agop (version 0.1-3)

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.

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

Arguments

x

a non-negative numeric vector

p

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

projection

function

Value

• result of projection(c($i, x_i$))

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 projection to identity to obtain the 2-dimensional index

If non-increasingly sorted vector is given, the function is O(n).

For historical reasons, this function is also available via its 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 algorithms for computing ''geometric'' scientific impact indices, Research Report of Systems Research Institute, Polish Academy of Sciences RB/1/2009, 2009b. 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.

See Also

Other impact_functions: index_g, index_g_zi, index_h, index_maxprod, index_rp, index_w, index.g, index.h, index.rp

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]