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Knee Of A Curve
In mathematics, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is particularly used in optimization, where a knee point is the optimum point for some decision, for example when there is an increasing function and a trade-off between the benefit (vertical ''y'' axis) and the cost (horizontal ''x'' axis): the knee is where the benefit is no longer increasing rapidly, and is no longer worth the cost of further increases – a cutoff point of diminishing returns. In heuristic use, the term may be used informally, and a knee point identified visually, but in more formal use an explicit objective function is used, and depends on the particular optimization problem. A knee may also be defined purely geometrically, in terms of the curvature or the second derivative. Definitions The knee of a curve can be defined as a vertex of the graph. This corresp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Power Point Tracking
Maximum power point tracking (MPPT) or sometimes just power point tracking (PPT), is a technique used with variable power sources to maximize energy extraction as conditions vary. The technique is most commonly used with photovoltaic (PV) solar systems, but can also be used with wind turbines, optical power transmission and thermophotovoltaics. PV solar systems have varying relationships to inverter systems, external grids, battery banks, and other electrical loads. The central problem addressed by MPPT is that the efficiency of power transfer from the solar cell depends on the amount of available sunlight, shading, solar panel temperature and the load's electrical characteristics. As these conditions vary, the load characteristic ( impedance) that gives the highest power transfer changes. The system is optimized when the load characteristic changes to keep power transfer at highest efficiency. This optimal load characteristic is called the ''maximum power point'' (MPP). MPPT is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elbow Method (clustering)
In cluster analysis, the elbow method is a heuristic used in determining the number of clusters in a data set. The method consists of plotting the explained variation as a function of the number of clusters and picking the elbow of the curve as the number of clusters to use. The same method can be used to choose the number of parameters in other data-driven models, such as the number of principal components to describe a data set. The method can be traced to speculation by Robert L. Thorndike in 1953. Intuition Using the "elbow" or "knee of a curve" as a cutoff point is a common heuristic in mathematical optimization to choose a point where diminishing returns are no longer worth the additional cost. In clustering, this means one should choose a number of clusters so that adding another cluster doesn't give much better modeling of the data. The intuition is that increasing the number of clusters will naturally improve the fit (explain more of the variation), since there are mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Webster's Dictionary
''Webster's Dictionary'' is any of the English language dictionaries edited in the early 19th century by American lexicographer Noah Webster (1758–1843), as well as numerous related or unrelated dictionaries that have adopted the Webster's name in honor. "''Webster's''" has since become a genericized trademark in the United States for English dictionaries, and is widely used in dictionary titles. Merriam-Webster is the corporate heir to Noah Webster's original works, which are in the public domain. Noah Webster's ''American Dictionary of the English Language'' Noah Webster (1758–1843), the author of the readers and spelling books which dominated the American market at the time, spent decades of research in compiling his dictionaries. His first dictionary, s:A Compendious Dictionary of the English Language, ''A Compendious Dictionary of the English Language'', appeared in 1806. In it, he popularized features which would become a hallmark of American English spelling (''c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Advisory Committee For Aeronautics
The National Advisory Committee for Aeronautics (NACA) was a United States federal agency founded on March 3, 1915, to undertake, promote, and institutionalize aeronautical research. On October 1, 1958, the agency was dissolved and its assets and personnel were transferred to the newly created National Aeronautics and Space Administration (NASA). NACA is an initialism, i.e., pronounced as individual letters, rather than as a whole word (as was NASA during the early years after being established). Among other advancements, NACA research and development produced the NACA duct, a type of air intake used in modern automotive applications, the NACA cowling, and several series of NACA airfoils, which are still used in aircraft manufacturing. During World War II, NACA was described as "The Force Behind Our Air Supremacy" due to its key role in producing working superchargers for high altitude bombers, and for producing the laminar wing profiles for the North American P-51 Mustang. NA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (curve)
In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature, and some authors define a vertex to be more specifically a local extremum of curvature. However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For space curves, on the other hand, a vertex is a point where the torsion vanishes. Examples A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry and in a quadratic of the form: :ax^2 + bx + c\,\! it can be found by completing the square or by differentiation., p. 127. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis. For a circle, which has constant curvature, every point is a verte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Derivative
In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: :\mathbf = \frac = \frac, where ''a'' is acceleration, ''v'' is velocity, ''t'' is time, ''x'' is position, and d is the instantaneous "delta" or change. The last expression \tfrac is the second derivative of position (x) with respect to time. On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Objective Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, short-term goal or approximation. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples that employ heuristics include using trial and error, a rule of thumb or an educated guess. Heuristics are the strategies derived from previous experiences with similar problems. These strategies depend on using readily accessible, though loosely applicable, information to control problem solving in human beings, machines and abstract issues. When an individual applies a heuristic in practice, it generally performs as expected. However it can alternatively cre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diminishing Returns
In economics, diminishing returns are the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal ( ceteris paribus). The law of diminishing returns (also known as the law of diminishing marginal productivity) states that in productive processes, increasing a factor of production by one unit, while holding all other production factors constant, will at some point return a lower unit of output per incremental unit of input. The law of diminishing returns does not cause a decrease in overall production capabilities, rather it defines a point on a production curve whereby producing an additional unit of output will result in a loss and is known as negative returns. Under diminishing returns, output remains positive, however productivity and efficiency decrease. The modern understanding of the law adds the dimension of holding other outputs equal, since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
