OpenText Analytics Database 26.2.x – Time series forecasting

Data-Analysis: Autoregression algorithms

Mon, 01 Jan 0001 00:00:00 +0000

Autoregessive models use past time series values to predict future values. These models assume that a future value in a time series is dependent on its past values, and attempt to capture this relationship in a set of coefficient values.

OpenText™ Analytics Database supports both autoregression (AR) and vector autoregression (VAR) models:

AR is a univariate autoregressive time series algorithm that predicts a variable's future values based on its preceding values. The user specifies the number of lagged timesteps taken into account during computation, and the model then predicts future values as a linear combination of the values at each lag.
VAR is a multivariate autoregressive time series algorithm that captures the relationship between multiple time series variables over time. Unlike AR, which only considers a single variable, VAR models incorporate feedback between different variables in the model, enabling the model to analyze how variables interact across lagged time steps. For example, with two variables—atmospheric pressure and rain accumulation—a VAR model could determine whether a drop in pressure tends to result in rain at a future date.

The AUTOREGRESSOR function automatically executes the algorithm that fits your input data:

One value column: the function executes autoregression and returns a trained AR model.
Multiple value columns: the function executes vector autoregression and returns a trained VAR model.

Data-Analysis: ARIMA model example

Mon, 01 Jan 0001 00:00:00 +0000

Autoregressive integrated moving average (ARIMA) models combine the abilities of AUTOREGRESSOR and MOVING_AVERAGE models by making predictions based on both preceding time series values and errors of previous predictions. ARIMA models also provide the option to apply a differencing operation to the input data, which can turn a non-stationary time series into a stationary time series. At model training time, you specify the differencing order and the number of preceding values and previous prediction errors that the model uses to calculate predictions.

You can use the following functions to train and make predictions with ARIMA models:

ARIMA: Creates and trains an ARIMA model
PREDICT_ARIMA: Applies a trained ARIMA model to an input relation or makes predictions using the in-sample data

These functions require time series data with consistent timesteps. To normalize a time series with inconsistent timesteps, see Gap filling and interpolation (GFI).

The following example trains three ARIMA models, two that use differencing and one that does not, and then makes predictions using the models.

Load the training data

Before you begin the example, load the Machine Learning sample data.

This example uses the following data:

daily-min-temperatures: provided in the machine learning sample data, this dataset contains data on the daily minimum temperature in Melbourne, Australia from 1981 through 1990. After you load the sample datasets, this data is available in the temp_data table.
db_size: a table that tracks the size of a database over consecutive months.

=> SELECT * FROM temp_data;
        time         | Temperature
---------------------+-------------
 1981-01-01 00:00:00 |        20.7
 1981-01-02 00:00:00 |        17.9
 1981-01-03 00:00:00 |        18.8
 1981-01-04 00:00:00 |        14.6
 1981-01-05 00:00:00 |        15.8
 ...
 1990-12-27 00:00:00 |          14
 1990-12-28 00:00:00 |        13.6
 1990-12-29 00:00:00 |        13.5
 1990-12-30 00:00:00 |        15.7
 1990-12-31 00:00:00 |          13
(3650 rows)

=> SELECT COUNT(*) FROM temp_data;
 COUNT
-------
3650
(1 row)

=> SELECT * FROM db_size;
 month | GB
-------+-----
     1 |   5
     2 |  10
     3 |  20
     4 |  35
     5 |  55
     6 |  80
     7 | 110
     8 | 145
     9 | 185
    10 | 230
(10 rows)

Train the ARIMA models

After you load the daily-min-temperatures data, you can use the ARIMA function to create and train an ARIMA model. For this example, the model is trained with lags of p=3 and q=3, taking the value and prediction error of three previous time steps into account for each prediction. Because the input time series is stationary, you don't need to apply differencing to the data:

=> SELECT ARIMA('arima_temp', 'temp_data', 'temperature', 'time' USING PARAMETERS p=3, d=0, q=3);
                             ARIMA
--------------------------------------------------------------
Finished in 20 iterations.
3650 elements accepted, 0 elements rejected.

(1 row)

You can view a summary of the model with the GET_MODEL_SUMMARY function:

=> SELECT GET_MODEL_SUMMARY(USING PARAMETERS model_name='arima_temp');
               GET_MODEL_SUMMARY
-----------------------------------------------

============
coefficients
============
parameter| value
---------+--------
  phi_1  | 0.64189
  phi_2  | 0.46667
  phi_3  |-0.11777
 theta_1 |-0.05109
 theta_2 |-0.58699
 theta_3 |-0.15882


==============
regularization
==============
none

===============
timeseries_name
===============
temperature

==============
timestamp_name
==============
time

==============
missing_method
==============
linear_interpolation

===========
call_string
===========
ARIMA('public.arima_temp', 'temp_data', 'temperature', 'time' USING PARAMETERS p=3, d=0, q=3, missing='linear_interpolation', init_method='Zero', epsilon=1e-06, max_iterations=100);

===============
Additional Info
===============
       Name       | Value
------------------+--------
        p         |   3
        q         |   3
        d         |   0
       mean       |11.17775
      lambda      | 1.00000
mean_squared_error| 5.80490
rejected_row_count|   0
accepted_row_count|  3650

(1 row)

Examining the db_size table, it is clear that there is an upward trend to the database size over time. Each month the database size increases five more gigabytes than the increase in the previous month. This trend indicates the time series is non-stationary.

To account for this in the ARIMA model, you must difference the data by setting a non-zero d parameter value. For comparison, two ARIMA models are trained on this data, the first with a d value of one and the second with a d value of two:

=> SELECT ARIMA('arima_d1', 'db_size', 'GB', 'month' USING PARAMETERS p=2, d=1, q=2);
                                 ARIMA
------------------------------------------------------------------------
 Finished in 9 iterations.
 10 elements accepted, 0 elements rejected.

(1 row)

=> SELECT ARIMA('arima_d2', 'db_size', 'GB', 'month' USING PARAMETERS p=2, d=2, q=2);
                                 ARIMA
------------------------------------------------------------------------
 Finished in 0 iterations.
 10 elements accepted, 0 elements rejected.

(1 row)

Make predictions

After you train the ARIMA models, you can call the PREDICT_ARIMA function to predict future time series values. This function supports making predictions using the in-sample data that the models were trained on or applying the model to an input relation.

Using in-sample data

The following PREIDCT_ARIMA call makes temperature predictions using the in-sample data that the arima_temp model was trained on. The model begins prediction at the end of the temp_data table and returns predicted values for ten timesteps:

=> SELECT PREDICT_ARIMA(USING PARAMETERS model_name='arima_temp', start=0, npredictions=10) OVER();
 index |   prediction
-------+------------------
     1 | 12.9745063293842
     2 | 13.4389080858551
     3 | 13.3955791360528
     4 | 13.3551146487462
     5 | 13.3149336514747
     6 | 13.2750516811057
     7 | 13.2354710353376
     8 | 13.1961939790513
     9 | 13.1572226788109
    10 | 13.1185592045127
(10 rows)

For both prediction methods, if you want the function to return the standard error of each prediction, you can set output_standard_errors to true:

=> SELECT PREDICT_ARIMA(USING PARAMETERS model_name='arima_temp', start=0, npredictions=10, output_standard_errors=true) OVER();
 index |    prediction    |     std_err
-------+------------------+------------------
     1 | 12.9745063293842 | 1.00621890780865
     2 | 13.4389080858551 | 1.45340836833232
     3 | 13.3955791360528 | 1.61041524562932
     4 | 13.3551146487462 | 1.76368421116143
     5 | 13.3149336514747 | 1.91223938476627
     6 | 13.2750516811057 | 2.05618464609977
     7 | 13.2354710353376 | 2.19561771498385
     8 | 13.1961939790513 | 2.33063553781651
     9 | 13.1572226788109 | 2.46133422924445
    10 | 13.1185592045127 | 2.58780904243988
(10 rows)

To make predictions with the two models trained on the db_size table, you only need to change the specified model_name in the above calls:

=> SELECT PREDICT_ARIMA(USING PARAMETERS model_name='arima_d1', start=0, npredictions=10) OVER();
 index |    prediction
-------+------------------
     1 | 279.882778508943
     2 | 334.398317856829
     3 | 393.204492820962
     4 | 455.909453114272
     5 | 522.076165355683
     6 | 591.227478668175
     7 | 662.851655189833
     8 | 736.408301395412
     9 | 811.334631481162
    10 | 887.051990217688
(10 rows)

=> SELECT PREDICT_ARIMA(USING PARAMETERS model_name='arima_d2', start=0, npredictions=10) OVER();
 index | prediction
-------+------------
     1 | 280
     2 | 335
     3 | 395
     4 | 460
     5 | 530
     6 | 605
     7 | 685
     8 | 770
     9 | 860
    10 | 955
(10 rows)

Comparing the outputs from the two models, you can see that the model trained with a d value of two correctly captures the trend in the data. Each month the rate of database growth increases by five gigabytes.

Applying to an input relation

You can also apply the model to an input relation. The following example makes predictions by applying the arima_temp model to the temp_data training set:

=> SELECT PREDICT_ARIMA(temperature USING PARAMETERS model_name='arima_temp', start=3651, npredictions=10, output_standard_errors=true) OVER(ORDER BY time) FROM temp_data;
 index |    prediction    |     std_err
-------+------------------+------------------
     1 | 12.9745063293842 | 1.00621890780865
     2 | 13.4389080858551 | 1.45340836833232
     3 | 13.3955791360528 | 1.61041524562932
     4 | 13.3551146487462 | 1.76368421116143
     5 | 13.3149336514747 | 1.91223938476627
     6 | 13.2750516811057 | 2.05618464609977
     7 | 13.2354710353376 | 2.19561771498385
     8 | 13.1961939790513 | 2.33063553781651
     9 | 13.1572226788109 | 2.46133422924445
    10 | 13.1185592045127 | 2.58780904243988
(10 rows)

Because the same data and relative start index were provided to both prediction methods, the arima_temp model predictions for each method are identical.

When applying a model to an input relation, you can set add_mean to false so that the function returns the predicted difference from the mean instead of the sum of the model mean and the predicted difference:

=> SELECT PREDICT_ARIMA(temperature USING PARAMETERS model_name='arima_temp', start=3680, npredictions=10, add_mean=false) OVER(ORDER BY time) FROM temp_data;
 index |  prediction
-------+------------------
     1 | 1.2026877112171
     2 | 1.17114068517961
     3 | 1.13992534953432
     4 | 1.10904183333367
     5 | 1.0784901998692
     6 | 1.04827044781798
     7 | 1.01838251238116
     8 | 0.98882626641461
     9 | 0.959601521551628
    10 | 0.93070802931751
(10 rows)

Data-Analysis: Moving-average model example

Mon, 01 Jan 0001 00:00:00 +0000

Moving average models use the errors of previous predictions to make future predictions. More specifically, the user-specified lag determines how many previous predictions and errors it takes into account during computation.

Use the following functions when training and predicting with moving-average models. Note that these functions require datasets with consistent timesteps.

MOVING_AVERAGE: trains a moving-average model
PREDICT_MOVING_AVERAGE: applies the model to a dataset to make predictions

To normalize datasets with inconsistent timesteps, see Gap filling and interpolation (GFI).

Example

Load the datasets from the Machine-Learning-Examples repository.

This example uses the daily-min-temperatures dataset, which contains data on the daily minimum temperature in Melbourne, Australia from 1981 through 1990:

=> SELECT * FROM temp_data;
        time         | Temperature
---------------------+-------------
 1981-01-01 00:00:00 |        20.7
 1981-01-02 00:00:00 |        17.9
 1981-01-03 00:00:00 |        18.8
 1981-01-04 00:00:00 |        14.6
 1981-01-05 00:00:00 |        15.8
...
 1990-12-27 00:00:00 |          14
 1990-12-28 00:00:00 |        13.6
 1990-12-29 00:00:00 |        13.5
 1990-12-30 00:00:00 |        15.7
 1990-12-31 00:00:00 |          13
(3650 rows)

Use MOVING_AVERAGE to create the moving-average model MA_temperature from the temp_data dataset. In this case, the model is trained with a lag of p=3, taking the error of 3 previous predictions into account for each estimation:

=> SELECT MOVING_AVERAGE('MA_temperature', 'temp_data', 'temperature', 'time' USING PARAMETERS q=3, missing='linear_interpolation', regularization='none', lambda=1);
                    MOVING_AVERAGE
---------------------------------------------------------
 Finished. 3650 elements accepted, 0 elements rejected.
(1 row)

You can view a summary of the model with GET_MODEL_SUMMARY:

=> SELECT GET_MODEL_SUMMARY(USING PARAMETERS model_name='MA_temperature');

 GET_MODEL_SUMMARY
-------------------

============
coefficients
============
parameter| value
---------+--------
phi_(t-0)|-0.90051
phi_(t-1)|-0.10621
phi_(t-2)| 0.07173


===============
timeseries_name
===============
temperature

==============
timestamp_name
==============
time

===========
call_string
===========
moving_average('public.MA_temperature', 'temp_data', 'temperature', 'time'
USING PARAMETERS q=3, missing=linear_interpolation, regularization='none', lambda=1);

===============
Additional Info
===============
       Name       | Value
------------------+--------
       mean       |11.17780
    lag_order     |   3
      lambda      | 1.00000
rejected_row_count|   0
accepted_row_count|  3650

(1 row)

Use PREDICT_MOVING_AVERAGE to predict future temperatures. The following query starts the prediction at the end of the dataset and returns 10 predictions.

=> SELECT PREDICT_MOVING_AVERAGE(Temperature USING PARAMETERS model_name='MA_temperature', npredictions=10) OVER(ORDER BY time) FROM temp_data;

 index |    prediction
-------+------------------
     1 | 13.1324365636272
     2 | 12.8071086272833
     3 | 12.7218966671721
     4 | 12.6011086656032
     5 | 12.506624729879
     6 | 12.4148247026733
     7 | 12.3307873804812
     8 | 12.2521385975133
     9 | 12.1789741993396
    10 | 12.1107640076638
(10 rows)

OpenText Analytics Database 26.2.x – Time series forecasting

Data-Analysis: Autoregression algorithms

Data-Analysis: ARIMA model example

Load the training data

Train the ARIMA models

Make predictions

Using in-sample data

Applying to an input relation

See also

Data-Analysis: Moving-average model example

Example