Transcript GEOGRAPHICAL STATISTICS GE 2110

GEOGRAPHICAL STATISTICS
GE 2110
Zakaria A. Khamis
PROBABILITY
• The field of Probability provides a foundation for inferential
statistics
• Probability is the study of uncertainty associated with possible
outcomes
• Probability may be thought of as a measure of the likelihood or
relative frequency of each possible outcome
• To be able to test for the uncertainty, the experiment is
performed (e.g. survey); in which the set of all possible outcome
is called SAMPLE SPACE, and the individual outcome from the
sample space is called SAMPLE EVENT/SAMPLE POINT
PROBABILITY
• Probability of an event e is always greater than or equal to 0 and
less than or equal to 1
• The sum of the probabilities over the sample space is equal to 1
• There are numerous ways to assign probabilities to the elements
of sample spaces
•
To assign probabilities on the basis of relative frequencies
•
Meteorologist may note that in 65 out of the last 100
observations that such a pattern prevailed, there was
measurable precipitation the next day
PROBABILITY
• The possible outcome – rain or no rain tomorrow – are assigned
probabilities of 0.65 and 0.35 respectively
•
On the basis of subjective beliefs
•
The description of the weather patterns is a simplification of
reality, and may be based upon only a small number of
variables, such as temp, wind speed and direction, pressure etc
•
The forecaster may, partly on the basis of other experience,
assess the likelihoods of precipitation and no precipitation as
0.6 and 0.4 respectively
PROBABILITY
• To assign each of the n possible outcomes a probability of 1/n
• This approach assumes that each sample point is equally likely,
and it is an appropriate way to assign probabilities to the
outcomes in the some experiments
• E.g. If the coin is tossed, the probability that the result will be
head is ½, and tail is ½
• If p is the probability that an event will occur, and p’ is the
probability that an event won’t occur, thus p + p’ = 1
RANDOM VARIABLES
• Random variables refer to the functions defined on a sample space
• Associated with each possible outcome is a quantity of interest
• The outcome of an experiment need not be a number, for example,
the outcome when a coin is tossed can be 'heads' or 'tails'.
However, we often want to represent outcomes as numbers.
• A random variable is a function that associates a unique numerical
value with every outcome of an experiment. The value of the
random variable will vary from trial to trial as the experiment is
repeated.
SAMPLING
• The collection of all elements/individuals that are the object of
our interest  Population
• The list of all elements in the population or sub-population from
which the sample will be drawn is referred to as the sampling
frame
• Sampling frame may consist of spatial elements – all shehias in
the Urban-West region
• What is SAMPLE?
• What is SAMPLING?
SAMPLING TECHNIQUES
• There are many ways to sample from a population
RANDOM SAMPLING
• This is the simplest way of sampling, in which each element has
equal probability of being selected
SYSTEMATIC SAMPLING
• Choosing a systematic sample of size n begins by selecting an
observation at random from among the first [N/n] elements 
note [] means the value should be integer
SAMPLING TECHNIQUES
• Once the first element is determined, the other elements will be
selected systematically following a certain predefined order
• Note that it was necessary to choose only one random number
STRATIFIED SAMPLING
• When it is known beforehand that there is likely to be variation
across certain sub-groups of the population, the sampling frame
may be stratified before sampling
• In some cases, we may need to make the sample proportions in
each strata equal  Proportional, stratified sampling
SAMPLING TECHNIQUES
Cluster Sampling
• If the population is widely dispersed, random and systematic
will involve a great deal of travel  It save an immense amount
of time to sample from carefully selected clusters
• A double sampling procedure is involved, first select
representative clusters (probably best done subjectively) and
then select the sample within each cluster (random, systematic or
stratified)
• In some studies we make use of SURROGATES instead of
SAMPLES
SAMPLING TECHNIQUES
SPATIAL/GEOGRAPHICAL SAMPLING
• When the sampling frame consist of all of the points located in a
geographical region of interest, there are again several
alternative sampling methods
• A random spatial sample consists of locations obtained by
choosing x-coordinates and y-coordinates at random
• If the pair of coordinates happens to correspond to a location
outside of the study region, the point is simply discarded
SAMPLING TECHNIQUES
• To ensure adequate coverage of the study area, the study region
may be broken into a number of mutually exclusive and
collectively exhaustive strata
• Divide a study region into a set of s = mn strata
• A stratified random spatial sample of size mnp is obtained by
taking a random sample of size p within each of the mn strata
• A stratified systematic spatial sample of size mnp is obtained by
taking a random sample of size p within any individual stratum
and then 
SAMPLING TECHNIQUES
• Using the sample spatial configuration of those p points within
that stratum within the other strata
• WHICH SPATIAL SAMPLING SCHEME IS BEST?
• Depends on the spatial characteristics of variability in the data.
• Practically, because values of variables at one location tend to be
strongly associated with values at nearby locations, random
spatial sampling can provide redundant information when
sample location are close to one another