Multiple Regression in GIS

1
Multiple Regression in GIS

• Review of Regression LabAmares neighbor problem

2
Multiple Regression(Chou p. 276)

• Explaining the distribution of a spatial
  phenomenon requires the analysis of relationships
  between the phenomenon and potential explanatory
  variables
• The two most useful methods in GIS spatial
  analysis are multiple regression analysis and
  logistic regression analysis
• Regression analysis is used to examine the
  relationship between the study phenomenon and
  multiple explanatory variables
• Y b0 b1X1 b2X2 bnXn e

3
Multiple Regression

• Y b0 b1X1 b2X2 bnXn e
• Where Y denotes the dependent variable and Xs are
  explanatory variables b0 represents the
  intercept, while bn denotes estimated parameter
  of the variable Xn and e is the randomly
  distributed error term
• Single most widely used statistical technique in
  the social sciences
• Multiple regression can be implemented in matrix
  form in the following MathCAD example
• see MathCAD example

4
Multiple Regression

• We are going to now kick things up a notch. In
  many cases, there is more than one independent
  variable to explain a dependent variable.
• For example, income is probably not just related
  to education level. It may also have something
  to do with the number of years a person works, or
  the industry a person is in.
• For example, in our sample plot, there were three
  points that were along the same X axis. How can
  this be possible, if we give one treatment, we
  yield? Its probably because there is some error
  as we discussed before, but, its very likely that
  there are some other explanatory variables that
  we havent accounted for.

5
Multiple Regression

• Multiple regression is simply an extension of the
  simple linear model. The difference is that
  there are more independent variables
• The assumptions for multiple regression are
  similar to simple linear regression
• The average value of the dependent variable Y is
  a linear combination of the independent
  variables.
• The only random component is the error term e ,
  and the independent variables are assumed to be
  fixed, and independent of e.
• Errors between observations are uncorrelated, and
  normally distributed with a mean of zero and a
  constant variance s2.

6
Multiple Regression

• While we were able to solve for a single
  explanatory variable using a spreadsheet, as we
  add more explanatory variables, it will be harder
  to solve. This is especially true when the
  number of independent variables is greater than
  3.
• However, through the use of matrices, the task is
  far less difficult. We can structure the
  regression equations into a matrix as follows
• where X is the observation matrix of independent
  variables, and b is the vector of unknown
  parameters.

7
Multiple Regression

• So, if we had 4 observations, and three
  explanatory variables, our matrices would look
  like the following
• The reason we have a 1 in the first column is
  because we have to include the intercept
  parameter. Therefore, we really have 4 unknowns
  to solve for (the coefficient for each
  explanatory variable, and the slope)
• Using basic least square with matrix algebra is
  fairly simple, once you have a computer program
  to do the work, we simply solve for the following

Matrix of dependent variables
8
Multiple Regression

• We will not be performing the math, but it will
  be useful to create the matrices, just to see how
  it all gets formed.
• In our example, suppose a gas utility company is
  trying to estimate revenue. They may have
  determined that heating cost is a function of the
  temperature, the amount of insulation in an
  attic, and the age of a furnace. They decided to
  look at 20 customer sites, and quantify the data
  as shown

9
Multiple Regression

• We now need to store this information in a matrix
  as follows (we are only going to do the first 4
  rows, just to make it simple
• You can see how for the first four rows, we have
  defined the monthly cost for the heating, the
  intercept, temperature, insulation, and age of
  furnace.
• Now, using least square principles with matrix
  algebra, we can come up with our unknown
  coefficients.

10
Multiple Regression

• Microsoft Excel has an excellent regression tool
  for relatively small problems. You will find it
  under the tools -gt data analysis tab.
• Once you select the tool, an interactive dialog
  will come up stepping you through the regression
  wizard.
• Here is where you will enter the range for the Y
  value (a single column), and the X values
  (multiple columns) as shown below.

11
Multiple Regression

• You should type the numbers into Excel, and
  attempt to perform the regression yourself.
  Check your answers against ours.
• What this tells us is that our R squre value is
  quite high (.80) representing a good fit, and we
  have a standard error of 51 (in dollars) for our
  20 observations.

12
Multiple Regression

• The next chart tells us our coefficient values
  (intercept, temperature, insulation, age of
  furnace). It also tells us our P-value, or a
  measure of significance. All the values except
  age of furnace are very low, meaning that they
  are all significant at the 95 level.
• So, what we now have is a formula
• Cost to Heat Home 427 -4.58(temperature)
  -14.83(attic insulation) 6.101(age of the
  furnace).
• Therefore, if a person with no attic insulation
  decided to add 12 inches, what would they save
  when the average temperature if 12 degrees?

13
What it means

• The intercept is 427.194. This the cost of
  heating when all the independent variables are
  equal to zero.
• The regression coefficients for the mean
  temperature and the amount of attic insulation
  are both negative. This is logical as the
  outside temperature increases, the cost of
  heating the house will go down.
• For each degree the mean temperature increases,
  we expect the heating cost to decrease 4.583 per
  month.
• P-value for all the coefficients are significant
  for ?0.05 except for the coefficient of the
  variable age of furnace(ß3). Hence, we can
  conclude that they are significantly different
  from zero.
• However, if we examine the p-value for the
  variable age of furnace, we see that it is not
  significant at ?0.05. Hence, we cannot conclude
  that it is significantly different from zero.
• In that case, we can drop this variable from the
  model. Lets see what happens if we drop the
  age of furnace variable from the model

14
Multiple Regression

• Rather than rerunning things, well go with the
  first conclusions
• Cost to Heat Home 427 -4.58(temperature)
  -14.83(attic insulation) 6.101(age of the
  furnace).
• Cost to Heat Home 427 -4.58(12) -14.83(0)
  6.101(6).
• 408
• Cost to Heat Home 427 -4.58(12) -14.83(12)
  6.101(6).
• 230
• A utility company could then use this information
  to determine how much revenue they would generate
  if they provided service to a neighborhood.

15
Using Geography in Multiple Regression

• GIS is a great tool for obtaining the explanatory
  variables. For example, consider the following
  problem to solve.
• Assume that an environmental remediation company
  wants to know how much phosphorous is being
  dumped in a lake.
• If they had all the data together, they could
  develop a regression model to predict the amount,
  and then prescribe different land use options for
  reducing the load.
• Lets assume that the company determined that the
  following information is a pretty good predictor
  of phosphorous loading
• The landuse developed land, and dairy farms
  will have more phosphorous than forest
• Distance to a stream areas near a stream will
  be more likely to load into the lake
• The soil type certain soil types and their
  erosion factors may play a role in the amount of
  loading.
• Slope areas uphill from the water will be more
  likely to load into the lake.

16
The data

• Here we have a picture of a number of phosphorous
  sampling sites, along streams right near the
  lake. And the landcover data.

17
The data

• The soil types and watershed boundaries for each
  sample site (that is, all the areas that pour
  into the sample site)

18
The data
Buffering the stream by 200 meters
19
The data

• Using basic GIS tools, we can determine the area
  of each watershed, and, we can overlay the land
  use, and figure out each of the following
• Area of each land use type per watershed
• Area of each land use type within each 200 meter
  buffer of the stream
• Area of each soil type per watershed
• The average slope of each watershed

20
Spatial analysis results cont.Land use () for
each sub-basin
21
Spatial Analysis results cont..Land use ()
within 200m buffer
22
Soil Type Coverage per sub-basin
23
Soil Type Coverage per sub-basin (200m buffer)
24
Slope Average (parameter for regression)

• Slope average for each sub-basin was calculated
  from the DEM
• This may give an indication of runoff, the
  steeper the slope the higher the runoff.

25
Putting it together

• Once all the tables are categorized for each
  sample site, the matrices can be formed with the
  dependent variable being the loading, and the
  independent variables being the land use,
  soil type, average slope, etc..
• Then, the company would just run a regression
  analysis like we did earlier, and obtain the
  coefficients.
• After the coefficients are obtained, the users
  can modify the values (assuming the coefficients
  yield good results) such as converting some
  agricultural land to forest, and then see how
  that may or may not impact the phosphorous
  loading in the lake.

26
Example

• Examining the relationship between bird
  distribution and selected environmental variables
• An example from Chou

27
Data
CLICK FOR EXCEL SHEET
28
Comments on Data

• The nominal variables (species and vegetation)
  not included in the model
• Can observation frequency of birds be explained
  by forest density, proximity to rivers, and
  slope?

29
Results of Analysis

• Which variables in the model are significant, and
  which ones are not necessary
• An efficient spatial model incorporates a small
  number of critical variables while generating a
  sufficiently accurate prediction

30
Results of Analysis

• According to statistical tables, the critical
  value of t is 2.101 (a .025 for d.f. 18)
• Therefore, only Forest Cover is significant in
  the distribution of birds in this study.

31
Theoretical Formation of Multiple Regression
using Matrices
Watch this one. With spatial data we often
violate this assumption. More about this in a
few weeks.
32
(No Transcript)
33
(No Transcript)
34
Here we are going to rerun the bird regression
example using matrices
35
(No Transcript)