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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox.

Contents

1 Examples

1.1 Basic example 1.2 Convex combination
Convex combination
example

2 Mathematical definition 3 Statistical properties 4 Dealing with variance

4.1 Correcting for over- or under-dispersion

5 Weighted sample variance

5.1 Frequency weights 5.2 Reliability weights

6 Weighted sample covariance

6.1 Frequency weights 6.2 Reliability weights

7 Vector-valued estimates 8 Accounting for correlations 9 Decreasing strength of interactions 10 Exponentially decreasing weights 11 Weighted averages of functions 12 See also 13 References 14 Further reading 15 External links

Examples[edit] Basic example[edit] Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:

Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98

Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99

The straight average for the morning class is 80 and the straight average of the afternoon class is 90. The straight average of 80 and 90 is 85, the mean of the two class means. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students):

x ¯

=

4300 50

= 86.

displaystyle bar x = frac 4300 50 =86.

Or, this can be accomplished by weighting the class means by the number of students in each class (using a weighted mean of the class means):

x ¯

=

( 20 × 80 ) + ( 30 × 90 )

20 + 30

= 86.

displaystyle bar x = frac (20times 80)+(30times 90) 20+30 =86.

Thus, the weighted mean makes it possible to find the average student grade in the case where only the class means and the number of students in each class are available. Convex combination
Convex combination
example[edit] Since only the relative weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such a linear combination is called a convex combination. Using the previous example, we would get the following weights:

20

20 + 30

= 0.4

displaystyle frac 20 20+30 =0.4

30

20 + 30

= 0.6

displaystyle frac 30 20+30 =0.6

Then, apply the weights like this:

x ¯

= ( 0.4 × 80 ) + ( 0.6 × 90 ) = 86.

displaystyle bar x =(0.4times 80)+(0.6times 90)=86.

Mathematical definition[edit] Formally, the weighted mean of a non-empty set of data

x

1

,

x

2

, … ,

x

n

,

displaystyle x_ 1 ,x_ 2 ,dots ,x_ n ,

(where x represents a set of mean values) with non-negative weights

x ¯

=

i = 1

n

w

i

x

i

i = 1

n

w

i

,

displaystyle bar x = frac sum limits _ i=1 ^ n w_ i x_ i sum limits _ i=1 ^ n w_ i ,

which means:

x ¯

=

w

1

x

1

+

w

2

x

2

+ ⋯ +

w

n

x

n

w

1

+

w

2

+ ⋯ +

w

n

.

displaystyle bar x = frac w_ 1 x_ 1 +w_ 2 x_ 2 +cdots +w_ n x_ n w_ 1 +w_ 2 +cdots +w_ n .

Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed). The formulas are simplified when the weights are normalized such that they sum up to

1

displaystyle 1

, i.e.

i = 1

n

w

i

= 1

displaystyle sum _ i=1 ^ n w_ i ' =1

. For such normalized weights the weighted mean is simply

x ¯

=

i = 1

n

w

i

x

i

displaystyle bar x =sum _ i=1 ^ n w_ i 'x_ i

. Note that one can always normalize the weights by making the following transformation on the original weights

w

i

=

w

i

j = 1

n

w

j

displaystyle w_ i '= frac w_ i sum _ j=1 ^ n w_ j

. Using the normalized weight yields the same results as when using the original weights. Indeed,

x ¯

=

i = 1

n

w

i

x

i

=

i = 1

n

w

i

j = 1

n

w

j

x

i

=

i = 1

n

w

i

x

i

j = 1

n

w

j

=

i = 1

n

w

i

x

i

i = 1

n

w

i

.

displaystyle begin aligned bar x &=sum _ i=1 ^ n w'_ i x_ i =sum _ i=1 ^ n frac w_ i sum _ j=1 ^ n w_ j x_ i = frac sum _ i=1 ^ n w_ i x_ i sum _ j=1 ^ n w_ j \&= frac sum _ i=1 ^ n w_ i x_ i sum _ i=1 ^ n w_ i .end aligned

The common mean

1 n

i = 1

n

x

i

displaystyle frac 1 n sum _ i=1 ^ n x_ i

is a special case of the weighted mean where all data have equal weights,

w

i

= w

displaystyle w_ i =w

. When the weights are normalized then

w

i

=

1 n

.

displaystyle w_ i '= frac 1 n .

Statistical properties[edit] The weighted sample mean,

X ¯

displaystyle bar X

, with normalized weights (weights summing to one) is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations as follows, If the observations have expected values

E (

X

i

) =

μ

i

¯

,

displaystyle E(X_ i )= bar mu _ i ,

then the weighted sample mean has expectation

E (

X ¯

) =

i = 1

n

w

i

μ

i

.

displaystyle E( bar X )=sum _ i=1 ^ n w_ i mu _ i .

In particular, if the means are equal,

μ

i

= μ

displaystyle mu _ i =mu

, then the expectation of the weighted sample mean will be that value,

E (

X ¯

) = μ .

displaystyle E( bar X )=mu .

For uncorrelated observations with variances

σ

i

2

displaystyle sigma _ i ^ 2

, the variance of the weighted sample mean is

σ

X ¯

2

=

i = 1

n

w

i

2

σ

i

2

displaystyle sigma _ bar X ^ 2 =sum _ i=1 ^ n w_ i ^ 2 sigma _ i ^ 2

whose square root

σ

X ¯

displaystyle sigma _ bar X

can be called the standard error of the weighted mean. Consequently, if all the observations have equal variance,

σ

i

2

=

σ

0

2

displaystyle sigma _ i ^ 2 =sigma _ 0 ^ 2

, the weighted sample mean will have variance

σ

X ¯

2

=

σ

0

2

i = 1

n

w

i

2

,

displaystyle sigma _ bar X ^ 2 =sigma _ 0 ^ 2 sum _ i=1 ^ n w_ i ^ 2 ,

where

1

/

n ≤

i = 1

n

w

i

2

≤ 1

displaystyle 1/nleq sum _ i=1 ^ n w_ i ^ 2 leq 1

. The variance attains its maximum value,

σ

0

2

displaystyle sigma _ 0 ^ 2

, when all weights except one are zero. Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have

σ

X ¯

=

σ

0

/

n

displaystyle sigma _ bar X =sigma _ 0 / sqrt n

, i.e., it degenerates into the standard error of the mean, squared. Note that because one can always transform non-normalized weights to normalized weights all formula in this section can be adapted to non-normalized weights by replacing all

w

i

displaystyle w_ i

by

w

i

=

w

i

i = 1

n

w

i

displaystyle w_ i '= frac w_ i sum _ i=1 ^ n w_ i

. Dealing with variance[edit] See also: Least squares § Weighted least squares, and Linear least squares (mathematics) § Weighted linear least squares For the weighted mean of a list of data for which each element

x

i

displaystyle x_ i

potentially comes from a different probability distribution with known variance

σ

i

2

displaystyle sigma _ i ^ 2

, one possible choice for the weights is given by:

w

i

=

1

σ

i

2

.

displaystyle w_ i = frac 1 sigma _ i ^ 2 .

The weighted mean in this case is:

x ¯

=

i = 1

n

(

x

i

σ

i

− 2

)

i = 1

n

σ

i

− 2

,

displaystyle bar x = frac sum _ i=1 ^ n left(x_ i sigma _ i ^ -2 right) sum _ i=1 ^ n sigma _ i ^ -2 ,

and the uncertainty on the weighted mean is:

σ

x ¯

=

1

i = 1

n

σ

i

− 2

,

displaystyle sigma _ bar x = sqrt frac 1 sum _ i=1 ^ n sigma _ i ^ -2 ,

which reduces to

σ

x ¯

2

=

σ

0

2

/

n

displaystyle sigma _ bar x ^ 2 =sigma _ 0 ^ 2 /n

when all

σ

i

=

σ

0

displaystyle sigma _ i =sigma _ 0

. The two equations above can be combined to obtain:

x ¯

=

σ

x ¯

2

i = 1

n

x

i

/

σ

i

2

.

displaystyle bar x =sigma _ bar x ^ 2 sum _ i=1 ^ n x_ i /sigma _ i ^ 2 .

The significance of this choice is that this weighted mean is the maximum likelihood estimator of the mean of the probability distributions under the assumption that they are independent and normally distributed with the same mean. Correcting for over- or under-dispersion[edit] Further information: Weighted sample variance Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that

χ

2

displaystyle chi ^ 2

is too large. The correction that must be made is

σ ^

x ¯

2

=

σ

x ¯

2

χ

ν

2

displaystyle hat sigma _ bar x ^ 2 =sigma _ bar x ^ 2 chi _ nu ^ 2

where

χ

ν

2

displaystyle chi _ nu ^ 2

is the reduced chi-squared. This gives the scaled variance in the weighted mean as:

σ ^

x ¯

2

=

1

i = 1

n

σ

i

− 2

×

1

( n − 1 )

i = 1

n

(

x

i

x ¯

)

2

σ

i

2

;

displaystyle hat sigma _ bar x ^ 2 = frac 1 sum _ i=1 ^ n sigma _ i ^ -2 times frac 1 (n-1) sum _ i=1 ^ n frac (x_ i - bar x )^ 2 sigma _ i ^ 2 ;

when all data variances are equal,

σ

i

=

σ

0

displaystyle sigma _ i =sigma _ 0

, they cancel out in the weighted mean variance,

σ

x ¯

2

displaystyle sigma _ bar x ^ 2

, which then reduces to the standard error of the mean (squared),

σ

x ¯

2

=

σ

2

/

n

displaystyle sigma _ bar x ^ 2 =sigma ^ 2 /n

, in terms of the sample standard deviation (squared),

σ

2

=

i = 1

n

(

x

i

x ¯

)

2

n − 1

.

displaystyle sigma ^ 2 = frac sum _ i=1 ^ n (x_ i - bar x )^ 2 n-1 .

Weighted sample variance[edit] See also: § Correcting for over- or under-dispersion Typically when a mean is calculated it is important to know the variance and standard deviation about that mean. When a weighted mean

μ

displaystyle mu ^ *

is used, the variance of the weighted sample is different from the variance of the unweighted sample. The biased weighted sample variance

σ ^

w

2

displaystyle hat sigma _ mathrm w ^ 2

is defined similarly to the normal biased sample variance

σ ^

2

displaystyle hat sigma ^ 2

:

σ ^

2

 

=

i = 1

N

(

x

i

− μ

)

2

N

σ ^

w

2

=

i = 1

N

w

i

(

x

i

μ

)

2

V

1

displaystyle begin aligned hat sigma ^ 2 &= frac sum _ i=1 ^ N left(x_ i -mu right)^ 2 N \ hat sigma _ mathrm w ^ 2 &= frac sum _ i=1 ^ N w_ i left(x_ i -mu ^ * right)^ 2 V_ 1 end aligned

where

V

1

=

i = 1

N

w

i

displaystyle V_ 1 =sum _ i=1 ^ N w_ i

, which is 1 for normalized weights. If the weights are frequency weights (and thus are random variables), it can be shown that

σ ^

w

2

displaystyle hat sigma _ mathrm w ^ 2

is the maximum likelihood estimator of

σ

2

displaystyle sigma ^ 2

for iid Gaussian observations. For small samples, it is customary to use an unbiased estimator for the population variance. In normal unweighted samples, the N in the denominator (corresponding to the sample size) is changed to N − 1 (see Bessel's correction). In the weighted setting, there are actually two different unbiased estimators, one for the case of frequency weights and another for the case of reliability weights. Frequency weights[edit] If the weights are frequency weights, then the unbiased estimator is:

s

2

 

=

i = 1

N

w

i

(

x

i

μ

)

2

V

1

− 1

displaystyle begin aligned s^ 2 &= frac sum _ i=1 ^ N w_ i left(x_ i -mu ^ * right)^ 2 V_ 1 -1 end aligned

This effectively applies Bessel's correction for frequency weights. For example, if values

2 , 2 , 4 , 5 , 5 , 5

displaystyle 2,2,4,5,5,5

are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample

2 , 4 , 5

displaystyle 2,4,5

with corresponding weights

2 , 1 , 3

displaystyle 2,1,3

, and we get the same result either way. Reliability weights[edit] If the weights are instead non-random (reliability weights), we can determine a correction factor to yield an unbiased estimator. Taking expectations we have,

E ⁡ [

σ ^

2

]

=

i = 1

N

E ⁡ [ (

x

i

− μ

)

2

]

N

= E ⁡ [ ( X − E ⁡ [ X ]

)

2

] −

1 N

E ⁡ [ ( X − E ⁡ [ X ]

)

2

]

=

(

N − 1

N

)

σ

actual

2

E ⁡ [

σ ^

w

2

]

=

i = 1

N

w

i

E ⁡ [ (

x

i

μ

)

2

]

V

1

= E ⁡ [ ( X − E ⁡ [ X ]

)

2

] −

V

2

V

1

2

E ⁡ [ ( X − E ⁡ [ X ]

)

2

]

=

(

1 −

V

2

V

1

2

)

σ

actual

2

displaystyle begin aligned operatorname E [ hat sigma ^ 2 ]&= frac sum _ i=1 ^ N operatorname E [(x_ i -mu )^ 2 ] N \&=operatorname E [(X-operatorname E [X])^ 2 ]- frac 1 N operatorname E [(X-operatorname E [X])^ 2 ]\&=left( frac N-1 N right)sigma _ text actual ^ 2 \operatorname E [ hat sigma _ mathrm w ^ 2 ]&= frac sum _ i=1 ^ N w_ i operatorname E [(x_ i -mu ^ * )^ 2 ] V_ 1 \&=operatorname E [(X-operatorname E [X])^ 2 ]- frac V_ 2 V_ 1 ^ 2 operatorname E [(X-operatorname E [X])^ 2 ]\&=left(1- frac V_ 2 V_ 1 ^ 2 right)sigma _ text actual ^ 2 end aligned

where

V

2

=

i = 1

N

w

i

2

displaystyle V_ 2 =sum _ i=1 ^ N w_ i ^ 2

. Therefore, the bias in our estimator is

(

1 −

V

2

V

1

2

)

displaystyle left(1- frac V_ 2 V_ 1 ^ 2 right)

, analogous to the

(

N − 1

N

)

displaystyle left( frac N-1 N right)

bias in the unweighted estimator. This means that to unbias our estimator we need to pre-divide by

1 −

(

V

2

/

V

1

2

)

displaystyle 1-left(V_ 2 /V_ 1 ^ 2 right)

, ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution. The final unbiased estimate of sample variance is:

s

2

 

=

σ ^

w

2

1 − (

V

2

/

V

1

2

)

=

i = 1

N

w

i

(

x

i

μ

)

2

V

1

− (

V

2

/

V

1

)

displaystyle begin aligned s^ 2 &= frac hat sigma _ mathrm w ^ 2 1-(V_ 2 /V_ 1 ^ 2 ) \&= frac sum _ i=1 ^ N w_ i (x_ i -mu ^ * )^ 2 V_ 1 -(V_ 2 /V_ 1 ) end aligned

,[1]

where

E ⁡ [

s

2

] =

σ

actual

2

displaystyle operatorname E [s^ 2 ]=sigma _ text actual ^ 2

. The degrees of freedom of the weighted, unbiased sample variance vary accordingly from N − 1 down to 0. The standard deviation is simply the square root of the variance above. As a side note, other approaches have been described to compute the weighted sample variance.[2] Weighted sample covariance[edit] In a weighted sample, each row vector

x

i

displaystyle textstyle textbf x _ i

(each set of single observations on each of the K random variables) is assigned a weight

w

i

≥ 0

displaystyle textstyle w_ i geq 0

. Then the weighted mean vector

μ

displaystyle textstyle mathbf mu ^ *

is given by

μ

=

i = 1

N

w

i

x

i

i = 1

N

w

i

.

displaystyle mathbf mu ^ * = frac sum _ i=1 ^ N w_ i mathbf x _ i sum _ i=1 ^ N w_ i .

And the weighted covariance matrix is given by:[3]

Σ

=

i = 1

N

w

i

(

x

i

μ

)

T

(

x

i

μ

)

V

1

.

displaystyle begin aligned Sigma &= frac sum _ i=1 ^ N w_ i left(mathbf x _ i -mu ^ * right)^ T left(mathbf x _ i -mu ^ * right) V_ 1 .end aligned

Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights. Frequency weights[edit] If the weights are frequency weights, the unbiased weighted estimate of the covariance matrix

Σ

displaystyle textstyle mathbf Sigma

, with Bessel's correction, is given by:[3]

Σ

=

i = 1

N

w

i

(

x

i

μ

)

T

(

x

i

μ

)

V

1

− 1

.

displaystyle begin aligned Sigma &= frac sum _ i=1 ^ N w_ i left(mathbf x _ i -mu ^ * right)^ T left(mathbf x _ i -mu ^ * right) V_ 1 -1 .end aligned

Reliability weights[edit] In the case of reliability weights, the weights are normalized:

V

1

=

i = 1

N

w

i

= 1.

displaystyle V_ 1 =sum _ i=1 ^ N w_ i =1.

(If they are not, divide the weights by their sum to normalize prior to calculating

V

1

displaystyle V_ 1

:

w

i

=

w

i

i = 1

N

w

i

displaystyle w_ i '= frac w_ i sum _ i=1 ^ N w_ i

Then the weighted mean vector

μ

displaystyle textstyle mathbf mu ^ *

can be simplified to

μ

=

i = 1

N

w

i

x

i

.

displaystyle mathbf mu ^ * =sum _ i=1 ^ N w_ i mathbf x _ i .

and the unbiased weighted estimate of the covariance matrix

Σ

displaystyle textstyle mathbf Sigma

is:[4]

Σ

=

i = 1

N

w

i

(

i = 1

N

w

i

)

2

i = 1

N

w

i

2

i = 1

N

w

i

(

x

i

μ

)

T

(

x

i

μ

)

=

i = 1

N

w

i

(

x

i

μ

)

T

(

x

i

μ

)

V

1

− (

V

2

/

V

1

)

.

displaystyle begin aligned Sigma &= frac sum _ i=1 ^ N w_ i left(sum _ i=1 ^ N w_ i right)^ 2 -sum _ i=1 ^ N w_ i ^ 2 sum _ i=1 ^ N w_ i left(mathbf x _ i -mu ^ * right)^ T left(mathbf x _ i -mu ^ * right)\&= frac sum _ i=1 ^ N w_ i left(mathbf x _ i -mu ^ * right)^ T left(mathbf x _ i -mu ^ * right) V_ 1 -(V_ 2 /V_ 1 ) .end aligned

The reasoning here is the same as in the previous section. Since we are assuming the weights are normalized, then

V

1

= 1

displaystyle V_ 1 =1

and this reduces to:

Σ =

i = 1

N

w

i

(

x

i

μ

)

T

(

x

i

μ

)

1 −

V

2

.

displaystyle Sigma = frac sum _ i=1 ^ N w_ i left(mathbf x _ i -mu ^ * right)^ T left(mathbf x _ i -mu ^ * right) 1-V_ 2 .

If all weights are the same, i.e.

w

i

/

V

1

= 1

/

N

displaystyle textstyle w_ i /V_ 1 =1/N

, then the weighted mean and covariance reduce to the unweighted sample mean and covariance above. Vector-valued estimates[edit] The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood estimate. We simply replace the variance

σ

2

displaystyle sigma ^ 2

by the covariance matrix

Σ

displaystyle Sigma

and the arithmetic inverse by the matrix inverse (both denoted in the same way, via superscripts); the weight matrix then reads:[5]

W

i

=

Σ

i

− 1

.

displaystyle text W _ i =Sigma _ i ^ -1 .

The weighted mean in this case is:

x

¯

=

Σ

x

¯

(

i = 1

n

W

i

x

i

)

,

displaystyle bar mathbf x =Sigma _ bar mathbf x left(sum _ i=1 ^ n text W _ i mathbf x _ i right),

(where the order of the matrix-vector product is not commutative), in terms of the covariance of the weighted mean:

Σ

x

¯

=

(

i = 1

n

W

i

)

− 1

,

displaystyle Sigma _ bar mathbf x =left(sum _ i=1 ^ n text W _ i right)^ -1 ,

For example, consider the weighted mean of the point [1 0] with high variance in the second component and [0 1] with high variance in the first component. Then

x

1

:=

[

1

0

]

,

Σ

1

:=

[

1

0

0

100

]

displaystyle mathbf x _ 1 := begin bmatrix 1&0end bmatrix ^ top ,qquad Sigma _ 1 := begin bmatrix 1&0\0&100end bmatrix

x

2

:=

[

0

1

]

,

Σ

2

:=

[

100

0

0

1

]

displaystyle mathbf x _ 2 := begin bmatrix 0&1end bmatrix ^ top ,qquad Sigma _ 2 := begin bmatrix 100&0\0&1end bmatrix

then the weighted mean is:

x

¯

=

(

Σ

1

− 1

+

Σ

2

− 1

)

− 1

(

Σ

1

− 1

x

1

+

Σ

2

− 1

x

2

)

=

[

0.9901

0

0

0.9901

]

[

1

1

]

=

[

0.9901

0.9901

]

displaystyle begin aligned bar mathbf x &=left(Sigma _ 1 ^ -1 +Sigma _ 2 ^ -1 right)^ -1 left(Sigma _ 1 ^ -1 mathbf x _ 1 +Sigma _ 2 ^ -1 mathbf x _ 2 right)\[5pt]&= begin bmatrix 0.9901&0\0&0.9901end bmatrix begin bmatrix 1\1end bmatrix = begin bmatrix 0.9901\0.9901end bmatrix end aligned

which makes sense: the [1 0] estimate is "compliant" in the second component and the [0 1] estimate is compliant in the first component, so the weighted mean is nearly [1 1]. Accounting for correlations[edit] See also: Generalized least squares
Generalized least squares
and Variance
Variance
§ Sum of correlated variables In the general case, suppose that

X

= [

x

1

, … ,

x

n

]

T

displaystyle mathbf X =[x_ 1 ,dots ,x_ n ]^ T

,

C

displaystyle mathbf C

is the covariance matrix relating the quantities

x

i

displaystyle x_ i

,

x ¯

displaystyle bar x

is the common mean to be estimated, and

W

displaystyle mathbf W

is the design matrix

[ 1 , . . . , 1

]

T

displaystyle [1,...,1]^ T

(of length

n

displaystyle n

). The Gauss–Markov theorem
Gauss–Markov theorem
states that the estimate of the mean having minimum variance is given by:

σ

x ¯

2

= (

W

T

C

− 1

W

)

− 1

,

displaystyle sigma _ bar x ^ 2 =(mathbf W ^ T mathbf C ^ -1 mathbf W )^ -1 ,

and

x ¯

=

σ

x ¯

2

(

W

T

C

− 1

X

) .

displaystyle bar x =sigma _ bar x ^ 2 (mathbf W ^ T mathbf C ^ -1 mathbf X ).

Decreasing strength of interactions[edit] Consider the time series of an independent variable

x

displaystyle x

and a dependent variable

y

displaystyle y

, with

n

displaystyle n

observations sampled at discrete times

t

i

displaystyle t_ i

. In many common situations, the value of

y

displaystyle y

at time

t

i

displaystyle t_ i

depends not only on

x

i

displaystyle x_ i

but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean

z

displaystyle z

for a window size

m

displaystyle m

.

z

k

=

i = 1

m

w

i

x

k + 1 − i

.

displaystyle z_ k =sum _ i=1 ^ m w_ i x_ k+1-i .

Exponentially decreasing weights[edit] See also: Exponentially weighted moving average In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction

0 < Δ < 1

displaystyle 0<Delta <1

at each time step. Setting

w = 1 − Δ

displaystyle w=1-Delta

we can define

m

displaystyle m

normalized weights by

w

i

=

w

i − 1

V

1

,

displaystyle w_ i = frac w^ i-1 V_ 1 ,

where

V

1

displaystyle V_ 1

is the sum of the unnormalized weights. In this case

V

1

displaystyle V_ 1

is simply

V

1

=

i = 1

m

w

i − 1

=

1 −

w

m

1 − w

,

displaystyle V_ 1 =sum _ i=1 ^ m w^ i-1 = frac 1-w^ m 1-w ,

approaching

V

1

= 1

/

( 1 − w )

displaystyle V_ 1 =1/(1-w)

for large values of

m

displaystyle m

. The damping constant

w

displaystyle w

must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then the following properties of exponentially decreasing weights are useful in making a suitable choice: at step

( 1 − w

)

− 1

displaystyle (1-w)^ -1

, the weight approximately equals

e

− 1

( 1 − w ) = 0.39 ( 1 − w )

displaystyle e^ -1 (1-w)=0.39(1-w)

, the tail area the value

e

− 1

displaystyle e^ -1

, the head area

1 −

e

− 1

= 0.61

displaystyle 1-e^ -1 =0.61

. The tail area at step

n

displaystyle n

is

e

− n ( 1 − w )

displaystyle leq e^ -n(1-w)

. Where primarily the closest

n

displaystyle n

observations matter and the effect of the remaining observations can be ignored safely, then choose

w

displaystyle w

such that the tail area is sufficiently small. Weighted averages of functions[edit] The concept of weighted average can be extended to functions.[6] Weighted averages of functions play an important role in the systems of weighted differential and integral calculus.[7] See also[edit]

Average Mean Summary statistics Central tendency Weight function Weighted least squares Weighted average cost of capital Weighting Weighted geometric mean Weighted harmonic mean Weighted median Standard deviation

References[edit]

^ "GNU Scientific Library – Reference Manual: Weighted Samples". Gnu.org. Retrieved 22 December 2017.  ^ "Weighted Standard Error and its Impact on Significance Testing (WinCross vs. Quantum & SPSS), Dr. Albert Madansky" (PDF). Analyticalgroup.com. Retrieved 22 December 2017.  ^ a b George R. Price (1972). "Ann. Hum. Genet., Lond, pp. 485-490, Extension of covariance selection mathematics" (PDF). Dynamics.org. Retrieved 22 December 2017.  ^ Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi. GNU Scientific Library - Reference manual, Version 1.15, 2011. Sec. 21.7 Weighted Samples ^ James, Frederick (2006). Statistical Methods in Experimental Physics (2nd ed.). Singapore: World Scientific. p. 324. ISBN 981-270-527-9.  ^ G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities (2nd ed.), Cambridge University Press, ISBN 978-0-521-35880-4, 1988. ^ Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN 0-9771170-1-4, 1980.

Further reading[edit]

Bevington, Philip R (1969). Data Reduction and Error Analysis for the Physical Sciences. New York, N.Y.: McGraw-Hill. OCLC 300283069.  Strutz, T. (2010). Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). Vieweg+Teubner. ISBN 978-3-8348-1022-9. 

External links[edit]

David Terr. "Weighted Mean". Math