statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

, and

econometrics Econometrics is an 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. ...

naturally occur in many industrial and scientific domains:

natural disaster A natural disaster is the very harmful impact on a society or community brought by natural phenomenon or Hazard#Natural hazard, hazard. Some examples of natural hazards include avalanches, droughts, earthquakes, floods, heat waves, landslides ...

s 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 Clinical trials are prospective biomedical or behavioral research studies on human subject research, human participants designed to answer specific questions about biomedical or behavioral interventions, including new treatments (such as novel v ...

(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 long periods of time (i.e., uses longitudinal data). It is often a type of observation ...

), 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 Internet of things (IoT) describes devices with sensors, processing ability, software and other technologies that connect and exchange data with other devices and systems over the Internet or other communication networks. The IoT encompasse ...

often transmit information only when a state changes to conserve battery life. There are many more examples in

climatology Climatology (from Greek , ''klima'', "slope"; 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. Climate concerns the atmospher ...

ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ...

, high-frequency finance,

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

, 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 In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...

- 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 inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...

, 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, 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. ...

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 matrix (mathemat ...

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 Spectral density estimation#Overview, frequency spectrum based on a least-squares fit of Sine wave, sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the ...

methods are commonly used for computing a

frequency spectrum In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed int ...

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 (prog ...

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 Data and information visualization, data visualization. It has been widely adopted in the fields of data mining, bioinformatics, data analysis, and data science. The core R language is ...

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 {{citation, 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 {{citation, 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 {{citation, 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 {{cite arXiv, author1=Mehmet Süzen , author2=Alper Yegenoglu , date=13 December 2021, title = Generalised learning of time-series: Ornstein-Uhlenbeck processes , class=stat.ML , eprint=1910.09394 Statistical signal processing Time series

Analysis

Software

See also

References