In statistics,

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, and

econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

, an unevenly (or unequally or irregularly) spaced time series is a sequence of observation time and value pairs (t_n, X_n) in which the spacing of observation times is not constant. Unevenly spaced

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...

naturally occur in many industrial and scientific domains: natural disasters such as earthquakes, floods, or volcanic eruptions typically occur at irregular time intervals. In

observational astronomy Observational astronomy is a division of astronomy that is concerned with recording data about the observable universe, in contrast with theoretical astronomy, which is mainly concerned with calculating the measurable implications of physical ...

, measurements such as spectra of celestial objects are taken at times determined by weather conditions, availability of observation time slots, and suitable planetary configurations. In clinical trials (or more generally,

longitudinal studies A longitudinal study (or longitudinal survey, or panel study) is a research design that involves repeated observations of the same variables (e.g., people) over short or long periods of time (i.e., uses longitudinal data). It is often a type of obs ...

), a patient's state of health may be observed only at irregular time intervals, and different patients are usually observed at different points in time. Wireless sensors in the Internet of things often transmit information only when a state changes to conserve battery life. There are many more examples in

climatology Climatology (from Greek , ''klima'', "place, zone"; and , ''-logia'') or climate science is the scientific study of Earth's climate, typically defined as weather conditions averaged over a period of at least 30 years. This modern field of study ...

ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...

, high-frequency finance,

geology Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ea ...

, and

Analysis

A common approach to analyzing unevenly spaced time series is to transform the data into equally spaced observations using some form of interpolation - most often linear - and then to apply existing methods for equally spaced data. However, transforming data in such a way can introduce a number of significant and hard to quantify

biases Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...

, especially if the spacing of observations is highly irregular. Ideally, unevenly spaced time series are analyzed in their unaltered form. However, most of the basic theory for

time series analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

was developed at a time when limitations in computing resources favored an analysis of equally spaced data, since in this case efficient

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

routines can be used and many problems have an explicit solution. As a result, fewer methods currently exist specifically for analyzing unevenly spaced time series data. The

least-squares spectral analysis Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...

methods are commonly used for computing a

frequency spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...

from such time series without any data alterations.

Software

Traces
is a

Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (pro ...

library for analysis of unevenly spaced time series in their unaltered form.
CRAN Task View: Time Series Analysis
is a list describing many

R (programming language) R is a programming language for statistical computing and graphics supported by the R Core Team and the R Foundation for Statistical Computing. Created by statisticians Ross Ihaka and Robert Gentleman, R is used among data miners, bioinfor ...

packages dealing with both unevenly (or irregularly) and evenly spaced time series and many related aspects, including uncertainty.
MessyTimeSeries
an
MessyTimeSeriesOptim
are Julia packages dedicated to incomplete time series.

References

{{Reflist, refs= {{cite journal , author1=Myron Scholes , author2=Joseph Williams , year = 1977 , title = Estimating betas from nonsynchronous data , journal = Journal of Financial Economics , volume = 5 , issue=3 , pages = 309–327 , doi=10.1016/0304-405X(77)90041-1 {{cite book , editor = Pierre Lequex , author1=Mark C. Lundin , author2=Michel M. Dacorogna , author3=Ulrich A. Müller , year = 1999 , title = The Financial Markets Tick by Tick , chapter = Chapter 5: Correlation of High Frequency Financial Time Series , pages = 91–126 {{cite journal , author1=Takaki Hayashi , author2=Nakahiro Yoshida , year = 2005 , title = On covariance estimation of non-synchronously observed diffusion processes , journal = Bernoulli , volume = 11 , issue=2 , pages = 359–379 , url = http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bj/1116340299 , doi=10.3150/bj/1116340299, doi-access = free {{cite journal , author1=K. Rehfeld , author2=N. Marwan , author3=J. Heitzig , author4=J. Kurths , year = 2011 , title = Comparison of correlation analysis techniques for irregularly sampled time series , journal = Nonlinear Processes in Geophysics , volume = 18 , issue=3 , pages = 389–404 , url = http://www.nonlin-processes-geophys.net/18/389/2011/npg-18-389-2011.pdf , doi=10.5194/npg-18-389-2011, doi-access = free {{cite journal , author = Ulrich A. Müller , year = 1991 , title = Specially Weighted Moving Averages with Repeated Application of the EMA Operator , journal = Working Paper, Olsen and Associates, Zurich, Switzerland , url = http://www.olsen.ch/fileadmin/Publications/Working_Papers/001207-emaOfEma.pdf {{cite journal , author1=Gilles Zumbach , author2=Ulrich A. Müller , year = 2001 , title = Operators on Inhomogeneous Time Series , journal = International Journal of Theoretical and Applied Finance , volume = 4 , pages = 147–178 , doi = 10.1142/S0219024901000900 }
Preprint
/ref> {{cite book , author1=Michel M. Dacorogna , author2=Ramazan Gençay , author3=Ulrich A. Müller , author4=Richard B. Olsen , author5=Olivier V. Pictet , year = 2001 , title = An Introduction to High-Frequency Finance , publisher = Academic Press, url=http://fxtrade.oanda.com/resources/hffbookchapter1.pdf {{cite journal , author = Andreas Eckner , year = 2014 , title = A Framework for the Analysis of Unevenly-Spaced Time Series Data , url = http://www.eckner.com/papers/unevenly_spaced_time_series_analysis.pdf {{cite journal , author = Andreas Eckner , year = 2017 , title = Algorithms for Unevenly-Spaced Time Series: Moving Averages and Other Rolling Operators , url = http://eckner.com/papers/Algorithms%20for%20Unevenly%20Spaced%20Time%20Series.pdf {{cite journal , author = Andreas Eckner , year = 2017 , title = A Note on Trend and Seasonality Estimation for Unevenly-Spaced Time Series , url = http://eckner.com/papers/Trend%20and%20Seasonality%20Estimation%20for%20Unevenly%20Spaced%20Time%20Series.pdf Statistical signal processing Time series

Analysis

Software

See also

References