GEOG 090 – Quantitative Methods in Geography The Scientific Method Exploratory methods (descriptive statistics) Confirmatory methods (inferential statistics) Mathematical Notation Summation notation Pi notation Factorial notation Combinations

The Scientific Method Both physical scientists and social scientists (in our context, physical and human geographers) often make use of the scientific method in their attempts to learn about the world organize surprise Concepts Description Hypothesis formalize validate Theory Laws Model

The Scientific Method The scientific method gives us a means by which to approach the problems we wish to solve The core of this method is the forming and testing of hypotheses A very loose definition of hypotheses is potential answers to questions Geographers use quantitative methods in the context of the scientific method in at least two distinct fashions:

Two Sorts of Approaches Exploratory methods of analysis focus on generating and suggesting hypotheses Confirmatory methods are applied in order to test the utility and validity of hypotheses Concepts Description Hypothesis Theory Laws Model organize surprise validate formalize

Two Sorts of Statistics Descriptive statistics To describe and summarize the characteristics of the sample Fall within the class of exploratory techniques Inferential statistics To infer something about the population from the sample Lie within the class of confirmatory methods

Mathematical Notation The mathematical notation used most often in this course is the summation notation The Greek letter is used as a shorthand way of indicating that a sum is to be taken: The expression is equivalent to:

Summation Notation: Components refers to where the sum of terms ends indicates what we are summing up indicates we are taking a sum refers to where the sum of terms begins

Summation Notation: Simplification A summation will often be written leaving out the upper and/or lower limits of the summation, assuming that all of the terms available are to be summed

Summation Notation: Examples Example I: All observations are included in the sum: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Example II: Only observations 3 through 5 are included in the sum:

Summation Notation: Rules Rule I: Summing a constant n times yields a result of na: Here we are simply using the summation notation to carry out a multiplication, e.g.:

Summation Notation: Rules Rule II: Constants may be taken outside of the summation sign

Rule II: Constants may be taken outside of the summation sign Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be: x1 = 4, x2 = 5, x3 = 6 y1 = 7, y2 = 8, y3 = 9

Summation Notation: Rules Rule III: The order in which addition operations are carried out is unimportant +

Rule III: The order in which addition operations are carried out is unimportant Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be: x1 = 4, x2 = 5, x3 = 6 y1 = 7, y2 = 8, y3 = 9

Summation Notation: Rules Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum

Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum Example: Now let the values of a set (n = 3) of x values be: x1 = 4, x2 = 5, x3 = 6

Summation Notation: Rules Rule V: Products are handled much like exponents

Rule V: Products are handled much like exponents Example: Now let the values of a set (n = 3) of x and y values be: x1 = 4, x2 = 5, x3 = 6 y1 = 7, y2 = 8, y3 = 9

Summation Notation: Compound Sums We frequently use tabular data (or data drawn from matrices), with which we can construct sums of both the rows and the columns (compound sums), using subscript i to denote the row index and the subscript j to denote the column index: Columns Rows

Pi Notation Whereas the summation notation refers to the addition of terms, the product notation applies to the multiplication of terms It is denoted by the following capital Green letter (pi), and is used in the same way as the summation notation

Factorial The factorial of a positive integer, n, is equal to the product of the first n integers Factorials can be denoted by an exclamation point There is also a convention that 0! = 1 Factorials are not defined for negative integers or nonintegers

Combinations Combinations refer to the number of possible outcomes that particular probability experiments may have Specifically, the number of ways that r items may be chosen from a group of n items is denoted by: or

Combinations Example – Suppose the landscape can be characterized by five land cover types: forest (F), grassland (G), shrubland (S), agriculture (A), and water (W). A region has only two land cover types, the number of possible combinations is:

Combinations Ten possible combinations: F – G, F – S, F – A, F – W G – S, G – A, G – W S – A, S – W A – W F (forest), G (grassland), S (shrubland), A (Agriculture), W (Water)

Assignment I Textbook, p39-40, #3 - #5 #3 is about summation notation #4 is about factorial #5 is about combinations Due: January 26th (Thursday) (preferably at the beginning of class, or put in my mailbox before 5pm – (Rm 315)) 

Mailboxes in Grad Workroom (315) . . . Jingfeng Xiao . . . . . . . . . . . .