The cross-lagged panel model is a type of discrete time structural equation model used to analyze panel data in which two or more variables are repeatedly measured at two or more different time points. This model aims to estimate the directional effects that one variable has on another at different points in time. [1] [2] This model was first introduced in 1963 by Donald T. Campbell and refined during the 1970s by David A. Kenny. [3] Kenny has described it as follows: "Two variables, X and Y, are measured at two times, 1 and 2, resulting in four measures, X1, Y1, X2, and Y2. With these four measures, there are six possible relations among them – two synchronous or cross‐sectional relations (see cross‐sectional design) (between X1 and Y1 and between X2 and Y2), two stability relations (between X1 and X2 and between Y1 and Y2), and two cross‐lagged relations (between X1 and Y2 and between Y1 and X2)." [4] Though this approach is commonly believed to be a valid technique to identify causal relationships from panel data, its use for this purpose has been criticized, as it depends on certain assumptions, such as synchronicity and stationarity, that may not be valid. [5] [6] [7]
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns.
In computer graphics, a line drawing algorithm is an algorithm for approximating a line segment on discrete graphical media, such as pixel-based displays and printers. On such media, line drawing requires an approximation. Basic algorithms rasterize lines in one color. A better representation with multiple color gradations requires an advanced process, spatial anti-aliasing.
In computer graphics, the Liang–Barsky algorithm is a line clipping algorithm. The Liang–Barsky algorithm uses the parametric equation of a line and inequalities describing the range of the clipping window to determine the intersections between the line and the clip window. With these intersections it knows which portion of the line should be drawn. So this algorithm is significantly more efficient than Cohen–Sutherland. The idea of the Liang–Barsky clipping algorithm is to do as much testing as possible before computing line intersections.
In statistics, canonical analysis (from Ancient Greek: κανων bar, measuring rod, ruler) belongs to the family of regression methods for data analysis. Regression analysis quantifies a relationship between a predictor variable and a criterion variable by the coefficient of correlation r, coefficient of determination r2, and the standard regression coefficient β. Multiple regression analysis expresses a relationship between a set of predictor variables and a single criterion variable by the multiple correlation R, multiple coefficient of determination R2, and a set of standard partial regression weights β1, β2, etc. Canonical variate analysis captures a relationship between a set of predictor variables and a set of criterion variables by the canonical correlations ρ1, ρ2, ..., and by the sets of canonical weights C and D.
Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregressive model by allowing for multivariate time series. VAR models are often used in economics and the natural sciences.
In geometry, the Hessian curve is a plane curve similar to folium of Descartes. It is named after the German mathematician Otto Hesse. This curve was suggested for application in elliptic curve cryptography, because arithmetic in this curve representation is faster and needs less memory than arithmetic in standard Weierstrass form.
As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method (FEM). In applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of these elements are then, through matrix mathematics, compiled into a single matrix equation which governs the behaviour of the entire idealized structure. The structure’s unknown displacements and forces can then be determined by solving this equation. The direct stiffness method forms the basis for most commercial and free source finite element software.
In thermodynamics and chemical engineering, the vapor–liquid equilibrium (VLE) describes the distribution of a chemical species between the vapor phase and a liquid phase.
In statistics, confirmatory factor analysis (CFA) is a special form of factor analysis, most commonly used in social science research. It is used to test whether measures of a construct are consistent with a researcher's understanding of the nature of that construct. As such, the objective of confirmatory factor analysis is to test whether the data fit a hypothesized measurement model. This hypothesized model is based on theory and/or previous analytic research. CFA was first developed by Jöreskog (1969) and has built upon and replaced older methods of analyzing construct validity such as the MTMM Matrix as described in Campbell & Fiske (1959).
In computer graphics, a digital differential analyzer (DDA) is hardware or software used for interpolation of variables over an interval between start and end point. DDAs are used for rasterization of lines, triangles and polygons. They can be extended to non linear functions, such as perspective correct texture mapping, quadratic curves, and traversing voxels.
In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials.
In game theory, a strong Nash equilibrium(SNE) is a combination of actions of the different players, in which no coalition of players can cooperatively deviate in a way that strictly benefits all of its members, given that the actions of the other players remain fixed. This is in contrast to simple Nash equilibrium, which considers only deviations by individual players. The concept was introduced by Israel Aumann in 1959. SNE is particularly useful in areas such as the study of voting systems, in which there are typically many more players than possible outcomes, and so plain Nash equilibria are far too abundant.
In model theory, interpretation of a structure M in another structure N is a technical notion that approximates the idea of representing M inside N. For example, every reduct or definitional expansion of a structure N has an interpretation in N.
In mathematics, Cayley's Ω process, introduced by Arthur Cayley, is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action.
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices.
A Rubinstein bargaining model refers to a class of bargaining games that feature alternating offers through an infinite time horizon. The original proof is due to Ariel Rubinstein in a 1982 paper. For a long time, the solution to this type of game was a mystery; thus, Rubinstein's solution is one of the most influential findings in game theory.
A Brownian surface is a fractal surface generated via a fractal elevation function.
In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a limiting operation on a parameter of the Lie algebra, altering the structure constants of this Lie algebra in a nontrivial singular manner, under suitable circumstances.
Ellen Louise "E.L." Hamaker is a Dutch-American psychologist, and statistician. Since 2018 she has been a full professor at Utrecht University, holding the chair Longitudinal Data Analysis at the Department of Methodology and Statistics. Her work focuses on the development of statistical models for the analysis of intensive longitudinal data in psychology, mainly within the frameworks of structural equation modeling and time series analysis.
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