agop (version 0.2.4)

index_lp: The l_p-index

Description

Given a sequence of \(n\) non-negative numbers \(x=(x_1,\dots,x_n)\), where \(x_i \ge x_j\) for \(i \le j\), the \(l_p\)-index for \(p=\infty\) equals to $$l_p(x)=\arg\max_{(i,x_i), i=1,\dots,n} \{ i x_i \}$$ if \(n \ge 1\), or \(l_\infty(x)=0\) otherwise. Note that if \((i,x_i)=l_\infty(x)\), then $$MAXPROD(x) = \mathtt{prod}(l_\infty(x)) = i x_i,$$ where \(MAXPROD\) is the index proposed in (Kosmulski, 2007), see index_maxprod. Moreover, this index corresponds to the Shilkret integral of \(x\) w.r.t. some monotone measure, cf. (Gagolewski, Debski, Nowakiewicz, 2013).

For the definition of the \(l_p\)-index for \(p < \infty\) we refer to (Gagolewski, Grzegorzewski, 2009a).

Usage

index_lp(x, p = Inf, projection = prod)
index.lp(x, p = Inf, projection = prod)  # deprecated alias

Value

result of projection(c(\(i, x_i\)))

Arguments

x

a non-negative numeric vector

p

index order, \(p \in [1,\infty]\); defaults \(\infty\) (Inf).

projection

function

Details

The \(l_p\)-index, by definition, is not an impact function, as it produces 2 numeric values. Thus, it should be projected to one dimension. However, you may set the projection argument to identity so as to obtain the 2-dimensional index

If a non-increasingly sorted vector is given, the function has O(n) run-time for any \(p\), see (Gagolewski, Debski, Nowakiewicz, 2013).

For historical reasons, this function is also available via an alias, index.lp [but its usage is deprecated].

References

Gagolewski M., Grzegorzewski P., A geometric approach to the construction of scientific impact indices, Scientometrics 81(3), 2009a, pp. 617-634.

Gagolewski M., Debski M., Nowakiewicz M., Efficient Algorithm for Computing Certain Graph-Based Monotone Integrals: the lp-Indices, In: Mesiar R., Bacigal T. (Eds.), Proc. Uncertainty Modelling, STU Bratislava, ISBN:978-80-227-4067-8, 2013, pp. 17-23.

Kosmulski M., MAXPROD - A new index for assessment of the scientific output of an individual, and a comparison with the h-index, Cybermetrics 11(1), 2007.

Shilkret, N., Maxitive measure and integration, Indag. Math. 33, 1971, pp. 109-116.

See Also

Other impact_functions: index_g(), index_h(), index_maxprod(), index_rp(), index_w(), pord_weakdom()

Examples

x <- runif(100, 0, 100)
index.lp(x, Inf, identity)  # two-dimensional value, can not be used
                            # directly in the analysis
index.lp(x, Inf, prod)      # the MAXPROD-index (one-dimensional) [default]