Zakaria A. Khamis GE 2110 GEOGRAPHICAL STATISTICS GE 2110

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 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 PROBABILITY

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. RANDOM VARIABLES

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

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 SAMPLING TECHNIQUES