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

Attribute Description extremes range 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. histogram 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. bell curve mean For a large number of records with random errors in their measurement, the histogram resembles a bell curve and is symmetrical about the mean.

Describing a classed raster grid % (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 Readings Data ExtremeDateTimeDMSDMS Elev Minimum 6/14/9510:47am Maximum 6/15/9510:47pm Range 1 Day12 hours

Mean Statistical average Sum of the values for one attribute divided by the number of records X i i1= 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.

l Average difference from the mean l 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 ) 2 i  n - 1 Standard Deviation

GPS Example Data: Elevation Standard 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 meters Elevation is most likely to lie between meters and meters. These limits are called the error band or margin of error.

Standard Deviations and the Bell Curve Mean One Std. Dev. below the mean One Std. Dev. above the mean

Testing Means (1) Mean elevation of meters Standard deviation meters What is the chance of a GPS reading of meters? is 25.3 meters above the mean 0.31 standard deviations ( Z-score) of the curve lies between the mean and this value beyond it

Mean % % Testing Means (2)

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 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 dependent variable independent variable observation best fit regression line y = a + bx intercept gradient y=a+bx

Testing the relation gr = 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... MapsStatistics Correlation is not causation!

Terrain Analysis Paula Messina

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 ?

Interpolation Using the Neighborhood Model 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

x Zx =Zx =  Z p d p -n P = 1 R  d p -n P = 1 R Z x = elevation at kernal (point x) Z p = elevation at known point p d p = distance from point x to point p n = “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 calls this 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 x 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 as a menu item in 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 x 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, 2/00Shuttle Radar Topography Mission SIR-C, 1994SIR-C Other Orbiters Magellan Mapping Mission of Venus, Click here to see an animation of the Venutian surface topographyMagellan Mapping Mission of VenusClick here Airborne Radar Mappers AirSAR/TopSARAirSAR/TopSAR GeoSAR: California mappingGeoSAR Click hereClick here to link to Hunter College’s Radar Mapping Web Site

How is Slope Computed? Slope = arctan [ ( ) 2 + ( ) 2 ] Grid cell = 100m x 100m dZ dX dZ dY Calculate the slope for the central pixel. Click here Click here for the solution.

How is Aspect Computed? Aspect A’ = arctan - ( ) ( ) Grid cell = 100m x 100m dZ dY dZ dX Calculate the aspect for the central pixel. Click here Click here for the solution. 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 dZ dX dZ dY dZ dY dZ dX dZ dX