Data fusion

Basic concepts

Information fusion is done with a specific goal: for instance parameter estimation according to various sensors or risk level evaluation according to different sources of information. The objective is to compare different alternatives, locations, sites or zones, in the case of spatial data, according to the whole information.

Only values in the same scale and with the same meaning can be aggregated. The most popular aggregation operator is the weighted mean. When the data are of same kind, like in sensor fusion, there is no problem. This is not true in the general case.

Main steps:

  • criterion: attribute with a preference relation
  • from raw data to satisfaction degrees
  • aggregation of satisfaction degrees: score
  • comparison: \mathbf{a \succ b \iff \psi(a) > \psi(b)}
\mathbf{(a_1, \ldots, a_n)} \mathbf{, (b_1, \ldots, b_n)} \mathbf{\overset{A}{\longrightarrow}} \mathbf{\psi (a)} \mathbf{, \psi (b)}
\uparrow \mathbf{\scshape{c} \downarrow}
\mathbf{(x_1, \ldots, x_n)} \mathbf{, (y_1, \ldots, y_n)} \mathbf{\underset{\sim}{\prec} (a,b)}
  • Criterion: if one wants to purchase a car, one of the attributes can be the time to reach a given speed, e. g. 100 km/h, from scratch. If the user prefers a fast car, then this time should be minimized. The acceleration time is an attribute of a car, the criterion of a fast car is better satisfied when this time is low.
  • Satisfaction degrees: the transformation of raw data into satisfaction degrees is done using a function. Figure 1 shows an example for the acceleration time.
    Figure 1

    To each of the possible values for the attribute corresponds a degree in a commensurable scale, meaning that all the degrees, for all the attributes, are in the same range, e. g. the unit interval [0,1], and have the same meaning, 0 when the criterion is not at all satisfied, for all t>20s in this example, 1 when it is fully satisfied, t<4s in the example.

  • Aggregation of all the degree for the same alternative. The values from the n information sources are now converted into n satisfaction degrees for each alternative, a car in our example. The next step is to summarize the n values into a single one to ease the comparison between alternatives. Among the suitable properties for the aggregation operators are the idempotence, monotony and compromise: \underset{i}{min} \ a_i \le \psi(a_1, \ldots, a_n) \le \underset{i}{max} \ a_i

    Two families of such operators are available in GeoFIS:

    • Weighted Arithmetic Mean (WAM)

          \[\psi(a_1, \ldots, a_n)= \sum_{i=1}^n w_ia_i, \quad w_i \in [0,1], \sum_{i=1}^n w_i=1\]

    • Ordered Weighted Average (OWA)

          \[\psi(a_1, \ldots, a_n)= \sum_{i=1}^n w_ia_{(i)}, \quad w_i \in [0,1], \sum_{i=1}^n w_i=1\]

      (.): permutation such as a_{(1)}\le \cdots \le a_{(n)}

With the WAM, the weights are assigned to the information sources. This is not the case for the OWA: as the degrees are ordered the weights are assigned to the locations in the distribution, from the minimum to the maximum, whatever the information sources.

GeoFIS implementation

In the current version all the attributes must belong to the same layer. To be used in the fusion module an attribute must be selected as an input and a function to turn raw data to satisfaction degree has to be defined.
Four types of membership functions (Mf) are proposed:

  • Semi trapezoidal inf: low values are preferred
  • Semi trapezoidal sup: high values are preferred
  • Trapezoidal: around an interval
  • Triangular: about a value

The aggregation is available as a new variable, which can also be used as an input for another aggregation step, yielding a hierarchical structure. An aggregation operator is defined for each aggregated variable. Two are currently available:

  • WAM: the weights are assigned to the information sources
  • OWA: the weights are given to the position in the distribution