Spectral density

The spectral density of a fluorescent light as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.

 

The voice waveform over time (left) has a broad audio power spectrum (right).

The power spectrum ${\displaystyle S_{xx}(f)}$ of a time series ${\displaystyle x(t)}$ describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.

When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over all time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating ${\displaystyle x^2(t)}$ over the time domain, as dictated by Parseval's theorem.

The spectrum of a physical process ${\displaystyle x(t)}$ often contains essential information about the nature of ${\displaystyle x}$. For instance, the pitch and timbre of a musical instrument are immediately determined from a spectral analysis. The color of a light source is determined by the spectrum of the electromagnetic wave's electric field ${\displaystyle E(t)}$ as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrograph, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.

However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in statistical signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics and engineering. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency.

Units

See also: Fourier transform § Units

In physics, the signal might be a wave, such as an electromagnetic wave, an acoustic wave, or the vibration of a mechanism. The power spectral density (PSD) of the signal describes the power present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in watts per hertz (W/Hz).

When a signal is defined in terms only of a voltage, for instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance. So one might use units of V² Hz⁻¹ for the PSD. Energy spectral density (ESD) would have units would be V² s Hz⁻¹, since energy has units of power multiplied by time (e.g., watt-hour).

In the general case, the units of PSD will be the ratio of units of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD in units of meters squared per hertz, m²/Hz. In the analysis of random vibrations, units of g² Hz⁻¹ are frequently used for the PSD of acceleration, where g denotes the g-force.

Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of x(t) will remain unspecified, but the independent variable will be assumed to be that of time.

Definition

Energy spectral density

Energy spectral density describes how the energy of a signal or a time series is distributed with frequency. Here, the term energy is used in the generalized sense of signal processing; that is, the energy ${\displaystyle E}$ of a signal ${\displaystyle x(t)}$ is:

${\displaystyle E\triangleq {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|x(t)\right|}^{2}dt.}$

The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. Finite or not, Parseval's theorem (or Plancherel's theorem) gives us an alternate expression for the energy of the signal:

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x(t){|}^{2}dt={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|\hat{x}(f)\right|}^{2}df,}$

where:

${\displaystyle \hat{x}(f)\triangleq {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i2\pi ft}x(t)dt}$

is the value of the Fourier transform of ${\displaystyle x(t)}$ at frequency ${\displaystyle f}$ (in Hz). The theorem also holds true in the discrete-time cases. Since the integral on the right-hand side is the energy of the signal, the value of${\displaystyle {\left|\hat{x}(f)\right|}^{2}df}$ can be interpreted as a density function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency ${\displaystyle f}$ in the frequency interval ${\displaystyle f+df}$.

Therefore, the energy spectral density of ${\displaystyle x(t)}$ is defined as:

${\displaystyle {\overline{S}}_{xx}(f)\triangleq {\left|\hat{x}(f)\right|}^2}$

(Eq.1)

The function ${\displaystyle {\overline{S}}_{xx}(f)}$ and the autocorrelation of ${\displaystyle x(t)}$ form a Fourier transform pair, a result also known as the Wiener–Khinchin theorem (see also Periodogram).

As a physical example of how one might measure the energy spectral density of a signal, suppose ${\displaystyle V(t)}$ represents the potential (in volts) of an electrical pulse propagating along a transmission line of impedance ${\displaystyle Z}$, and suppose the line is terminated with a matched resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By Ohm's law, the power delivered to the resistor at time ${\displaystyle t}$ is equal to ${\displaystyle V(t{)}^2/Z}$, so the total energy is found by integrating ${\displaystyle V(t{)}^2/Z}$ with respect to time over the duration of the pulse. To find the value of the energy spectral density ${\displaystyle {\overline{S}}_{xx}(f)}$ at frequency ${\displaystyle f}$, one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies (${\displaystyle \mathrm{\Delta }f}$, say) near the frequency of interest and then measure the total energy ${\displaystyle E(f)}$ dissipated across the resistor. The value of the energy spectral density at ${\displaystyle f}$ is then estimated to be ${\displaystyle E(f)/\mathrm{\Delta }f}$. In this example, since the power ${\displaystyle V(t{)}^2/Z}$ has units of V² Ω⁻¹, the energy ${\displaystyle E(f)}$ has units of V² s Ω⁻¹ = J, and hence the estimate ${\displaystyle E(f)/\mathrm{\Delta }f}$ of the energy spectral density has units of J Hz⁻¹, as required. In many situations, it is common to forget the step of dividing by ${\displaystyle Z}$ so that the energy spectral density instead has units of V² Hz⁻².

This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values ${\displaystyle x_n}$ such as a signal sampled at discrete times ${\displaystyle t_n=t_0+(n\mathrm{\Delta }t)}$:

${\displaystyle {\overline{S}}_{xx}(f)=\underset{N\to \mathrm{\infty }}{lim}(\mathrm{\Delta }t{)}^2\underset{{\left|{\hat{x}}_d(f)\right|}^2}{\underbrace{{\left|\sum _{n=-N}^N{x}_n{e}^{-i2\pi fn\mathrm{\Delta }t}\right|}^2}},}$

where ${\displaystyle {\hat{x}}_d(f)}$ is the discrete-time Fourier transform of ${\displaystyle x_n.}$  The sampling interval ${\displaystyle \mathrm{\Delta }t}$ is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit ${\displaystyle \mathrm{\Delta }t\to 0.}$  But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see normalized frequency)

Power spectral density

Not to be confused with spectral power distribution.

 

The power spectrum of the measured cosmic microwave background radiation temperature anisotropy in terms of the angular scale. The solid line is a theoretical model, for comparison.

The above definition of energy spectral density is suitable for transients (pulse-like signals) whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist. For continuous signals over all time, one must rather define the power spectral density (PSD) which exists for stationary processes; this describes how the power of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. For example, statisticians study the variance of a function over time ${\displaystyle x(t)}$ (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as the power spectrum even when there is no physical power involved. If one were to create a physical voltage source which followed ${\displaystyle x(t)}$ and applied it to the terminals of a one ohm resistor, then indeed the instantaneous power dissipated in that resistor would be given by ${\displaystyle x(t{)}^2}$ watts.

The average power ${\displaystyle P}$ of a signal ${\displaystyle x(t)}$ over all time is therefore given by the following time average, where the period ${\displaystyle T}$ is centered about some arbitrary time ${\displaystyle t=t_0}$:

${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{t_0-T/2}^{t_0+T/2}|x(t){|}^{2}dt}$

However, for the sake of dealing with the math that follows, it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral. As such, we have an alternative representation of the average power, where ${\displaystyle x_T(t)=x(t)w_T(t)}$ and ${\displaystyle w_T(t)}$ is unity within the arbitrary period and zero elsewhere.

${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x_T(t){|}^{2}dt}$

Clearly in cases where the above expression for P is non-zero (even as T grows without bound) the integral itself must also grow without bound. That is the reason that we cannot use the energy spectral density itself, which is that diverging integral, in such cases.

In analyzing the frequency content of the signal ${\displaystyle x(t)}$, one might like to compute the ordinary Fourier transform ${\displaystyle \hat{x}(f)}$; however, for many signals of interest the Fourier transform does not formally exist. Regardless, Parseval's theorem tells us that we can re-write the average power as follows.

${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|{\hat{x}}_T(f){|}^{2}df}$

Then the power spectral density is simply defined as the integrand above.

${\displaystyle S_{xx}(f)=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}|{\hat{x}}_T(f){|}^2}$

(Eq.2)

From here, we can also view ${\displaystyle |{\hat{x}}_T(f){|}^2}$ as the Fourier transform of the time convolution of ${\displaystyle x_T^{\ast }(-t)}$ and ${\displaystyle x_T(t)}$

${\displaystyle {\left|{\hat{x}}_T(f)\right|}^2=\mathcal{F}\left\{x_T^{\ast }(-t)\mathbf{\ast }{x}_T(t)\right\}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau }$

Now, if we divide the time convolution above by the period ${\displaystyle T}$ and take the limit as ${\displaystyle T\to \mathrm{\infty }}$, it becomes the autocorrelation function of the non-windowed signal ${\displaystyle x(t)}$, which is denoted as ${\displaystyle R_{xx}(\tau )}$, provided that ${\displaystyle x(t)}$ is ergodic, which is true in most, but not all, practical cases.

${\displaystyle \underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\left|{\hat{x}}_T(f)\right|}^2={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xx}(\tau )e^{-i2\pi f\tau }d\tau }$

From here we see, again assuming the ergodicity of ${\displaystyle x(t)}$, that the power spectral density can be found as the Fourier transform of the autocorrelation function (Wiener–Khinchin theorem).

${\displaystyle S_{xx}(f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xx}(\tau )e^{-i2\pi f\tau }d\tau ={\hat{R}}_{xx}(f)}$

(Eq.3)

Many authors use this equality to actually define the power spectral density.

The power of the signal in a given frequency band ${\displaystyle [f_1,f_2]}$, where ${\displaystyle 0<f_1<f_2}$, can be calculated by integrating over frequency. Since ${\displaystyle S_{xx}(-f)=S_{xx}(f)}$, an equal amount of power can be attributed to positive and negative frequency bands, which accounts for the factor of 2 in the following form (such trivial factors depend on the conventions used):

${\displaystyle P_{\mathsf{\text{bandlimited}}}=2{\int }_{f_1}^{f_2}{S}_{xx}(f)df}$

More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval ${\displaystyle T}$ is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than ${\displaystyle 1/T}$ are not sampled, and results at frequencies which are not an integer multiple of ${\displaystyle 1/T}$ are not independent. Just using a single such time series, the estimated power spectrum will be very "noisy"; however this can be alleviated if it is possible to evaluate the expected value (in the above equation) using a large (or infinite) number of short-term spectra corresponding to statistical ensembles of realizations of ${\displaystyle x(t)}$ evaluated over the specified time window.

Just as with the energy spectral density, the definition of the power spectral density can be generalized to discrete time variables ${\displaystyle x_n}$. As before, we can consider a window of ${\displaystyle -N\le n\le N}$ with the signal sampled at discrete times ${\displaystyle x_n=x_0+(n\mathrm{\Delta }t)}$ for a total measurement period ${\displaystyle T=(2N+1)\mathrm{\Delta }t}$.

${\displaystyle S_{xx}(f)=\underset{N\to \mathrm{\infty }}{lim}\frac{(\mathrm{\Delta }t{)}^2}{T}{\left|\sum _{n=-N}^N{x}_n{e}^{-i2\pi fn\mathrm{\Delta }t}\right|}^2}$

Note that a single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved when ${\displaystyle N}$ (and thus ${\displaystyle T}$) approaches infinity and the expected value is formally applied. In a real-world application, one would typically average a finite-measurement PSD over many trials to obtain a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a periodogram. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval ${\displaystyle T}$ approach infinity (Brown & Hwang).

If two signals both possess power spectral densities, then the cross-spectral density can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the cross-correlation.

Properties of the power spectral density

Some properties of the PSD include:

• The power spectrum is always real and non-negative, and the spectrum of a real valued process is also an even function of frequency: ${\displaystyle S_{xx}(-f)=S_{xx}(f)}$.
• For a continuous stochastic process x(t), the autocorrelation function R_xx(t) can be reconstructed from its power spectrum S_xx(f) by using the inverse Fourier transform
• Using Parseval's theorem, one can compute the variance (average power) of a process by integrating the power spectrum over all frequency:

  ${\displaystyle P=\mathrm{Var}(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{S}_{xx}(f)df}$

• For a real process x(t) with power spectral density ${\displaystyle S_{xx}(f)}$, one can compute the integrated spectrum or power spectral distribution ${\displaystyle F(f)}$, which specifies the average bandlimited power contained in frequencies from DC to f using:
  ${\displaystyle F(f)=2{\int }_0^f{S}_{xx}(f^{\prime })d{f}^{\prime }.}$
  Note that the previous expression for total power (signal variance) is a special case where ƒ → ∞.

Cross power spectral density

See also: Coherence (signal processing)

Given two signals ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$, each of which possess power spectral densities ${\displaystyle S_{xx}(f)}$ and ${\displaystyle S_{yy}(f)}$, it is possible to define a cross power spectral density (CPSD) or cross spectral density (CSD). To begin, let us consider the average power of such a combined signal.

${\displaystyle \begin{array}{rl}P& =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left[x_T(t)+y_T(t)\right]}^{\ast }\left[x_T(t)+y_T(t)\right]dt\\ & =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x_T(t){|}^2+x_T^{\ast }(t)y_T(t)+y_T^{\ast }(t)x_T(t)+|y_T(t){|}^{2}dt\end{array}}$

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain

${\displaystyle \begin{array}{rlrl}{S}_{xy}(f)& =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}\left[{\hat{x}}_T^{\ast }(f){\hat{y}}_T(f)\right]& S_{yx}(f)& =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}\left[{\hat{y}}_T^{\ast }(f){\hat{x}}_T(f)\right]\end{array}}$

where, again, the contributions of ${\displaystyle S_{xx}(f)}$ and ${\displaystyle S_{yy}(f)}$ are already understood. Note that ${\displaystyle S_{xy}^{\ast }(f)=S_{yx}(f)}$, so the full contribution to the cross power is, generally, from twice the real part of either individual CPSD. Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit ${\displaystyle T\to \mathrm{\infty }}$ becomes the Fourier transform of a cross-correlation function.

${\displaystyle \begin{array}{rl}{S}_{xy}(f)& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}_T^{\ast }(t-\tau )y_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\ S_{yx}(f)& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{y}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{yx}(\tau )e^{-i2\pi f\tau }d\tau \end{array}}$

where ${\displaystyle R_{xy}(\tau )}$ is the cross-correlation of ${\displaystyle x(t)}$ with ${\displaystyle y(t)}$ and ${\displaystyle R_{yx}(\tau )}$ is the cross-correlation of ${\displaystyle y(t)}$ with ${\displaystyle x(t)}$. In light of this, the PSD is seen to be a special case of the CSD for ${\displaystyle x(t)=y(t)}$. If ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$ are real signals (e.g. voltage or current), their Fourier transforms ${\displaystyle \hat{x}(f)}$ and ${\displaystyle \hat{y}(f)}$ are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two.

${\displaystyle {\mathrm{CPSD}}_{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}$

For discrete signals x_n and y_n, the relationship between the cross-spectral density and the cross-covariance is

${\displaystyle S_{xy}(f)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xy}({\tau }_n)e^{-i2\pi f{\tau }_n}\mathrm{\Delta }\tau }$

Estimation

Main article: Spectral density estimation

The goal of spectral density estimation is to estimate the spectral density of a random signal from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve parametric or non-parametric approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an autoregressive model. A common non-parametric technique is the periodogram.

The spectral density is usually estimated using Fourier transform methods (such as the Welch method), but other techniques such as the maximum entropy method can also be used.

Related concepts

Not to be confused with spectral density (physical science).

• The spectral centroid of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.
• The spectral edge frequency (SEF), usually expressed as "SEF x", represents the frequency below which x percent of the total power of a given signal are located; typically, x is in the range 75 to 95. It is more particularly a popular measure used in EEG monitoring, in which case SEF has variously been used to estimate the depth of anesthesia and stages of sleep.
• A spectral envelope is the envelope curve of the spectrum density. It describes one point in time (one window, to be precise). For example, in remote sensing using a spectrometer, the spectral envelope of a feature is the boundary of its spectral properties, as defined by the range of brightness levels in each of the spectral bands of interest.
• The spectral density is a function of frequency, not a function of time. However, the spectral density of a small window of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a spectrogram. This is the basis of a number of spectral analysis techniques such as the short-time Fourier transform and wavelets.
• A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. For transfer functions (e.g., Bode plot, chirp) the complete frequency response may be graphed in two parts: power versus frequency and phase versus frequency—the phase spectral density, phase spectrum, or spectral phase. Less commonly, the two parts may be the real and imaginary parts of the transfer function. This is not to be confused with the frequency response of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domain impulse response ${\displaystyle h(t)}$ cannot generally be uniquely recovered from the power spectral density alone without the phase part. Although these are also Fourier transform pairs, there is no symmetry (as there is for the autocorrelation) forcing the Fourier transform to be real-valued. See Ultrashort pulse#Spectral phase, phase noise, group delay.
• Sometimes one encounters an amplitude spectral density (ASD), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz^−1/2. This is useful when the shape of the spectrum is rather constant, since variations in the ASD will then be proportional to variations in the signal's voltage level itself. But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency or over a specified bandwidth.

Applications

Further information: Spectrum

Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as visible light (perceived as color), musical notes (perceived as pitch), radio/TV (specified by their frequency, or sometimes wavelength) and even the regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component. And additionally there may be peaks corresponding to harmonics of a fundamental peak, indicating a periodic signal which is not simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter.

Electrical engineering

Spectrogram of an FM radio signal with frequency on the horizontal axis and time increasing upwards on the vertical axis.

The concept and use of the power spectrum of a signal is fundamental in electrical engineering, especially in electronic communication systems, including radio communications, radars, and related systems, plus passive remote sensing technology. Electronic instruments called spectrum analyzers are used to observe and measure the power spectra of signals.

The spectrum analyzer measures the magnitude of the short-time Fourier transform (STFT) of an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density.

Cosmology

Primordial fluctuations, density variations in the early universe, are quantified by a power spectrum which gives the power of the variations as a function of spatial scale.

Climate Science

Power spectral-analysis have been used to examine the spatial structures for climate research. These results suggests atmospheric turbulence link climate change to more local regional volatility in weather conditions.

Time series

From Wikipedia, the free encyclopedia

 

Time series: random data plus trend, with best-fit line and different applied filters.

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. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.

Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series.

Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility).

Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).

Methods for analysis

Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain.

Additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.

Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate.

Panel data

A time series is one type of panel data. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a cross-sectional dataset). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to ask what makes one data record unique from the other records. If the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier which is unrelated to time (e.g. student ID, stock symbol, country code), then it is panel data candidate. If the differentiation lies on the non-time identifier, then the data set is a cross-sectional data set candidate.

Analysis

There are several types of motivation and data analysis available for time series which are appropriate for different purposes.

Motivation

In the context of statistics, econometrics, quantitative finance, seismology, meteorology, and geophysics the primary goal of time series analysis is forecasting. In the context of signal processing, control engineering and communication engineering it is used for signal detection. Other applications are in data mining, pattern recognition and machine learning, where time series analysis can be used for clustering, classification, query by content, anomaly detection as well as forecasting.

Exploratory analysis

Tuberculosis incidence US 1953-2009

 

Further information: Exploratory analysis

A straightforward way to examine a regular time series is manually with a line chart. An example chart is shown on the right for tuberculosis incidence in the United States, made with a spreadsheet program. The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly steadily dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/- 10%, with 'surges' in 1975 and around the early 1990s. The use of both vertical axes allows the comparison of two time series in one graphic.

A study of corporate data analysts found two challenges to exploratory time series analysis: discovering the shape of interesting patterns, and finding an explanation for these patterns. Visual tools that represent time series data as heat map matrices can help overcome these challenges.

Other techniques include:

• Autocorrelation analysis to examine serial dependence
• Spectral analysis to examine cyclic behavior which need not be related to seasonality. For example, sunspot activity varies over 11 year cycles. Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
• Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see trend estimation and decomposition of time series

Curve fitting

Main article: Curve fitting

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

The construction of economic time series involves the estimation of some components for some dates by interpolation between values ("benchmarks") for earlier and later dates. Interpolation is estimation of an unknown quantity between two known quantities (historical data), or drawing conclusions about missing information from the available information ("reading between the lines"). Interpolation is useful where the data surrounding the missing data is available and its trend, seasonality, and longer-term cycles are known. This is often done by using a related series known for all relevant dates. Alternatively polynomial interpolation or spline interpolation is used where piecewise polynomial functions are fit into time intervals such that they fit smoothly together. A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function (also called regression). The main difference between regression and interpolation is that polynomial regression gives a single polynomial that models the entire data set. Spline interpolation, however, yield a piecewise continuous function composed of many polynomials to model the data set.

Extrapolation is the process of estimating, beyond the original observation range, the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results.

Function approximation

Main article: Function approximation

In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way. One can distinguish two major classes of function approximation problems: First, for known target functions, approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).

Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points (a time series) of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead. A related problem of online time series approximation is to summarize the data in one-pass and construct an approximate representation that can support a variety of time series queries with bounds on worst-case error.

To some extent, the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.

Prediction and forecasting

In statistics, prediction is a part of statistical inference. One particular approach to such inference is known as predictive inference, but the prediction can be undertaken within any of the several approaches to statistical inference. Indeed, one description of statistics is that it provides a means of transferring knowledge about a sample of a population to the whole population, and to other related populations, which is not necessarily the same as prediction over time. When information is transferred across time, often to specific points in time, the process is known as forecasting.

• Fully formed statistical models for stochastic simulation purposes, so as to generate alternative versions of the time series, representing what might happen over non-specific time-periods in the future
• Simple or fully formed statistical models to describe the likely outcome of the time series in the immediate future, given knowledge of the most recent outcomes (forecasting).
• Forecasting on time series is usually done using automated statistical software packages and programming languages, such as Julia, Python, R, SAS, SPSS and many others.
• Forecasting on large scale data can be done with Apache Spark using the Spark-TS library, a third-party package.

Classification

Main article: Statistical classification

Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in sign language.

Signal estimation

See also: Signal processing and Estimation theory

This approach is based on harmonic analysis and filtering of signals in the frequency domain using the Fourier transform, and spectral density estimation, the development of which was significantly accelerated during World War II by mathematician Norbert Wiener, electrical engineers Rudolf E. Kálmán, Dennis Gabor and others for filtering signals from noise and predicting signal values at a certain point in time. See Kalman filter, Estimation theory, and Digital signal processing

Segmentation

Main article: Time-series segmentation

Splitting a time-series into a sequence of segments. It is often the case that a time-series can be represented as a sequence of individual segments, each with its own characteristic properties. For example, the audio signal from a conference call can be partitioned into pieces corresponding to the times during which each person was speaking. In time-series segmentation, the goal is to identify the segment boundary points in the time-series, and to characterize the dynamical properties associated with each segment. One can approach this problem using change-point detection, or by modeling the time-series as a more sophisticated system, such as a Markov jump linear system.

Models

Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. These three classes depend linearly on previous data points. Combinations of these ideas produce autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial "V" for "vector", as in VAR for vector autoregression. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous".

Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a chaotic time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models, over those from linear models, as for example in nonlinear autoregressive exogenous models. Further references on nonlinear time series analysis: (Kantz and Schreiber), and (Abarbanel)

Among other types of non-linear time series models, there are models to represent the changes of variance over time (heteroskedasticity). These models represent autoregressive conditional heteroskedasticity (ARCH) and the collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc.). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a doubly stochastic model.

In recent work on model-free analyses, wavelet transform based methods (for example locally stationary wavelets and wavelet decomposed neural networks) have gained favor. Multiscale (often referred to as multiresolution) techniques decompose a given time series, attempting to illustrate time dependence at multiple scales. See also Markov switching multifractal (MSMF) techniques for modeling volatility evolution.

A Hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (hidden) states. An HMM can be considered as the simplest dynamic Bayesian network. HMM models are widely used in speech recognition, for translating a time series of spoken words into text.

Notation

A number of different notations are in use for time-series analysis. A common notation specifying a time series X that is indexed by the natural numbers is written

X = (X₁, X₂, ...).

Another common notation is

Y = (Y_t: t ∈ T),

where T is the index set.

Conditions

There are two sets of conditions under which much of the theory is built:

• Stationary process
• Ergodic process

Ergodicity implies stationarity, but the converse is not necessarily the case. Stationarity is usually classified into strict stationarity and wide-sense or second-order stationarity. Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified.

In addition, time-series analysis can be applied where the series are seasonally stationary or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in time-frequency analysis which makes use of a time–frequency representation of a time-series or signal.

Tools

Tools for investigating time-series data include:

• Consideration of the autocorrelation function and the spectral density function (also cross-correlation functions and cross-spectral density functions)
• Scaled cross- and auto-correlation functions to remove contributions of slow components
• Performing a Fourier transform to investigate the series in the frequency domain
• Discrete, continuous or mixed spectra of time series, depending on whether the time series contains a (generalized) harmonic signal or not
• Use of a filter to remove unwanted noise
• Principal component analysis (or empirical orthogonal function analysis)
• Singular spectrum analysis
• "Structural" models:
  • General State Space Models
  • Unobserved Components Models
• Machine Learning
  • Artificial neural networks
  • Support vector machine
  • Fuzzy logic
  • Gaussian process
  • Genetic Programming
  • Gene expression programming
  • Hidden Markov model
  • Multi expression programming
• Queueing theory analysis
• Control chart
  • Shewhart individuals control chart
  • CUSUM chart
  • EWMA chart
• Detrended fluctuation analysis
• Nonlinear mixed-effects modeling
• Dynamic time warping
• Dynamic Bayesian network
• Time-frequency analysis techniques:
  • Fast Fourier transform
  • Continuous wavelet transform
  • Short-time Fourier transform
  • Chirplet transform
  • Fractional Fourier transform
• Chaotic analysis
  • Correlation dimension
  • Recurrence plots
  • Recurrence quantification analysis
  • Lyapunov exponents
  • Entropy encoding

Measures

Time series metrics or features that can be used for time series classification or regression analysis:

• Univariate linear measures
  • Moment (mathematics)
  • Spectral band power
  • Spectral edge frequency
  • Accumulated Energy (signal processing)
  • Characteristics of the autocorrelation function
  • Hjorth parameters
  • FFT parameters
  • Autoregressive model parameters
  • Mann–Kendall test
• Univariate non-linear measures
  • Measures based on the correlation sum
  • Correlation dimension
  • Correlation integral
  • Correlation density
  • Correlation entropy
  • Approximate entropy
  • Sample entropy
  • Fourier entropyuk
  • Wavelet entropy
  • Dispersion entropy
  • Fluctuation dispersion entropy
  • Rényi entropy
  • Higher-order methods
  • Marginal predictability
  • Dynamical similarity index
  • State space dissimilarity measures
  • Lyapunov exponent
  • Permutation methods
  • Local flow
• Other univariate measures
  • Algorithmic complexity
  • Kolmogorov complexity estimates
  • Hidden Markov Model states
  • Rough path signature
  • Surrogate time series and surrogate correction
  • Loss of recurrence (degree of non-stationarity)
• Bivariate linear measures
  • Maximum linear cross-correlation
  • Linear Coherence (signal processing)
• Bivariate non-linear measures
  • Non-linear interdependence
  • Dynamical Entrainment (physics)
  • Measures for Phase synchronization
  • Measures for Phase locking
• Similarity measures:
  • Cross-correlation
  • Dynamic Time Warping
  • Hidden Markov Models
  • Edit distance
  • Total correlation
  • Newey–West estimator
  • Prais–Winsten transformation
  • Data as Vectors in a Metrizable Space
    • Minkowski distance
    • Mahalanobis distance
  • Data as time series with envelopes
    • Global standard deviation
    • Local standard deviation
    • Windowed standard deviation
  • Data interpreted as stochastic series
    • Pearson product-moment correlation coefficient
    • Spearman's rank correlation coefficient
  • Data interpreted as a probability distribution function
    • Kolmogorov–Smirnov test
    • Cramér–von Mises criterion

Visualization

Time series can be visualized with two categories of chart: Overlapping Charts and Separated Charts. Overlapping Charts display all-time series on the same layout while Separated Charts presents them on different layouts (but aligned for comparison purpose).

Overlapping charts

• Braided graphs
• Line charts
• Slope graphs
• GapChartfr

Separated charts

• Horizon graphs
• Reduced line chart (small multiples)
• Silhouette graph
• Circular silhouette graph