AnalySeries 1.0: a Macintosh software for the analysis of geophysical time-
series.
Didier PAILLARD1,3, Laurent LABEYRIE2, Pascal YIOU1
1Laboratoire de Modélisation du Climat et de l'Environnement, Centre
d'Etudes de Saclay, 91191 Gif-sur-Yvette, FRANCE
2Centre des Faibles Radioactivités, Domaine du CNRS, 91198 Gif-sur-Yvette,
FRANCE
also at Département des Sciences de la Terre, Université d'Orsay, 91405
Orsay, FRANCE
3currently visiting NCAR, P.O.Box 3000, Boulder CO 80307, USA
version submitted to EOS in june 1996.
Note:
final version was published in: Eos, Transactions, AGU, 77, Sept. 24, 1996,
page 379.
Post-scriptum (january 2015):
The latest release of AnalySeries (currently AnalySeries 2.0.8) can be
found on the LSCE web site at:
http://www.lsce.ipsl.fr/Phocea/Page/index.php?id=3
Part 1: (EOS newspaper)
The AnalySeries software has just been released. It is a freely available
program, designed for Apple Macintosh computers, that can perform many
different time-series analysis procedures. Much programming efforts have
been aimed at designing a 'user-friendly' graphical user interface (GUI) in
order to allow an easy access even to people unfamiliar with computer
calculations. Previous preliminary versions are already used by hundreds of
scientists worldwide. Though originally designed for paleoclimatic time-
series, it can be useful to most fields of Earth Sciences.
The essential features of AnalySeries are: (1) graphical and interactive
tools for stratigraphic correlation between sedimentary records and age-
model development. Several possible age models are available (piecewise-
linear, spline, harmonic), following either a tie-point [Prell et al.,
1986] or an optimization procedure [Martinson et al., 1987], or both; (2)
basic time-series tools including interpolation, integral-sampling,
smoothing, filtering, fitting; (3) an algorithm for the calculation of
astronomical and insolation time-series, following Berger [1978]; (4)
several spectral analysis tools, including the Blackman-Tukey, maximum
entropy, multi-taper methods, as well as singular spectrum analysis
[Vautard and Ghil, 1989].
Input of data and output of results can easily be performed using the cut
and paste commands to and from any spreadsheet-like software, though more
basic text file input and output is, of course, also possible. All commands
are menu-driven and results appear graphically on screen. A simple on-line
help is also included in the application.
AnalySeries is available on the Internet at the two following locations,
one in Europe: ftp://ftp-lmce.cea.fr/incoming/paillard/AnalySeries, and one
in the United States: http://www.ngdc.noaa.gov/paleo/softlib.html. It is a
stand-alone application working on all Macintosh systems. An optimized
PowerMacintosh version and an optimized coprocessor version are also
available. For more information, contact Didier Paillard at
[email protected]. This is LMCE contribution number 00375 and
CFR contribution number 1882.
References
Berger, A., Long-term variations of daily insolation and Quaternary
climatic change, J. Atmos. Sci., 35, 2362-2367, 1978.
Martinson, D.G., N.G. Pisias, J.D. Hays, J. Imbrie, T.C. Moore and N.J.
Shackleton, Age dating and the orbital theory of the ice ages: development
of a high-resolution 0-300,000 year chronostratigraphy, Quat. Res., 27, 1-
30, 1987.
Prell, W.L., J. Imbrie, D.G. Martinson, J.J. Morley, N.G. Pisias, N.J.
Shackleton, and H.F. Streeter, Graphic correlation of oxygen isotope
stratigraphy: application to the late Quaternary, Paleoceanography, 1, 137-
162, 1986.
Vautard, R. and M. Ghil, Singular spectrum analysis in nonlinear dynamics,
with applications to paleoclimatic time series, Physica D35, 295-424,
1989.
Part 2: (EOS electronic supplement)
AnalySeries 1.0 has just been released. It is available free of charge on
the Internet, for european users at location: ftp://ftp-
lmce.cea.fr/incoming/paillard/AnalySeries, and for american users at
location: http://www.ngdc.noaa.gov/paleo/softlib.html (or equivalently
ftp://ftp.ngdc.noaa.gov/paleo/softlib).
Previous preliminary versions of the software have been already used by
many scientists throughout the world, particularly in the field of
paleoclimatology for which it was designed. However, many tools available
in AnalySeries are more generaly useful, such as the spectral analysis
methods and most of the other time-series mathematical treatments.
Similarly, the methods of stratigraphic correlation are widely used in
Earth sciences.
AnalySeries is available for Macintoshes only (For spectral analysis on
UNIX workstations, you may use the 'user-friendly' SSAToolkit package
[Dettinger et al., 1995], also available on the Internet). AnalySeries was
designed to provide powerful mathematical tools to a broad range of
potential users, and most programming efforts were aimed at defining an
intuitive graphical user interface. The software requires at least about
1MB of available RAM and about 1.5MB of disk space. The number of data
points in series is limited only by available RAM and computing speed.
Depending of the computer, a few tens or hundreds of thousands of points is
probably a practical limit.
A brief discussion of the AnalySeries software features is given in the
following.
General features.
AnalySeries is an Apple Macintosh application that follows the general
requirements of graphically oriented, menu driven applications. Input and
output of data are possible either through file opening and saving, or more
simply through copying and pasting columns from a spreadsheet software. It
is therefore easy to work at the same time on AnalySeries and on some
spreadsheet, while exchanging data between the two applications. All
operations performed during an AnalySeries session are listed on its main
window, which may be saved independently of the data series to provide a
permanent record. Data series appear on the AnalySeries window as
rectangles, that can be selected and edited by mouse clicks. Once selected,
any series or group of series can be manipulated by one of the commands
available in the menu bar, eventually creating new result series. Series or
groups of series can be plotted on screen, with several possible options
(zooming, x- and y-scaling, ...). Output series include a small header
explaining how it was obtained. The general environment thus provided by
AnalySeries enables a fast and easy access to sophisticated and powerful
mathematical time-series analysis methods, as well as to other useful
simple tools. A simple on-line help is also included in the application.
AnalySeries has been specially designed to facilitate the study of
paleoclimatic records using the approach and some of the methods defined by
the SPECMAP group [Martinson et al., 1987; Imbrie et al., 1984, 1989]. Two
main problems are thus addressed: (1) the transformation of 'data versus
depth' records into 'data versus age' records; (2) the spectral analysis of
the paleo records, to study their relationships with insolation, ice
volume, and other climatic parameters in the frequency domain.
Constructing age-depth relations by correlating sedimentary records.
A classical method for establishing an age-scale on a sedimentary record is
to use a comparable well-dated signal as a reference signal, and then to
optimize some measurement of the similarity between the two series, while
changing the depth-scale of the first one into the age-scale of the second
one. Two main questions come then into mind. First, what is a good measure
of the similarity between the two time-series, and second, what kind of an
age-depth relation is desirable. Very often, when many quantities are
measured on the same sedimentary record, a third problem is then to use as
much as possible of the available information to construct a final
consistent age-depth relation.
With regard to the first question, two classical approaches are commonly
used. The first and simplest one is to 'put together the corresponding
remarkable features of both signals' [e.g. Prell et al. 1986]. Though very
appealing by its simplicity, this method may give subjective 'user-
dependent' results, for the identification of 'remarkable features' may
sometimes be somewhat arbitrary. The second possibility is to use a
mathematical measure of the similarity between both signals, for example a
correlation coefficient, and then optimize this measure when adjusting the
age-depth relation [Martinson et al. 1987]. This procedure is likely to
give a more objective result. Unfortunately, the 'fit' does not always
appear as good as with the simple visual correlation. A mathematical
measure such as a correlation coefficient will indeed give much more weight
to the large time scale signal fluctuations (low-frequency variations)
where much of the variance is located, than to the rapid ones that usually
account for a small amount of the total variance. As a result, the rapid
transitions or spikes are not exactly in phase, as they should be according
to the underlying simultaneity hypothesis (see Figure 1). This second
approach is therefore more objective, but often less precise.
In addition to offering both classical methods, AnalySeries also provides a
trade-off by allowing the simultaneous use of both methods, associating
remarkable features identified on both signals, and optimizing a
correlation coefficient in other regions of the record where no such clear
features have been identified.
Furthermore, the age-depth relationship cannot be fully arbitrary, for the
number of available constraints on the age-scale is always much smaller
than the number of data points. Most commonly, when using tie-points to
constraint the age of a record, a constant sedimentation rate is assumed
between the tie-points (piecewise-linear age-depth relation). This is
probably the method of choice in AnalySeries (command Linage), the only
clear drawback of this method being to introduce discontinuities in the
sedimentation rate. When computing fluxes to the sediment, this results in
undesirable artificial transitions. The simplest alternative is probably to
use a spline interpolation between tie-points, instead of a piecewise-
linear one. But one must be careful that the corresponding age-depth
relation remains strictly increasing, as required by basic stratigraphy.
This is automatically enforced in AnalySeries (command Splinage) by using a
special class of spline, where the continuity of the second derivative is
eventually relaxed in order to enforce monotony. A third possibility
offered by AnalySeries is to use a linear + sinusoidal relation, for
compatibility reasons with the Martinson et al. [1987] algorithm.
But one main feature of AnalySeries is the possibility to perform this kind
of stratigraphic adjustment (tie-points or optimization, piecewise-linear
or other) simultaneously using several proxies of the same sedimentary
record. One example of such a situation is when comparable records are
obtained from two nearby sites. It is then desirable to build a common
stratigraphic framework, using not one, but eventually many proxies in both
records. AnalySeries allows the user to put tie-points on one pair of
signals, while interactively showing the effect on all other pairs. By
changing pairs appearing on screen, adding and removing tie-points while
controlling the result, it is thus possible to establish quickly and easily
a consistent common stratigraphic scale.
Astronomical and insolation series.
AnalySeries calculates the astronomical and daily insolation time series,
following Berger [1978]. The results are accurate up to about a million
years in the past [Berger and Loutre, 1991]. The astronomical series are
the eccentricity of the Earth orbit, the obliquity (the tilt) of the Earth
axis, and the precessional parameter, classically defined as e sin ω,
where e is the eccentricity and ω the longitude of the perihelion. These
three parameters govern the amount of solar energy received by the Earth at
the top of the atmosphere, or insolation.
AnalySeries offers two different possible calculations for the insolation.
First, the "daily insolation", which is defined as the instantaneous amount
of energy received at a given latitude and a given orbital position (i.e.,
a given pseudo-calendar date), and averaged over one Earth rotation. The
second option is to average this previous value between two orbital
positions (two pseudo-calendar dates), taking into account the varying
speed of the Earth on its orbit. It is worth mentioning that the present-
day calendar cannot be applied directly to define the Earth position on its
orbit, for the number of days between each astronomical season (solstices
and equinoxes) is different at different geological time periods. For
example, winters are approximately 90 days and summers 92 days long in our
present calendar, but the opposite was true 11 kyrs ago. Therefore, only
the true orbital position (the angular sector defined between the Earth and
the vernal point) has a real signification. The pseuso-calendar provided in
AnalySeries is an averaged calendar, made of 360 days (12 months of 30
days) corresponding to the 360 angular degrees on the Earth orbit. The days
thus defined are not 24 hours time lapses, but just a convienient way to
define the Earth position on its orbit.
Spectral analysis.
Given a time series, one of the first concern is often to identify
recurrent features or periodicities, and spectral analysis is then the tool
of choice. Many different methods have been suggested for identifying
periodicities in time series, and each has its associated advantages and
drawbacks. AnalySeries provides a set of classical spectral analysis
methods that are often complementary in terms of robustness versus
resolution.
First, the periodogramme method, which is the most straightforward and
unstable method for doing spectral analysis. The data (eventually
multiplied by a window, so as to minimize side effects, or leakage) is just
Fourier transformed, and the spectrum calculated directly from the Fourier
coefficients. This method should not be used for real (noisy) data, because
results are not stable with respect to small changes in the input signal.
It is present in AnalySeries mainly for pedagogical reasons, as the
simplest possible method, and helps understand the necessity of more robust
and sophisticated methods.
The Blackman-Tukey method [Blackman and Tukey, 1958] is the classical
method for doing spectral analysis. The algorithm computes first the
autocovariance of the data, then applies a window, and finally fourier-
transforms it to compute the spectrum. It is a very robust method, unlikely
to present spurious spectral features. The main drawback is its poor
resolution in the spectral domain: Sharp features are most of the time
considerably smoothed. This method requires you to choose a resolution
versus confidence parameter: The length of the autocovariance series. You
can choose it in terms of a number of lags (number of autocovariance values
calculated), percentage of the series length (number of lags divided by the
length of the series) or some predefined levels. It is a good idea to
perform several computations, with different resolution/confidence levels.
Different types of window are also available, though this should not affect
considerably the results for typical (short and noisy) geophysical time
series (except for the square window, which may cause considerable
aliasing). This method also provides an error bar on the spectrum, as well
as a band width (error bar on the frequency). The confidence level
associated with the error bar on the spectrum can be adjusted as desired.
Cross-spectral analysis is also provided with this method.
The maximum entropy method [see e.g. Haykin, 1983] is useful for its high
resolution, its main drawback being the lack of any statistical confidence
estimate. As in the Blackman-Tukey method, you need to choose a
resolution/confidence parameter. The conventional wisdom is to make several
calculations, increasing this parameter, so as to increase the resolution,
but to stop the procedure when to many spectral lines pop out of the
background spectrum. It is strongly advised not to use this method alone,
but in conjunction with a more robust method, like Blackman-Tukey.
The multi-taper method [Thomson, 1982] is a relatively new method offering
some very interesting features: a high resolution and statistical estimates
that are independent of the spectral power (small amplitude oscillations
may have a high significance level). Some caution is nevertheless advised:
the statistical confidence levels given by this method are often (much)
more optimistic than the ones given by more classical methods. This is due
to different statistical null-hypothesis, therefore the confidence levels
are not directly comparable to the ones provided by other methods.
The singular spectral analysis method [Vautard and Ghil, 1989] is not truly
a spectral analysis method. It performs an empirical orthogonal function
(EOF) analysis in the time domain, and thus represents the signal as a sum
of components which are not necessarily oscillations, but more general,
data adaptive, functions. It can thus not only be used to identify spectral
lines (which appears as a pair of nearly sine and cosine functions in the
signal decomposition), but also as a very powerful noise filter, through
its ability to separate self-coherent features from random ones.
Other tools
Several other commonly used tools are also included in AnalySeries. First,
a powerful 'Resampling' command enables to resample any input signal using
any other x-scale (a user defined evenly spaced one, or a combination of
scales in other series). The resampling can be done by interpolation (using
a piecewise-constant, piecewise-linear or spline function), by integral-
sampling (new samples are averages of previous ones) or by fitting to some
model function (piecewise-constant, piecewise-linear or spline). It is
therefore extremely easy to put several records on the same scale, while
under- or over-sampling them. Smoothing and filtering are also provided.
The smoothing function is a simple n-points least square low-pass filter.
Filtering is performed using a band-pass gaussian filter, whose center and
width are chosen by the user.
For more information, comments or suggestions on AnalySeries, contact
Didier Paillard at
[email protected].
This is LMCE contribution number 00375 and CFR contribution number 1882.
Acknowledgements: The AnalySeries software is derived from a previous
Macintosh program written on MPW/Fortran format by E. Chol, L. Jodet and F.
Lecoat under the direction of L. Labeyrie, and financial support from the
French Programme National d'Etude de la Dynamique du Climat. We thank J.
Overpeck for helpful suggestions on the manuscript, as well as the numerous
users of AnalySeries who have made comments and reported bugs.
References
Berger, A., Long-term variations of daily insolation and Quaternary
climatic change, J. Atmos. Sci., 35, 2362-2367, 1978.
Berger, A. and M.-F. Loutre, Insolation values for the climate of the last
10 million years, Quaternary Science Reviews, 10, 297-317, 1991.
Blackman R.B. and J.W. Tukey, The measurement of power spectra from the
point of view of communication engineering, 190 pp., Dover Publications,
New York, 1958.
Dettinger, M.D., M. Ghil, C.M. Strong, W. Weibel and P. Yiou, Software
expedites singular-spectrum analysis of noisy time series, Eos Trans.,
AGU, 76, 12, 1995.
Haykin S., Nonlinear methods of spectral analysis, 2nd edition, Springer-
Verlag, Berlin, 1983.
Imbrie, J., J.D. Hays, D.G. Martinson, A. McIntyre, A.C. Mix, J.J. Morley,
N.G. Pisias, W.L. Prell, and N.J. Shackleton, The orbital theory of
Pleistocene climate: support from a revised chronology of the marine δ18O
record, in Milankovitch and Climate, Part 1, edited by A.L. Berger et
al., pp. 269-305, D. Riedel, Hingham, MA, 1984.
Imbrie, J., A. McIntyre, and A. Mix, Oceanic response to orbital forcing in
the late Quaternary: observational and experimental strategies, in
Climate and Geosciences, NATO ASI Ser. C edited by A. Berger et al., pp.
121-164, Kluwer Academic, Dordrecht, 1989.
Martinson, D.G., N.G. Pisias, J.D. Hays, J. Imbrie, T.C. Moore and N.J.
Shackleton, Age dating and the orbital theory of the ice ages:
development of a high-resolution 0-300,000 year chronostratigraphy, Quat.
Res., 27, 1-30, 1987.
Prell, W.L., J. Imbrie, D.G. Martinson, J.J. Morley, N.G. Pisias, N.J.
Shackleton, and H.F. Streeter, Graphic correlation of oxygen isotope
stratigraphy: application to the late Quaternary, Paleoceanography, 1,
137-162, 1986.
Thomson, D.J., Spectrum estimation and harmonic analysis, IEEE Proc. 70
(9), 1055-1096, 1982.
Vautard, R. and M. Ghil, Singular spectrum analysis in nonlinear dynamics,
with applications to paleoclimatic time series, Physica D35, 295-424,
1989.
Figure Captions
Figure 1: The two different methods for stratigraphic correlation. Here,
the lower (blue) curve is shifted so as to optimize (a) visual alignment
of maxima or (b) the correlation coefficient c. Though the correlation
coefficient is larger in (b) (0.930 versus 0.856), it can be argued that
(a) is better, for the events are simultaneous. Similarly, though the
correlation coefficient is larger in (d) than in (c), the transitions are
simultaneous in (c) while they are slightly offset in (d).
