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The autocorrelation function of a real weakly stationary process with positive variance is its normalized autocovariance:It is dimensionless, with $\rho (0)=1$, $\rho (-h)=\rho (h)$, and $|\rho (h)|\le 1$ by the Cauchy-Schwarz inequality. It is unchanged by replacing $X_t$ with $a{X}_t+b$ for fixed $a\ne 0$. This makes autocorrelation useful when comparing records measured on different scales. The ACF is a property of the model; the sample autocorrelation function estimates it and will generally not reproduce theoretical zeros exactly.

An AR(1) with coefficient $\varphi =0.7$ has autocorrelations $0.7,0.49,0.343$ at lags one through three. With $\varphi =-0.7$ these become $-0.7,0.49,-0.343$: the same diminishing magnitude, alternating in sign. An MA(1) with $\theta =0.7$ instead has lag-one autocorrelation $0.7/1.49=70/149\simeq 0.4698$ and zero at every higher lag. Equal numerical coefficients in an autoregressive model and a moving-average model do not mean equal dependence. See Bartlett, lecture 3: theoretical ACFs and prediction.

The bars show exact model autocorrelations, not estimates from the illustrated sample paths in time series. The AR(1) retains a diminishing trace of a disturbance at arbitrarily large lags; the MA(1) stops sharing driving noise after lag one. A zero autocorrelation means uncorrelated random variables, and does not generally imply independent random variables. For a Gaussian process it does imply independence of the two coordinates.

There is a direct prediction interpretation. Among affine predictions based on just $X_t$, the best linear predictor of $X_{t+h}$ is $\mu +\rho (h)(X_t-\mu )$, with mean squared prediction error $\gamma (0)[1-\rho (h{)}^2]$. Thus negative autocorrelation is also useful predictive information. A prediction using the whole observed past can do better; best linear prediction from a finite past describes that problem. Interpreting the affine prediction as a conditional expectation requires additional assumptions, such as a joint normal distribution.

A causal time-series representation expresses a time series using present and past driving white noise. Its coefficients vanish at negative lags. Square summability ensures mean-square convergence for a white-noise input; absolute summability is the stronger stable-filter convention. Causality refers to the particular driving sequence, not merely to stationary existence.

For two real centered time series $X_t,Y_t$, require joint weak stationarity: all auto- and cross-covariances depend only on the time lag. Individual weak stationarity alone does not impose this condition on their joint behavior. With the convention ${\gamma }_{XY}(h)=\mathbb{E}[X_t{Y}_{t+h}]$, define the cross-spectrum byAssume absolute summability of the auto- and cross-covariance sequences, so the displayed Fourier series and the two individual spectral densities are continuous. Reality and joint weak stationarity giveConsequently the spectral matrix $F(\lambda )=\left(\begin{array}{cc}{f}_{XX}& f_{XY}\\ f_{YX}& f_{YY}\end{array}\right)$ is a Hermitian matrix. It is positive semidefinite, because its quadratic forms are nonnegative spectral densities of complex linear combinations of the jointly stationary series. Its nonnegative determinant therefore yields $|f_{XY}(\lambda ){|}^2\le f_{XX}(\lambda )f_{YY}(\lambda )$.

The imaginary part records directional lag information. For white noise $X_t$ of variance $v$ and $Y_t=X_{t-1}$, only ${\gamma }_{XY}(1)=v$ is nonzero. Hence $f_{XY}(\lambda )=v{e}^{-i\lambda }/(2\pi )$, although both individual spectral densities equal $v/(2\pi )$. The cross-spectrum is generally complex, rather than real and even like a real series' own spectral density.

Cross-validation estimates predictive performance by repeatedly fitting on one part of the data and evaluating on held-out observations. In K-fold cross-validation, divide training data into disjoint folds $I_1,\dots ,I_K$; $k(i)$ identifies observation $i$'s fold, and ${\hat{f}}^{(-k)}$ is fitted without fold $I_k$. The displayed average scores each observation using a rule that did not train on it. Weight by observation count when fold sizes differ. Leave-one-out cross-validation uses one observation per fold. ^[ref]

For the training error example, three folds contain indices $i$ with equal remainder modulo 3. Each fit uses eight observations. The squared-error loss averages for degrees 1, 2, and 7 are $0.724132$, $0.027192$, and $1.532790$ respectively. Select degree 2 and refit on all twelve training data observations. The folds reuse scarce data while challenging each candidate to predict unseen responses; they do not touch the final test set.

Any learned preprocessing, including variable selection, must also be fitted within each training fold to avoid data leakage. ^[ref] Repeated tuning can itself overfit the cross-validation score; nested cross-validation uses inner folds for model selection and outer folds for assessment of that entire procedure. ^[ref]

Folds use fewer training observations than the final refit, and their scores are dependent because training observations overlap. ^[ref] Splitting must match deployment: for time series forecasting, rolling-origin cross-validation trains on the past and evaluates later observations, preventing future information from entering a forecast. Ordinary random folds need not answer that forecasting question. ^[ref]

Culturomics investigates historical patterns through large-scale quantitative analysis of recorded language, especially digitized books. A text corpus turns occurrences of words or n-grams into measurable time series. Jean-Baptiste Michel and collaborators demonstrated the approach with a corpus of 5,195,769 books, selecting material partly by the quality of optical character recognition and publication metadata.^[ref]

For an original example, 500 occurrences in ten million word tokens give $500/10^7=5\times 10^{-5}$. A later collection with 800 occurrences in twenty million tokens gives $4\times 10^{-5}$: the count rose, but the normalized rate fell. A changing name can also change the measured series without an equivalent change in its referent. Culturomics therefore connects computational counting to hypotheses about language change, while requiring checks on publication selection, transcription errors and changing terminology.

The equipartition theorem says that each independent quadratic term in a Hamiltonian described by classical mechanics has mean energy $k_{B}T/2$ in a canonical ensemble. Here $k_B$ is the Boltzmann constant and $T>0$ the absolute temperature. Count terms in the energy, rather than vaguely counting everything that can move: one harmonic oscillator has one position degree of freedom but TWO quadratic terms, one in momentum and one in displacement.

A precise sufficient setting is $H(y,\xi )=a{y}^2/2+H_0(\xi )$ with $a>0$, where $y$ is one of the canonical variables, $\xi$ denotes all remaining variables, and the phase space integration measure factors as $dyd\xi$. The range of $y$ is the whole real line, independently of $\xi$; there is no constraint coupling its allowed values to other variables. The partition function must be finite, and the relevant expected values must exist. The Boltzmann factor then givesThe probability density function ${\rho }_y$ is a normal distribution with variance $k_{B}T/a$. A larger coefficient narrows the fluctuations enough to cancel its greater energy cost. This is why the mean contribution is independent of mass or spring constant.^[ref]

For a proof outline, the one-variable Gaussian integral is $Z_y={\int }_{-\infty }^{\infty }{e}^{-\beta a{y}^2/2}dy=\sqrt{2\pi /(\beta a)}$, so $-{\partial }_{\beta }\ln Z_y=1/(2\beta )$. More generally, integration by parts in a canonical variable gives $⟨y{\partial }_{y}H⟩=k_{B}T$ when its boundary term $[y{e}^{-\beta H}]$ vanishes and the integrals converge. For $a{y}^2/2$ this is twice its mean energy. At a finite hard boundary the boundary term need not vanish; hard-wall correction to harmonic equipartition illustrates the resulting departure. A constraint, a nonquadratic potential energy, or a noncanonical integration measure therefore requires its own calculation. The theorem is an ensemble average, not equality of instantaneous energies, and applying it to a recorded trajectory requires equilibration and sufficient sampling.

For a classical one-dimensional harmonic oscillator, with positive mass $m$ and spring constant $k$,The origin is the equilibrium position. Its heat capacity is $k_B$ at fixed $m,k$, not $k_B/2$. Although the two mean energies agree, their measured amplitudes do not: $⟨p^2⟩=m{k}_{B}T$ and $⟨x^2⟩=k_{B}T/k$. In a harmonic optical trap, the position measurement observes the potential energy contribution; it does not measure both contributions by observing $x$ alone.^[ref]

For $N$ particles in a three-dimensional classical monatomic gas, neglect interactions and thermally accessible internal excitations. Its three translational momentum terms give internal energy $U=3N{k}_{B}T/2$, heat capacity at constant volume $C_V=3N{k}_B/2$, and heat capacity at constant pressure $C_p=C_V+N{k}_B=5N{k}_B/2$. Per mole, replace $N{k}_B$ by the molar gas constant $R$. For example, using stipulated $R=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, the molar heat capacity at fixed volume is $12.471\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. These results assume a dilute, nonrelativistic ideal gas with classical translational statistics, rather than a quantum degeneracy regime.

For an ideal gas of diatomic molecules, modeled as two chemical atoms with a bond, there are three translational quadratic terms, two accessible rotational kinetic energy terms for changes of bond orientation, and one bond-stretching harmonic oscillator with two quadratic terms. Rotation about the bond axis is not an additional active mode in this linear-molecule model. The three familiar limits of heat capacity at constant volume areCorresponding molar heat capacities are $3R/2$, $5R/2$ and $7R/2$. Counting the stretching harmonic oscillator only once as $k_{B}T/2$ would give the wrong last value.^[ref]

The frozen regimes require quantum mechanics. In the simple molecular rotation model, $E_j=k_B{\theta }_{r}j(j+1)$ with state degeneracy $2j+1$, ${\theta }_r={\hslash }^2/(2I{k}_B)$, and moment of inertia $I$. For the bond-stretching quantum harmonic oscillator, ${\theta }_v=\hslash \omega /k_B$ with angular frequency $\omega$ and $\hslash =h/(2\pi )$ from the Planck constant. When ${\theta }_r\ll {\theta }_v$, the three limits above correspond to $T\ll {\theta }_r$, ${\theta }_r\ll T\ll {\theta }_v$, and $T\gg {\theta }_v$, provided translational motion remains classical. They are smooth crossovers, not abrupt switches. Real molecules can have different allowed rotational populations; at high temperature, thermal dissociation, internal excitations and nonharmonic bond motion can invalidate the fixed-molecule model before its last limit is reached.^[ref]

For a quantum harmonic oscillator with fixed $\omega$, put $z={\theta }_v/T$. Its exact mean energy and heat capacity areAt $T\gg {\theta }_v$, the mean energy approaches $k_{B}T$ and $C_{\mathrm{osc}}$ approaches $k_B$. At $T\ll {\theta }_v$, thermal excitation is exponentially suppressed and $C_{\mathrm{osc}}$ tends to zero, although the zero-point energy $\hslash \omega /2$ survives. Frozen thus means little thermal response, not zero motion or zero energy.^[ref]

A motivating measurement is thermal calibration of an optical trap. A focused laser through a microscope objective confines a bead in water; imaging or a quadrant photodetector records a position time series. Gieseler and colleagues describe a polystyrene bead of radius $1.03\mu \mathrm{m}$ at $293.15\mathrm{K}$, a $1064\mathrm{nm}$ trapping laser, and photodetector acquisition at $10^5\mathrm{Hz}$. Their recorded bead continues undergoing Brownian motion; higher trapping power produces a narrower equilibrium position distribution. Infer the centered variance, not a squared distance from an arbitrary camera origin:The normal distribution position histogram and agreement with dynamical calibration methods support the local harmonic optical trap interpretation. For a stipulated example, $T=300\mathrm{K}$ and ${\sigma }_x=20\mathrm{nm}$ imply $k\simeq 1.04\times 10^{-5}N/m=0.0104pN/nm$. This calculated scale is not an experimental reading.^[ref]

Position calibration is essential: if detector output $v$ has conversion $x=s(v-v_0)$, its physical variance is $s^2\mathrm{Var}(v)$. An unknown scale $s$ cannot be determined together with $k$ from that one variance. A complementary calibration demonstrated by Tolić-Nørrelykke and colleagues combines the power spectrum of thermal motion with the bead response to known sinusoidal motion of a translation stage. The driven response supplies an additional length scale, and their method does not require bead size, liquid dynamic viscosity or wall distance as inputs. It still uses a thermal dynamical model; agreement between methods is a consistency check, not a theory-free proof of equipartition theorem.^[ref]

The readout itself can hide fluctuations. Wong and Halvorsen used a $1064\mathrm{nm}$ optical trap and adjustable-exposure CCD imaging of a bead in water. They varied exposure and trapping power, held the bead away from the nearest wall, adjusted illumination to maintain comparable images, and measured tracking noise using a stuck bead. The measured variance fell as exposure increased; corrected results agreed with their time-averaging model. For a stationary overdamped harmonic optical trap, with optical trap relaxation time $\tau$, an exposure of length $W$ measures ${\bar{x}}_W=W^{-1}{\int }_0^{W}x(t)dt$. Its ideal noise-free variance isFor $\alpha >0$, $S(\alpha )<1$: using the blurred variance in the uncorrected formula overestimates $k$ by $1/S(\alpha )$. At $W=\tau$, the calculated factor is $e/2\simeq 1.36$. Additive tracking noise instead raises the measured variance and biases the inferred $k$ downward if left uncorrected.^[ref]

Useful controls are to record over many optical trap relaxation times, compare blocks for drift, calibrate the position scale, measure fixed-bead noise, vary exposure, and check the position histogram for departures from a normal distribution. A sparse sequence of accurate instantaneous positions can estimate an equilibrium variance; a long exposure averages away motion even if many frames are collected. Flow, appreciable laser heating, escape from the harmonic region or nonconservative forces can break the assumed thermal equilibrium. Inferring $k$ by imposing equipartition theorem is a calibration, whereas testing the relation requires an independently established stiffness and position scale. The beautiful practical connection is that microscopic randomness can set a force scale: once calibrated, a small constant applied force displaces the mean by $f/k$. The same quadratic-term counting predicts macroscopic heat capacity and makes microscopic fluctuations useful measurements.

Estradiol is an estrogen steroid hormone involved in reproductive development and regulation. It is one of the estrogens produced by the ovary and participates in the interacting hormonal signals of the menstrual cycle.^[ref][ref]

A concentration measured once cannot show the shape or ordering of a changing hormonal signal. In 1983, J. D. Hoff and collaborators sampled five women at two-hour intervals for five days around midcycle. The onset of the luteinizing hormone and follicle-stimulating hormone surges was temporally associated with peak estradiol levels.^[ref]

Plotting these measurements as time series distinguishes an increase, a peak and a subsequent decline. The observations support a temporal relationship in the measured cohort; their timing alone does not isolate every causal link in the feedback system. This makes sampling interval and the chosen time reference part of an interpretable hormone measurement.

Estrogens are a class of steroid hormones that includes estradiol, estrone and estriol. They participate in several physiological processes, including development of reproductive tissues and regulation of the menstrual cycle. The class name identifies a family of signals, rather than one uniquely specified chemical.^[ref]

Estrogens also contribute to the growth of long bones, illustrating why their effects extend beyond reproductive tissues.^[ref] An experimental measurement consequently needs to identify both the substance being measured and the response of interest. An estradiol assay does not automatically measure every estrogen or summarize all of their biological effects.

For example, a 1983 study sampled five women at two-hour intervals over five days around midcycle. Peak estradiol levels were temporally associated with the onset of luteinizing hormone and follicle-stimulating hormone surges.^[ref] This connects a specified member of the class to a measured time series; temporal association alone does not establish the direction of causation.

Let $X_t$ be a centered real stationary Gaussian time series with absolutely summable autocovariances and two-sided angular-frequency spectral density $f$. The parent periodogram uses $I_T(\omega )=(2\pi T{)}^{-1}|{\sum }_{t=1}^T{X}_t{e}^{-it\omega }{|}^2$. At an interior positive frequency define the one-sided quantitiesThus the doubling applies to both the statistic and its target spectral density.

For Fourier-grid frequencies ${\omega }_T=2\pi k_T/T\to \omega \in (0,\pi )$ with $f(\omega )>0$, setThen $I_T^{+}({\omega }_T)=A_T^2+B_T^2$. Summability makes their variances tend to $f(\omega )=s(\omega )/2$ and their covariance tend to zero. Joint Gaussianity therefore givesUnder these Gaussian assumptions the fourth moments also converge, so $\mathrm{var}(I_T^{+}({\omega }_T))\to s(\omega {)}^2$. An unsmoothed periodogram is therefore not a consistent estimator of a positive spectral density. At zero and the Nyquist frequency $\pi$, the sine coordinate vanishes and the real-only calibration differs; the interior doubling and two-coordinate argument do not apply there.^[ref]

For $t\ge 1$, iterating an AR(1) from a fixed value $X_0=x_0$ gives the displayed finite sum. With centered white noise of variance ${\sigma }^2$ and $|\varphi |<1$,For $h\ge 0$, its covariance is $\mathrm{Cov}(X_{t+h},X_t)={\varphi }^h\mathrm{Var}(X_t)$, which depends on $t$. Even setting $x_0=\mu$ leaves a changing variance. At $\varphi =0.7$, ${\sigma }^2=1$, and $x_0=\mu$, the variances at times zero through three are $0,1,1.49,1.7301$, approaching $100/51$; this finite initialized record is not weakly stationary.

To initialize in weak stationarity, give $X_0$ mean $\mu$, variance ${\sigma }^2/(1-{\varphi }^2)$, and zero covariance with every future driving noise value. The full stationary construction sets $X_0-\mu ={\sum }_{j\ge 0}{\varphi }^j{\epsilon }_{-j}$. For discrete Gaussian white noise, equivalently draw $X_0$ from $N(\mu ,{\sigma }^2/(1-{\varphi }^2))$ independently of future noise. This gives exact strict stationarity from the first plotted point; it is the initialization used in the figure in time series. Matching these first two moments alone does not guarantee strict stationarity for non-Gaussian inputs. See also stationary initialization of a scalar linear state-space model.

A functional time series is a time series whose observation at each time is a whole function, often modeled as a random element of the Hilbert space $L^2[0,1]$. For example, $X_j(s)$ might be the temperature profile within day $j$, with $s$ marking time of day. Dependence across days is different from dependence between nearby positions within one curve.

For a simple model, let $X_j(s)=A_j+B_{j}s$, where $(A_j,B_j)$ is a stationary bivariate time series. Its mean curve is $\mu (s)=\mathbb{E}{A}_j+s\mathbb{E}{B}_j$. The covariance between curves at lag $h$ is a kernel $C_h(s,t)=\mathrm{Cov}(X_0(s),X_h(t))$, retaining both within-curve coordinates.

Estimating $\mu$ by $n^{-1}{\sum }_{j=1}^n{X}_j$ resembles scalar averaging, but its uncertainty depends on serial covariance operators rather than only the variance of one curve. Treating every densely measured point as an independent observation can severely overstate precision.^[ref]

Update the level and slope by ${\ell }_t=\alpha X_t+(1-\alpha )({\ell }_{t-1}+b_{t-1})$ and $b_t=\beta ({\ell }_t-{\ell }_{t-1})+(1-\beta )b_{t-1}$, with gains in $(0,1)$. The first update compares the new observation with the previous level extrapolated one step; the second smooths the latest level increment. The displayed forecast extrapolates that local slope. This is a double exponential smoothing method for a nonseasonal trending time series.

A causal square-integrable linear time series can be written $X_t=\mu +{\sum }_{j\ge 0}{\psi }_j{\epsilon }_{t-j}$. Square summability of the coefficients ensures mean-square convergence for white noise. An innovation representation additionally identifies its driving noise as the innovation process.

A lag of a time series is a separation between observation indices. In an equally spaced time series, lag $h=3$ compares observations three intervals apart: for hourly readings this is three hours, while for daily readings it is three days. Positive and negative lags describe the same separation in opposite directions. The autocovariance and autocorrelation of a real weakly stationary process are even functions of lag.

Lajos Horváth works on mathematical statistics, time series and econometrics. His University of Utah profile identifies these research interests.^[ref]

His joint paper with Piotr Kokoszka and Ron Reeder studies comparisons of mean curves in a functional time series. Its Hilbert space formulation yields Gaussian limits and a test based on the distance between two estimated mean functions, retaining dependence within each sample.^[ref]

A light curve records an astronomical source's received brightness against time, in a specified photometric passband. Its measured form is a time series of timestamps, brightness estimates, and uncertainties; a smooth $F(t)$ is a model of that record. Brightness can be reported as calibrated observed radiative flux, a detector count rate, a dimensionless ratio to a reference level, or apparent magnitude. These quantities require different units and calibration conventions. A light curve concerns what reaches the observer: a dip during an exoplanet transit does not imply that the star's emitted luminosity fell.

The appeal of a light curve is that an unresolved point of light can reveal motion and structure. In exoplanet transit photometry, repeated dips constrain the orbital period; under the uniform-disc, dark-planet, full-overlap assumptions, the exoplanet transit depth is the blocked area fraction $(R_p/R_{\star }{)}^2$. A toy one-percent dip therefore corresponds to a radius ratio of $0.10$. Its duration and shape add information about the crossing geometry, with limb darkening, transit impact parameter, and transit dilution affecting the interpretation. See exoplanet transit photometry for the geometry. The first figure instead follows the measurement: an instrumental trend and sky contribution can obscure a small physical dip.

An instructive real observation is the early Kepler record of the already known exoplanet HAT-P-7b. Welsh and colleagues analysed 33.5 days from 13 May to 15 June 2009, containing 15 complete exoplanet transits at an orbital period of about $2.20473$ days. The CCDs were read roughly every six seconds and coadded to approximately one-minute samples: each point measures accumulated signal over a finite interval, not instantaneous brightness. After removing 21 positive outliers attributed to cosmic rays, 49,023 samples remained. The authors estimated a scatter of about $150$ parts per million per one-minute sample. A gradual $0.13\%$ count decrease, suspected to arise from focus drift with a fixed extraction aperture, was removed with a cubic polynomial fitted while masking the exoplanet transits. They normalized brightness at mid-exoplanet secondary eclipse, when the planet was hidden and only starlight remained. ^[ref].

What does the instrument actually measure? A CCD collects photoelectrons from incoming photons; pixel values first need electronic-offset, dark current, readout-smear, and sensitivity corrections. ^[ref]. Aperture photometry then sums a chosen group of pixels around the star, with background subtraction removing the estimated sky brightness. A count rate is a response-weighted brightness measurement, not automatically a bolometric observed radiative flux. Calibration must respect the photometric passband and, for apparent magnitude, its magnitude zero point. Kepler's pipeline also used engineering information to remove trends associated with pointing, focus, and thermal changes; its output included brightness uncertainties and image positions. ^[ref].

Calibration and interpretation need separate checks. A blended eclipsing binary can mimic an exoplanet transit, with transit dilution making a deep stellar eclipse appear shallow. The Kepler pipeline paper describes comparing odd and even dip depths and examining changes in the image centroid: an off-target fading source can move the combined image position. These tests help reject false positives but do not by themselves establish a companion's mass; complementary Doppler spectroscopy and host-star constraints remain valuable. ^[ref].

Even a real exoplanet's signal can have several contributors. Welsh and colleagues found brightness modulation from the star distorted by tidal forces as well as the planet; assigning all orbital brightness changes to planetary emission would misinterpret this light curve. ^[ref]. Starspots, limb darkening, and transit dilution likewise belong in the relevant physical model. A flexible trend fitted through a dip could absorb that dip; the masked fit in the real example illustrates why the fitting procedure is part of the measurement. Exposure-averaged light curves and phase folding of a light curve make two further limitations explicit.

For a square-integrable time series, let ${\mathcal{H}}_{t-1}$ be the closed linear span of its past. The innovation ${\epsilon }_t=X_t-{\mathrm{proj}}_{{\mathcal{H}}_{t-1}}{X}_t$ is orthogonal to every past linear observation. A causal invertible autoregressive moving-average model is driven by these innovations, which are weak white noise for a weakly stationary process.

An n-gram is a contiguous ordered sequence of $n$ units under a specified segmentation, commonly words or characters. A one-unit sequence is a unigram and a two-unit sequence a bigram. In a text corpus, counting n-grams retains local ordering that individual-word totals discard.^[ref]

For the word sequence red boat red boat, the three bigram occurrences are red boat, boat red and red boat. Thus red boat accounts for $2/3$ of these bigram windows, although the same sequence occurs twice. For $L$ units and $1\le n\le L$, the number of unpadded windows is $L-n+1$.

The units and denominator are part of the definition of a measured rate. Character bigrams differ from word bigrams, and word segmentation, capitalization and sentence boundaries can change the count. A reproducible analysis states those choices before comparing time series or language varieties.

Phonetics studies the physical production, transmission and perception of speech. Articulatory phonetics examines the moving structures that shape the airflow; acoustic phonetics examines the resulting sound waves; auditory phonetics examines how a listener perceives them. These are different measurements along the same communication chain.

During voiced speech, airflow from the lungs drives vocal fold vibration. Movements of the tongue, lips and other parts of the vocal tract shape the emerging sound. This separates the production of a vibrating source from the articulation of recognizable speech, and both from the language expressed by it.^[ref]

A simple acoustic comparison records the same spoken phrase twice, once as a statement and once as a question. The sound pressure time series and its spectrogram allow changes in fundamental frequency, segment duration and pauses to be compared. The comparison investigates physical differences in prosody; the written words alone do not specify those measurements.

Piotr Kokoszka studies functional data analysis, time series and spatial statistical methods. His Colorado State University homepage describes inference for observations that can themselves be curves.^[ref]

With Lajos Horváth and Ron Reeder, he developed estimation and statistical hypothesis testing for mean curves under temporal dependence. Averaging many curves requires a covariance accounting for correlation between successive observations; treating the curves as independent can misstate uncertainty.^[ref]