The l_p-index
Given a sequence of n non-negative numbers x=(x_1,…,x_n), where x_i ≥ x_j for i ≤ j, the l_p-index for p=∞ equals to
l_p(x) = arg max_(i,x_i) { i*x_i } for i=1,…,n
if n ≥ 1, or l_∞(x)=0 otherwise. Note that if (i,x_i)=l_∞(x), then
MAXPROD(x) = prod(l_∞(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 < ∞ we refer to (Gagolewski, Grzegorzewski, 2009a).
index_lp(x, p = Inf, projection = prod) index.lp(x, p = Inf, projection = prod) # deprecated alias
x |
a non-negative numeric vector |
p |
index order, p in [1,∞]; defaults ∞ ( |
projection |
function |
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].
result of projection(c(i, x_i))
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.
Other impact_functions: index_g,
index_h, index_maxprod,
index_rp, index_w,
pord_weakdom
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]Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.