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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. organize. surprise. validate.

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GEOG 090 – Quantitative Methods in Geography

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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


organize

surprise

validate

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

Concepts

Description

Hypothesis

formalize

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 quantitativemethods in the context of the scientific method in at least two distinct fashions:


organize

surprise

Concepts

Description

Hypothesis

formalize

validate

Theory

Laws

Model

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


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 constantn 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


Example: Now let the values of a set (n = 3) of x values be:

x1 = 4, x2 = 5, x3 = 6

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


Summation Notation: Rules

  • 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

  • Rule V: Products are handled much like exponents


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 nitems 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

. . .

. . .

. . .

. . .


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