White Noise
White Noise
There are uncertainties in dynamics of social and natural processes. Basic approaches of statistical analysis model these processes based on theoretical derivations or empirical observations. The primary goal in statistical modeling is to extract as much underlying information of the processes as possible and let the residuals approximate a realization of white noise. White noise is one of the fundamental stochastic processes in many fields. Mathematically, it has a constant spectrum, which is the same as the white light we observe through our eyes.
White noise has been utilized to mask distraction of undesirable sound in the environment. Financial analysts have applied white noise to model stock markets. In fact white noise has been used for audio synthesis, impulse response, art, sensory deprivation, sleeping aid, and more. White noise is a basic form of stochastic process that provides the foundation for almost all useful statistical models used in natural and social sciences.
Let {Z_{t} } be an equally spaced time series (a sequence of random variables) with a mean of zero and finite autocovariance
where E (X ) is the expected value (mean) of random variable X and t, s donate the time. If {Z_{t} } has a nonzero mean, without loss generality, it can be subtracted from the time series to get a time series with a zero mean. The expected value of a discrete random variable X is defined as
where P (X = x ) is the probability mass function of X. For a continuous random variable, its expected value is
where f (x ) is the density function of X.
If the autocovariance function γ (t, s ) of time series {Z_{t} } is only a function of t – s , then {Z_{t} } is a (weak) stationary process. White noise is the simplest stationary process with
where is the variance of Z_{t}.
White noise plays an important role not only in physical sciences such as in signal processing, but also in almost all statistical analysis of time series observed from social and economic activities. For example, according to George Box, Gwilym M. Jenkins, and Gregory C. Reinsel in their 1994 book Time Series Analysis: Forecasting and Control, the widely used Autoregressive and Integrated Moving Average (ARIMA(p,d,q)) models in time series analysis are generated by the white noise innovations {Z_{t} } through the following expression:
where Y_{t} is the time series under study, φ (B ) = 1 – is the back shift operator defined as BY_{t} = Y_{t} – Y_{t – 1}, d is an integer, and Z_{t} is the Gaussian white noise with mean zero and variance σ ^{2}.
From expression (5) we can see that Y_{t} of ARIMA(0,0,0) is the white noise.
ARIMA(0,1,0) denotes a random walk model, which is a model of unit root and widely used in modeling financial markets. ARIMA(p,0,0) becomes an autoregressive model of order p (AR(p)), ARIMA(0,0,q) is a moving average model of order q (MA(q)), and ARIMA(p,0,q) is ARMA(p,q). For a proper range of the parameters a_{l} and β_{j}, the Y_{t} in ARMA(p,q) is stationary and invertible (Box, Jenkins, and Reinsel 1994).
Many methods have been proposed to estimate the parameters in ARIMA models from observed time series. A crucial step in model diagnosis is to check, through a battery of tests and plots, if the residuals from the models are consistent with white noise (Box, Jenkins, and Reinsel 1994).
In general there are two types of techniques in analyzing time series. The first type is based on direct modeling such as ARIMA models, which are called time domain techniques (Box, Jenkins, and Reinsel 1994). The second type, according to M. B. Priestley in the 1981 publication Spectral Analysis and Time Series, is in the frequency
domain that utilizes the spectrum of a stationary process. The spectrum is defined as
where γ_{k} = γ (t, t + k ) and i =. For a stationary ARMA(p,q) process, the spectrum is f (ω ) = σ ^{2}θ (e ^{–i ω})^{2}φ (e ^{–i ω})^{–2}, where . is the norm of a complex number.
For white noise, its spectrum is
a constant for all value of frequency ω. That is, the plot of the spectrum of white noise is flat against ω. The spectrum of white noise can be estimated through the periodogram of finite observations with length n. Let g_{k} be the sample autocovariance computed from the n observations of a stationary time series {Y_{t} }:
where . Then, the periodogram ordinates are
usually computed at the Fourier frequencies ω_{j} = 2πj /n. The plot of the periodogram ordinates gives a visual examination of the underlying spectrum of the process. For a sample of white noise, its I (ω ) is distributed as , where is a chi square random variable with 2 degrees of freedom. Note that I (ω ) is an unbiased estimator of f (ω ), but not consistent (Priestley 1981). Many techniques were proposed to construct unbiased and consistent estimators of f (ω ) using functions of I (ω ), and many tests on white noise can also be constructed via periodogram ordinates I (ω ) (Priestley 1981).
White noise is not only the driving force for classic ARIMA models, it also plays a fundamental role in the popular Generalized Autoregressive and Conditional Heteroscedastic (GARCH) models in financial analysis, in vector cointegration models for economic time series, and in long memory time series models such as in Autoregressive Fractional Integrated Moving Average (ARFIMA(p,d,q)) models with d being a noninteger. White noise is the core of statistical analysis. However, some realizations of simple deterministic chaotic systems may exhibit white noise like sequences. The study of chaos requires new techniques and concepts that are beyond the classic approaches of time series analysis.
SEE ALSO Autoregressive Models; Cointegration; Randomness; Regression; Regression Analysis; Residuals; Unit Root and Cointegration Regression
BIBLIOGRAPHY
Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. 1994. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall.
Priestley, M. B. 1981. Spectral Analysis and Time Series. London: Academic Press.
Dejian Lai
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white noise
white noise Noise occurring in a channel and regarded as continuous in time and continuous in amplitude, the noise being uniform in energy over equal intervals of frequency. (Note that, by contrast, white light is uniform in energy over equal intervals of wavelength.) Compare impulse noise.
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white noise
white noise • n. Physics noise containing many frequencies with equal intensities.
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