Phenomenon: a fact or situation that is observed to exist or happen, especially one whose cause or explanation is in question.

A lightning strike
A coastline
A country
A dog on a kayak!
Anything and everything are phenomena!

Discrete Objects

Distinct boundaries

Chat can be exactly measured

Finite

They are countable and cannot be infinitely subdivided

Continuous Fields

No distinct boundaries

Everywhere has a value

Infinitely divisible

They are not countable and can be infinitely subdivided

When is a phenomenon discrete or continuous?

To an extent, it depends on our perspective and the scale of our analysis.
Many phenomenon are a bit of both.

A strike is a discrete object
What about a lighting bolt ...?

Doesn't really have clearly defined boundaries?
Sort of continuous

Strike frequency is a continuous field

Everywhere has a frequency of lighting strikes
Even the absence of lighting strikes, is still a frequency of strikes

Continuous field at large scale

Tides & waves
Where is the exact boundary?

Discrete object at small scale

Zoom out and the tides/waves don't really matter
Its easy to draw a static boundary

Unless you change the time scale

Until you start thinking about the long term :(

Most things don't fall perfectly into one category or the other.

That said, it is a helpful framework as long as we recognize the discrete vs. continuous dichotomy is not a perfect classification

TopHat Question 1

Discrete objects: (select all that apply)

Are countable
Do not have distinct boundaries
Are infinitely divisible
Have well defined boundaries

Buildings are a great example.

Concrete boundaries
Countable
Real physical object

Political Boundaries are also a great example.

Distinct boundaries
Countable
Not a physical object

Elevation is a great example.

Everywhere on Earth
No "number of elevations"
A physical property

Density of tweets is also a great example.

Everywhere has this too
Derived from something countable

But not countable itself

Not a physical property

Frequently we'll end up working with both discrete objects and continuous fields.

In Module 1, you worked with:

Cholera deaths (Discrete objects)
Kernel density (continuous field)

We use spatial data models to represent information in a GIS

Vector Data Model = discrete objects
Raster Data Model = continuous fields

We'll talk more about spatial data models later. For now, lets think about data more broadly. How do we represent data in a computer?

Digital information is represented as bits (0's and 1's)
We typically quantify data as bytes (8 bits):

Kilobyte (kB) = 1,000 bytes
Megabyte (MB) = 1,000,000 bytes
Gigabyte (GB) = 1,000,000,000 bytes

There are numerous ways to translate human readable data to binary, such as ASCII.

Each character is represented as one byte
There are 2⁸ = 256 unique combinations of 0's and 1's in a byte
Some examples:

"A" : 01000001
"CAT": 01000011 01000001 01010100
"31": 00110011 00110001

Modern computers use 64-bit "architecture". That is, the central processing unit (CPU) can handle 64 bits (8 bytes) of information at a time.

"Word" length is 64 bits

A "Word" is a unit of data

i.e., an individual piece of information

2⁶⁴ possible unique values

>18 quintillion possible unique combinations of 1's and 0's to describe something

CPUs can be stacked in parallel to handle more information at one time

Within the context of a GIS, every piece of information describing a phenomenon is referred to as an Attribute.

Broadly speaking each attribute can address one of three questions:

Where?
What?
When?

There are multiple ways to classify/think about attributes. One important distinction we must make

Non-Spatial Attributes: describe what or when
Spatial Attributes: describe where

Puts the Geographic in GIS
Requires some special considerations

We have already talked a bit about these consideration (map projections)
We'll discuss more considerations in the next module

All data (attributes), spatial and non-spatial, can be either qualitative or quantitative.

The types of analysis we can do with qualitative data are more limited
That does not make quantitative data “better”
Measurement scales: both qualitative and quantitative can be measured on different scales

Qualitative: Nominal or Ordinal Sales
Quantitative: Interval or Ratio Sales

Qualitative data is Categorical. It is strictly descriptive and lacks any meaningful numeric value.

Textual, coded numerals, pictures, sounds, etc

Typically working with textual & coded numerals most frequently in GIS

Limited number of computational options, often requires careful consideration when analyzing
Measured on either a Nominal or Ordinal scale.

Names or categories with no ranking or direction. Categories are not more/less, better/worse, they just different. Some examples include:

Flower Species
Zoning Categories
Land cover Classification

With nominal data we can:

Check equivalency
Count frequencies
Nothing else

Names or categories with a ranking or direction. Categories are more/less, better/worse, etc. But the differences are relative, there is no way to say by "how much". Some examples include:

Spice levels
Relative heights
Compass Direction

With ordinal data we can:

Check equivalency
Count frequencies
Check order/rank

All the same operations as nominal data + more.

Sometimes we can calculate the median.

Odd sets the median is the middle.
Even sets, average of the middle two.
One solution, arbitrarily assign a numeric score.

All the same operations as nominal data + more.

Exceptions that blur the lines.

Grade membership to assign categories
Where to draw the line between forest/alpine?
Winner take all: alpine meadow

45% alpine meadow
40% forest
5% bare rock

In practice, lots of qualitative data we work with, especially for natural phenomena, are actually graded membership.

The downside: variability within the area is lost.

TopHat Question 2

Which of the following would be examples of Nominal Data? (select all that apply)

Air temperature
Ice cream flavors
Tree height
Colors
Drink sizes

Quantitative data is Numeric. It describe the quantities associated with a phenomenon. Key properties include:

Values separated by a meaningful unit.
More arithmetic operations possible.
Can be Discrete or Continuous numbers.
Measured on either a Ratio or Interval scale.

Discrete

Whole numbers
Counts
Not infinitely divisible
Names in ArcGIS Pro:

Integer, Long, etc.

Continuous

Decimals
Measurements
Infinitely divisible
Names in ArcGIS Pro:

Float, Double, etc.

Discrete

Countable
Examples:

Population
Year
"Age"

Continuous

Non-countable
Examples:

Temperature
Height
Speed

Both Interval and Ratio data can consist of discrete or continuous numbers. These types of quantitative data are closely related, but have one important distinction.

Interval scales have an arbitrarily zero point, which means:

Can have negative values
Cannot multiply/divide ...

To compare the magnitude of one value to another
Can still multiply/divide for other calculations, i.e. calculating an average

Ratio scales have a fixed, absolute zero point, which means:

Cannot have negative values

Absolute zero point does not imply an end point, can still have infinitely many values

Can multiply/divide

Celsius (interval) vs. Kelvin (ratio).

°C = K-273.15.
0 °C: Freezing point of water

Drops below 0 °C all the time

0 K: "Absolute Zero"

Physically cannot get any colder

100 °C is not ∞% warmer than as 0 °C

It's actually ~ 36% warmer
(373.15 K - 273.15 K) ⁄ 273.15 K ~ 0.36

Interval data has an arbitrarily zero point. Examples include:

Calendar years

Discrete interval data

Temperature (in celsius)

Continuous interval data

Other examples:

ph scale (continuous)
IQ scores (continuous)
Times (discrete-ish)

Ratio data has a fixed, absolute zero point. Examples include:

Population

Discrete ratio data

Tree height

Continuous ratio data

Other examples:

Precipitation (Continuous)
Area (Continuous)
Units of time (Continuous)
Popular Vote Totals (Discrete)

TopHat Question 3

Match the value to the type measurement scale and type of number:

Length a hiking trail	Interval (Discrete)
Temperature in Fahrenheit	Ratio (Discrete)
Global Orca Population	Ratio (Continuous)
Change in Global Orca Population from 2000 to 2022	Interval (Continuous)

Derived Ratio

Sometimes called normalizing or standardizing, we calculate derived ratios to account for the influence of a confounding variable over a variable of interest. e.x. Housing affordability (H_a):

You need to account for income (I) to figure out how affordable rent (R) is:

$H_a = \frac{R}{I}$

R: My rent $1,450/mo
I: I make ~$4600/mo
H_a: 31.5% of my income goes to rent
Income and rent ($) are both discrete, housing affordability (%) is continuous.

Derived Ratio

In Lab, you are going to work with two derived ratios:

Income and Food expenditures are correlated

Need to account for income if you analyze other factors

Population and Area are not highly correlated

But area definitely influences population
Need to account for area to analyze other factors

TopHat Question 4

Speed is another example of a derived ratio. If a line of thunderstorm takes 5 hours to travel from Brandon, MB to Winnipeg, MB (200 km), what is the storm's speed in km/hr?

Operation	Nominal	Ordinal	Interval	Ratio
Equality	x	x	x	x
Counts/Mode	x	x	x	x
Rank/Order		x	x	x
Median		~	x	x
Add/Subtract			x	x
Mean			x	x
Multiply/Divide				x

What is Data?