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A Computer Program for General Recursive Time-Series Analysis
Copyright © IFAC Identifica tion and System Para mete r Estimation 1982 , W as hington D.C ., USA 1982
A COMPUTER PROGRAM FOR GENERAL RECURSIVE TIME-SERIES ANALYSIS A. J . Jakeman*, P. C. Young** and A. J. Bayes* ·Centre for R esource and Environmental Studies, Australian National University, Australia "Department of Environmental Sciences, University of Lancaster, UK
Abstract. The paper presents a computer program for recursive time series analysis of data which may come from MISO or SISO systems with discrete or continuous-time representations. The major novel features are the incorporation of powerful model structure identification criteria and the provision to obtain asymptotically efficient estimates when the stochastic disturbances have rational spectral density. Some runs of the central algorithms are displayed to demonstrate the program's visual interactive mode of operation and flexibility. Keywords. Discrete-time systems, continuous-time systems, time-varying systems, multivariable systems, linear differential equations, difference equations, identification, recursive parameter estimation, modeling, transfer functions, filtering, data processing, data analysis. INTRODUCTION
algorithm which yields statistically efficient estimates for a specific class of stochastic disturbances; extensions to handle multiple inputsingle output (MISO) transfer function models; extensions to estimate parameters in continuous-time transfer function models from SISO discrete-time data using a refined IV algorithm; extens ions to smooth or different i ate any time series during pre-processing of data.
The suite of programs described in this paper form an extended version of the CAPTAI N (£omputer ~i ded .E!'0gram for !.i me series analysis and the identification of noisy systems) package whTCh was developed primarily to obtain parametrically efficient mathematical relationships between stochastic time series data (see e.g. Young and Jakeman, 1979). Although CAPTAIN contains useful subsidiary routines performing functions such as prewhitening, auto-, cross- and partial correlations, impulse respons e simulations and forecasting for any time series, the determination of causal relationships between time series is bas i ca lly rest ri cted to di screte-t i me representations of single input - single output (SISO) systems. The central tool employed for this is the basic instrumental variable (IV) algorithm-----wn1ch provides statistically consistent estimates of parameters in the Transfer Function (TF) time series model.
The aim of the present paper is to describe how the original package and these extensions have been incorporated into this more general time series analYSis package. The actual mathemat i ca 1 background is omitted here. However, the model forms used are based upon the SISO discrete-time Transfer Function model used originally by Box and Jenk i ns (l970). These forms and their associated recursive algorifhms a~e covered in Young and Jakeman (1979 • 1979 • 1980). Jakeman and Young (1979) and Jakeman and others (1980). The method of model order identification used in this package is based primarily on an error variance norm EVN (the average parameter variance) and/or NEVN (the vari ances are normal i sed by absolute parameter values before averaging) for each possible model in conjuncti~n with the total correlation coefficient R which measures the degree of fit to the output data (Young and others. 1980).
Young and Jakeman (1979 1 ) discuss a number of proposed enhancements to this original form and demonst rate the ut il i ty of these enhancements with practical examples. Of these, the actua 1 extens ions that ha ve now been made in the new CAPTAIN package are as follows: incorporation of more objective forms of model structure identification; improvements to the basic IV estimation for SISO data by using a refined IV 567
568
A. J. Jakeman, P . C. Young and A. J . Bayes
ORGANISATION OF THE PACKAGE This new CAPTAIN package is composed of 4 separately executable programs, each of which requires less than 30K words of 36 bit storage on a Uni vac 1100 series computer. Thus, it is designed to be suitable for computers with fairly modest memory sizes. As is shown in Fig. 1, the initial program conta ins most featu res. These can all be overlaid so that our imposed 30K storage limit is not exceeded. This first program must be entered prior to the other three, even if only to read the input-output data. The co-ordination among all four programs and among all routines is via temporary fi 1es contai ni ng the computer output of previously used routines. Thus it is necessary to know the function of all routines as well as the hidden computer output produced by them which is required for communication between routines.
1.
READ I/O DATA simply reads the time series data. Included with this is the number of samples and the number of inputs. When the package is initially used in any run, this routine must be called first. If no other routines are ca 11 ed in the fi rst program, then the tempora ry fil e created and read by any of the other 3 programs is just the ori gi na 1 data.
2.
SET TITLE AND PLOT SIZES requests an identifying title for general computer output and plots as well as the number of units per inch required on the horizontal time axis. Scale on the vertical a xis is automatically calculated. Defaults are set if this routine is not called.
3.
PREPROCESS constructs transformed input and/or output series which replace on tempora ry fil e the ori gi na 1 raw input/output data. The current transformation options are Kalman filtering/smoothing, low/high pass filtering, mean removal; and also a non-linear filter option (used for the compensation of specific soil moisture effects in certain hydrological app 1 i cat ions). When a sequence of the above options is used, PREPROCESS updates the 1ast temporary fi 1 e created followi~mplementation of the last option, not the original raw data. However-:- there is a further option for the latter to be reintroduced (REFRESH) at any stage in this routine.
4.
SISO IV is the basic instrumental variable routine which performs identification/estimation between the output time-series and anyone of the input series. To the (pre-processed) I/O data on tempora ry fil es, it adds the model output, model residuals and impulse response. All of these can be
used in PLOT, AML or CORR. It also stores on fil e the fi na 1 pa rameter estimates for potential use ln initialising TVAR, REFINED IV or CONT IV. 5.
MISO IV is the basic instrumental variable routine for multiple input data. It requests whi ch of the input series are required for estimation against the output. There is also the facility to initialise the multiple transfer function parameters at specified values but, without accurate estimates, the user is advised to obtain them by using SISO IV separately on each input series. The routine produces a model output and residuals for potential use in AML, CORR or PLOT.
6.
TVAR allows the pattern of Time VARiation of any parameters estimated in SISO IV to be checked when these parameters are allowed to vary as a simple random walk. It cannot be called without prior use of SISO IV. computer output is the time The variation for any specified parameter(s), the time varying model output and the corresponding residuals. These can be used in PLOT.
7.
REFINED IV is the refined instrumental variable (and co-ordinated refined AML) algorithm which performs more statistically efficien t estimation between the output time series and any specified input series. It can be called without prior use of SISO IV as it can perform its own initialisation. The model output and residuals are again available after passing through this instrumental variable routine.
8.
AML is the approximate maximum 1 ikel ihood al gorithm used to estimate the parameters in an ARMA model of specified order for either (i) the noise residuals computed in SISO IV, MISO IV, REFINED IV, TVAR or CONT IV; or (ii) any (preprocessed) input series. To aid identification of order, Akaike's (1974) information criterion (AIC) is given for each model structure selected.
9.
FINISH allows the user to exit from the fi rst program and 1eaves hi m free to enter programs 10,11 or 12.
10.
CONT IV is the instrumental variable algorithm for estimation of the continuous- time model parameters in (2). If initial parameter estimates are not known, then SISO IV must be called prior to this. As for all the instrumental variable routines, model output and residual series are produced here.
569
A Computer Program for General Recursive Time-Series Analysis
INITIAL
1REA D' I/O DATA
11 S ET
~LOT ~IIPRE-~E' I SIS~
TITLE SI ZES
'0 CONT IV
IV
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Fig. 1 The four programs now available in CAPTAIN
11.
CORR will cross. auto-. partially correl ate or prewhiten any of the series available on temporary files. e. g. raw or processed inputs and outputs. model outputs. res i dua 1 s; it can be used for impul se response estimation based on a prewhitened input series.
12.
PLOT wi 11 graph the series avai lable from temporary files. These series are displayed immediately upon accessing Initially the raw input the program. and output series are available for plotting. After routine 3 has been utilized. the preprocessed I/O is ready for plotting. In addition to these. the following are available after the indicated routines have been used: 4:SISO IV model output. residuals and impulse response 5:MISO IV model output and residuals 6:TVAR model output. residuals. and parameter variations 7:REFINED IV model output and residuals 10:CONT IV model output and residuals 11:cross-. autoand partial corre 1at i on of any seri es so processed in 11.
Note also that these files. once created by the last call of the routines. are all available (e.g. for plotting) until Signing off the computer. The on ly except ions to this rule are the raw I/O data which must be plotted before preprocessing. EXAMPLES To demonstrate the ease of use of the package. a simple example has been taken from an air quality application in Jakeman and ~thers (1980) and Young and Jakeman (1979). It is most useful in that it
demonstrates all the major uses of the package. i.e. all the IV routines: SISO IV. REFINED IV. CONT IV and MISO IV. The data consist of two inputs. the 'upstream' concentrations of ozone at two different locations in the San Joaquin Valley of California. and an output which is the downwind concentrati on of ozone at another location in the valley. Either input can be used wi th the output to i dent i fy and estimate a good transfer function model; or a lternat i ve ly both inputs can be used. The computer outputs shown in this section are the results of the following operations: enter package via program 1 read I/O data. select plot title and sizes and exit (c) plot I/O data by enteri ng program PLOT and exit (d) re-enter program 1 and preprocess input 1 by lagging it one time interval (determined from the previous plots. use of the CORR program and the identification statistics from SISO IV using no lag. lag 1 and lag 2 - this is not shown explicitly) (e) call routine SISO IV for input 1 invoking identification criteria through the D (display criteria) option (f) repeat (e) for input 2 (g) call MISO IV using the 2 input series (one of them has been preprocessed) and initialise a model of order nl = n2 = 1. ml+l = m2+1 = 2 with initialisatlon from rounded" estimates obtai ned in (e) and (f) (h) change plot title and call REFINED IV for input 1 using model order identified in (e) and exit (i ) enter program CONT IV. use estimates in (e) for initialisation and exit (j) enter PLOT and obtain graphs of output, mode 1 output and res i dua 1s from CONT I V algorithm. then exit.
(a)
(b)
570
A. J. Jakeman, P. C. Young and A. J. Bayes
XOT AJ8852*MD.ABSl RELEASE 13/2/81
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CONCLUSIONS Thi s paper has out 1i ned the 1atest vers i on of the CAPTAIN time-series analysis computer package and demonstrated how it can be used for the SISO or MISO modeling of discrete data in discrete or continuous-time model form. Such model i ng is ai ded cons i derab ly by the novel structure identification strategies available in CAPTAIN and the sophistication of the visual interactive, conversational mode of operation. It is fe It that the package represents the most flexible and complete form of recursive estimation so far developed. MIMO estimation using similar estimation techniques is possible (Jakeman and Young, 1979) but has not yet been implemented in the package because of its complexity. REFERENCES Akaike, H. (1974). A new look at statistical model identification, IEEE Truns. Auto. ContT'oZ, AC-l9, 718-722. Box, G.E.P., and G.M. Jenkins (l970). Time Series AnaZY8i8, FOT'eca8ting and ContT'oZ, Holden Day: San Francisco. Jakeman, A.J., and P.C. Young (l979). Refined instrumental variable methods of recursive time series analysis. Part Il: multivariable systems, Int. J. ContT'oZ, ~, 621-644. Jakeman, A.J., Steele, L.P. and P.C. Young (l980). Instrumental variable algorithms for I1lJltiple input systems described by multiple transfer funct ions, IEEE TT'an8 SY8tem8, Man and CybeT'netic8, SMC-lO, 593-602.
Wellstead, P.E. (1978). An instrumental product moment test for model order estimation, Automatica, ~ B9-91. Young, P.C. (l974). Recursive approaches to time series analysis, BuZZ. In8t. Math8 and AppZ., la-, 209-224. Young, P.C., and A.J. Jakeman (1979 1 ). The development of CAPTAIN: a computer aided program for time series analysis and the identification of noisy systems. In: ComputeT' Aided De8ign of ContT'OZ SY8tem8, M.A. Cuenod (ed.), IFAC Symposium, Zurich, Switzerland, Aug. 29-31, 391-400. Young, P.C., and A.J. Jakeman (l979 2 ). Refined instrumental variable methods of recursi ve time seri es analysi s. Part I: single input-single output systems, Int. J. ContT'oZ,~, 1-30. Young, P.C., and A.J. Jakeman (1980). Refined instrumental variable methods of recursive time series analysis. Part Ill: Extensions, Int. J. ContT'oZ, ~ 741-764. A.J. and R.E. Young, P.C., Jakeman, McMurtrie (1980). An instrumental variable method for model order identification, Autol7ntica, l.!..! 281294.
A COMPUTER PROGRAM FOR GENERAL RECURSIVE TIME-SERIES ANALYSIS A. J . Jakeman*, P. C. Young** and A. J. Bayes* ·Centre for R esource and Environmental Studies, Australian National University, Australia "Department of Environmental Sciences, University of Lancaster, UK
Abstract. The paper presents a computer program for recursive time series analysis of data which may come from MISO or SISO systems with discrete or continuous-time representations. The major novel features are the incorporation of powerful model structure identification criteria and the provision to obtain asymptotically efficient estimates when the stochastic disturbances have rational spectral density. Some runs of the central algorithms are displayed to demonstrate the program's visual interactive mode of operation and flexibility. Keywords. Discrete-time systems, continuous-time systems, time-varying systems, multivariable systems, linear differential equations, difference equations, identification, recursive parameter estimation, modeling, transfer functions, filtering, data processing, data analysis. INTRODUCTION
algorithm which yields statistically efficient estimates for a specific class of stochastic disturbances; extensions to handle multiple inputsingle output (MISO) transfer function models; extensions to estimate parameters in continuous-time transfer function models from SISO discrete-time data using a refined IV algorithm; extens ions to smooth or different i ate any time series during pre-processing of data.
The suite of programs described in this paper form an extended version of the CAPTAI N (£omputer ~i ded .E!'0gram for !.i me series analysis and the identification of noisy systems) package whTCh was developed primarily to obtain parametrically efficient mathematical relationships between stochastic time series data (see e.g. Young and Jakeman, 1979). Although CAPTAIN contains useful subsidiary routines performing functions such as prewhitening, auto-, cross- and partial correlations, impulse respons e simulations and forecasting for any time series, the determination of causal relationships between time series is bas i ca lly rest ri cted to di screte-t i me representations of single input - single output (SISO) systems. The central tool employed for this is the basic instrumental variable (IV) algorithm-----wn1ch provides statistically consistent estimates of parameters in the Transfer Function (TF) time series model.
The aim of the present paper is to describe how the original package and these extensions have been incorporated into this more general time series analYSis package. The actual mathemat i ca 1 background is omitted here. However, the model forms used are based upon the SISO discrete-time Transfer Function model used originally by Box and Jenk i ns (l970). These forms and their associated recursive algorifhms a~e covered in Young and Jakeman (1979 • 1979 • 1980). Jakeman and Young (1979) and Jakeman and others (1980). The method of model order identification used in this package is based primarily on an error variance norm EVN (the average parameter variance) and/or NEVN (the vari ances are normal i sed by absolute parameter values before averaging) for each possible model in conjuncti~n with the total correlation coefficient R which measures the degree of fit to the output data (Young and others. 1980).
Young and Jakeman (1979 1 ) discuss a number of proposed enhancements to this original form and demonst rate the ut il i ty of these enhancements with practical examples. Of these, the actua 1 extens ions that ha ve now been made in the new CAPTAIN package are as follows: incorporation of more objective forms of model structure identification; improvements to the basic IV estimation for SISO data by using a refined IV 567
568
A. J. Jakeman, P . C. Young and A. J . Bayes
ORGANISATION OF THE PACKAGE This new CAPTAIN package is composed of 4 separately executable programs, each of which requires less than 30K words of 36 bit storage on a Uni vac 1100 series computer. Thus, it is designed to be suitable for computers with fairly modest memory sizes. As is shown in Fig. 1, the initial program conta ins most featu res. These can all be overlaid so that our imposed 30K storage limit is not exceeded. This first program must be entered prior to the other three, even if only to read the input-output data. The co-ordination among all four programs and among all routines is via temporary fi 1es contai ni ng the computer output of previously used routines. Thus it is necessary to know the function of all routines as well as the hidden computer output produced by them which is required for communication between routines.
1.
READ I/O DATA simply reads the time series data. Included with this is the number of samples and the number of inputs. When the package is initially used in any run, this routine must be called first. If no other routines are ca 11 ed in the fi rst program, then the tempora ry fil e created and read by any of the other 3 programs is just the ori gi na 1 data.
2.
SET TITLE AND PLOT SIZES requests an identifying title for general computer output and plots as well as the number of units per inch required on the horizontal time axis. Scale on the vertical a xis is automatically calculated. Defaults are set if this routine is not called.
3.
PREPROCESS constructs transformed input and/or output series which replace on tempora ry fil e the ori gi na 1 raw input/output data. The current transformation options are Kalman filtering/smoothing, low/high pass filtering, mean removal; and also a non-linear filter option (used for the compensation of specific soil moisture effects in certain hydrological app 1 i cat ions). When a sequence of the above options is used, PREPROCESS updates the 1ast temporary fi 1 e created followi~mplementation of the last option, not the original raw data. However-:- there is a further option for the latter to be reintroduced (REFRESH) at any stage in this routine.
4.
SISO IV is the basic instrumental variable routine which performs identification/estimation between the output time-series and anyone of the input series. To the (pre-processed) I/O data on tempora ry fil es, it adds the model output, model residuals and impulse response. All of these can be
used in PLOT, AML or CORR. It also stores on fil e the fi na 1 pa rameter estimates for potential use ln initialising TVAR, REFINED IV or CONT IV. 5.
MISO IV is the basic instrumental variable routine for multiple input data. It requests whi ch of the input series are required for estimation against the output. There is also the facility to initialise the multiple transfer function parameters at specified values but, without accurate estimates, the user is advised to obtain them by using SISO IV separately on each input series. The routine produces a model output and residuals for potential use in AML, CORR or PLOT.
6.
TVAR allows the pattern of Time VARiation of any parameters estimated in SISO IV to be checked when these parameters are allowed to vary as a simple random walk. It cannot be called without prior use of SISO IV. computer output is the time The variation for any specified parameter(s), the time varying model output and the corresponding residuals. These can be used in PLOT.
7.
REFINED IV is the refined instrumental variable (and co-ordinated refined AML) algorithm which performs more statistically efficien t estimation between the output time series and any specified input series. It can be called without prior use of SISO IV as it can perform its own initialisation. The model output and residuals are again available after passing through this instrumental variable routine.
8.
AML is the approximate maximum 1 ikel ihood al gorithm used to estimate the parameters in an ARMA model of specified order for either (i) the noise residuals computed in SISO IV, MISO IV, REFINED IV, TVAR or CONT IV; or (ii) any (preprocessed) input series. To aid identification of order, Akaike's (1974) information criterion (AIC) is given for each model structure selected.
9.
FINISH allows the user to exit from the fi rst program and 1eaves hi m free to enter programs 10,11 or 12.
10.
CONT IV is the instrumental variable algorithm for estimation of the continuous- time model parameters in (2). If initial parameter estimates are not known, then SISO IV must be called prior to this. As for all the instrumental variable routines, model output and residual series are produced here.
569
A Computer Program for General Recursive Time-Series Analysis
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11.
CORR will cross. auto-. partially correl ate or prewhiten any of the series available on temporary files. e. g. raw or processed inputs and outputs. model outputs. res i dua 1 s; it can be used for impul se response estimation based on a prewhitened input series.
12.
PLOT wi 11 graph the series avai lable from temporary files. These series are displayed immediately upon accessing Initially the raw input the program. and output series are available for plotting. After routine 3 has been utilized. the preprocessed I/O is ready for plotting. In addition to these. the following are available after the indicated routines have been used: 4:SISO IV model output. residuals and impulse response 5:MISO IV model output and residuals 6:TVAR model output. residuals. and parameter variations 7:REFINED IV model output and residuals 10:CONT IV model output and residuals 11:cross-. autoand partial corre 1at i on of any seri es so processed in 11.
Note also that these files. once created by the last call of the routines. are all available (e.g. for plotting) until Signing off the computer. The on ly except ions to this rule are the raw I/O data which must be plotted before preprocessing. EXAMPLES To demonstrate the ease of use of the package. a simple example has been taken from an air quality application in Jakeman and ~thers (1980) and Young and Jakeman (1979). It is most useful in that it
demonstrates all the major uses of the package. i.e. all the IV routines: SISO IV. REFINED IV. CONT IV and MISO IV. The data consist of two inputs. the 'upstream' concentrations of ozone at two different locations in the San Joaquin Valley of California. and an output which is the downwind concentrati on of ozone at another location in the valley. Either input can be used wi th the output to i dent i fy and estimate a good transfer function model; or a lternat i ve ly both inputs can be used. The computer outputs shown in this section are the results of the following operations: enter package via program 1 read I/O data. select plot title and sizes and exit (c) plot I/O data by enteri ng program PLOT and exit (d) re-enter program 1 and preprocess input 1 by lagging it one time interval (determined from the previous plots. use of the CORR program and the identification statistics from SISO IV using no lag. lag 1 and lag 2 - this is not shown explicitly) (e) call routine SISO IV for input 1 invoking identification criteria through the D (display criteria) option (f) repeat (e) for input 2 (g) call MISO IV using the 2 input series (one of them has been preprocessed) and initialise a model of order nl = n2 = 1. ml+l = m2+1 = 2 with initialisatlon from rounded" estimates obtai ned in (e) and (f) (h) change plot title and call REFINED IV for input 1 using model order identified in (e) and exit (i ) enter program CONT IV. use estimates in (e) for initialisation and exit (j) enter PLOT and obtain graphs of output, mode 1 output and res i dua 1s from CONT I V algorithm. then exit.
(a)
(b)
570
A. J. Jakeman, P. C. Young and A. J. Bayes
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PROCESSED INPUT 1 PROCESSED INPUT 2
OUTPUT OUTVEC SXHAT
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SPSI MXHAT
MPSl
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EXECUTION TIME:
112
128
OR E FOR END
4 MILLISECONDS .
CONCLUSIONS Thi s paper has out 1i ned the 1atest vers i on of the CAPTAIN time-series analysis computer package and demonstrated how it can be used for the SISO or MISO modeling of discrete data in discrete or continuous-time model form. Such model i ng is ai ded cons i derab ly by the novel structure identification strategies available in CAPTAIN and the sophistication of the visual interactive, conversational mode of operation. It is fe It that the package represents the most flexible and complete form of recursive estimation so far developed. MIMO estimation using similar estimation techniques is possible (Jakeman and Young, 1979) but has not yet been implemented in the package because of its complexity. REFERENCES Akaike, H. (1974). A new look at statistical model identification, IEEE Truns. Auto. ContT'oZ, AC-l9, 718-722. Box, G.E.P., and G.M. Jenkins (l970). Time Series AnaZY8i8, FOT'eca8ting and ContT'oZ, Holden Day: San Francisco. Jakeman, A.J., and P.C. Young (l979). Refined instrumental variable methods of recursive time series analysis. Part Il: multivariable systems, Int. J. ContT'oZ, ~, 621-644. Jakeman, A.J., Steele, L.P. and P.C. Young (l980). Instrumental variable algorithms for I1lJltiple input systems described by multiple transfer funct ions, IEEE TT'an8 SY8tem8, Man and CybeT'netic8, SMC-lO, 593-602.
Wellstead, P.E. (1978). An instrumental product moment test for model order estimation, Automatica, ~ B9-91. Young, P.C. (l974). Recursive approaches to time series analysis, BuZZ. In8t. Math8 and AppZ., la-, 209-224. Young, P.C., and A.J. Jakeman (1979 1 ). The development of CAPTAIN: a computer aided program for time series analysis and the identification of noisy systems. In: ComputeT' Aided De8ign of ContT'OZ SY8tem8, M.A. Cuenod (ed.), IFAC Symposium, Zurich, Switzerland, Aug. 29-31, 391-400. Young, P.C., and A.J. Jakeman (l979 2 ). Refined instrumental variable methods of recursi ve time seri es analysi s. Part I: single input-single output systems, Int. J. ContT'oZ,~, 1-30. Young, P.C., and A.J. Jakeman (1980). Refined instrumental variable methods of recursive time series analysis. Part Ill: Extensions, Int. J. ContT'oZ, ~ 741-764. A.J. and R.E. Young, P.C., Jakeman, McMurtrie (1980). An instrumental variable method for model order identification, Autol7ntica, l.!..! 281294.