index_lp {agop} R Documentation

## The l_p-index

### Description

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).

### 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,∞]; defaults ∞ (`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].

### Value

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

### 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.

Other impact_functions: `index_g`, `index_h`, `index_maxprod`, `index_rp`, `index_w`, `pord_weakdom`
```x <- runif(100, 0, 100)