White Noise: A Series of Uncorrelated Random Variables with Constant Mean and Variance

White noise refers to a stochastic process where each value is an independently generated random variable with a fixed mean and variance, often used in signal processing and time series analysis.

White noise refers to a stochastic process, often used in the context of time series analysis and signal processing. It is characterized by a set of uncorrelated random variables, each having a constant mean (usually zero) and constant variance. In practical terms, white noise appears as a random sequence of data points that shows no predictable pattern or trend.

Mathematical Definition§

Mathematically, a time series ${X_t}$ is considered white noise if it satisfies two conditions:

• Uncorrelated Random Variables: $E[X_t X_s] = 0$ for any $t \neq s$, where $E$ denotes the expected value.
• Constant Mean and Variance: $E[X_t] = \mu$ and $\text{Var}(X_t) = \sigma^2$ for all $t$, where $\mu$ is the mean and $\sigma^2$ is the variance.

Typically, for simplicity, $\mu$ is often taken to be zero, giving $E[X_t] = 0$. The auto-covariance function $\gamma(k)$ of a white noise process is:

$$\gamma(k) = \begin{cases} \sigma^2, & \text{if } k = 0 \\ 0, & \text{if } k \neq 0 \end{cases}$$

Types of White Noise§

Gaussian White Noise§

If each $X_t$ follows a normal distribution $N(\mu, \sigma^2)$, the process is called Gaussian white noise.

Non-Gaussian White Noise§

When the random variables $X_t$ come from other distributions (e.g., uniform, binomial), the process is referred to as non-Gaussian white noise.

Special Considerations§

Stationarity§

White noise inherently satisfies weak stationarity, as its mean, variance, and covariance are consistent over time.

Independence§

While white noise sequences are uncorrelated, they are not necessarily independent. For independence, each $X_t$ must not only be uncorrelated but also satisfy the higher-order moments.

Examples of White Noise§

• In Finance: The returns of an asset if modeled without serial correlation.
• In Signal Processing: Background electrical noise at different frequencies.

Example Calculation§

Consider a white noise series ${X_t}$ with mean $\mu = 0$ and variance $\sigma^2 = 1$. Some possible values might be $X_1 = 0.3, X_2 = -0.8, X_3 = 1.2$, and so on. Each value appears to be unpredictable and does not exhibit any correlation with the others.

Historical Context§

The concept of white noise has roots in 20th-century statistical methods and signal processing. It is named after white light in physics, which contains all frequencies and hence forms a ’noisy’ blend of colors.

Applicability§

In Time Series Analysis§

White noise is the basic building block for more complex time series models, like ARIMA (AutoRegressive Integrated Moving Average).

In Control Systems§

Used to model the effect of random disturbances or noise affecting the system.

In Econometrics§

Helps in diagnosing the presence of autocorrelation in residual analysis of models.

Comparisons§

White Noise vs. Pink Noise§

While white noise has equal intensity across frequencies, pink noise’s power decreases with increasing frequency.

White Noise vs. Random Walk§

A random walk $Y_t = Y_{t-1} + \epsilon_t$ with $\epsilon_t$ as white noise is a non-stationary process, whereas white noise itself is stationary.

Related Terms§

• Autocorrelation: Measures the correlation of a signal with a delayed copy of itself as a function of delay.
• Stationarity: A stationary process has statistical properties that do not change over time.
• Moving Average Process: A generalization of white noise where today’s value is a weighted sum of past white noise terms.

FAQs§

What is the significance of white noise in modeling?

White noise is crucial for model diagnostics and understanding random perturbations within data.

Can white noise be observed in nature?

Yes, white noise can be observed in various contexts like environmental measurements and economics.

References§

1. Shumway, R.H., & Stoffer, D.S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer.
2. Box, G. E., & Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control. Holden-Day.

Summary§

White noise is a fundamental concept in statistics and signal processing, representing a series of uncorrelated random variables with constant mean and variance. It serves as a critical element in time series analysis and various practical applications ranging from finance to control systems. Understanding white noise helps in building more complex models and diagnosing issues within datasets.

By ensuring clarity and statistical rigor, this entry aims to provide a comprehensive understanding of white noise, its properties, and applications.