Mathematics for GIS Professionals

Number Systems Lesson

The alphabet of mathematics is composed of numbers and symbols and it is worthwhile to know the different types of numbers in use.A number is a mathematical object used to count, measure and label.Numbers can be classified into sets, called number systems, such as the natural numbers and the real numbers.

  • [Natural Numbers or Positive Integers] also known as counting numbers. $1, 2, 3, 4, … $ or $1, 2, 3, 4, …$
  • [Integers] whole numbers including zero $…, {-5}, {-4}, {-3}, {-2}, {-1}, 0, 1, 2, 3, 4, 5, …$
  • [Rational Numbers] can be written as a:
  1. Fraction in the form $\frac{a}{b}$ where both $a$ and $b$ are real numbers with $b \neq 0$
  2. Terminating decimal: e.g. $0.5 ; 0.246$
  3. Non-terminating and recurring decimal e.g. $0.33333…$ which is equivalent to $\frac{1}{3}$
  • [Irrational Numbers] e.g $\sqrt{2}$, $\sqrt{3}$, $e$, $\pi$ these numbers CANNOT be written as:
  1. Fractions in the form $\frac{a}{b}$ where both $a$ and $b$ are real numbers with $b \neq 0$
  2. Terminating decimals: e.g. $$0.5 ; 0.246$$
  3. Non-terminating and recurring decimals e.g. $$0.33333…$$ which is equivalent to $\frac{1}{3}$ i.e. the decimal is non-terminating and non-recurring e.g $\pi = 3.142…$
  • [Real numbers] the set that contains all rational and irrational numbers i.e all the numbers on the number line.

There are many ways that numbers can be categorized. For example the whole number line can be divided into even numbers and odd numbers.

Even number – any number that is a multiple of two i.e. any number that can be divided by two without leaving a remainder e.g. $S= (2,4,6,8,10,…)$ is a set of even numbers which can also be written as $$2n$$ where $n$ is an integer.

Odd number – any number that is not a multiple of two i.e. any number that cannot be divided by two i.e. they leave a remainder e.g. $Q= (1,3,5,7,9,…)$ is a set of even numbers which can also be written as $2n$ where $n$ is an integer.

Data can also be divided into categorical or numerical.

Categorical data – information placed in a category or codified according to a classification system. Such data is also called nominal data and have no numerical value. e.g. Males = 1; Females =2 .

Data can also be divided into discrete and continuous.

Discrete data – can only be distinct values (whole numbers) e.g. natural numbers

Continuous data –  can be a  decimal or a  whole number e.g. rational numbers.

 
Feature Points Lines Areas
Physical objects Corner of buiding road network Planning zone
Statistical values Sampling point isoline layer tints
Areas Central point Boundary line polygon
Surfaces Height point contour hill shading
Text House numbers street names district names

Geographical data have one particular characteristic that distinguishes them from all forms of data, namely location.

Graphical data can be plotted on a map and represented by points, lines and areas.

A point is dimensionless, a line has one dimension (length), and an area has two and volume has three.

A point on a map is is a blob or very small  area while a line has thickness and also direction.

Each has a category representing some attribute or attributes associated with it, and each has a location.

Data and Measurement Scales

Each scale of measurement satisfies one or more of the following conditions of measurement.

  • Identity. Each value on the measurement scale has a unique meaning, there are no two items with the same meaning.
  • Magnitude. Values on the measurement scale have an ordered relationship to one another. That is, some values are larger and some are smaller.
  • Equal intervals. Scale units along the scale are equal to one another. This means, for example, that the difference between 1 and 2 would be equal to the difference between 19 and 20.
  •  A minimum value of zero. The scale has a true zero point, below which no values exist.

[Ratio data] Has a natural zero point and can take any value upwards. Mathematical operations can be used on these values with predictable and meaningful results. Examples of ratio measurements are age, distance, weight, and volume. Difference and ratio between any two scores is meaningful. Ratio data is continuous.

[Interval data] Can take any value upwards, but has no natural zero point.

Difference between any two scores is meaningful, but not their ratio. Time of day, calendar years, the Fahrenheit temperature scale, and pH values are all examples of interval measurements. Interval data is continuous.

[Ordinal data] Indicates only rank in a series.

Differences are not mathematically meaningful. An ordinal variable is a categorical variable. Observations can take a value that can be logically ordered or ranked. An example of ordinal data is a group of polygons coloured lighter to darker to represent less to more densely populated areas.

[Nominal data] Data-point is a number that represents membership of a category

Values assigned to variables represent a descriptive category, but have no inherent numerical value with respect to magnitude.

e.g. Gender (1=male, 2=female)

Why store data as a raster?

Many features (such as points) and measurements (such as rainfall) can be stored as a feature (vector) data type (or both) and imagery can only be stored as a raster. The advantages of storing data as a raster are:

A simple data structure—A matrix of cells with values representing a coordinate and sometimes linked to an attribute table.

A powerful format for advanced spatial and statistical analysis.

The ability to represent continuous surfaces and to perform surface analysis.

The ability to uniformly store points, lines, polygons, and surfaces.

The ability to perform fast overlays with complex datasets.

The disadvantages of using raster storage are:

  • Possibility of spatial inaccuracies due to the limits imposed by the raster data-set cell dimensions.
  • Raster data-sets are potentially very large data-sets increasing cost in both disk space and processing speeds. For a given area, changing cells to one-half the current size requires as much as four times the storage space, depending on the type of data and storage techniques used.
  • There is also a loss of precision that accompanies restructuring data to a regularly spaced raster-cell boundary.

General characteristics of raster data

In raster datasets, each cell (pixel) has a value. The cell values represent either a category, magnitude, height, or spectral value. The category could be a land-use class such as grassland, forest, or road. A magnitude might represent gravity, noise pollution, or percent rainfall. Height (distance) could represent surface elevation above mean sea level, which can be used to derive slope, aspect, and watershed properties. Spectral values are used in satellite imagery and aerial photography to represent light reflectance and colour.

Cell values can be either positive or negative, integer, or floating point. Integer values are best used to represent categorical (discrete) data, and floating-point values to represent continuous surfaces. Cells can also have a NoData value to represent the absence of data.

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