AUTO_PARM

FNLStat : Time Series Analysis and Forecasting : AUTO_PARM

AUTO_PARM

Required Arguments

Estimates structural breaks in non‑stationary univariate time series.

Required Arguments

Y — Array containing the time series. (Input)

NPCS — Number of requested/estimated pieces or segments of the time series. NPCS is considered input when IFITONLY = 1. (Input/Output)

ARP — A pointer to an array of size NPCS × 2. ARP is considered input when IFITONLY = 1. (Input/Output)

Column Index	Description
1	Requested/estimated break points
2	AR order for each segment

ARFIT — A pointer to an array of size NPCS × MAXARORDER containing the AR coefficient estimates for each segment. ARFIT (i,j) is the j‑th coefficient for segment i where i=1, NPCS and j=l, ARP(i,2). (Output)
Note that the intercept is not reported.

ARSTAT — A pointer to an array of size NPCS × 2. (Output)

Column Index	Description
1	Likelihood values for each of the fitted AR models
2	Residual variances for each of the fitted AR models

SCVAL — Final value of the selection criterion. (Output)

Optional

MAXARORDER—Maximum order to consider for each AR model. (Input)
Default: MAXARORDER = 20.

IMTH — Method of estimation. (Input)

Value	Method
0	Yule‑Walker
1	Least Squares
2	Burg

Default: IMTH = 0.

ISELCRI — Selection criterion. (Input)

Value	Method
0	Minimum Description Length (MDL)
1	Akaike’s Information Criterion (AIC)

Default: ISELCRI = 0.

ILIKE — Likelihood computation method. (Input)

Value	Method
0	Exact
1	Approximate

Default: ILIKE = 0.

IFITONLY — Option to only fit the specified model. (Input)

Value	Action
0	No, do all the computations
1	Yes, only fit the specified model

Default: IFITONLY = 0.

IPRINT — Printing option. (Input)

Value	Action
0	No printing
1	Prints final results only
2	Prints intermediate and final results

Default: IPRINT = 0.

ISEED — Seed of the random number generator. (Input)
For the same data and parameter settings, AUTO_PARM will return the same results each time if ISEED = 1, 2 … 2147483646. If ISEED = 0, the system clock will be used to generate a seed. The result will be nondeterministic.
Default: ISEED = 0.

Note: The following arguments are for setting up and running the embedded Genetic Algorithm. In most situations, the default values should be used for these arguments. Users may wish to change some or all for testing or research purposes.

PDISTN — Array of length MAXARORDER + 1 giving the probability distribution over the AR order variable p = 0,…, MAXARORDER. (Input)
j = 1,…, MAXARORDER + 1 is used to randomly assign an AR order to breakpoint position j for a given chromosome. PDISTN(j) > = 0 and if SUM(PDISTN) is not equal to 1, the values will be normalized, i.e., PDISTN(j) = PDISTN(j)/SUM(PDISTN).
Default: PDISTN (j) = 1/(MAXARORDER + 1) for all j.

MSPAN — Array of length MAXARORDER + 1 containing minimum number of observations required for valid estimates of AR model with order p = 0, …, MAXARORDER. (Input)
Default: MSPAN (p+ 1) = 2 * (number of parameters) + 2 = 2 * (p + 2) + 2.

GAPARM — Array of length 4 containing parameters that control the behavior of the genetic algorithm. These values should be strictly greater than zero and less than one to avoid unexpected results. (Input)

GAPARM (1) — Probability used to set initial break points in a chromosome.
Default: MIN (MSPAN) / size(Y).

GAPARM (2) — Probability of crossover used to decide between a crossover and a mutation.
Default: 1 – MIN (MSPAN) / size(Y).

GAPARM (3) — In the mutation operation, probability an AR(p) model is enforced at the current position.
Default: 0.4.

GAPARM (4) — In the mutation operation, probability a break point is disallowed at the current position.
Default: 0.3.

Note: GAPARM(3) and GAPARM(4) must be valid probabilities and their sum must be between 0 and 1. 1 – GAPARM(3)– GAPARM(4) is the probability that the chromosome j inherits the parent’s chromosome j.

ISLAND — Array of length 5 containing the migration policy parameters. (Input)

ISLAND (1) — Number of islands.
Default: 40.

ISLAND (2) — Number of generations that pass before migration occurs. Note that the convergence of the algorithm is revised whether migrations take place or not (see argument ISLAND(5)).
Default: 5.

ISLAND (3) — Number of subjects that migrate at each migration event.
Default: 2.

ISLAND (4) — Population size (number of chromosomes) per island.
Default: 40.

ISLAND (5) = Migration flag. If 1, migration is performed.
Default: 1.

MAXMIG — Maximum number of times that migrations may take place before the routine is stopped if convergence has not occurred. (Input)
Default: MAXMIG = 20.

STOPITERS — Number of iterations. The routine will declare convergence and stop the iterations if the criterion value (MDL/AIC) has not changed after STOPITERS consecutive migrations. Otherwise, the algorithm will declare non‑convergence and stop after MAXMIG migrations have taken place. See also MAXMIG and ISLAND(2). Note that logically, STOPITERS < MAXMIG. (Input)
Default: STOPITERS = 10.

Interface

Generic: CALL AUTO_PARM (Y, NPCS, ARP, ARFIT, ARSTAT, SCVAL [, …])

Specific: The specific interface names are S_AUTO_PARM and D_AUTO_PARM.

Description

Routine AUTO_PARM estimates the structural breaks of a non‑stationary time series using, with permission from the Authors, the method developed by Davis et al (2006). AUTO_PARM estimates a partition of the time index and models the time series in each segment as a separate auto‑regressive (AR(p)) process. The routine returns the estimated breakpoints, the estimated AR(p) models, and supporting statistics.

For the observed time series ,

the problem is to find m(m+1 = NPCS), the number of breaks, their locations,

(ARP(:,1)), and Pj(ARP(:,2)), j = 1, …, m + 1, the order of the AR process in which the j-th segment is modeled. That is,

for

(for convenience,

and

) where {Xt,j} is an AR process of order Pj.

and ɛt, the noise sequence, is i.i.d. with mean 0 and variance 1. Note that a series with m breaks will have m + 1 segments.

The vector

completely specifies a piecewise AR model. To estimate this vector AUTO_PARM optimizes, with respect to this vector, one of two selection critieria: the first is a Minimum Description Length (MDL) criterion, and the second is the Akaike’s Information Criterion (AIC). The MDL is defined as

while the AIC criterion is given by

where, given a candidate value of the vector

, L is the likelihood of the fitted piecewise AR model evaluated at the parameter estimates,

The parameters

of the j-th AR segment are estimated by the choice of one of three estimation methods: Yule-Walker, Burg, or Least Squares.

For simplicity, assume the mean of each series

is 0 and that the errors are Gaussian. Then, the piecewise AR model has Gaussian likelihood

where

is the variance-covariance of the j-th AR segment (of order

) and

is the vector of observations of the j-th segment, i.e.,

To find the minimizer {m,

j, pj}* of either MDL or AIC, AUTO_PARM employs a Genetic Algorithm with islands, migration, cross‑over and mutations. See Davis et.al. (2006) for further details.

Comments

1. AUTO_PARM approximates locally stationary time series by independent auto‑regressive processes. Experimental results suggest that AUTO_PARM gives reasonable estimates of the structural breaks of a given time series, even if the segment series are not autoregressive. Also, based on experimental results, MDL gives better results than AIC as a selection criterion.

2. Informational Error

Type	Code	Description
3	1	ISEED has been set out of range. ISEED is being reset to 123457.
3	2	MAXMIG migrations were reached in the genetic algorithm before the selection criterion value converged. Try increasing MAXMIG or using the double precision routine.

Examples

The examples below illustrate different scenarios using AUTO_PARM. The example series used in each case is the airline demand data (Box, Jenkins and Reinsel, 1994), which gives monthly total demand for the period January 1949 through December 1960. Each scenario sets the optional argument, ISEED = 123457.

use gdata_int

use auto_parm_int

use diff_int

use umach_int

implicit none

! Specifications for local variables

! arguments for auto_parm

integer :: npcs, maxarorder, ifitonly, iprint, island(5), iseed

integer, pointer :: arp(:,:)

real(kind(1e0)) :: x(144,1), scval

real(kind(1e0)), pointer, dimension(:,:) :: arfit, arstat

! Other locals

integer :: n, nvar, idata, nlost, iper(1), iord(1), nout

real(kind(1e0)) :: dx(144,1)

call umach(2, nout)

! Get the data

iseed = 123457

idata = 4

nullify(arp)

nullify(arfit)

nullify(arstat)

call gdata(idata, x, n, nvar)

! Example 1: Use defaults and print

! final results

write(nout,*)'Example 1: Use defaults'

write(nout,*)

call auto_parm(x(:,1), npcs, arp, arfit, arstat, scval, &

iseed=iseed, iprint=1)

! Example 2: Differenced series

! set period for the difference.

! iper is in years for this data set

write(nout,*)

write(nout,*)'Example 2: Differenced series'

write(nout,*)

! Set the order for the difference.

iper(1) = 1

iord(1) = 1

! Get differenced series dx

call diff(x(:,1),iper,iord,n,dx(:,1),nlost=nlost)

! Compare results on the differenced

! series

deallocate(arp)

deallocate(arfit)

deallocate(arstat)

call auto_parm(dx((1+nlost):n,1), npcs, arp, arfit, arstat, &

scval, iseed=iseed, iprint=1)

! Example 3: Original series

! lower maximum AR order

write(nout,*)

write(nout,*)'Example 3: Original series, lower order allowed'

write(nout,*)

maxarorder = 5

deallocate(arp)

deallocate(arfit)

deallocate(arstat)

call auto_parm(x(:,1), npcs, arp, arfit, arstat, scval, &

maxarorder=maxarorder, iseed=iseed, iprint=1)

! Example 4: differenced series, lower

! maximum AR order

write(nout,*)

write(nout,*)'Example 4: Differenced series, lower maximum'// &

' AR order'

write(nout,*)

deallocate(arp)

deallocate(arfit)

deallocate(arstat)

call auto_parm(dx((1+nlost):n,1), npcs, arp, arfit, arstat, &

scval, maxarorder=maxarorder, iseed=iseed, &

iprint=1)

! Example 5: Original series, force

! fit the segments

! Fit the specified model only at the

! break points

write(nout,*)

write(nout,*)'Example 5: Original series'

write(nout,*)'Force fit the segments at the break points'

write(nout,*)

npcs = 2

deallocate(arp)

allocate(arp(npcs,2))

arp(1,2) = 2

arp(2,2) = 1

arp(1,1) = 1

arp(2,1) = 60

deallocate(arfit)

deallocate(arstat)

call auto_parm(x(:,1), npcs, arp, arfit, arstat, scval, &

ifitonly=1, iseed=iseed, iprint=1)

end

Output

Example 1: Use defaults

============== final results ===============

number of pieces: 2

selection criteria value: 684.242

total time: 0.7198125 conv: 1

==================== final model estimates =====================

break point order est. coeff. likelihood resid. var

ARP( 1,1) ARP( 1,2) ARFIT( 1,:) ARSTAT( 1,1) ARSTAT( 1,2)

1 1 0.77542

186.945 355.025

ARP( 2,1) ARP( 2,2) ARFIT( 2,:) ARSTAT( 2,1) ARSTAT( 2,2)

44 13 1.03701

-0.07802

-0.03890

-0.03452

0.11961

-0.12852

0.01990

-0.04886

0.08090

-0.13118

0.22122

0.53863

-0.61515

486.665 691.471

Example 2: Differenced series

============== final results ===============

number of pieces: 1

selection criteria value: 624.284

total time: 0.7204375 conv: 1

==================== final model estimates =====================

break point order est. coeff. likelihood resid. var

ARP( 1,1) ARP( 1,2) ARFIT( 1,:) ARSTAT( 1,1) ARSTAT( 1,2)

1 12 -0.02842

-0.22436

-0.16846

-0.24267

-0.10573

-0.22429

-0.12126

-0.26446

-0.07087

-0.24327

-0.07136

0.57129

619.321 297.351

Example 3: Original series, lower order allowed

============== final results ===============

number of pieces: 2

selection criteria value: 705.297

total time: 0.3136875 conv: 1

==================== final model estimates =====================

break point order est. coeff. likelihood resid. var

ARP( 1,1) ARP( 1,2) ARFIT( 1,:) ARSTAT( 1,1) ARSTAT( 1,2)

1 1 0.89533

270.393 333.564

ARP( 2,1) ARP( 2,2) ARFIT( 2,:) ARSTAT( 2,1) ARSTAT( 2,2)

63 2 1.19788

-0.35922

424.270 1632.337

Example 4: Differenced series, lower maximum AR order

============== final results ===============

number of pieces: 2

selection criteria value: 698.359

total time: 0.2981875 conv: 1

==================== final model estimates =====================

break point order est. coeff. likelihood resid. var

ARP( 1,1) ARP( 1,2) ARFIT( 1,:) ARSTAT( 1,1) ARSTAT( 1,2)

1 0 -------

335.565 357.388

ARP( 2,1) ARP( 2,2) ARFIT( 2,:) ARSTAT( 2,1) ARSTAT( 2,2)

77 1 0.33310

352.175 1786.345

Example 5: Original series

Force fit the segments at the break points

============== final results ===============

number of pieces: 2

selection criteria value: 712.521

==================== final model estimates =====================

break point order est. coeff. likelihood resid. var

ARP( 1,1) ARP( 1,2) ARFIT( 1,:) ARSTAT( 1,1) ARSTAT( 1,2)

1 2 1.12156

-0.24876

258.192 313.889

ARP( 2,1) ARP( 2,2) ARFIT( 2,:) ARSTAT( 2,1) ARSTAT( 2,2)

60 1 0.88605

443.696 1937.635

Published date: 03/19/2020

Last modified date: 03/19/2020