Autocovariance measures how a time series varies jointly with lagged versions of itself. Lag 0 is the variance, while nonzero lags show the direction and scale of linear dependence between observations separated in time, expressed in the squared units of the original series...
The autocorrelation function measures the unit-free linear association between a time series and lagged versions of itself. The partial autocorrelation function asks a narrower question: how much direct linear relationship remains at a given lag after the intervening lags have been accounted for...
ARMA models describe stationary linear dependence through a combination of autoregressive terms, which use past observations, and moving-average terms, which use current and past innovations. ARIMA extends that structure by differencing a series with a stochastic trend, while SARIMA adds seasonal di...
Trend and seasonality are systematic forms of time structure that operate on different scales. A trend is a persistent long-run movement in level or slope, while seasonality is a pattern that repeats at a fixed and known period such as day of week, month of year, or quarter...
Regression with ARMA errors separates two kinds of structure that often appear together in time series. The regression explains how observed predictors shift the conditional mean, while the ARMA component models serial dependence that remains in the unexplained part...
A time-series forecast is a conditional statement about a future observation given the information available at a specific forecast origin. The horizon, target functional, and information set are part of the forecast itself, so a point prediction without its timing and uncertainty is an incomplete d...
A moving-average model represents the current observation as a finite weighted combination of the current innovation and a fixed number of past innovations. Because each shock enters the model for only a limited number of periods, an MA($q$) process has a finite direct shock response and a theoretic...
A time series is an ordered set of observations together with an information structure: timestamps, sampling rules, and sequence position determine what can depend on what and what information is available for prediction. Treating the same values as an unordered sample would discard much of the stru...
The Yule-Walker equations connect an autoregressive model's coefficients to its autocovariances and autocorrelations. They translate a recursion written in terms of lagged observations into a set of moment relationships that can be studied theoretically or estimated from data...
The backward shift operator is a compact notation for referring to earlier observations in a time series. Instead of writing each lag separately, it lets lagged values, differences, and model equations be expressed as polynomials in a single operator...
Sequences and series provide the convergence language behind many time-series representations. A sequence describes an ordered set of terms, while a series asks whether the cumulative effect of those terms approaches a finite limit...
Time-series modeling is an iterative process of specifying structure, estimating parameters, diagnosing what remains unexplained, and testing forecasts on future-like data. Choosing an equation is only one step; transformations, initial conditions, residual behavior, and the forecast information set...
Statistical moments summarize features such as the center, spread, and joint variation of a distribution. In time series, those same quantities acquire a time dimension: the mean and variance may change across the record, and covariance becomes a function of lag as well as scale...
Randomness tests look for specific kinds of structure that should not appear under a stated null model. In time-series work, they are most useful as diagnostics: one test may target linear autocorrelation, another unusual turning-point behavior, another monotone trend, and another dependence in squa...
Probability theory is built on a small set of principles, called axioms, that define how probability measures behave. These axioms, formalized by the Russian mathematician Andrey Kolmogorov, provide the foundation for the rules used throughout probability theory...
Geometric probability is a fascinating branch of probability theory where outcomes are associated with geometric figures and their measures—such as lengths, areas, and volumes—rather than discrete numerical outcomes. It often deals with continuous random variables and employs integral calculus to ca...
The law of total probability allows for the computation of the probability of an event A based on a set of mutually exclusive and exhaustive events. It's particularly useful when the overall sample space is divided into several distinct scenarios, or partitions, that cover all possible outcomes. The...
Bayes' theorem provides a way to update the probability of an event when new evidence becomes available. It connects conditional probabilities and allows us to revise an initial probability, or prior, in light of additional information...
Conditional Probability is the likelihood of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$, representing the probability of event $A$ happening, assuming event $B$ has already taken place. This concept is crucial in understanding dependent events in prob...
Probability theory offers a structured approach to assessing the probability of events, allowing for logical and systematic reasoning about their likelihood...
Probability trees are a visual representation of all possible outcomes of a probabilistic experiment and the paths leading to these outcomes. They are especially helpful in understanding sequences of events, particularly when these events are conditional on previous outcomes...
Descriptive statistics summarize the main characteristics of a dataset or sample. They help us understand data by describing its frequency, center, spread, and overall distribution...
Statistics is the science of learning from data. It provides methods for collecting, summarizing, analyzing, and interpreting observations so that we can describe patterns, quantify uncertainty, and make informed decisions...
The F-distribution, also known as the Fisher-Snedecor distribution, is a continuous probability distribution that arises in hypothesis testing when comparing the variances of two normally distributed populations. The F-distribution is denoted as $X \sim F(d_1, d_2)$, where $d_1$ and $d_2$ are the de...
Student's t-distribution is a continuous probability distribution that arises when a normally distributed quantity is standardized using an estimated standard deviation rather than a known population standard deviation. It has heavier tails than the normal distribution and approaches the normal dist...
A statistic is computed from a sample, but before the sample is observed the statistic is itself a random variable. Its probability distribution over hypothetical repeated samples is called its sampling distribution...
A chi-square distribution is a continuous probability distribution of the sum of the squares of k independent standard normal random variables. The chi-square distribution is denoted as $X \sim \chi^2(k)$, where k is the number of degrees of freedom...
The Central Limit Theorem (CLT) is a fundamental result in statistics. It explains why the distribution of sample means often approaches a normal distribution as the sample size increases, even when the population itself is not normally distributed...
Expected Value (E), also known as the mean, is the long-run average of a random variable, representing the value we anticipate on average from repeated random draws from a population...
Bayesian and frequentist statistics are two major approaches to statistical inference. Both use sample data to learn about an underlying population or data-generating process, but they differ in how they interpret probability, represent uncertainty, and draw conclusions about unknown parameters...
Multiple linear regression is a statistical technique used to model the relationship between a single dependent variable and two or more independent variables. It extends the concept of simple linear regression by incorporating multiple predictors to explain the variability in the dependent variable...
Logistic regression is a statistical method used for modeling the probability of a binary outcome based on one or more predictor variables. It is widely used in various fields such as medicine, social sciences, and machine learning for classification problems where the dependent variable is dichotom...
Simple linear regression is a statistical method used to model the relationship between a single dependent variable and one independent variable. It aims to find the best-fitting straight line through the data points, which can be used to predict the dependent variable based on the independent varia...
Does peer assessment enhance student learning...
In both statistics and mechanics the word moment measures how much "leverage" the values of a quantity exert about a chosen reference point. In statistics the leverage is exerted by probability mass, in mechanics by physical mass, but the mathematics is identical: take a distance from the reference ...