Why is it there? (How can a GIS analyze data?) Getting Started, Chapter 6 Paula Messina.

Why is it there?

(How can a GIS analyze data?)

Getting Started, Chapter 6

Paula Messina

GIS is capable of data analysis

Attribute Data Describe with statistics Analyze with hypothesis testing

Spatial Data Describe with maps Analyze with spatial analysis

Describing one attribute

Flat File Database

Record Value Value Value

Attribute Attribute Attribute

Record Value Value Value

Record Value Value Value

Attribute Description

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

A histogramhistogram 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.

For a large number of records with random errors in their measurement, the histogram resembles a bell curvebell curve and is symmetrical about the meanmean.

Describing a classed raster grid

5

10

15

20

% (blue) = 19/48

If the attributes are:

Numbers statistical description min, max, range variance standard deviation

Statistical description

Range : max-min Central tendency : mode, median,

mean Variation : variance, standard

deviation

Statistical description

Range : outliers mode, median, mean Variation : variance, standard deviation

Elevation (book example)

GPS Example Data: Elevation

Table 6.2: Sample GPS ReadingsData Extreme DateTime D M S D M S ElevMinimum 6/14/95 10:47am 42 30 54.8 75 41 13.8 247Maximum 6/15/95 10:47pm 42 31 03.3 75 41 20.0 610Range 1 Day 12 hours 00 8.5 00 6.2 363

Mean

Statistical average Sum of the values

for one attribute divided by the number of records

X i

i 1=

n

= X / n

Variance

The total variance is the sum of each record with its mean subtracted and then multiplied by itself.

The standard deviation is the square root of the variance divided by the number of records less one.

Average difference from the mean

Sum of the mean subtracted from the value for each record, squared, divided by the number of records-1, square rooted.

st.dev. =(X - X )

2i

n - 1

Standard DeviationStandard Deviation

GPS Example Data: ElevationStandard Deviation

Same units as the values of the records, in this case meters.

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

meters 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.

Standard Deviations and the Bell Curve

Mean

459.

2

542.

1

376.

3

One Std. Dev.below the mean

One Std. Dev.above the mean

Testing Means (1)

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

Mean

12.17 %

37.83 %

Testing Means (2)

459.

2

484

.5

Accuracy

Determined by testing measurements against an independent source of higher fidelity and reliability.

Must pay attention to units and significant digits.

Not to be confused with precision!

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 is different from statistics because the results can be placed onto a map for visual analysis.

Spatial Statistical Description For coordinates, the means and

standard deviations correspond to the mean center and the standard distance

A centroid is any point chosen to represent a higher dimension geographic feature, of which the mean center is only one choice.

Spatial Statistical Description For coordinates, data extremes

define the two corners of a bounding rectangle.

Geographic extremes

Southernmost point in the continental United States.

Range: e.g. elevation difference; map extent

Depends on projection, datum etc.

Mean Center

mean y

mean x

Centroid: mean center of a feature

Mean center?

Comparing spatial means

Spatial Analysis

Lower 48 United States 1996 Data from the U.S. Census on

gender Gender Ratio = # females per 100

males Range is 96.4 - 114.4 What does the spatial distribution

look like?

Gender Ratio by State: 1996

Searching for Spatial Pattern A linear relation is a predictable straight-

line link between the values of a dependent and an independent variable. (y = a + bx) It is a simple model of correlation.

A linear relation can be tested for goodness of fit with least squares methods. The coefficient of determination r-squared is a measure of the degree of fit, and the amount of variance explained.

Simple linear relation

dependentvariable

independent variable

observationbest fitregression liney = a + bx

intercept

gradient

y=a+bx

Testing the relation

gr = 117.46 + 0.138 long.

GIS and Spatial Analysis

Geographic inquiry examines the relationships between geographic features collectively to help describe and understand the real-world phenomena that the map represents.

Spatial analysis compares maps, investigates variation over space, and predicts future or unknown maps.

Many GIS systems have to be coaxed to generate a full set of spatial statistics.

You can lie with...

MapsMaps

StatisticsStatisticsCorrelation is not causation!

Terrain Analysis

Paula Paula MessinaMessina

Introduction to Terrain Analysis What is terrain analysis? How are data points interpolated to

a grid? How are topographic data sets

produced from non-point data? How are derivative data sets (i.e.,

slope and aspect maps) produced by ArcView?

What is Terrain Analysis? Terrain Analysis: the study of ground-

surface relief and pattern by numerical methods (a.k.a geomorphometry).

Geomorphology qualitative

Geomorphometry = quantitative

Interpolation to a Grid

Assumptions: Elevations are continuously distributed The influence of one known point over an

unknown point increases as distance between them decreases

58

46

97

86

70

58

46

86

7097

?

Interpolation Using the Neighborhood Model

58

46

86

7097

Inverse-Distance theory dictates: The value of X > 58 The value of X < 97 The value of X is

closer to 58 than 97

x

58

46

86

7097

x

Zx =

Zp dp-n

P = 1

R

dp-n

P = 1

R

Zx= elevation at kernal (point x)

Zp = elevation at known point pdp = distance from point x to point pn = “friction of distance” value; usually between 1 and 6

Neighborhood Interpolation Using Inverse Distance Weighting

When n=2, the technique is called “inverse-squared distance weighting.”

ArcGIS callsthis IDW

Types of “Neighborhoods” used with IDW

Nearest n Neighbors in this example, n = 3 this method isn’t effective when

there are clusters of points “nearest in quadrants,” and

“nearest in octants” searches can help

Fixed Radius a radius is selected points are selected only if they

lie within that fixed radius

58

46

86

7097

x

46

86

97

58

70

x

Interpolation using the Spline Method

The spline interpolator fits a minimum-curvature surface through input points. “Rubber sheet fit”

The spline interpolator fits a mathematical function to a specified number of nearest points

Interpolation Using Kriging Based on regionalized variable

theory Drift, random correlated component,

noise This method produces a

statistically optimal surface, but it is very computationally intensive

Kriging is used frequently in soil science and geology

Trend Interpolator Fits a mathematical function (a

polynomial of specified order) to input points Points may be chosen by nearest neighbor or radius

searches --or-- All points may be used

Uses a least-squares regression fit The surface produced does not

necessarily pass through the points used This is an excellent choice when data points are sparse

Not available asa menu itemin ArcGIS

Which ArcView menu interpolator is better?

IDW Assumption: The variable being mapped

decreases in influence with distance• Example: interpolating consumer purchasing

power for a retail site analysis

Spline Assumption: The variable being mapped is a

smooth, continuous surface; it is not particularly good for surfaces with

large variability over small horizontal distances

• Examples: terrain, water table heights, pollution concentration, etc.

The Finished Grid

The Messina “Eyeball” Interpolator was used

58

46

86

7097

x

56 58 65 74

46 56 54

86 84 80

70 75 78 86 94 94 80

66 69 73 80 90 88 86

72 76 80 84 90 89 84

50 52 60 64 68 80 80

48 50 54 56

46 48 50 52 46 46 44

Grids are subject to the “layer cake effect”

Point Data Collection in the Field It is critical to obtain data at the

corners of the grid extent It is advisable to obtain the VIPs

(Very Important Points) such as the highest and lowest elevations

Other Continuous Surface Sources

USGS DEMs produced directly from USGS Topographic Maps Elevations of an area are averaged within the grid cell

(pixel) High and low points can never be saved as a grid cell

value Various techniques (i.e. stereograms) were used to

accomplish this process Original datum (i.e. NAD27, NAD83) is preserved in the

DEM Spatial resolution: 30m (7.5 minute data), 1 arc-second (1 degree data), 10m*, 5m* *(limited

coverage)

Other Continuous Surface Sources

Synthetic Aperture Radar, Side-looking Airborne Radar Shuttle Missions:

• Shuttle Radar Topography Mission, 2/00• SIR-C , 1994

Other Orbiters• Magellan Mapping Mission of Venus,

1990-1994 Click here to see an animation of the Venutian surface topography

Airborne Radar Mappers• AirSAR/TopSAR• GeoSAR: California mapping

Click here to link to Hunter College’s Radar Mapping Web Site

How is Slope Computed?

Slope = arctan [( )2+( )2]

100 130 140

120 150 160

160 170 200

Grid cell = 100m x 100m

dZdX

dZdY

Calculate the slopefor the central pixel.Click here for thesolution.

How is Aspect Computed?

Aspect A’ = arctan -( ) ( )

100 130 140

120 150 160

160 170 200

Grid cell = 100m x 100m

dZdY

dZdX

Calculate the aspectfor the central pixel.Click here for thesolution.

If is negative, add 90 to A’

If is positive, and is negative: add 270 to A’

If is positive, and is positive: subtract A’ from 270

dZdX

dZdY dZdY

dZdXdZdX