum()</a> in cases when a sum may overflow the element type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a13c0b0c53e8b0dc4e08c21bb8152ee7d" title="This function finds the sum of abs of the values in the array. NULLs are ignored. Return type is the ...">array_abs_sum()</a></th><td><p class="starttd">This function finds the sum of abs of the values in the array. NULLs are ignored. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#ac14e74c10b58f5518cd0e3e56067e5ba" title="This function takes an array as the input and finds absolute value of each element in the array...">array_abs()</a></th><td><p class="starttd">This function takes an array as the input and finds abs of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a407598f9eb70637798b02fd731bfca2c" title="This function finds the mean of the values in the array. NULLs are ignored. ">array_mean()</a></th><td><p class="starttd">This function finds the mean of the values in the array. NULLs are ignored. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a3b6c2d173a611e6d6b184d825c2b336d" title="This function finds the standard deviation of the values in the array. NULLs are ignored. ">array_stddev()</a></th><td><p class="starttd">This function finds the standard deviation of the values in the array. NULLs are ignored. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#ab066e65a41db78b00b4532996b2a6efc" title="This function creates an array of set size (the argument value) of FLOAT8, initializing the values to...">array_of_float()</a></th><td><p class="starttd">This function creates an array of set size (the argument value) of FLOAT8, initializing the values to 0.0. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#ab7d8550e66d2e0bd54b8f0997d93880c" title="This function creates an array of set size (the argument value) of BIGINT, initializing the values to...">array_of_bigint()</a></th><td><p class="starttd">This function creates an array of set size (the argument value) of BIGINT, initializing the values to 0. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a065a5323f3b742be47e39ad8b4c90fc2" title="This functions set every values in the array to some desired value (provided as the argument)...">array_fill()</a></th><td><p class="starttd">This functions set every value in the array to some desired value (provided as the argument). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#acc295a568878940ffc3e2c9a75990efb" title="This function takes an array as the input and keep only elements that satisfy the operator on specifi...">array_filter()</a></th><td><p class="starttd">This function takes an array as the input and keep only elements that satisfy the operator on specified scalar. It requires that the array is 1-D and all the values are NON-NULL. Return type is the same as the input type. By default, this function removes all zeros. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#ae6881cc5c86941b6ffca35d7f3cd5c12" title="This function takes an array as the input and executes element-wise multiplication by the scalar prov...">array_scalar_mult()</a></th><td><p class="starttd">This function takes an array as the input and executes element-wise multiplication by the scalar provided as the second argument, returning the resulting array. It requires that all the values are NON-NULL. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a0b6ffe59b12c3dee076c3059f9ab363f" title="This function takes an array as the input and executes element-wise addition by the scalar provided a...">array_scalar_add()</a></th><td><p class="starttd">This function takes an array as the input and executes element-wise addition of the scalar provided as the second argument, returning the resulting array. It requires that all the values are NON-NULL. Return type is the same as the input type. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a83451ed0c3ca5a9c62751dba47e45df7" title="This function takes an array as the input and finds square root of each element in the array...">array_sqrt()</a></th><td><p class="starttd">This function takes an array as the input and finds square root of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#a761e7ca753a5e1acf26896b37ed8b0bd" title="This function takes an array and a float8 as the input and finds power of each element in the array...">array_pow()</a></th><td><p class="starttd">This function takes an array and a float8 as the input and finds power of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#aff60f4091bed6374683f047c8a70ef9a" title="This function takes an array as the input and finds square of each element in the array...">array_square()</a></th><td><p class="starttd">This function takes an array as the input and finds square of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#acb57ea4521dcb717f9e3148e0acccc74" title="This function normalizes an array as sum of squares to be 1. ">normalize()</a></th><td><p class="starttd">This function normalizes an array as sum of squares to be 1. It requires that the array is 1-D and all the values are NON-NULL. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="array__ops_8sql__in.html#af057b589f2a2cb1095caa99feaeb3d70" title="This function takes a 2-D array as the input and unnests it by one level. It returns a set of 1-D arr...">array_unnest_2d_to_1d()</a></th><td><p class="starttd">This function takes a 2-D array as the input and unnests it by one level. It returns a set of 1-D arrays that correspond to rows of the input array as well as an ID column with values corresponding to row positions occupied by those 1-D arrays within the 2-D array. </p>
+<p class="endtd"></p>
+</td></tr>
+</table>
+</dd></dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Create a database table with two integer array columns and add some data. <pre class="example">
+CREATE TABLE array_tbl ( id integer,
+ array1 integer[],
+ array2 integer[]
+ );
+INSERT INTO array_tbl VALUES
+ ( 1, '{1,2,3,4,5,6,7,8,9}', '{9,8,7,6,5,4,3,2,1}' ),
+ ( 2, '{1,1,0,1,1,2,3,99,8}','{0,0,0,-5,4,1,1,7,6}' );
+</pre></li>
+<li>Find the minimum, maximum, mean, and standard deviation of the <code>array1</code> column. <pre class="example">
+SELECT id, madlib.array_min(array1), madlib.array_max(array1),
+ madlib.array_min_index(array1), madlib.array_max_index(array1),
+ madlib.array_mean(array1), madlib.array_stddev(array1)
+FROM array_tbl;
+</pre> Result: <pre class="result">
+id | array_min | array_max | array_min_index | array_max_index | array_mean | array_stddev
+----+-----------+-----------+---------------+---------------+------------------+------------------
+ 1 | 1 | 9 | {1,1} | {9,9} | 5 | 2.73861278752583
+ 2 | 0 | 99 | {0,3} | {99,8} | 12.8888888888889 | 32.3784050118457(2 rows)
+</pre></li>
+<li>Perform array addition and subtraction. <pre class="example">
+SELECT id, madlib.array_add(array1,array2),
+ madlib.array_sub(array1,array2)
+FROM array_tbl;
+</pre> Result: <pre class="result">
+ id | array_add | array_sub
+ ---+------------------------------+-------------------------
+ 2 | {1,1,0,-4,5,3,4,106,14} | {1,1,0,6,-3,1,2,92,2}
+ 1 | {10,10,10,10,10,10,10,10,10} | {-8,-6,-4,-2,0,2,4,6,8}
+(2 rows)
+</pre></li>
+<li>Perform element-wise array multiplication and division. The row with <code>id=2</code> is excluded because the divisor array contains zero, which would cause a divide-by-zero error. <pre class="example">
+SELECT id, madlib.array_mult(array1,array2),
+ madlib.array_div(array1,array2)
+FROM array_tbl
+WHERE 0 != ALL(array2);</pre> Result: <pre class="result">
+ id | array_mult | array_div
+ ---+----------------------------+---------------------
+ 1 | {9,16,21,24,25,24,21,16,9} | {0,0,0,0,1,1,2,4,9}
+(1 row)
+</pre></li>
+<li>Calculate the dot product of the arrays. <pre class="example">
+SELECT id, madlib.array_dot(array1, array2)
+FROM array_tbl;
+</pre> Result: <pre class="result">
+ id | array_dot
+ ---+----------
+ 2 | 745
+ 1 | 165
+(2 rows)
+</pre></li>
+<li>Multiply an array by a scalar 3. <pre class="example">
+SELECT id, madlib.array_scalar_mult(array1,3)
+FROM array_tbl;
+</pre> Result: <pre class="result">
+ id | array_scalar_mult
+ ---+--------------------------
+ 1 | {3,6,9,12,15,18,21,24,27}
+ 2 | {3,3,0,3,3,6,9,297,24}
+(2 rows)
+</pre></li>
+<li>Construct a nine-element array with each element set to the value 1.3. <pre class="example">
+SELECT madlib.array_fill(madlib.array_of_float(9), 1.3::float);
+</pre> Result: <pre class="result">
+ array_fill
+ --------------------------------------
+ {1.3,1.3,1.3,1.3,1.3,1.3,1.3,1.3,1.3}
+(1 row)
+</pre></li>
+<li>Unnest a column of 2-D arrays into sets of 1-D arrays. <pre class="example">
+SELECT id, (madlib.array_unnest_2d_to_1d(val)).*
+FROM (
+ SELECT 1::INT AS id, ARRAY[[1.3,2.0,3.2],[10.3,20.0,32.2]]::FLOAT8[][] AS val
+ UNION ALL
+ SELECT 2, ARRAY[[pi(),pi()/2],[2*pi(),pi()],[pi()/4,4*pi()]]::FLOAT8[][]
+) t
+ORDER BY 1,2;
+</pre> Result: <pre class="result">
+ id | unnest_row_id | unnest_result
+----+---------------+--------------------------------------
+ 1 | 1 | {1.3,2,3.2}
+ 1 | 2 | {10.3,20,32.2}
+ 2 | 1 | {3.14159265358979,1.5707963267949}
+ 2 | 2 | {6.28318530717959,3.14159265358979}
+ 2 | 3 | {0.785398163397448,12.5663706143592}
+(5 rows)
+</pre> If the function is called without the .* notation then it will return a composite record type with two attributes: the row ID and corresponding unnested array result.</li>
+</ol>
+<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="array__ops_8sql__in.html" title="implementation of array operations in SQL ">array_ops.sql_in</a> for list of functions and usage. </p>
+</div><!-- contents -->
+</div><!-- doc-content -->
+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
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+ <li class="footer">Generated on Wed Dec 27 2017 19:05:57 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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