library.tsa Submodule

Module Summary

Interfaces for the NAG Mark 30.3 tsa Chapter.

tsa - Time Series Analysis

This module provides facilities for investigating and modelling the statistical structure of series of observations collected at points in time. The models may then be used to forecast the series.

The module covers the following models and approaches:

1. univariate time series analysis, including autocorrelation functions and autoregressive moving average (ARMA) models;

2. univariate spectral analysis;

3. transfer function (multi-input) modelling, in which one time series is dependent on other time series;

4. bivariate spectral methods including coherency, gain and input response functions;

5. vector ARMA models for multivariate time series;

6. Kalman filter models (linear and nonlinear);

7. GARCH models for volatility;

8. inhomogeneous time series.

See Also

naginterfaces.library.examples.tsa :

This subpackage contains examples for the tsa module. See also the Examples subsection.

Functionality Index

ARMA modelling

ACF: uni_autocorr()

diagnostic checking: uni_arima_resid()

Dickey–Fuller unit root test: uni_dickey_fuller_unit()

differencing: uni_diff()

estimation (comprehensive): uni_arima_estim()

estimation (easy-to-use): uni_arima_estim_easy()

forecasting from fully specified model: uni_arima_forcecast()

forecasting from state set: uni_arima_forecast_state()

mean/range: uni_means()

PACF: uni_autocorr_part()

preliminary estimation: uni_arima_prelim()

update state set: uni_arima_update()

Change point

detection

binary segmentation: cp_binary()

PELT: cp_pelt()

Exponential smoothing: uni_smooth_exp()

GARCH

EGARCH

fitting: uni_garch_exp_estim()

forecasting: uni_garch_exp_forecast()

GJR GARCH

fitting: uni_garch_gjr_estim()

forecasting: uni_garch_gjr_forecast()

symmetric or type I AGARCH

fitting: uni_garch_asym1_estim()

forecasting: uni_garch_asym1_forecast()

type II AGARCH

fitting: uni_garch_asym2_estim()

forecasting: uni_garch_asym2_forecast()

Inhomogeneous series

iterated exponential moving average

final value only returned: inhom_iema()

intermediate values returned: inhom_iema_all()

moving average: inhom_ma()

Kalman

filter

time invariant

square root covariance: multi_kalman_sqrt_invar()

time varying

square root covariance: multi_kalman_sqrt_var()

unscented: kalman_unscented_state()

unscented (reverse communication): kalman_unscented_state_revcom()

Spectral analysis

Bivariate

Bartlett, Tukey, Parzen windows: multi_spectrum_lag()

cross amplitude spectrum: multi_spectrum_bivar()

direct smoothing: multi_spectrum_daniell()

gain and phase: multi_gain_bivar()

noise spectrum: multi_noise_bivar()

Univariate

Bartlett, Tukey, Parzen windows: uni_spectrum_lag()

direct smoothing: uni_spectrum_daniell()

Transfer function modelling

cross-correlations: multi_xcorr()

filtering: multi_filter_transf()

fitting: multi_inputmod_estim()

forecasting from fully specified model: multi_inputmod_forecast()

forecasting from state set: multi_inputmod_forecast_state()

preliminary estimation: multi_transf_prelim()

pre-whitening: multi_filter_arima()

update state set: multi_inputmod_update()

Vector ARMA

cross-correlations: multi_corrmat_cross()

diagnostic checks: multi_varma_diag()

differencing: multi_diff()

fitting: multi_varma_estimate()

forecasting: multi_varma_forecast()

partial autoregression matrices: multi_regmat_partial()

partial correlation matrices: multi_corrmat_partlag()

squared partial autocorrelations: multi_autocorr_part()

update forecast: multi_varma_update()

zeros of ARIMA operator: uni_arma_roots()

For full information please refer to the NAG Library document

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13intro.html

Examples

naginterfaces.library.examples.tsa.cp_pelt_ex.main()

Example for naginterfaces.library.tsa.cp_pelt().

Change point detection, using the PELT algorithm.

>>> main()
naginterfaces.library.tsa.cp_pelt Python Example Results.
Change point detection, using the PELT algorithm.
  -- Change Points --        --- Distribution ---
  Number    Positions             Parameters
    1          12            0.34            1.00
    2          32            2.57            1.00
    3          49            1.45            1.00
    4          52           -0.48            1.00
    5          70            1.20            1.00
    6         100           -0.23            1.00

naginterfaces.library.examples.tsa.inhom_iema_ex.main()

Example for naginterfaces.library.tsa.inhom_iema().

Iterated exponential moving average for a univariate inhomogeneous time series.

>>> main()
naginterfaces.library.tsa.inhom_iema Python Example Results.
IEMA for an inhomogeneous time series.
 Time    IEMA
______________
 7.500   0.531
 8.200   0.544
18.100   0.754
22.800   0.406
25.800   0.232
26.800   0.217
31.100   0.357
38.400   0.630
45.900   0.263
48.200   0.241
48.900   0.279
57.900   0.713
58.500   0.717
63.900   0.385
65.200   0.346
66.600   0.330
67.400   0.315
69.300   0.409
69.900   0.459
73.000   0.377
75.600   0.411
77.000   0.536
84.700   0.632
86.800   0.538
88.000   0.444
88.500   0.401
91.000   0.331
93.000   0.495
93.700   0.585
94.000   0.612

naginterfaces.library.examples.tsa.kalman_unscented_state_ex.main()

Example for naginterfaces.library.tsa.kalman_unscented_state().

Combined time and measurement update, one iteration of the Unscented Kalman Filter for a nonlinear state space model, with additive noise.

Demonstrates passing user-defined communication data to callback functions.

>>> main()
naginterfaces.library.tsa.kalman_unscented_state Python Example Results.
Applying the UKF to a nonlinear state space model, with additive noise.
At time point 0, state estimate is:
(     0.664,     -0.092,      0.104)
At time point 1, state estimate is:
(     1.598,      0.081,      0.314)
At time point 2, state estimate is:
(     2.128,      0.213,      0.378)
At time point 3, state estimate is:
(     3.134,      0.674,      0.660)
At time point 4, state estimate is:
(     3.809,      1.181,      0.906)
At time point 5, state estimate is:
(     4.730,      2.000,      1.298)
At time point 6, state estimate is:
(     4.429,      2.474,      1.762)
At time point 7, state estimate is:
(     4.357,      3.246,      2.162)
At time point 8, state estimate is:
(     3.907,      3.852,      2.246)
At time point 9, state estimate is:
(     3.360,      4.398,      2.504)
At time point 10, state estimate is:
(     2.552,      4.741,      2.750)
At time point 11, state estimate is:
(     2.191,      5.193,      3.281)
At time point 12, state estimate is:
(     1.309,      5.018,      3.610)
At time point 13, state estimate is:
(     1.071,      4.894,      4.031)
At time point 14, state estimate is:
(     0.618,      4.322,      4.124)
Estimate of the Cholesky factorization of the state
covariance matrix at the last time point:
[
    1.92e-01
   -3.82e-01,   2.22e-02
    1.58e-06,   2.23e-07,   9.95e-03
]

naginterfaces.library.examples.tsa.multi_inputmod_ex.main()

Example for naginterfaces.library.tsa.multi_inputmod_estim() and naginterfaces.library.tsa.multi_inputmod_forecast_state().

Multivariate time series: estimation of multi-input model and forecasting.

>>> main()
naginterfaces.library.tsa.multi_inputmod Python Example Results.
Multivariate time series: estimate and forecast.
Begin estimation.
Begin forecasting.
The forecast values and their standard errors:
No.   Value   Std error
  1   97.306    4.788
  2  101.785    6.328
  3  103.812    7.145
  4  102.073    7.360
Forecast components z(t):
Series No.: 1:
[175.413, 177.712, 179.373, 178.372]
Output noise n(t):
[-78.10695358 -75.92741605 -75.56034026 -76.29929963]

naginterfaces.library.examples.tsa.uni_dickey_fuller_unit_ex.main()

Example for naginterfaces.library.tsa.uni_dickey_fuller_unit().

(Augmented) Dickey–Fuller unit root test statistic.

>>> main()
naginterfaces.library.tsa.uni_dickey_fuller_unit Python Example Results.
Dickey--Fuller unit root test.
Dickey--Fuller test statistic = -2.538

naginterfaces.library.examples.tsa.uni_garch_asym1_ex.main()

Example for naginterfaces.library.tsa.uni_garch_asym1().

Type-I AGARCH parameter estimation and forecasting.

>>> main()
naginterfaces.library.tsa.uni_garch_asym1 Python Example Results.
Fit a GARCH(1,1) model with Student's t-distributed residuals.
                Parameter        Standard
                estimates         errors
alpha_0             0.00            0.06
alpha_1             0.11            0.13
 beta_1             0.66            0.23
  gamma            -0.62            0.62
     df             6.25            4.70
    b_0             3.85           24.11
    b_1             1.48            1.82
    b_2             2.15           10.16
Volatility forecast =         0.09