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| <div class="title">ARIMA<div class="ingroups"><a class="el" href="group__grp__tsa.html">Time Series Analysis</a></div></div> </div> |
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| <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> |
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