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| Time series analysis | |
|---|---|
| Name | Time series analysis |
| Field | Statistics, Econometrics, Signal processing |
| Introduced | 19th century |
| Notable figures | Norbert Wiener, W. Edwards Deming, George E. P. Box, Gwilym Jenkins, Sir Clive Granger, Robert Engle, Herman Wold, Andrey Kolmogorov, Yule, G. U., Fisher, Ronald A. |
Time series analysis is the study of ordered sequences of data points measured typically at successive times, emphasizing temporal dependence and structure for description, modeling, forecasting, and control. It combines ideas from Statistics, Econometrics, Signal processing, Control theory, Information theory and Computer science to extract patterns such as trends, cycles, seasonality, and autocorrelation for prediction and inference.
Time series analysis addresses problems of modeling observations indexed by time from sources like U.S. Bureau of Labor Statistics, Federal Reserve System, European Central Bank, National Aeronautics and Space Administration, National Weather Service and World Health Organization. Core objectives include forecasting for entities such as International Monetary Fund, World Bank, Organisation for Economic Co-operation and Development, and policy institutions like Bank of England and Bank of Japan. Practical tasks involve preprocessing by agencies including U.S. Census Bureau and Eurostat, while theoretical foundations draw on work from scholars at Princeton University, Harvard University, University of Chicago, London School of Economics, MIT, Stanford University and University of Cambridge.
Early formal work traces to studies by Francis Galton and Karl Pearson in statistics and forecasting at institutions like University College London; later foundational advances came from Andrey Kolmogorov in probability theory and Norbert Wiener in prediction theory at MIT. Mid-20th century contributions include George E. P. Box and Gwilym Jenkins who developed the Box–Jenkins methodology, while Herman Wold influenced structural modeling at Stockholm University. Developments in Econometrics by Clive Granger and Robert Engle at University of California, San Diego and University of California, Berkeley introduced cointegration and ARCH/GARCH models, shaping policy work at Federal Reserve Bank of St. Louis and research at National Bureau of Economic Research. Advances in computing from IBM, Bell Labs, Microsoft Research and Google enabled large-scale empirical work using algorithms inspired by Alan Turing and implemented on platforms from UNIVAC to modern supercomputers at Lawrence Livermore National Laboratory.
Classical linear models include autoregressive (AR), moving average (MA), autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) popularized by Box–Jenkins methods; extensions include seasonal ARIMA used by NOAA and Met Office. Multivariate methods include vector autoregression (VAR) applied in studies by Christopher Sims and structural VAR used by Olivier Blanchard. Long-memory models trace to Mandelbrot, Benoit and fractional integration; state-space models and the Kalman filter were advanced by Rudolf E. Kálmán and used in control at NASA. Nonlinear and heteroskedastic models include autoregressive conditional heteroskedasticity (ARCH) and generalized ARCH (GARCH) developed by Tim Bollerslev and Robert Engle, with stochastic volatility models used in Federal Reserve research. Spectral analysis was formalized by John Tukey and Herman Wold; wavelet-based time-frequency methods were promoted by Stéphane Mallat and Ingrid Daubechies. Machine learning approaches integrate tools from Yann LeCun, Geoffrey Hinton, Yoshua Bengio, and architectures like recurrent neural networks (RNNs), long short-term memory (LSTM) by Sepp Hochreiter and Jürgen Schmidhuber, and transformers introduced at Google Brain.
Parameter estimation techniques include maximum likelihood (inspired by Ronald Fisher), method of moments with roots in Karl Pearson, and Bayesian inference popularized by work at University of Cambridge and Harvard. Inference about correlation and causality leverages tests such as Granger causality by Clive Granger and cointegration tests by Robert Engle and Clive Granger used in macroeconomics at International Monetary Fund. Asymptotic theory uses results from Andrey Kolmogorov and Sergei N. Bernstein; bootstrap methods were developed by Bradley Efron and applied by researchers at Stanford University. Robust estimation and shrinkage methods borrow from work at University of California, Berkeley and Yale University on penalized likelihood and regularization.
Model diagnostics use residual analysis, autocorrelation function (ACF) and partial autocorrelation function (PACF) with tools referenced in textbooks by Box, George E. P. and Tiao, George C.. Information criteria such as Akaike information criterion (AIC) by Hirotugu Akaike and Bayesian information criterion (BIC) by Gideon Schwarz guide selection and are employed in applied research at OECD and central banks. Hypothesis tests include Ljung–Box developed by Greta Ljung and George Box, augmented Dickey–Fuller test by David A. Dickey and Wayne A. Fuller for unit roots, and Hansen’s tests for structural breaks developed by Bruce E. Hansen. Cross-validation schemes adapted for time series are used in industry by teams at Amazon, Netflix, and Apple.
Applications span macroeconomic forecasting for International Monetary Fund and World Bank, financial risk modeling at Goldman Sachs and J.P. Morgan Chase, and volatility forecasting in markets overseen by Securities and Exchange Commission. Meteorological forecasting at National Oceanic and Atmospheric Administration and Met Office, epidemiological modeling by Centers for Disease Control and Prevention and World Health Organization, and energy load forecasting for Edison Electric Institute use time series techniques. Other domains include speech processing at Bell Labs and Dolby Laboratories, seismology at United States Geological Survey, climatology at Intergovernmental Panel on Climate Change, neuroscience at National Institutes of Health, and transportation demand forecasting for Transport for London.
Common software environments include R (programming language) packages such as forecast and vars maintained by CRAN contributors, Python (programming language) libraries like statsmodels and Prophet (by Facebook) and machine learning frameworks from TensorFlow and PyTorch developed by Google and Facebook AI Research. Commercial systems include SAS procedures, MATLAB toolboxes from MathWorks, and EViews used in University of Cambridge and London School of Economics research. Open-source projects on platforms such as GitHub host implementations by contributors affiliated with Carnegie Mellon University, Massachusetts Institute of Technology and ETH Zurich.