Regression analysisIn statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion.
Multivariate adaptive regression splineIn statistics, multivariate adaptive regression splines (MARS) is a form of regression analysis introduced by Jerome H. Friedman in 1991. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models nonlinearities and interactions between variables. The term "MARS" is trademarked and licensed to Salford Systems. In order to avoid trademark infringements, many open-source implementations of MARS are called "Earth". This section introduces MARS using a few examples.
Linear regressionIn statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.
Multivariate statisticsMultivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable, i.e., multivariate random variables. Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied.
Segmented regressionSegmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable is partitioned into intervals and a separate line segment is fit to each interval. Segmented regression analysis can also be performed on multivariate data by partitioning the various independent variables. Segmented regression is useful when the independent variables, clustered into different groups, exhibit different relationships between the variables in these regions.
TunnelA tunnel is an underground passageway, dug through surrounding soil, earth or rock, and enclosed except for the entrance and exit, commonly at each end. A pipeline is not a tunnel, though some recent tunnels have used immersed tube construction techniques rather than traditional tunnel boring methods. A tunnel may be for foot or vehicular road traffic, for rail traffic, or for a canal. The central portions of a rapid transit network are usually in the tunnel.
Bayesian linear regressionBayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often labelled ) conditional on observed values of the regressors (usually ).
BasaltBasalt (UKˈbæsɔːlt,_-əlt; USbəˈsɔːlt,_ˈbeɪsɔːlt) is an aphanitic (fine-grained) extrusive igneous rock formed from the rapid cooling of low-viscosity lava rich in magnesium and iron (mafic lava) exposed at or very near the surface of a rocky planet or moon. More than 90% of all volcanic rock on Earth is basalt. Rapid-cooling, fine-grained basalt is chemically equivalent to slow-cooling, coarse-grained gabbro. The eruption of basalt lava is observed by geologists at about 20 volcanoes per year.
Nonparametric regressionNonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship between predictors and dependent variable. Nonparametric regression requires larger sample sizes than regression based on parametric models because the data must supply the model structure as well as the model estimates.
General linear modelThe general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as where Y is a matrix with series of multivariate measurements (each column being a set of measurements on one of the dependent variables), X is a matrix of observations on independent variables that might be a design matrix (each column being a set of observations on one of the independent variables), B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors (noise).
Paramount PicturesParamount Pictures Corporation is an American film and television production and distribution company and the namesake division of Paramount Global (formerly ViacomCBS). It is the fifth-oldest film studio in the world, the second-oldest film studio in the United States (behind Universal Pictures), and the sole member of the "Big Five" film studios located within the city limits of Los Angeles. In 1916, film producer Adolph Zukor put 24 actors and actresses under contract and honored each with a star on the logo.
Paramount GlobalParamount Global (doing business as Paramount) is an American multinational mass media and entertainment conglomerate controlled by National Amusements and headquartered at One Astor Plaza in Midtown Manhattan in New York City. The company was formed on December 4, 2019, as ViacomCBS Inc. through the merger of the second incarnations of CBS Corporation and Viacom (which were split from the original Viacom on December 31, 2005). The company changed its name to Paramount Global on February 16, 2022, the day after its Q4 earnings presentation.
Cauchy stress tensorIn continuum mechanics, the Cauchy stress tensor , true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector e to the traction vector T(e) across an imaginary surface perpendicular to e: or, The SI units of both stress tensor and traction vector are N/m2, corresponding to the stress scalar.
Phreatomagmatic eruptionPhreatomagmatic eruptions are volcanic eruptions resulting from interaction between magma and water. They differ from exclusively magmatic eruptions and phreatic eruptions. Unlike phreatic eruptions, the products of phreatomagmatic eruptions contain juvenile (magmatic) clasts. It is common for a large explosive eruption to have magmatic and phreatomagmatic components. Several competing theories exist as to the exact mechanism of ash formation. The most common is the theory of explosive thermal contraction of particles under rapid cooling from contact with water.
Stress (mechanics)In continuum mechanics, stress is a physical quantity that describes forces present during deformation. An object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation. An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has units of force per area, such as newtons per square meter (N/m2) or pascal (Pa).
Refractive indexIn optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or refracted, when entering a material. This is described by Snell's law of refraction, n1 sin θ1 = n2 sin θ2, where θ1 and θ2 are the angle of incidence and angle of refraction, respectively, of a ray crossing the interface between two media with refractive indices n1 and n2.
LavaLava is molten or partially molten rock (magma) that has been expelled from the interior of a terrestrial planet (such as Earth) or a moon onto its surface. Lava may be erupted at a volcano or through a fracture in the crust, on land or underwater, usually at temperatures from . The volcanic rock resulting from subsequent cooling is also often called lava. A lava flow is an outpouring of lava during an effusive eruption. (An explosive eruption, by contrast, produces a mixture of volcanic ash and other fragments called tephra, not lava flows.
Frequency domainIn mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids (or other basis waveforms) at the frequency components of the signal.
Simplon TunnelThe Simplon Tunnel (Simplontunnel, Traforo del Sempione or Galleria del Sempione) is a railway tunnel on the Simplon railway that connects Brig, Switzerland and Domodossola, Italy, through the Alps, providing a shortcut under the Simplon Pass route. It is straight except for short curves at either end. It consists of two single-track tunnels built nearly 15 years apart. The first to be opened is long; the second is long, making it the longest railway tunnel in the world for most of the twentieth century, from 1906 until 1982, when the Daishimizu Tunnel opened.
Stress–strain analysisStress–strain analysis (or stress analysis) is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. In simple terms we can define stress as the force of resistance per unit area, offered by a body against deformation.
