qmdaa:moro

# Spatial Data, Complexity, and Introduction to Uncertainty

Antonio Moro
Facoltà di Ingegneria, Università di Firenze
moro@dicea.unifi.it

## Lecture #1

Maths for Archaeology → Dwight D. Read, S. E. Van Der Leeuw

Perfect knowledge is reductionist. Statistics try to find if there's an order in disorder, but explanations are just valid at specific scales of study. Entropy (Shannon) is a measure of disorder. In disordered structures, measurements change a lot with respect to the chosen unit and scale.

Modeling can be determined (while not necessarily deterministic) or undetermined. Determined modeling deals with predictable future. Indetermination can be a result of absence of information, accuracy, arbitration, comprehension.

Complex systems are more than the sum of their parts. The properties are subjective, and small changes can cause great effects.

StatisticsProbability
set of rules to deal with indetermination mathematical theory to deal with indetermination

Dyes are a simple random system, with independence, and the probability of drawing a number stays the same.

Estimators are functions of data. They need to be unbiased and consistent. Distributions can be normal, gamma or beta. Remember that not everything is normal.

Production of data (numbers) and extraction of information is done by descriptive statistics. Forecast is done by (probabilistic) models. The last step is represented by fit tests.

Bear in mind that data can be censored, have outliers, have a multimodal distribution and other problems.

The required sample size is a way to determine the minimum amount of required data. It is important to consider the correlation between variables.

Basic statistics do not deal with causality. Introducing the time parameter, things get more complex. Time-series involve fixed intervals of time, and at each point you have some quantity.

Spatial processes can be stationary, e.g. the distributions are invariant, given the originating point. If it is invariant also with respect to direction, it is isotropic.

Spatial data examples:

• spatial point patterns
• geostatistical continuous data (kriging, tessellation)
• lattice, area data (subregions)
• spatial interaction data (i.e. transport systems), relationships between data

Subjective: each one's degree of belief about a fact is different. Coherence is obtained through the “bet” clause. A subjective approach can give us a way to handle situations where we have little data and a great theory. The model can be updated, based upon experience, during time.

Measures

Generating processes

Philosophical positions

## Lecture #2

Complex ≠ Complicated
Simple ≠ Easy
Multitude → complexity but also simplicity

1. Chaotic systems
2. Synergy systems
3. Mesoscopic systems

Complexity depends on the context of study.

<m>y = x^3 + a·x</m>

if <m>a=1</m> or <m>a=-1</m> the 2 resulting systems are qualitatively completely different. Mathematics catastrophe theory knows just 7 types of catastrophes described in mathematical terms.

C. Renfrew 1978

Cellular Automata <m>{S_i}^{t+1} = R ({S^t}_{i-1},{S^t}_{i},{S^t}_{i+1})</m>

Power laws

Independence → No prediction

## Lecture #3

Fractals

• not a single statistics
• not only descriptive
• self-similar (scaling factors)
• defined by recursive algorithms
• fractional dimension compared to integer geometries (line=1, square=2, cube=3)

Fractional dimension D characterizes how the mean depends on the size of the sample.

Zipf 1949

Strange attractors are fractals (Lorentz)

Self-organized Criticality 