Seasonality and trends are fundamental components in time series data that significantly impact analysis and forecasting. Understanding and correctly modeling these elements are crucial for accurate predictions and effective time series modeling...

Autocovariance functions describe how values of a time series relate to their lagged counterparts, measuring the joint variability between a series at time $t$ and its value at a previous time $t-k$ (where $k$ is the lag). In autoregressive models, these relationships are expressed through coefficie...

A sequence is an ordered list of numbers that can be viewed as a function mapping each natural number $n$ to a specific value $a_n$. More formally, a sequence ${a_n}$ is a function whose domain is the set of natural numbers, and the values are called the terms of the sequence...

A difference equation (also known as a recurrence relation) defines each term of a sequence based on previous terms. In some cases, the general term of a sequence is given explicitly (e.g., $a_n = 3n + 2$, resulting in the sequence $5, 8, 11, \dots$). However, more commonly, a difference equation pr...

In time series analysis, understanding the relationships between observations at different time lags is crucial for model identification and forecasting. Two essential tools for analyzing these relationships are the Autocorrelation Function (ACF) and the Partial Autocorrelation Function (PACF)...

The Yule-Walker equations are a set of linear equations that relate the autocorrelations of an autoregressive (AR) process to its parameters. These equations are crucial for estimating the parameters of AR models and for understanding the autocorrelation structure of the process...

Autoregressive (AR) models are fundamental tools in time series analysis, used to describe and forecast time-dependent data. An AR model predicts future values based on a linear combination of past observations. The order of an AR model, denoted as ( p ), indicates how many lagged past values are us...

The random walk is a fundamental and widely used time series model, often applied in finance to represent stock prices and other economic indicators. The idea behind the random walk is that the value of the process at time $t$ is the sum of its value at time $t-1$ and a random shock (or noise). Esse...

The backward shift operator (denoted by $B$) is a powerful tool in time series analysis, used to simplify the notation and manipulation of time series models. The operator shifts the time index of a time series back by one period, making it useful in autoregressive, moving average, and mixed models...

Time series data consists of sequential observations collected over a period of time. This kind of data is prevalent in a range of fields such as finance, economics, climatology, and more. Time series analysis involves the exploration of this data to identify inherent structures such as patterns or ...

In time series modeling, invertibility is the property of a model that allows the innovation process (also called the noise or disturbance process) to be expressed as a function of the observed series and its past values. This is particularly relevant for Moving Average (MA) models...

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...

Correlation is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is a fundamental concept in statistics, enabling researchers and analysts to understand how one variable may predict or relate to another. The most commonly used corre...

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...

Descriptive statistics offer a summary of the main characteristics of a dataset or sample. They facilitate the understanding and interpretation of data by providing measures of central tendency, dispersion, and shape. In this section, we will discuss the essential concepts and measures in descriptiv...

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...

Bayes' theorem provides a way to update our probability estimates for an event based on new evidence. It connects the conditional and marginal probabilities of events, allowing us to revise our predictions or hypotheses in light of additional information. The theorem is stated mathematically as...

Probability theory is based on a set of principles, or axioms, that define the properties of the probability measure. These axioms, first formalized by the Russian mathematician Andrey Kolmogorov, are the foundation upon which the entire framework of probability is built...

One of the fundamental skills is to navigate and manage files and directories effectively. Here, we focus on the crucial concepts that will facilitate your work within the file system...

Choosing the most effective methodology for creating and merging branches in a Git repository can significantly impact your development workflow. The right branching strategy often depends on several variables, such as organizational structure, project size and complexity, as well as the team's pref...

The find, locate, and which commands are commonly used for file search operations. The find command performs a comprehensive search using attributes such as name, size, and type. locate provides a faster, albeit periodically updated, search by filename. which locates the path of a program's executab...

In NumPy, manipulating the structure of arrays is a common operation. Whether combining multiple arrays into one or splitting a single array into several parts, NumPy provides a set of intuitive functions to achieve these tasks efficiently. This guide will cover various methods to join and split arr...

Systems of linear equations are a cornerstone of linear algebra and play a crucial role in various fields such as engineering, physics, computer science, and economics. These systems involve multiple linear equations that share common variables. By utilizing matrix notation, we can represent and sol...

Designing parallel programs involves breaking down computational tasks into smaller, concurrent units to be executed simultaneously. This approach leverages the power of multiple processors to enhance performance and efficiency. Key steps in this process include partitioning, communication, agglomer...

Processes are fundamental elements in any computing system. They represent an instance of a running program and are essential for the execution of various tasks. A process is more than just the program code (often referred to as the text section in Unix); it also includes the current activity, inclu...

A matrix is a systematic arrangement of numbers (or elements) in rows and columns. An m Ã— n matrix has m rows and n columns. The dimensions of the matrix are represented as m Ã— n...

VTK offers a set of tools to create animations and visualize time-varying data. This is particularly useful in scenarios such as...

Evaluating the performance of parallel computing systems is crucial for understanding their efficiency and identifying potential bottlenecks. Here are key metrics and concepts for evaluating performance...

Asynchronous programming is a technique used to achieve concurrency, where tasks can be executed independently without waiting for other tasks to finish. It allows for nonblocking behavior, in contrast to synchronous execution that waits for one task to complete before starting the next task...

Linux is a versatile and powerful open-source operating system that forms the backbone of countless technological infrastructures, from servers and desktops to mobile devices and embedded systems. Known for its stability, security, and flexibility, Linux provides a robust platform that can be custom...

Multiprocessing involves running multiple processes simultaneously. Each process has its own memory space, making them more isolated from each other compared to threads, which share the same memory. This isolation means that multiprocessing can be more robust and less prone to errors from shared sta...

Debian and Ubuntu are popular Linux distributions for home users. These distributions and their derivatives use the Advanced Package Tool (APT). Other distributions use alternative package managers, like DNF, YUM, Pacman, which have unique functionalities and syntax...

Git's powerful suite of commands offers an insightful look into your codebase's progression. By probing changes, tracking progress, identifying anomalies, and fostering effective collaboration becomes easier...

Let's start by defining some key terms and emphasizing the distinctions between related concepts. In general those concepts are universal and may be applied to any programming language. The differences between the languages will be discussed in greater detail later, when we attempt to explain the sp...