# Mahalanobis distance - PowerPoint PPT Presentation

1 / 27

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

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

Mahalanobis distance

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 - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

• 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 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

z

w

p

Ecological

Distance

x

y

r

### 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 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

• 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

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

• 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

• 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)

### Variables

• Human density (SSB)

• number of humans per square kilometers

• Finland: humans linked to buildings

• Sweden: humans linked to estates

• Infrastructure

• harmonization of public roads and private roads in Sweden and Norway

• Railway

### 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

• 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)

• 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)

• 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)

• A circular window of 5 km radius ≈ 80km2

• Smallest core area

• Species perception of space (Salvatori 2003)

• Smoothing executed with FOCALMEAN (Tomlin 1990)

### Focalmean

Example with human density around Indre Oslofjord

### 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

• 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

• 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

• 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