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Mahalanobis distance

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Mahalanobis distance

A theoretical and practical approach

- An introduction of Mahalanobis distance
- Our project:
- Methodolgy
- Results

- 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)

- 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

z

w

p

Ecological

Distance

x

y

r

- 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

- 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

- 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

Reports for the Large Predator Policy Statement.

Potential habitat for large carnivores in

Scandinavia; a GIS analysis at the

ecoregion level.- NINA Fagrapport 064

- 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

- 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)

- Reclassified into 6 classes

- 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

- 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

- 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.

- All data were transformed into raster from vector (polygrid)
- Grids with 1km x 1km resolution
- Either constant (0/1) or continous
- 16 bit

- 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

- A circular window of 5 km radius ≈ 80km2
- Smallest core area
- Species perception of space (Salvatori 2003)
- Smoothing executed with FOCALMEAN (Tomlin 1990)

Example with human density around Indre Oslofjord

- 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)

- 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

- 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

- 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