Objectives of Lecture:
Truly comprehensive results come from a further set of operations:
A Family of Problems; a common approach
Some problems involve a global measure or constraint, phrased
as a minimum overall cost or as a measure of equality. Many seemingly different problems
can be reformulated into a small number of basic techniques.
The simple statement involves:
For example: Area Education Authority Offices serve the School Districts of Iowa...
NOTE: This means a RELATIONSHIP, a higher-order of information than a simple overlay or indeed most of the prior operations...
The problem can be constrained in a number of ways (after (Rushton 1979, p.33)):
Additional complexity involves finding the best number of 'suppliers' (rather than assuming that a certain number must exist.
These comprehensive solutions seem to provide the 'best' location, but in every case, they must use a 'heuristic', not a global optimum. These problems skirt very close to the edge of the BIG problems for computation. (NP-Complete).
For example:
- The Traveling Salesperson Problem requires the shortest path to visit a set of 'cities' over some network without visiting any city twice.
- The Knapsack Problem requires the best fit of a set of 'parcels' (integers) into a set of 'knapsacks' (containers, or regions). [Participatory experiment...]
These problems belong to a group of graph problems that may
not ever be solved in 'polynomial' time (hence they are Non-Polynomial
or NP). The reason is that you cannot find an iterative
solution that allows you to say that the salesman MUST follow
this particular path or that this particular parcel fits in that
particular knapsack until everything has been fit into place.
Essentially, these problems require examining all the possible
combinations, a set of numbers that rise very very fast. [all
the possible combinations of 59 items would require a number
as large as all the baryons in the universe...]
NP problems can be approximated rather closely using iterative
heuristics, just without a guarantee that the solution is optimal.
A statistical model set out a framework of relationships that
are then confirmed or rejected by some measure of 'goodness of
fit'. To discover these relationships, a series of axioms are
implied (homogeneity of 'population', source of error, etc.)
The estimation for the model is based on a model of error, usually
based on the error in sampling from a 'population'. Yet, most
spatial data is an exhaustive partitioning of a region (as with
census tracts, counties, etc.). The connections between these
two assumptions are not as direct as some hope... The calculations
are totally global: means and deviations are abstracted from their
neighborhoods totally.
Data model for statistics: a 'case' as a replicate in an experiment;
matrix of cases by variables (the old 'geographical
matrix' of Berry) : THE SPACE PART CAN BE LOST...
Example: Gold deposits for Nova Scotia [overheads]: pixels
or watersheds are not indivisible 'cases'
Resources about statistics
and GIS