- Weight function
A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more "weight" or influence on the result than other elements in the same set. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" and "meta-calculus".
In the discrete setting, a weight function is a positive function defined on a discrete set A, which is typically finite or countable. The weight function w(a): = 1 corresponds to the unweighted situation in which all elements have equal weight. One can then apply this weight to various concepts.
but given a weight function , the weighted sum is defined as
One common application of weighted sums arises in numerical integration.
by the weighted mean or weighted average
In this case only the relative weights are relevant.
Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity f measured multiple independent times fi with variance , the best estimate of the signal is obtained by averaging all the measurements with weight , and the resulting variance is smaller than each of the independent measurements . The Maximum likelihood method weights the difference between fit and data using the same weights wi .
The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations :, then the lever will be in balance if the fulcrum of the lever is at the center of mass
which is also the weighted average of the positions .
In the continuous setting, a weight is a positive measure such as w(x)dx on some domain Ω,which is typically a subset of a Euclidean space , for instance Ω could be an interval [a,b]. Here dx is Lebesgue measure and is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.
can be generalized to the weighted integral
Note that one may need to require f to be absolutely integrable with respect to the weight w(x)dx in order for this integral to be finite.
If E is a subset of Ω, then the volume vol(E) of E can be generalized to the weighted volume
If Ω has finite non-zero weighted volume, then we can replace the unweighted average
by the weighted average
If and are two functions, one can generalize the unweighted inner product
to a weighted inner product
See the entry on Orthogonality for more details.
- Center of mass
- numerical integration
- Weighted average
- Weighted mean
- Kernel (statistics)
Wikimedia Foundation. 2010.
Look at other dictionaries:
Weight function — Источник: ГОСТ 21878 76: Случайные процессы и динамические системы. Термины и определения … Словарь-справочник терминов нормативно-технической документации
weight function — svorinė funkcija statusas T sritis fizika atitikmenys: angl. weight function; weighting function vok. Belegungsfunktion, f; Gewichtsfunktion, f rus. весовая функция, f pranc. fonction de pondération, f … Fizikos terminų žodynas
Weight (disambiguation) — Weight is a measurement of the gravitational force acting on an object, or of an object s mass.Weight may also refer to:* Weight (representation theory), a type of function. * Weight (strings), the number of times a letter occurs in a string. *… … Wikipedia
Weight for Age — (WFA) is a term in thoroughbred horse racing which is one of the conditions for a race. It means that a horse will carry a set weight in accordance with the Weight for Age Scale. This weight varies depending on the horse’s age, its sex, the race… … Wikipedia
Function problem — In computational complexity theory, a function problem is a problem other than a decision problem, that is, a problem requiring a more complex answer than just YES or NO.Notable examples include the travelling salesman problem, which asks for the … Wikipedia
Weight Watchers — Infobox Company company name = Weight Watchers International, Inc. company company type = Public company slogan = “Stop Dieting. Start Living.” foundation = 1963 location = 11 Madison Avenue New York, New York 10010 key people = David P.… … Wikipedia
Function Drinks — Infobox Company company name = Function Drinks company type = Privately Owned foundation = 2004 founder = Alex Hughes, Dayton Miller, Josh Simon, Will Harbin cite web |url = http://www.beverageworld.com/content/view/34131/ |title = Beverage World … Wikipedia
weighting function — svorinė funkcija statusas T sritis fizika atitikmenys: angl. weight function; weighting function vok. Belegungsfunktion, f; Gewichtsfunktion, f rus. весовая функция, f pranc. fonction de pondération, f … Fizikos terminų žodynas
Renal function — Diagram showing the basic physiologic mechanisms of the kidney Renal function, in nephrology, is an indication of the state of the kidney and its role in renal physiology. Glomerular filtration rate (GFR) describes the flow rate of filtered fluid … Wikipedia
Cardinal function — In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. Contents 1 Cardinal functions in set theory 2 Cardinal functions in topology 2.1 Basic inequalities … Wikipedia