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The difference $\Delta X_t=X_t-X_{t-1}$ can remove a unit-root stochastic trend. A seasonal difference operator instead subtracts an observation a full seasonal period earlier. Neither operation automatically removes a changing variance.

The seasonal difference operator has $b_0=1$, $b_{12}=-1$ and all other coefficients zero. Its frequency response is $1-e^{12i\lambda }$, soThe right sketch shows zeros at $\lambda =j\pi /6$ for $j=0,\dots ,6$, and maxima of height two at $(2j+1)\pi /12$ for $j=0,\dots ,5$. The seasonal difference operator removes every exactly twelve-periodic component, including a constant. Its filter gain has a comb pattern: it selectively suppresses seasonal frequencies, rather than increasing monotonically with frequency.

Series 1 wanders over a changing level rather than fluctuating around a stable local mean. Its sample autocorrelation function is strongly positive and decreases very slowly. This is the usual diagnostic evidence for an ordinary unit root: an autoregressive polynomial containing $1-z$, with a zero at $z=1$, and a stationary model after first differencing. The plots support an integrated model, rather than specifying the number of its remaining stationary autoregressive or moving-average terms.

Series 2 has a pronounced oscillation with period about six observations. Its sample autocorrelation alternates between large positive and negative values with little damping: approximately positive at multiples of six and negative halfway between. Together with the changing amplitude, this suggests a conjugate pair of unit-circle zeros nearThe associated real autoregressive factor is $1-2\cos (\pi /3)z+z^2=1-z+z^2$. A targeted filter $1-B+B^2$ removes this pair; the broader seasonal difference operator $1-B^6$ also contains it but introduces additional differencing factors. This is the oscillatory unit-root diagnosis from an undamped sample autocorrelation.

Thus Series 1 suggests a zero at 1; Series 2 suggests a conjugate pair on the unit circle at a seasonal frequency. These are model diagnoses, not deductions of exact roots from a finite sample. A stationary model very close to a unit root can look similar, and an undamped periodic covariance can also arise from a stationary random sinusoid. The figure does not identify exact orders or prove nonstationarity by itself.

Put $c_t=\cos (2\pi t/S)$ and assume ${\sigma }^2>0$. ThenThus $Y$ generally has a periodic mean, and $Z$ generally has a periodic variance. The edge cases allowed by $S\in {\mathbb{N}}^{\ast }$ matter: $Y$ is stationary when $S=1$, because the mean is then constant, and $Z$ is stationary for $S=1$ or $S=2$. In the latter case multiplying iid centered Gaussian white noise by $(-1{)}^t$ leaves its iid distribution unchanged. For every $S\ge 3$, $c_0^2=1$ but $c_1^2<1$, so $Z$ is nonstationary. For $S>1$, $Y$ is nonstationary because $c_t$ is not constant.

Use the seasonal difference operator ${\Delta }_S=1-B^S$. Since $c_t=c_{t-S}$,This is a stationary moving-average model of iid Gaussian white noise. Its covariance is $2{\sigma }^2$ at lag $0$, $-{\sigma }^2$ at lags $\pm S$, and zero otherwise.

For the variance-modulated process the same operation givesIts variance is $2{\sigma }^2{c}_t^2$ and its covariance at lag $S$ is $-{\sigma }^2{c}_t^2$. Therefore it remains nonstationary for $S\ge 3$, and the operation doubles the marginal variance at nonzero seasons. Seasonal differencing does not remove periodic variance: it removes a periodic mean, but the periodic variance generally remains. Practical alternatives are a periodic model or seasonal variance standardization; at seasons with $c_t=0$ the observations are deterministically zero, so division by $c_t$ is not possible there. For the already stationary special cases $S=1,2$, filtering preserves stationarity.

A periodically correlated process with period $S$ has its mean and two-time covariance unchanged when both times are shifted by $S$. For $Z$, the mean is zero andHence $Z$ is a periodically correlated process. Its covariance period can be smaller than $S$: when $S$ is even, $c_{t+S/2}^2=c_t^2$, so $S/2$ is already a covariance period.

Seasonality is systematic repetition at a calendar period. It can appear in a deterministic mean, in a periodic variance, or in dependence across seasons. A seasonal difference operator removes a fixed periodic mean but seasonal differencing does not remove periodic variance. A periodic-looking path alone does not prove a nonstationary process.

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 Peter Bartlett, Introduction to Time Series Analysis, lecture 1: objectives and models.

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 G. Udny Yule, On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers (1927) and Eugen Slutsky, The Summation of Random Causes as the Source of Cyclic Processes (1937), including the editorial note on the 1927 original.