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valsi 
cnanfadi 
type 
fu'ivla 
creator 
krtisfranks 
time entered 
Sat Mar 31 21:54:55 2018 
Examples


English


Definition #70579
 Preferred
[edit]


definition 
x_{1} (li; number/quantity) is the weighted quasiarithmetic mean/generalized fmean of/on data x_{2} (completely specified ordered multiset/list) using function x_{3} (defaults according to the notes; if it is an extendedreal number, then it has a particular interpretation according to the Notes) with weights x_{4} (completely specified ordered multiset/list with same cardinality/length as x_{2}; defaults according to Notes). 
notes 
Potentially dimensionful. Make sure to convert x_{3} from an operator to a sumti; x_{3} is the 'f' in "fmean" and must be a complexvalued, singlevalued function which is defined and continuous on x_{2} and which is injective; it defaults to the pthpower function (z^{p}) for some nonzero p (note that it need not be positive or an integer) and indeterminate/variable/input z, or log, or exp (as functions); culture or context can further constrain the default. If x_{2} is set to "
z^{+.}" (the exponent is positive infinity, given by "ma'uci'i") for indeterminate/variable z (the function is the functional limit of the monic, singleterm polynomial as the degree increases without bound), then the result (x_{1}) is the weightsumscaled maximum of the products of the data (terms of x_{2}) with their corresponding weights (terms of x_{4}) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering; likewise, if x_{2} is set to "
z^{.}" (the exponent is negative infinity, given by "ni'uci'i") for indeterminate/variable z (the function is the functional limit of the monic, singleterm reciprocalpolynomial as the reciprocaldegree increases without bound (or the degree decreases without bound)), then the result (x_{1}) is the weightsumscaled minimum of the products of the data (terms of x_{2}) with their corresponding weights (terms of x_{4}) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering. The default of x_{4} is the ordered set of n terms with each term equal identically to 1/n, where the cardinality of x_{2} is n. Let "f" denote the sumti in x_{3}, "y_{i}" denote the ith term in x_{2} for all i, n denote the cardinality of x_{2} (thus also x_{4}), and w_{i} denote the ith term in x_{4} for any i; then the result x_{1} is equal to: f^{(1)}(Sum
(w_{i}f (y_{i}), i in Set
(1,..., n))/Sum(w_{i}, i in Set
(1,..., n))). Note that if the weights are all 1 and x_{2} is set equal to not the pthpower function, but instead the pthpower function leftcomposed with the absolute value function (or the forward difference function), then the result is the pnorm on x_{2} scaled by
n^{(1/p)} for integer n being the cardinality of x_{2}. This should typically not refer to the mean of a function ( https://en.wikipedia.org/wiki/Mean_of_a_function ), although it generalizes easily; alternatively, with appropriate weighting, allow x_{2} to be the image (set) of the function whose average is desired over the entire relevant subset of its domain  notice that the weights will have to themselves be functions of the data or the domain of the function; in this context, the function is not necessarily x_{3}. If x_{3} is a single extendedreal number p (not a function), then this word refers to the weighted powermean and it is equivalent to letting x_{2} equal the pthpower function as before iff p is nonzero real, the max or min as before if p is infinite (according to its signum as before), and log if p = 0 (thus making the overall mean refer to the geometric mean); this overloading is for convenience of usage and will not cause confusion because constant functions are very much so not injective.

gloss words 

created by 
krtisfranks 
vote information 
2

time 
Sun Aug 12 21:36:08 2018 



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