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The gradient of F is then normal to the hypersurface. Similarly, an affine algebraic hypersurface may be defined by an equation F(x 1, ..., x n) = 0, where F is a polynomial. The gradient of F is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the central limit theorem.
Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near ...
For normal goods, the Engel curve has a positive gradient. That is, as income increases, the quantity demanded increases. That is, as income increases, the quantity demanded increases. Amongst normal goods, there are two possibilities.
Numerous methods exist to compute descent directions, all with differing merits, such as gradient descent or the conjugate gradient method. More generally, if is a positive definite matrix, then = is a descent direction at . [1]
A selection gradient describes the relationship between a character trait and a species' relative fitness. [1] A trait may be a physical characteristic, such as height or eye color, or behavioral, such as flying or vocalizing. Changes in a trait, such as the amount of seeds a plant produces or the length of a bird's beak, may improve or reduce ...
where n is the number of observations and Σ is a (symmetric positive semi-definite) covariance matrix. Suppose we want to estimate the variance of a scalar-valued function h of the estimator B. Keeping only the first two terms of the Taylor series, and using vector notation for the gradient, we can estimate h(B) as