AskDefine | Define weighted

The Collaborative Dictionary

Weight \Weight\, v. t. [imp. & p. p. Weighted; p. pr. & vb. n. Weighting.] [1913 Webster]
To load with a weight or weights; to load down; to make heavy; to attach weights to; as, to weight a horse or a jockey at a race; to weight a whip handle. [1913 Webster] The arrows of satire, . . . weighted with sense. --Coleridge. [1913 Webster]
(Astron. & Physics) To assign a weight to; to express by a number the probable accuracy of, as an observation. See Weight of observations, under Weight. [1913 Webster]
(Dyeing) To load (fabrics) as with barite, to increase the weight, etc. [Webster 1913 Suppl.]
(Math.) to assign a numerical value expressing relative importance to (a measurement), to be multiplied by the value of the measurement in determining averages or other aggregate quantities; as, they weighted part one of the test twice as heavily as part
[PJC] [1913 Webster]

Word Net

weighted adj
1 made heavy or weighted down with weariness; "his leaden arms"; "weighted eyelids" [syn: leaden]
2 adjusted to reflect value or proportion; "votes weighted according to the size of constituencies"; "a law weighted in favor of landlords"; "a weighted average"

English

Pronunciation

  • /ˈweɪˌtɛd/

Homophones

Adjective

  1. Having weights on it
    She wore a weighted dress so it woldn't blow in the wind.
  2. biased, so as to favour one party.
    The competition was weighted so he'd be the clear favourite to win.

Translations

Verb

weighted
  1. past of weight
A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a "weight" than others. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be constructed in both discrete and continuous settings.

Discrete weights

General definition

In the discrete setting, a weight function \scriptstyle w: A \to ^+ 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.
If the function \scriptstyle f: A \to is a real-valued function, then the unweighted sum of f on A is defined as
\sum_ f(a);
but given a weight function \scriptstyle w: A \to ^+, the weighted sum is defined as
\sum_ f(a) w(a).
One common application of weighted sums arises in numerical integration.
If B is a finite subset of A, one can replace the unweighted cardinality |B| of B by the weighted cardinality
\sum_ w(a).
If A is a finite non-empty set, one can replace the unweighted mean or average
\frac \sum_ f(a)
\frac.
In this case only the relative weights are relevant.

Statistics

Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity f measured multiple independent times f_i with variance \scriptstyle\sigma^2_i, the best estimate of the signal is obtained by averaging all the measurements with weight \scriptstyle w_i=\frac 1 , and the resulting variance is smaller than each of the independent measurements \scriptstyle\sigma^2=1/\sum w_i. The Maximum likelihood method weights the difference between fit and data using the same weights w_i .

Mechanics

The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights \scriptstyle w_1, \ldots, w_n (where weight is now interpreted in the physical sense) and locations :\scriptstyle\boldsymbol_1,\ldots,\boldsymbol_n, then the lever will be in balance if the fulcrum of the lever is at the center of mass
\frac,
which is also the weighted average of the positions \scriptstyle\boldsymbol_i.

Continuous weights

In the continuous setting, a weight is a positive measure such as w(x) dx on some domain \Omega,which is typically a subset of an Euclidean space \scriptstyle^n, for instance \Omega could be an interval [a,b]. Here dx is Lebesgue measure and \scriptstyle w: \Omega \to \R^+ is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.

General definition

If f: \Omega \to is a real-valued function, then the unweighted integral
\int_\Omega f(x)\ dx
can be generalized to the weighted integral
\int_\Omega f(x) w(x)\, dx
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.

Weighted volume

If E is a subset of \Omega, then the volume vol(E) of E can be generalized to the weighted volume
\int_E w(x)\ dx.

Weighted average

If \Omega has finite non-zero weighted volume, then we can replace the unweighted average
\frac \int_\Omega f(x)\ dx
by the weighted average
\frac

Inner product

If \scriptstyle f: \Omega \to and \scriptstyle g: \Omega \to are two functions, one can generalize the unweighted inner product
\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx
to a weighted inner product
\langle f, g \rangle := \int_\Omega f(x) g(x)\ w(x) dx
See the entry on Orthogonality for more details.
weighted in German: Gewichtung
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