# Concentration

## Definition

Concentration. Denotes an excess (richness, abundance, the opposite of paucity) of a Quantity in a certain context. A state of being concentrated as opposed to diluted.

Quantity is here understood in the most general sense, as a measured Value expressed in some Measurement Unit or simply a Count

## Context

There are various contexts in which concentration becomes a useful concept:

• Spatial Concentration: the preponderance of a quantity in a spatial domain (one, two or three dimensional).
• Temporal Concentration: the preponderance of a quantity in a certain period of time (frequency).
• Concentration along some abstract system dimension

## Examples

• The concentration of individuals in a given land area (e.g. a city) is an example of spatial concentration
• The clustering of commercial transactions during some part of the week is an example of temporal concentration
• Abstract dimensions are very common: The concentration of substances in a container, the market share concentration in a business sector

## Measurement

The measurement of concentration is based in the first instance on the regular measurement of underlying Quantity Value. This leads to a Value Distribution of a Variable along some of the aforementioned dimensions. The relevant dimension can be completely abstract. E.g. a Categorical Variable may be classifying a collection of instances of a classes into sub-classes, thereby creating the concept of concentration among the different subclasses.

A very useful concept in concentration measurement is the Concentration Index, a function that converts a potentially complex distribution of observed values into a single number / indicator

### Absolute Concentration

Absolute concentration measurement concerns the production of a metric that can be used on a standalone basis. The threshold(s) of what constitutes high concentration are provided exogenously.

### Relative Concentration

Relative concentration measurement focuses insted in the comparative analysis between e.g. two entities, portfolios or other systems being analyzed. In this mode, the objective is to produce a concentration ranking (i.e., X is more concentrated that Y).

## Risk Management

In Risk Management context, concentration may be indicative or Risk. It may describe, for example, which percentage one buyer represents on the vendor's total accounts receivable. It is normally expressed in percent.

The term Concentration Risk is usually associated with high concentration, but clearly one can imagine cases where the risk is due to insufficient concentration (e.g., lack of critical mass)