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Katinas G.S.
Scientific Committee “Chronobiology and Chronomedicine”RAMSci, Saint-Petersburg
LOGISTIC INFORMATIVE COMPLEX OF TIME SERIES ANALYSIS
Most of biological and medical time series are non-equidistant. The reasons are: inevitable gaps of records during night spans using manual registrations, errors in records at the automating monitoring, occasional errors with manual data input and so on. Most of technical program applications deal with equidistant series, and that is why using them for biological series is restricted.
Most of specialized chronobiological programs developed in F.Halberg’s Chronobiological Center work with non-equidistant series, but are used only inside his lab. Only COSINOR became widespread and now has a lot of modifications.
We began elaborating a specialized complex for time series analysis in 1970-th and now it represents a group of several programs which allow using the output data of any of them as the input data for another ones. This paper presents a general description of the possibilities of the programs. The general chronobi-ological and statistical terms, as well as the elements of the theory of time series analysis are considered to be known for the reader.
The complex provides the following possibilities: sampling, accumulating, storage and processing (analysis) both un-equidistant and equidistant time series, storage of intermediate and final results, commutation of different links of analytical ways in any sequence, presenting the accounts of results in listings and graphs.
Both consequent processing of a single series by different programs and comparison of the results after several different series analyses are provided. The possibility of using intermediate results and commuting the ways of analyses admits considering the complex as a logistic informative one. Choosing the ways for commuting is facilitated by the united system of file nomination.
The input parameters are set by the user.
It is important, since the results of processing, and thus, their following interpretation, depend on them. All the programs usually must be used carefully, and their input parameters should be tuned for the solution of the concrete task. In this term the user might be compared with a surgeon, and a program -with a surgical knife: for amputation of a leg it must have some specific qualities (size, shape of blade and so on), and for eye lens cataract removing - another
one. The results of the program processing will also be obtained differently depending on the input settings. The program is not a cliché for strictly adjusted for making pastry pets for pies, but it is a chisel for sculpture grinding out - for revealing the regularities hidden for the naked eye.
The main input parameters used by different programs of the complex, which should be set by the user are:
A - phase reference point,
B - length of the series to be analyzed (interval), C - shift of the interval (increment),
D - the minimal trial period,
E - the maximal trial period,
F - smoothing polynomial order,
G - critical value of probability,
H - resolving power of the spectrum (specification of a periodogram curve: amount of trial periods in a single harmonic limits),
I - the mode of abscissa presenting in the output files: arithmetic progression (linear in periods), harmonic (linear in frequencies), or geometric (linear by logarithm of frequency),
J - the measure of a specifying signal shape description (gliding interval length divided by trial period length).
The series to be analyzed is put in too. It may be any output product of other links of complex.
An analysis of a single series includes the following main steps:
1 - Weeping out outliers (Fig. 1).
The data, whose deviations from the middle level of the process or from the trend line are outside the limits of the critical probability, are eliminated. Input settings: primary data series, G.
Output data: series, where outliers are replaced by some conditional symbol,
before using another programs this symbol would be eliminated.
2 - High-frequency noise bolting out.
Series is smoothed by a gliding 1-st order polynomial by 3 consequent points.
This procedure eliminates hidden periodicities which are out of the resolution power of spectral analysis.
Input settings: series which contains high-frequency components to be excluded.
Output data: 1 - smoothed series having the same abscissas.
3 - Computing preliminary spectrum.
Periodogram is plotted, which demonstrates
peaks distribution in the trial period window. It allows a decision whether the spectral band of frequencies was well chosen, and which frequencies would be chosen for filtering data, if necessary.
Input settings: A, D, E, H.
Output data: 1 - periodogram, middle level of the process, 3 - for each peak its frequency (3a), power (3b), amplitude (3c) and acrophase (3d).
4 - Low-frequency trend elimination and/or filtering series (see Fig. 1 and Fig. 2).
File is separated in 2 sub-series. One of them contains long periods (low frequencies), another one
- residuals containing short periods (high frequencies).
Input settings: A, B, C, F.
Output data: 1 - trend abscissas and ordinates, 2
- residual abscissas and ordinates, 3 - general power of the process, 4 - trend and residuals power, 5 - determination coefficient.
5. Computing global spectrum of oscillations for elimination of false periodogram peaks (Fig. 3 and 4).
Periodogram of the total series is plotted, which makes it possible not only to reveal peaks inside the trial period band, but also to eliminate those, which appear because of eliasing and/or do not increase the useful variance of approximated signal.
Input settings: A, D, E, G, H, I.
Output data: 1 -the length of detected trial periods, 2 - midlevel of the process (mesor), 3 - amplitude, 4 - acrophase, 5 - 8 - their confidence limits, 9
- each trial period power, 10 - residual sum of squares, 11 - statistical significance of spectral components, 12 - critical values of power for significance levels set, 13 - ordinates of the process approximating curve are computed along the total series length.
6 - Gliding spectrum and serial time sections computing, quasi-stationary and stationary spans detecting (Fig. 5).
Input settings: A, B, C, D, H, I.
Output data: matrices (tables). First line includes trial periods rehearsal (1), the following ones -
abscissa of the mid interval value (2), and parameter values of each trial period.
Separate matrices are created for each parameter of spectral component:
3 - уровня (мезора, 3), 4 - амплитуды, 5 - ак-рофазы, 6-8 - их погрешностей, midlevel/mesor (3), amplitude(4), acrophase (5), their statistical errors (68), power (9), statistical significance (9 -10).
7 - Signal shape approximation at quasi- stationary spans (Fig. 6).
Detection of hidden periodic signals having any wave form. Reconstruction (approximation) of the wave form without using summarizing trigonometric functions. Statistic evaluation of the curve profile parameters.
Input settings: A, B, C, E, F, J.
Output data: 1 - midlevel, 2 and 3 - maximal and minimal values of input data, 4 - their timing. 5 -swing of input data, 6 and 7 - peak and bottom values (maximal and minimal values of approximating curve), 8 and 9 - phase of their appearance (clock time if the period is equal to 24 h), 10 - swing of approximated values, 11 and 12 - value of intermediate hills and delves between peak and bottom, 13 - their statistical significance, 14 - their timing (phase of the cycle), 15 - confidence limits of approximated midlevel, as well as of absissas and ordinates of peak, bottom, hills and delves with the statistical significance set, 16 - determination coefficient, 17 - approximation quality evaluation, 18 - Velocity (first derivative) of the process at various cycle phases, acceleration of the process (second derivative) at various cycle phases.
Analyses of the data obtained as results of processing several separate series are provided by correlation and regression methods, confidence ellipses computing, estimation of possible resonances and coherence of processes.
All stages of time series are now realized in special programs and macroses.
All persons interested in the following developing complex are welcome: [email protected], matter -series.
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Систолическое артериальное давление (К-БсИ025, м 22 г), первичная регистрация в течение 7 сут
Время наблюдения
Fig 1. Eliminating outliers.
Chronogram of automatic systolic blood pressure (SBP) monitoring.
Abscissa: time (calendar date); ordinate: SBP (mm Hg).
Dots - results of primary recording.
Lines - broken blue line connects residuals after outliers elimination, orange line - outliers which would be eliminated before the following analyses,
blue smooth line - about-weekly (circaceptan) trend, which may be removed at the following processing stages (see stage 3)
Слабо выраженный тренд
Fig. 2. Trend of the primary series recorded by automatic diastolic blood pressure (DBP) monitoring.
Abscissa - time (calendar dates); ordinate - DBP (mm Hg).
Lines: blue - primary records, red - about-weekly (circaceptan) trend.
Outliers are not eliminated, it can be done after trend deleting, but processing in the opposite sequence leads to more precise analyzing at the following stages.
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Рис. 3. Периодограмма систолического артериального давления (пациент К, м 56)
Fig. з. Periodogram of systolic blood pressure (SBP): circadian and ultradian spectral bands.
Operative version of the graph.
Abscissa: frequency (cycles per 72 h); ordinate: power of spectral components (determination coefficient).
Horizontal blue lines - critical probability levels. Period values corresponding to the peaks are shown in frames on the main field of the graph.
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LEAST SQUARE PERIODOGRAM: DETERM. COEFF.
Cycles per 9 years
Fig. 4. Periodogram of geomagnetic Ap-index (2ооз - 2оіі); spectral band і.з years - 2з days.
Operative version of the graph.
Abscissa: frequency (cycles per 9 years); ordinate: power of spectral components (determination coefficient).
Period lengths are framed on the main field of the graph, they correspond to peaks whose superposition increases the general power of the useful signal.
Mass of smaller side peaks situated around a main one is one of the signs of frequency modulation.
Graph was plotted using automatic BP monitoring data in the patient suffering from heart failure and intensively treated therapeutically and (after it) surgically.
Dynamics of spectrum detected disturbances in 24-and 12-hour rhythms interrelation and their amplitude modulation.
SPECTRAL DISTURBANCES
Oscillations (Periods, h) 96
Interval 48 h
Increment 12 h
Harmonic resolution 0.1
SBP
Amplitude (mm Hg)
■ >21
■ 18 - < 21
■ 15-< 18
G 12-< 15
G 09-< 12
G 06 -< 09
G 03 - < 06
5 10 15 20 25 30
Time (calendar dates in May 2010)
1- 2 Cardiac failure treatment, 3 - pacemaker implanted, 4 - atrio-ventricular ablation. 5 - end of treatment
Fig. 5. Gliding spectrum of systolic blood pressure (SBP).
Abscissa: time (calendar dates); ordinate: period length (h); right (framed) scale of colors of applicate axis, corresponding to the amplitude at the given moment and frequency.
Above the main field of the graph - approximation parameters (framed).
The graph is like a geographic map of surface height distribution (amplitude) according to width (ordinates) and longitude (abscissa).
СИСТОЛИЧЕСКОЕ АРТЕРИАЛЬНОЕ ДАВЛЕНИЕ ( Ч, ж 54, 21 - 24 мая 2012)
9 12 15
Время суток (ч)
18
21
24
Fig. 6. Wave shape of the 24-hour rhythm component.
Operative graph plot using the 3-day monitoring of systolic blood pressure (SBP) of patient Ch.
Abscissa: clock time (h); ordinate: SBP (mm Hg).
Black dots - separate SBP records.
Horizontal lines: black - midlevel at night rest time and day wake (brown lines - 95% confidential intervals). Black thick curve - approximated SBP profile.
Green line around it - 95% confidential interval of the profile values.
Blue thin lines enveloping black dots - 95% confidential interval of the total SBP values.
Blue thick line - velocity of SBP dynamics, horizontal blue one - o-level of velocity (conditional scale, the separate graph is plotted to demonstrate the real values, not shown here).
Green thick line - accelerations of SBP dynamics, horizontal green one - o-level of accelerations (conditional scale, the separate graph is plotted to demonstrate the real values, not shown here).
Orange arrows lead to the initial moment of time when physiological mechanisms are included, resulting in SBP elevation.
Red vertical lines with arrows are necessary for computing optimal hypotensive treatment timing (taking into account their pharmacokinetics).
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