Articles

Moving Average Models 🇺🇸

March 23, 2026
Category: Statistics Notes

Moving Average (MA) models are a fundamental class of univariate time series models used for forecasting and understanding temporal data. Unlike Autoregressive (AR) models, which rely on past values of the series itself, MA models utilize past forecast errors to model the current value of the series...

Time Series 🇺🇸

March 23, 2026
Category: Statistics Notes

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 ...

Yule Walker Equations 🇺🇸

March 23, 2026
Category: Statistics Notes

The Yule-Walker equations are a set of linear relationships that tie the autocovariances/autocorrelations of a stationary autoregressive (AR $p$) process to its parameters. They are the work-horse for parameter estimation, diagnostic checking, and theoretical analysis of AR models...

Backward Shift Operator 🇺🇸

March 23, 2026
Category: Statistics Notes

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...

Series 🇺🇸

March 23, 2026
Category: Statistics Notes

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...

Time Series Modeling 🇺🇸

March 23, 2026
Category: Statistics Notes

Time series modeling involves analyzing data points collected or recorded at specific time intervals to understand underlying structures and make forecasts. Various models, such as Autoregressive (AR), Moving Average (MA), and their combinations (ARMA, ARIMA), are employed to capture different aspec...

Statistical Moments and Time Series 🇺🇸

March 23, 2026
Category: Statistics Notes

Understanding the behavior of time series data is crucial across various fields such as finance, economics, and engineering. Statistical moments, especially the mean and standard deviation, are essential tools in summarizing and analyzing time series data. This section explores how these statistical...

Randomness Tests 🇺🇸

March 23, 2026
Category: Statistics Notes

When a series looks noisy, it is still useful to check whether the noise is random or whether weak structure (trend or dependence) is present. The tests below are lightweight diagnostics for an IID or weak-dependence null...

Taylor Series 🇺🇸

March 23, 2026
Category: Numerical Methods

The Taylor series is a fundamental tool in calculus and mathematical analysis, offering a powerful way to represent and approximate functions. By expanding a function around a specific point, known as the "center" or "point of expansion," we can express it as an infinite sum of polynomial terms deri...

Thin Plate Spline Interpolation 🇺🇸

March 23, 2026
Category: Numerical Methods

Thin Plate Spline (TPS) interpolation is a non‑parametric, spline‑based technique for fitting a smooth surface through scattered data in two or more spatial dimensions. In its classical 2‑D form one seeks a function $f\colon\mathbb R^{2}\to\mathbb R$ that passes through specified data points while m...

Gaussian Interpolation 🇺🇸

March 23, 2026
Category: Numerical Methods

Gaussian Interpolation, often associated with Gauss’s forward and backward interpolation formulas, is a technique that refines polynomial interpolation for equally spaced data points. Rather than building the interpolating polynomial from one end of the data interval (as Newton’s forward or backward...

Cubic Spline Interpolation 🇺🇸

March 23, 2026
Category: Numerical Methods

Cubic spline interpolation is a refined mathematical tool frequently used within numerical analysis. It's an approximation technique that employs piecewise cubic polynomials, collectively forming a cubic spline. These cubic polynomials are specifically engineered to pass through a defined set of dat...

Least Squares 🇺🇸

March 23, 2026
Category: Numerical Methods

Least Squares Regression is a fundamental technique in statistical modeling and data analysis used for fitting a model to observed data. The primary goal is to find a set of parameters that minimize the discrepancies (residuals) between the model’s predictions and the actual observed data. The "leas...

Linear Interpolation 🇺🇸

March 23, 2026
Category: Numerical Methods

Linear interpolation is one of the most basic and commonly used interpolation methods. The idea is to approximate the value of a function between two known data points by assuming that the function behaves linearly (like a straight line) between these points. Although this assumption may be simplist...

Newton Polynomial 🇺🇸

March 23, 2026
Category: Numerical Methods

Newton’s Polynomial, often referred to as Newton’s Interpolation Formula, is another classical approach to polynomial interpolation. Given a set of data points $(x_0,y_0),(x_1,y_1),\dots,(x_n,y_n)$ with distinct $x_i$ values, Newton’s method constructs an interpolating polynomial in a form that make...

Regression 🇺🇸

March 23, 2026
Category: Numerical Methods

Regression analysis and curve fitting are important tools in statistics, econometrics, engineering, and modern machine-learning pipelines. At their core they seek a deterministic (or probabilistic) mapping $\widehat f: \mathcal X \longrightarrow \mathcal Y$ that minim...

Lagrange Polynomial Interpolation 🇺🇸

March 23, 2026
Category: Numerical Methods

Lagrange Polynomial Interpolation is a widely used technique for determining a polynomial that passes exactly through a given set of data points. Suppose we have a set of $(n+1)$ data points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$ where all $x_i$ are distinct. The aim is to find a polynomial $L...

Liczby Losowe 🇵🇱

March 19, 2026
Category: Od C Do Cpp

W języku C++ liczby losowe generuje się za pomocą standardowej biblioteki . Proces losowania zaczyna się od utworzenia generatora liczb pseudolosowych, np. std::mt19937, który bazuje na algorytmie Mersenne Twister. Aby uzyskać bardziej losowe wyniki, generator inicjalizuje się za pomocą unik...

L Wartosci R Wartosci 🇵🇱

March 19, 2026
Category: Od C Do Cpp

W C++ bardzo dużo rzeczy kręci się wokół pytania: czy dane wyrażenie wskazuje na „konkretny obiekt w pamięci”, czy jest tylko tymczasowym wynikiem obliczeń. Z tego biorą się L-wartości (lvalues) i R-wartości (rvalues). Zrozumienie tego tematu odblokowuje m.in....

Typ Wyliczeniowy 🇵🇱

March 19, 2026
Category: Od C Do Cpp

Typ wyliczeniowy (enum) pozwala opisać zamknięty zbiór możliwych wartości pod czytelnymi nazwami. Zamiast “magicznych liczb” (np. 0,1,2) używasz sensownych identyfikatorów (Poniedzialek, Wtorek), co poprawia czytelność i zmniejsza liczbę błędów...

Filters and Algorithms 🇺🇸

March 19, 2026
Category: Vtk Examples

VTK’s filters and algorithms allow you to convert your data from “a static dataset” to a dynamic pipeline: you generate something, clean it up, extract meaning, and reshape it into a form that’s easier to analyze or visualize. Think of it like a workshop line: raw material comes in, tools operate on...

Data Types and Structures 🇺🇸

March 19, 2026
Category: Vtk Examples

VTK is built to carry real-world 2D/3D data all the way from “numbers in memory” to “something you can see and reason about.” That means it needs data types that store values, but also store where those values live in space and how they connect. If you pick the right structure early, everything down...

Backward Difference 🇺🇸

March 19, 2026
Category: Numerical Methods

The backward difference method is a finite difference technique employed to approximate the derivatives of functions. Unlike the forward difference method, which uses information from points ahead of the target point, the backward difference method relies on function values from points preceding the...

Forward Difference 🇺🇸

March 19, 2026
Category: Numerical Methods

The forward difference method is a fundamental finite difference technique utilized for approximating the derivatives of functions. Unlike the central and backward difference methods, which use information from both sides or preceding points, respectively, the forward difference method relies solely...

Central Difference 🇺🇸

March 19, 2026
Category: Numerical Methods

The central‐difference method is a finite‐difference scheme for estimating derivatives that combines forward and backward differences via Taylor‐series expansions. By evaluating the function at points symmetrically placed around the target, it cancels out many of the lower‐order error terms, yieldin...

Messaging System Integration 🇺🇸

January 24, 2026
Category: Backend Engineers Guide

In modern distributed architectures, messaging systems form an essential backbone for decoupling services, handling asynchronous communication, and enabling more resilient data flows. They allow separate applications or microservices to interact by sending and receiving messages through well-defined...

Optimistic vs Pessimistic Locking 🇺🇸

January 24, 2026
Category: Backend Engineers Guide

Locking is about managing concurrent access to shared data. Engineers often make it sound harder than it is, but the core idea is simple: choose between optimistic or pessimistic approaches depending on how costly retries are...

Data Warehousing 🇺🇸

January 24, 2026
Category: Backend Engineers Guide

Data warehousing unifies large volumes of information from different sources into a centralized repository that supports analytics, reporting, and strategic decision-making. By collecting operational data, transforming it, and then loading it into one or more specialized databases, data warehouses a...

Replication 🇺🇸

January 24, 2026
Category: Backend Engineers Guide

Replication is a method of maintaining copies of data across multiple nodes in distributed systems, making it useful for improving availability, reducing latency, and distributing load. Below are detailed notes, organized in bullet points, each containing one highlighted word in the middle to emphas...

Types of Databases 🇺🇸

January 24, 2026
Category: Backend Engineers Guide

Databases store and organize data so that applications and users can retrieve, manage, and manipulate information efficiently. The choice of database often depends on data structure requirements, scale, performance expectations, and the nature of the workload. Over the years, numerous types of datab...

Distributed Database Systems 🇺🇸

January 24, 2026
Category: Databases Notes

A distributed database system (DDS) is a collection of logically interrelated databases distributed across multiple physical locations, connected by a network. The data in these systems might be replicated and/or partitioned among different sites, but the system should ideally appear to the user as ...

Graphs 🇺🇸

January 23, 2026
Category: Algorithms And Data Structures

In many areas of life, we come across systems where elements are deeply interconnected, whether through physical routes, digital networks, or abstract relationships. Graphs offer a flexible way to represent and make sense of these connections...

Mpi 🇺🇸

January 23, 2026
Category: Parallel And Concurrent Programming

The Message Passing Interface (MPI) is a standardized and portable message-passing system designed to function on a wide variety of parallel computing architectures. It provides a set of library routines that can be called from programming languages like C, C++, and Fortran to write parallel applica...

Searching 🇺🇸

January 05, 2026
Category: Algorithms And Data Structures

Searching is the task of finding whether a particular value exists in a collection and, if it does, where it lives (its index, pointer, node, or associated value). It shows up everywhere: checking if a username is taken, locating a record in a database, finding a file in an index, routing packets, o...

Custom Filters and Algorithms 🇺🇸

January 05, 2026
Category: Vtk Examples

Creating custom filters and algorithms opens up a world of possibilities for tailored data processing and visualization. By extending VTK's capabilities, specialized techniques can be introduced that meet the unique needs of scientific research, engineering, medical imaging, or data analysis...