njit-etd2003-081 - New Jersey Institute of Technology
njit-etd2003-081 - New Jersey Institute of Technology njit-etd2003-081 - New Jersey Institute of Technology
115 or multiple dimensions. For example, if fast foods were clustered, one could take into account the number of calories they contain, their price, subjective ratings of taste, etc. The most straightforward way of computing distances between objects in a multidimensional space is to compute Euclidian distances. If one had a two- or threedimensional space this measure is the actual geometric distance between objects in the space (i.e., as if measured with a ruler). However, the joining algorithm does not "care" whether the distances that are "fed" to it are actual real distances, or some other derived measure of distance that is more meaningful to the researcher; and it is up to the researcher to select the right method for his/her specific application. 3.15.4 Euclidian Distance This is probably the most commonly chosen type of distance. It simply is the geometric distance in the multidimensional space. It is computed as: Note that Euclidian (and squared Euclidian) distances are usually computed from raw data, and not from standardized data. This method has certain advantages (e.g., the distance between any two objects is not affected by the addition of new objects to the analysis, which may be outliers). However, the distances can be greatly affected by differences in scale among the dimensions from which the distances are computed. For example, if one of the dimensions denotes a measured length in centimeters, and one then converts it to millimeters (by multiplying the values by 10), the resulting Euclidian or squared Euclidian distances (computed from multiple dimensions) can be greatly affected, and consequently, the results of cluster analyses may be very different.
116 3.15.5 Squared Euclidian Distance One may want to square the standard Euclidian distance in order to place progressively greater weight on objects that are further apart. This distance is computed as (see also the note in the previous paragraph): 3.15.6 City-block (Manhattan) Distance This distance is simply the average difference across dimensions. In most cases, this distance measure yields results similar to the simple Euclidian distance. However, note that in this measure, the effect of single large differences (outliers) is dampened (since they are not squared). The city-block distance is computed as: 3.15.7 Chebychev Distance This distance measure may be appropriate in cases when one wants to define two objects as "different" if they are different on any one of the dimensions. The Chebychev distance is computed as: 3.15.8 Power Distance Sometimes one may want to increase or decrease the progressive weight that is placed on dimensions on which the respective objects are very different. This can be accomplished via the power distance. The power distance is computed as:
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116<br />
3.15.5 Squared Euclidian Distance<br />
One may want to square the standard Euclidian distance in order to place progressively<br />
greater weight on objects that are further apart. This distance is computed as (see also<br />
the note in the previous paragraph):<br />
3.15.6 City-block (Manhattan) Distance<br />
This distance is simply the average difference across dimensions. In most cases, this<br />
distance measure yields results similar to the simple Euclidian distance. However, note<br />
that in this measure, the effect <strong>of</strong> single large differences (outliers) is dampened (since<br />
they are not squared). The city-block distance is computed as:<br />
3.15.7 Chebychev Distance<br />
This distance measure may be appropriate in cases when one wants to define two objects<br />
as "different" if they are different on any one <strong>of</strong> the dimensions. The Chebychev<br />
distance is computed as:<br />
3.15.8 Power Distance<br />
Sometimes one may want to increase or decrease the progressive weight that is placed<br />
on dimensions on which the respective objects are very different. This can be<br />
accomplished via the power distance. The power distance is computed as:
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