GEO 465/565 - Lecture 11
Why is it There?

Recap....

models/structures (Maps as Numbers)

formats (Maps as Numbers)

data input (Map into Computer)

geocoding (Map into Computer)

retrieval & display (What is where?)

construction/mgmt ----> use

ANALYSIS (Why is there?)
description - arithmetic and statistics
modelling - encoding a model or algorithm (formula)

Fundamental Questions:
Why is it There?

How can two maps be compared with other??

How can variations in these geographic properties over a single area or GIS data set be described and analyzed??

How can we use what we have learned using the analysis to explain and therefore predict future maps of the geography in question

GIS is capable of data analysis

Attribute Data
Describe with statistics Analyze with hypothesis testing

Spatial Data
Describe with maps
Analyze with spatial analysis
statistics that have visual and geographic meaning

Describing one attribute
statistically (non-spatial) - GRAPHIC

Attribute Description
What are the extremes of the data?

The extremes of an attribute are the highest and lowest values, and the range is the difference between them in the units of the attribute.

A histogram is a two-dimensional plot of attribute values grouped by magnitude and the frequency of records in that group, shown as a variable-length bar.

Describing a classed raster grid - GRAPHIC

The Mean in a
Normal Distribution
Looking for a single representative value but random errors give us scatter - GRAPHIC

Alternative attribute histograms - GRAPHIC

The Mean
If errors are consistent, we can correct for them...

A representative value, and for measurements with normally distributed error, converges on the true reading.

More accurate than the median which is just one value, however it is actual data

A value lacking sufficient data for computation is called a missing value.

Mean

Statistical average

Sum of the values for one attribute divided by the number of records

How much variation can we we expect in our data?
How reliable is any one measurement?
How much does a value differ from the average?

The total variance is the sum of each record with its mean subtracted and then multiplied by itself.
shows how much #s disagree

The standard deviation is the square root of the variance divided by the number of records less one.
averages variance among records
square root so you end up with same units

Statistical description

Range (max - min, min, max,)

Central tendency (mean, median)

Variation (variance, standard deviation)


Statistical Testing:
How to go from sample to population?

A sample is a set of measurements taken from a larger group or population.

Sample means and variances can serve as estimates for their populations.

How likely is it that an estimated measurement will be correct?

GPS Example Data:
What is the Elevation? - GRAPHIC

GPS Example Data:
Elevation Standard Deviation

Elevation is the mean (459.2 meters) plus or minus the standard deviation or expected error of 82.92 meters

Average amount readings differ from the average

Can be above or below the mean

Elevation is most likely to lie between 376.28 meters and 542.12 meters.

These limits are called the error band or margin of error.

Is there a difference in measured elevation between 2 kinds of GPS receiver?
Testing Means

Mean elevation of 459.2 meters

standard deviation 82.92 meters

What is the chance of a GPS reading of 484.5 meters?

484.5 is 25.3 meters above the mean

0.31 standard deviations ( Z-score)

0.1217 of the curve lies between the mean and this value

0.3783 beyond it

As # of samples is different (16 vs. 19), estimate stdev of overall mean as sq root of 2 variances added together.
116 m difference - GRAPHIC

Testing the Mean

Mathematical version of the normal distribution can be used to compute probabilities associated with measurements with known means and standard deviations.

A test of means can establish whether two samples from a population are different from each other, or whether the different measures they have are the result of random variation.

Spatial (versus statistical) description:
The difference is the map

GIS data description answers the question: Where?

GIS data analysis answers the question: Why is it there?

GIS data description (spatial description) is different from elementary statistics because TWO attributes are involved, not ONE

Spatial Description

For coordinates, data extremes (range) are defined by the two corners of a bounding rectangle.

Mean Center
center of a geographic distribution
map equivalent of the mean - GRAPHIC

Centroid: mean center of a feature - GRAPHIC

Mean center?
point furthest from coastline?
center of all points making up a coastline?
center of bounding rectangle?
center of bounding circle? - GRAPHIC

Comparing spatial means
Standard distance = map equivalent of standard deviation
sq. root of sum of sq. distances - GRAPHIC

Spatial Statistical Description

For coordinates (2 attributes) the map or spatial equivalent of the range is the bounding rectangle

Mean and standard deviation correspond to the mean center and standard distance

A centroid is any point chosen to represent a higher dimensional geographic feature

The standard distance for a set of point spatial measurements is the expected spatial error.

GIS and Spatial Analysis

Descriptions of geographic properties are often verbal
points are sparse, shapes rounded, lines squiggly Quantitative measure can be devised, although few are computed by GIS.

Points - bounding rectangle, distance
Lines - # of points, length
Areas - area, perimeter, area of bounding rectangle, number of holes, etc.

Quantitative descriptions
of features (Fig. 6.8)

http://dusk.geo.orst.edu/gis/lec11.html

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