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<title>MADlib: ARIMA</title>
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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 (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 \( d&gt;0 \) 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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