Logistic Regression

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Logistic Regression

• Often, the spatial phenomenon under investigation
  can only be described by a categorical variable.
• Wild fires typically depicted with polygons
  showing burned vs. not burned
• Or, bird distribution indicating presence or
  absence of birds
• Previous regression technique is not suitable
  because the dependent variable is neither
  interval or ratio
• Logistic regression treats the distribution in a
  probabilistic manner, that is, the occurrence of
  the study phenomenon is evaluated in terms of
  probability

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Logistic Regression

• If the probability of presence of a phenomenon is
  Pa, then Pb represents the absence of the
  phenomenon and
• Pa Pb 1
• Ua b0 b1X1 b2X2 bnXn e
• Ua is the utility function of event a expressed
  as a linear combination of a number of
  explanatory variables X1, X2, .., and bn is the
  estimated parameter of variable Xn

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Logistic Regression

• A greater value of Ua implies a greater
  probability for the event to take place. When Ua
  approaches infinity, Pa approaches 1, indicating
  a high likelihood for the event to occur. When
  Ua approaches negative infinity, Pa approaches 0.
• When Ua equals zero, the probability is .50,
  implying a 50/50 chance for the event to occur.

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Logistic Regression Example

• Example from Chou
• Fires in San Jacinto Ranger District of the San
  Bernardino National Forest were examined to map
  the distribution of fire occurrence probability.
  The basic model consisted of eight independent
  variables
• Area, perimeter, vegetation, proximity to
  buildings, proximity to campgrounds, proximity to
  roads, maximum temperature in July, and annual
  precipitation

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Variables in Fire Distribution Study

• X1 Area area of geographic unitX2 Perimeter
  perimeter of geographic unitX3 Vegetation
  vegetation computed by rotation
  periodX4 Building proximity to
  structuresX5 Campground proximity to
  campgroundsX6 Road proximity to
  roadsX7 Temperature maximum temperature in
  JulyX8 Precipitation annual precipitation
• Dependent variable is a code indicating whether
  or not a geographic unit is burned or not. Area
  and perimeter provide general geometric
  characteristics. Vegetation, precipitation, and
  temperature represent environmental factors,
  while building, campground, and road represent
  human-related factors

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Results of Logistic Regression

• The model indicatesthat perimeter, vegetation,
  campground, road, and temperature are variables
  to be included in the model. Other variables are
  not included as they are not statistically
  different from 0

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Results of Logistic Regression

• Percentage-correctly-estimated (PCE) index shows
  the maximum level of estimation accuracy of a
  model.
• In this example, PCE is 60, not much better than
  a random 50/50 chance.
• Therefore, another parameter was evaluated

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Alternative Model

• Included an additional variable to determine
  whether it makes any significant difference in
  model performance
• New variable represents neighborhood effects, or
  conditions of the surrounding geographic units
• Assumes that fire occurrence probability is not
  only affected by the environmental and
  human-related variables listed in the basic
  model, but by the distribution of fire occurrence
  probability of adjacent units
• The new spatial term X9 is defined by the
  percentage of neighboring units that were burned
  during the study period

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New Results

• Results from the new study are quite different
• Only two variables are statistically significant
  vegetation and neighborhood effects
• Vegetation appears to be the determining
  environmental variable in the distribution of
  wildfires in the study area
• Finally, wildfires are influenced by neighborhood
  conditions

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Testing Statistical Signficance

• Did the neighborhood effects significantly change
  the model? Need to test the chi-square test of
  likelihood ratio
• Where L0 denotes the likelihood of the basic
  model and L1 denotes the likelihood of the study
  model
• Statistical testing suggests that the
  neighborhood variable significantly improved the
  performance of the model

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Procedure for Regression Analysis (Barber, p. 448)

• Specify the variables in the model and the exact
  form of the relationship between them
• Collect data
• Estimate the parameters of the model
• Statistically test the utility of the developed
  model, and check whether the assumptions of the
  simple linear regression model are satisfied
• Use the model for prediction

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Example of Data Manipulation and Programming in
ArcView

• Manipulating Yield Data with DataManipulation.ave

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Spatial Prediction of Landslide Hazard Using
Logistic Regression and GIS

• Art Lembo
• 620 Presentation
• Based on paper by Gorsevski, Gessler, and Folz

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Introduction

• Landslides are natural geologic processes that
  cause different types of damage, causing billions
  of dollars in damage and thousands of deaths each
  year
• 95 of landslides occur in developing countries

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Causes of Landslides

• Human activities, such as deforestation and urban
  expansion, accelerate the process of landslides
• Roads and harvest activities in timberlands
  increase the occurrence of landslides
• In undisturbed forest, soil erosion is generally
  negligible

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Clearwater National Forest

• 1995-1996
• Major landslides occurred during the winter
  following heavy rains, snowmelt, and high river
  flow
• Over 900 landslides were recorded on the unstable
  slopes of the forest
• Landslide occurrence was widely distributed and
  included artificial slopes such as road cuts and
  fills, or natural slopes in clearcut areas

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Landslide Data

• Within the large remote area, a DEM was used to
  generate quantitative topographic attributes
• Slope, elevation, aspect, profile, curvature,
  tangent curvature, plan curvature, flow path, and
  contributing area
• Photo interpretation and field inventory
  identified landslide areas

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Considerations in Creating Hazard Models

• Datasets combined and stored in a GIS database
• Hazard Model assumptions
• Strength of a model depends on the quality of the
  data collected
• Data driven models are not appropriate to
  extrapolate to neighboring areas
• Climatic conditions may change so that the past
  is not an indicator of the future
• Uncertainty exists when a hazard map is derived
  from a statistically based model

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Models Used in Study

• Logistic regression was used, which correlated
  the environmental attributes and landslide
  distribution
• Because of the existence of uncertainty, a
  Receiver-Operating Curve curve plots the
  proportion of false positives against the true
  positives at each level of the criterion

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Assessing Landslide Hazard

• Field inspection using a check list to identify
  sites susceptible to landsliding
• Projection of future patterns of instability from
  analysis of landslide inventories
• Multivariate analysis of factors characterizing
  observed sites of slope instability
• Stability ranking based on criteria such as
  slope, land forms, or geologic structure
• Failure probability analysis based on slope
  stability models with stochastic hydrologic
  simulation

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Preparing the Data

• Primary and secondary attributes are derived from
  a DEM, reducing the high cost of collecting the
  data (30m)
• Landslides assessed through aerial reconnaissance
• Landslide hazard area are then identified based
  on spatial correlation between the attributes
• Identifying landslide hazard is based on spatial
  correlation between the attributes derived from
  the DEM
• ROC curves used for decision making

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Data Sampling

• 15 of non-landslide cells were randomly sampled
  for an absence of landslides
• Multivariate subset was derived from the
  coverages where landslides were absent
• The landslide coverage was a point data set
  sampled grid cells where landslides were present
• Both samples were joined together where the
  dependent variable had a binary response (present
  or absent)
• Final output stored in ASCII and used in SAS

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Statistical Analysis

• Normal plot of data to determine if the data
  followed a normal distribution
• Plot showed that data points do not fall along a
  straight line. The data is not multivariate
  normal
• Logistic regression is used when the predictor
  variables are not normally distributed, and
  some predictor variables are categorical
• Factor analysis was applied to determine the
  number of underlying variables
• Only significantly loaded variables were
  considered

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Statistical Analysis

• The form of the logistic regression model is
  defined as
• Where x is the data vector for a randomly
  selected experimental unit and y is the value of
  the binary outcome variable. Maximum likelihood
  was used to estimate B for the predictive
  equation
• Variables not significant at the .1 level were
  eliminated

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Logit Results

• Logit showed that the most important variables
  contributing to the slope instability were Flow
  Path and mean slope of upland area
• log (p/(1-p)) (-2.2642 FACTOR8 0.4969
  FLPATH 0.6039)          or p exp (-2.2642
  FACTOR8 0.4969 FLPATH 0.6039)/(1
  exp(-2.2642 FACTOR8 0.4969 FLPATH
  0.6039)__________________________________________
  ____________________p probability of
  landslide hazard FACTOR8 factor with
  underlying characteristics of aspectFLPATH
  Maximum distance of water to the point in the
  catchment

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Logit Results

• Coefficients of Logit model included positive
  coefficients. Therefore, higher scores would
  increase the probability of landslide hazard.
• Logit model assumes a nonlinear relationship
  between the probability and the explanatory
  variables
• Hazard map based on ROC curve technique groups
  the hazard into two classes Low Hazard and High
  Hazard, showing five classes of probabilities of
  landslide hazard

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Final Results

• 59.1 of the landslides and 69.8 of non
  landslides were correctly determined
• Model can be applied to large geographic areas
• ROC curves are incorporated as a sophisticated
  tool for decision makers for the spatial
  prediction of landslide hazard

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a) Cut-off based on ROC curve technique b)
Probability of landslide hazard