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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
Nonlinear functional analysis is a branch of mathematical analysis that deals with nonlinear mappings.
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Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
Nonlinear functional analysis is a branch of mathematical analysis that deals with nonlinear mappings.
In time series modeling, a nonlinear autoregressive exogenous model (NARX) is a nonlinear autoregressive model which has exogenous inputs. This means that the model relates the current value of a time series to both: past values of the same series; and current and past values of the driving (exogenous) series — that is, of the externally determined series that influences the series of interest. In addition, the model contains an error term which relates to the fact that knowledge of other terms will not enable the current value of the time series to be predicted exactly. Such a model can be stated algebraically as y t = F ( y t − 1 , y t − 2 , y t − 3 , … , u t , u t − 1 , u t − 2 , u t − 3 , … ) + ε t {\displaystyle y_{t}=F(y_{t-1},y_{t-2},y_{t-3},\ldots ,u_{t},u_{t-1},u_{t-2},u_{t-3},\ldots )+\varepsilon _{t}} Here y is the variable of interest, and u is the externally determined variable. In this scheme, information about u helps predict y, as do previous values of y itself. Here ε is the error term (sometimes called noise). For example, y may be air temperature at noon, and u may be the day of the year (day-number within year). The function F is some nonlinear function, such as a polynomial. F can be a neural network, a wavelet network, a sigmoid network and so on. To test for non-linearity in a time series, the BDS test (Brock-Dechert-Scheinkman test) developed for econometrics can be used.
In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function f ( x ) {\displaystyle \displaystyle f(x)} f ( x ) = ‖ A x − b ‖ 2 , {\displaystyle \displaystyle f(x)=\|Ax-b\|^{2},} the minimum of f {\displaystyle f} is obtained when the gradient is 0: ∇ x f = 2 A T ( A x − b ) = 0 {\displaystyle \nabla _{x}f=2A^{T}(Ax-b)=0} . Whereas linear conjugate gradient seeks a solution to the linear equation A T A x = A T b {\displaystyle \displaystyle A^{T}Ax=A^{T}b} , the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient ∇ x f {\displaystyle \nabla _{x}f} alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum and the second derivative is non-singular there. Given a function f ( x ) {\displaystyle \displaystyle f(x)} of N {\displaystyle N} variables to minimize, its gradient ∇ x f {\displaystyle \nabla _{x}f} indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction: Δ x 0 = − ∇ x f ( x 0 ) {\displaystyle \Delta x_{0}=-\nabla _{x}f(x_{0})} with an adjustable step length α {\displaystyle \displaystyle \alpha } and performs a line search in this direction until it reaches the minimum of f {\displaystyle \displaystyle f} : α 0 := arg min α f ( x 0 + α Δ x 0 ) {\displaystyle \displaystyle \alpha _{0}:=\arg \min _{\alpha }f(x_{0}+\alpha \Delta x_{0})} , x 1 = x 0 + α 0 Δ x 0 {\displaystyle \displaystyle x_{1}=x_{0}+\alpha _{0}\Delta x_{0}} After this first iteration in the steepest direction Δ x 0 {\displaystyle \displaystyle \Delta x_{0}} , the following steps constitute one iteration of moving along a subsequent conjugate direction s n {\displaystyle \displaystyle s_{n}} , where s 0 = Δ x 0 {\displaystyle \displaystyle s_{0}=\Delta x_{0}} : Calculate the steepest direction: Δ x n = − ∇ x f ( x n ) {\displaystyle \Delta x_{n}=-\nabla _{x}f(x_{n})} , Compute β n {\displaystyle \displaystyle \beta _{n}} according to one of the formulas below, Update the conjugate direction: s n = Δ x n + β n s n − 1 {\displaystyle \displaystyle s_{n}=\Delta x_{n}+\beta _{n}s_{n-1}} Perform a line search: optimize α n = arg min α f ( x n + α s n ) {\displaystyle \displaystyle \alpha _{n}=\arg \min _{\alpha }f(x_{n}+\alpha s_{n})} , Update the position: x n + 1 = x n + α n s n {\displaystyle \displaystyle x_{n+1}=x_{n}+\alpha _{n}s_{n}} , With a pure quadratic function the minimum is reached within N iterations (excepting roundoff error), but a non-quadratic function will make slower progress. Subsequent search directions lose conjugacy requiring the search direction to be reset to the steepest descent direction at least every N iterations, or sooner if progress stops. However, resetting every iteration turns the method into steepest descent. The algorithm stops when it finds the minimum, determined when no progress is made after a direction reset (i.e. in the steepest descent direction), or when some tolerance criterion is reached. Within a linear approximation, the parameters α {\displaystyle \displaystyle \alpha } and β {\displaystyle \displaystyle \beta } are the same as in the linear conjugate gradient method but have been obtained with line searches. The conjugate gradient method can follow narrow (ill-conditioned) valleys, where the steepest descent method slows down and follows a criss-cross pattern. Four of the best known formulas for β n {\displaystyle \displaystyle \beta _{n}} are named after their developers: Fletcher–Reeves: β n F R = Δ x n T Δ x n Δ x n − 1 T Δ x n − 1 . {\displaystyle \beta _{n}^{FR}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.} Polak–Ribière: β n P R = Δ x n T ( Δ x n − Δ x n − 1 ) Δ x n − 1 T Δ x n − 1 . {\displaystyle \beta _{n}^{PR}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{\Delta x_{n-1}^{T}\Delta x_{n-1}}}.} Hestenes–Stiefel: β n H S = Δ x n T ( Δ x n − Δ x n − 1 ) − s n − 1 T ( Δ x n − Δ x n − 1 ) . {\displaystyle \beta _{n}^{HS}={\frac {\Delta x_{n}^{T}(\Delta x_{n}-\Delta x_{n-1})}{-s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.} Dai–Yuan: β n D Y = Δ x n T Δ x n − s n − 1 T ( Δ x n − Δ x n − 1 ) . {\displaystyle \beta _{n}^{DY}={\frac {\Delta x_{n}^{T}\Delta x_{n}}{-s_{n-1}^{T}(\Delta x_{n}-\Delta x_{n-1})}}.} . These formulas are equivalent for a quadratic function, but for nonlinear optimization the preferred formula is a matter of heuristics or taste. A popular choice is β = max { 0 , β P R } {\displaystyle \displaystyle \beta =\max\{0,\beta ^{PR}\}} , which provides a direction reset automatically. Algorithms based on Newton's method potentially converge much faster. There, both step direction and length are computed from the gradient as the solution of a linear system of equations, with the coefficient matrix being the exact Hessian matrix (for Newton's method proper) or an estimate thereof (in the quasi-Newton methods, where the observed change in the gradient during the iterations is used to update the Hessian estimate). For high-dimensional problems, the exact computation of the Hessian is usually prohibitively expensive, and even its storage can be problematic, requiring O ( N 2 ) {\displaystyle O(N^{2})} memory (but see the limited-memory L-BFGS quasi-Newton method). The conjugate gradient method can also be derived using optimal control theory. In this accelerated optimization theory, the conjugate gradient method falls out as a nonlinear optimal feedback controller, u = k ( x , x ˙ ) := − γ a ∇ x f ( x ) − γ b x ˙ {\displaystyle u=k(x,{\dot {x}}):=-\gamma _{a}\nabla _{x}f(x)-\gamma _{b}{\dot {x}}} for the double integrator system, x ¨ = u {\displaystyle {\ddot {x}}=u} The quantities γ a > 0 {\displaystyle \gamma _{a}>0} and γ b > 0 {\displaystyle \gamma _{b}>0} are variable feedback gains.
In statistics, correlation is a type of statistical relationship between two random variables or bivariate data. It usually refers to the extent to which a pair of quantities are linearly related. More generally, an arbitrary relationship between variables is called an association, meaning the degree to which the variability in one can be accounted for by the other. The presence of a correlation is not sufficient to infer the presence of a causal relationship, and this is often stated as "correlation does not imply causation". Furthermore, the concept of correlation is not the same as dependence: if two variables are independent, then they are uncorrelated, but the opposite is not necessarily true – even if two variables are uncorrelated, they might be dependent on each other. Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship because extreme weather causes people to use more electricity for heating or cooling. There are several correlation coefficients that may be used to measure correlation, often denoted ρ {\displaystyle \rho } or r {\displaystyle r} . The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which in turn may be present even when one variable is a nonlinear function of the other). Other correlation coefficients, such as Spearman's rank correlation coefficient, have been developed to be more robust than Pearson's and to detect less structured relationships between variables. The concept has been generalized to other forms of association between two variables, such as mutual information and distance covariance.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Mae Jemison, aboard Space Shuttle Endeavour in 1992.
