= Time series
{wiki}
A <time series> is a sequence of observations indexed by time: hourly temperatures, daily sales, or successive measurements of a fluctuating signal. A probabilistic model treats these observations as one <sample path> of a <stochastic process> $(X_t)$. For equally spaced observations, take $t\in\mathbb Z$ and measure a <lag of a time series> in observation intervals. The order matters: shuffling a record preserves its observed values but destroys information about how one value follows another.
Two basic ways to model temporal dependence are an <autoregressive model>, which feeds past observations into the next value, and a <moving-average model>, which combines present and past <white noise>. An <AR(1)> carries a disturbance forward with diminishing weights; an <MA(1)> carries it into just two observations. Their <autocorrelation functions> make this difference visible. A <weakly stationary process> supplies a common <mean> and a lag-dependent <autocovariance> with which to describe that dependence.
These models help forecast future observations and quantify the uncertainty of averages. Successive measurements need not provide as much information as the same number of <independent random variables>: the <long-run variance of a stationary process> accounts for their accumulated <covariances>. Trend and <seasonality> require attention before fitting a stationary model; <differencing> and a <seasonal difference operator> can help, but do not automatically make every process stationary. See https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/1.pdf[Peter Bartlett, Introduction to Time Series Analysis, lecture 1: objectives and models].
\Image[/time-series-persistence-examples.png]
{title=Stationary Gaussian examples: positive AR(1) persistence, negative AR(1) alternation, and MA(1) short memory}
The three illustrated <sample paths> have zero <mean> and unit <variance>, so the visual comparison concerns dependence rather than scale. The <AR(1)> paths start from their stationary <normal distribution>, independently of future <discrete Gaussian white noise>; the <MA(1)> path includes the noise preceding its first observation. These are finite random examples, not exact pictures of an <ACF>. Compare their theoretical <autocorrelation functions> in <autocorrelation>.
Historically, Yule's 1927 study investigated disturbed periodicities in sunspot observations; it should not be read as the discovery of the particular <AR(1)> examples here. Slutsky studied how summing random inputs could produce wave-like records, a warning against inferring a periodic cause from appearance alone. The editorial note to his 1937 English paper identifies the 1927 Russian original. See https://doi.org/10.1098/rsta.1927.0007[G. Udny Yule, On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers (1927)] and https://www.e-m-h.org/Slutzky1937.pdf[Eugen Slutsky, The Summation of Random Causes as the Source of Cyclic Processes (1937), including the editorial note on the 1927 original].
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