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 <div class="header">
   <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 (not currently implemented) </dt>
 <dd><p class="startdd">TEXT, default: NULL.</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>
 <dl class="section note"><dt>Note</dt><dd>Grouping is not currently implemented for ARIMA, but will be added in the future. Any non-NULL value for this parameter is ignored.</dd></dl>
 </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 \( d&gt;0 \) 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>
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	<div class="header">
	<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 (not currently implemented) </dt>
	<dd><p class="startdd">TEXT, default: NULL.</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>
	<dl class="section note"><dt>Note</dt><dd>Grouping is not currently implemented for ARIMA, but will be added in the future. Any non-NULL value for this parameter is ignored.</dd></dl>
	</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>
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