mahalanobis distance
Download
Skip this Video
Download Presentation
Mahalanobis distance

Loading in 2 Seconds...

play fullscreen
1 / 27

Mahalanobis distance - PowerPoint PPT Presentation


  • 215 Views
  • Uploaded on

Mahalanobis distance. A theoretical and practical approach. Preview. An introduction of Mahalanobis distance Our project: Methodolgy Results A demonstration in how to use Mahalanobis distance. Mahalanobis distance. Introduced by P. C. Mahalanobis in 1936

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Mahalanobis distance' - conroy


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
mahalanobis distance

Mahalanobis distance

A theoretical and practical approach

preview
Preview
  • An introduction of Mahalanobis distance
  • Our project:
      • Methodolgy
      • Results
  • A demonstration in how to use Mahalanobis distance
mahalanobis distance1
Mahalanobis distance
  • Introduced by P. C. Mahalanobis in 1936
  • A distance measure: based on correlations between the variables and by which different patterns could be identified and analyzed with respect to base or reference point (Taguchi & Jugulum, 2002)
mahalanobis distance2
Mahalanobis distance
  • M.D. is a very useful way of determining the ”similarity” of a set of values from an ”unknown”: sample to a set of values measured from a collection of ”known” samples
  • Superior to Euclidean distance because it takes distribution of the points (correlations) into account
  • Traditionally to classify observations into different groups
mahalanobis distance3

z

w

p

Ecological

Distance

x

y

r

Mahalanobis distance
mahalanobis distance4
Mahalanobis distance
  • D2t(x) = (x – mt)S-1t(x – mt)`
      • Dt is the generalized squared distance of each pixel from the t group
      • St represents the within-group covariance matrix
      • mt is the vector of the means of the variables of the t group
      • X is the vector containing the values of the environmental variables observed at location x
mahalanobis distance5
Mahalanobis distance
  • The result of using this algorithm (with GIS) is a single raster with the value of ecological distance from the species’ ”optimal” conditions; the higher the distance, the less suitable the pixel’s ecological conditions
mahalanobis vs other classical statistical approaches
Mahalanobis vs. other classical statistical approaches
  • 1. It takes into account not only the average value but also its variance and the covariance of the variables measured
  • 2. It accounts for ranges of acceptability (variance) between variables
  • 3. It compensates for interactions (covariance) between variables
  • 4. It is dimensionless
  • 5.If the variables are normally distributed they can be converted to probabilities using the x2 density function
our project
Our project

Reports for the Large Predator Policy Statement.

Potential habitat for large carnivores in

Scandinavia; a GIS analysis at the

ecoregion level.- NINA Fagrapport 064

methodology
Methodology
  • Potential suitability maps
  • Species involved: bear (Ursus arctos), wolf (Canis lupus), lynx (Lynx lynx), and wolverine (Gulo gulo)
  • Two datasets

1. A given set of environmental variables in which we thought to influence the large carnivores distribution

2. A training set consisting of known data on the presence of the large carnivores today

variables
Variables
  • Landcover (1km x 1km): derived from an AVHRR image and put together with an elevationmodel (100m x 100m)
    • Reclassified into 6 classes
      • Water
      • Forest
      • Cultivated land
      • Mountain
      • Alpine tundra (above 550 meters classified to mountain)
      • Ice/snow/bare mountain)
variables1
Variables
  • Human density (SSB)
    • number of humans per square kilometers
    • Finland: humans linked to buildings
    • Norway: humans linked to adresses (GAB)
    • Sweden: humans linked to estates
  • Infrastructure
    • harmonization of public roads and private roads in Sweden and Norway
    • Railway
variables2
Variables
  • Prey density
    • Based on maps with average shot moose, roe deer and deer per county (kommune) and wild reindeer per wildreindeer management area
    • Created an index based on each species preference for the prey species (Solberg et al. 2003)
    • Example: lynx – 20% deer – 100% roedeer – 80% wild reindeer
training data
Training data
  • Core homeranges
      • Multiannual fixes from of radio-collared female bears older than 2years, from Sarek and Dalarna
      • Multiannual fixes from radio-collared female lynx older than 2 years, from Sarek, Grimsø, Nord Trøndelag, Hedmark and Østfold
      • Multiannual fixes from radio-collared female wolverines older than 2 years, from Sarek, Troms and the Snøhetta Plateau
      • Packranges of both radio-collared and snow-tracked wolves.
  • The point coordinates were estimated to homeranges with Ranges 6, using a minimum konveks polygon method 75 %
  • The core homeranges for each species were transformed to masks (a mask grid formes the outerlining of our reference area).
pre processing arcinfo
Pre-processing (ArcInfo)
  • All data were transformed into raster from vector (polygrid)
  • Grids with 1km x 1km resolution
  • Either constant (0/1) or continous
  • 16 bit
preprocessing arcinfo
Preprocessing (ArcInfo)
  • One projection!
  • Parameter til Lambert Azimuthal Equal Area
  • Units of Measure: meters
  • Pixel Size: 1000 meters
  • Radius of sphere: 6370997 m
  • Longitude of origin: 20 00 00 E
  • Latitude of origin: 55 00 00 N
  • False easting: 0.0
  • False northing: 0.0
focal operation neighboorhood
Focal operation (neighboorhood)
  • A circular window of 5 km radius ≈ 80km2
  • Smallest core area
  • Species perception of space (Salvatori 2003)
  • Smoothing executed with FOCALMEAN (Tomlin 1990)
focalmean
Focalmean

Example with human density around Indre Oslofjord

results
Results
  • One single grid with values from 0 – 900 001
  • The homerangemask is used to cut the reference dataset
  • The dataset is treated in S –plus (0 values are deleted, .33 and .66 quantiles)
  • The result grid is reclassified into these classes:
          • 1. 0 – 33%
          • 2. 33% - 66%
          • 3. 66% - max (inside the homerange)
          • 4. Max – ∞ (900 001)
validation of the results
Validation of the results
  • Overlay with pointdata on shot femalebears, lynx and wolverines, also observed lynx familygroups and registered wolverine natal dens
  • No available independent data on wolves
  • A historical dataset on bounty payments (skuddpremier) showed presence of large carnivores over the whole Scandinavian peninsula
conclusion
Conclusion
  • The result shows large non fragmented areas suitable for large carnivores
  • Over 90% of the total area is potentially suitable for reproductive females of the species; bear, wolf and lynx
  • About 48% of the total area is potentially suitable for wolverines
recomended references
Recomended references
  • Clark, J.D., Dunn, J.E. & Smith, K.G. 1993. A multivariate model of female black bear habitat use for a geographic information system. The Journal of Wildlife Management 57(3):519 – 526
  • Corsi, F., Sinibaldi, I. & Boitani, L. 1998 Large carnivores conservation areas in Europe; discussion paper for the Large Carnivore Initiative IEA – Istituto Ecologia Applicata, Rome
  • Corsi, F., Dupre, E. & Boitani, L. 1999. A large-scale model of wolf distribution in Italy for conservation planning. Conservation Biology 13:150 - 159
  • Knick, S. T. & Dyer, D. L. 1997. Distribution of black tailed jackrabbit habitat determined by GIS in Southwestern Idaho. Journal of Wildlife Management 61(1):75 – 85
  • Salvatori in prep. 2003
ad