**3. Quantifying spatial-temporal change in home ranges**

An overview of the three methods presented is provided in Fig. 5.

Fig. 5. An overview of the three methods presented: STAMP, kernel density estimation (KDE) change detection, and the integration of STAMP and KDE change detection. T1 and T2 indicate time period 1 and time period 2, respectively.

Quantifying Wildlife Home Range Changes 275

Table 1. Area and proportion of spatial-temporal patterns of home range change from 2006

Traditional methods demonstrate that the Swan Lake caribou's home range declined from 2006 (1999.31 km2) to 2007 (1231.28 km2). The STAMP analysis indicates that while a decline in home range area dominates, in some regions new habitat was used. For instance, Fig. 2 indicates that caribou were using new habitat to the east. In addition to providing a more complete spatial representation of space-time habitat use, the results of STAMP are mappable. Mapped spatial-temporal patterns can be related to additional data sets in order to evaluate hypotheses associated with home range change. For example, associations between resources and space-time patterns may be hypothesized and tested by integrating

A method of change detection designed specifically for use with kernel density estimated surfaces is well suited to characterizing change in the intensity of habitat use within home ranges (Nelson et al., 2008). Kernel density estimation change detection identifies locations of statistically significant positive and negative changes, and enables the rate of change, considered significant, to vary spatially (Bowman & Azzalini, 1997, pp. 112-117). This method is a square root variance stabilizing transformation of the difference between two kernel density estimated surfaces, and is most appropriate for use when kernel estimates are generated using the same bandwidth (Bowman & Azzalini, 1997, pg. 114). The difference between the square root kernel density estimates at location *i,* for two time periods *t* and

> , ,1 2 2 1

∧ ∧

 λ

+

(3)

<sup>+</sup> is the kernel density

λ*i t*, 1 ∧

+

*se se t t*

+

*t*+1, change*iΔt*, is measured in terms of pooled standard deviations by calculating

Δ

is the kernel density estimate at location *i* in year *t*, and

over space. Therefore, the standard error is a constant defined as

change *it it i t*

λ

<sup>−</sup> <sup>=</sup>

estimate at the same location in the following year. *<sup>t</sup> se* and *<sup>t</sup>* <sup>1</sup> *se* <sup>+</sup> are the standard errors in

The standard error is a measure of the variance of the kernel function. Kernel density variance is dependent on the shape or curvature of the kernel, the search radius, and the total sample size. For traditional kernel density estimators, these parameters are invariant

the spatial-temporal patterns with resource availability data.

**3.2 Kernel density estimation change detection (method 2)** 

to 2007.

where

λ*i t*, ∧

the respective years.

 **Spatial-Temporal Patterns**  km2 % **Stable** 781.48 31.91 **Disappearance** 299.38 12.22 **Contraction** 918.45 37.50 **Generation** 142.63 5.82 **Expansion** 307.17 12.54 **Total** 2449.11 100.00

### **3.1 Spatial-temporal analysis of moving polygons (method 1)**

STAMP employs topological relationships of polygons to characterize spatial-temporal patterns of home range change between two time periods (*t* and *t*+1) (Sadahiro, 2001; Sadahiro & Umemura, 2001; Robertson et al., 2007). By intersecting home range polygons for two time periods, within a GIS, polygon relationships may be used to categorize space-time patterns of change. New polygons are produced by the intersection, and each is classified based on the polygon state (home range or not) in both time periods and the space-time patterns of adjacent polygons. Polygons are assigned to one of five pattern categories: stable, disappearance, contraction, generation, and expansion (Fig. 6). Stable patterns are locations where the home range is present in *t* and *t*+1. In stable locations there is consistent habitat use or site fidelity (e.g., Edwards et al., 2009). Disappearance and contraction patterns indicate that a location is part of a home range in *t* but not *t*+1. Disappearance patterns are spatially isolated, as opposed to contraction patterns which are spatially adjacent to other home range areas that have changed in a different way. Generation and expansion patterns both indicate that a location was not part of a home range in *t,* but became part of a home range in *t*+1. While generation patterns are spatially isolated, expansion events are spatially adjacent to home range areas that have changed in other ways. Disappearance, contraction, generation, and expansion all indicate different types of home range drift (e.g., Edwards et al., 2009).

Fig. 6. Spatial-temporal patterns in 2006 to 2007 caribou home ranges. Spatial-temporal patterns are defined by STAMP or topological relationships between home range polygons.

For the Swan Lake caribou, all five spatial-temporal patterns were identified (Fig. 6, Table 1). Contraction was the dominant pattern (37.50%), while generation was least common (5.82%). Stable patterns occurred for 31.91% of the home range area, and expansion and disappearance occurred in similar proportions, 12.54% and 12.22% respectively.

STAMP employs topological relationships of polygons to characterize spatial-temporal patterns of home range change between two time periods (*t* and *t*+1) (Sadahiro, 2001; Sadahiro & Umemura, 2001; Robertson et al., 2007). By intersecting home range polygons for two time periods, within a GIS, polygon relationships may be used to categorize space-time patterns of change. New polygons are produced by the intersection, and each is classified based on the polygon state (home range or not) in both time periods and the space-time patterns of adjacent polygons. Polygons are assigned to one of five pattern categories: stable, disappearance, contraction, generation, and expansion (Fig. 6). Stable patterns are locations where the home range is present in *t* and *t*+1. In stable locations there is consistent habitat use or site fidelity (e.g., Edwards et al., 2009). Disappearance and contraction patterns indicate that a location is part of a home range in *t* but not *t*+1. Disappearance patterns are spatially isolated, as opposed to contraction patterns which are spatially adjacent to other home range areas that have changed in a different way. Generation and expansion patterns both indicate that a location was not part of a home range in *t,* but became part of a home range in *t*+1. While generation patterns are spatially isolated, expansion events are spatially adjacent to home range areas that have changed in other ways. Disappearance, contraction, generation, and expansion all indicate different types of home range drift (e.g., Edwards et

Fig. 6. Spatial-temporal patterns in 2006 to 2007 caribou home ranges. Spatial-temporal patterns are defined by STAMP or topological relationships between home range

disappearance occurred in similar proportions, 12.54% and 12.22% respectively.

For the Swan Lake caribou, all five spatial-temporal patterns were identified (Fig. 6, Table 1). Contraction was the dominant pattern (37.50%), while generation was least common (5.82%). Stable patterns occurred for 31.91% of the home range area, and expansion and

**3.1 Spatial-temporal analysis of moving polygons (method 1)** 

al., 2009).

polygons.


Table 1. Area and proportion of spatial-temporal patterns of home range change from 2006 to 2007.

Traditional methods demonstrate that the Swan Lake caribou's home range declined from 2006 (1999.31 km2) to 2007 (1231.28 km2). The STAMP analysis indicates that while a decline in home range area dominates, in some regions new habitat was used. For instance, Fig. 2 indicates that caribou were using new habitat to the east. In addition to providing a more complete spatial representation of space-time habitat use, the results of STAMP are mappable. Mapped spatial-temporal patterns can be related to additional data sets in order to evaluate hypotheses associated with home range change. For example, associations between resources and space-time patterns may be hypothesized and tested by integrating the spatial-temporal patterns with resource availability data.

#### **3.2 Kernel density estimation change detection (method 2)**

A method of change detection designed specifically for use with kernel density estimated surfaces is well suited to characterizing change in the intensity of habitat use within home ranges (Nelson et al., 2008). Kernel density estimation change detection identifies locations of statistically significant positive and negative changes, and enables the rate of change, considered significant, to vary spatially (Bowman & Azzalini, 1997, pp. 112-117). This method is a square root variance stabilizing transformation of the difference between two kernel density estimated surfaces, and is most appropriate for use when kernel estimates are generated using the same bandwidth (Bowman & Azzalini, 1997, pg. 114). The difference between the square root kernel density estimates at location *i,* for two time periods *t* and *t*+1, change*iΔt*, is measured in terms of pooled standard deviations by calculating

$$\text{change}\_{i\Delta t} = \frac{\sqrt{\hat{\boldsymbol{\lambda}}\_{i,t}^{\hat{\boldsymbol{\alpha}}}} - \sqrt{\hat{\boldsymbol{\lambda}}\_{i,t+1}^{\hat{\boldsymbol{\alpha}}}}}{\sqrt{\mathbf{s}\boldsymbol{\varepsilon}\_{t}^{\hat{\boldsymbol{\alpha}}} + \mathbf{s}\boldsymbol{\varepsilon}\_{t+1}}^{2}} \tag{3}$$

where λ*i t*, ∧ is the kernel density estimate at location *i* in year *t*, and λ*i t*, 1 ∧ <sup>+</sup> is the kernel density estimate at the same location in the following year. *<sup>t</sup> se* and *<sup>t</sup>* <sup>1</sup> *se* <sup>+</sup> are the standard errors in the respective years.

The standard error is a measure of the variance of the kernel function. Kernel density variance is dependent on the shape or curvature of the kernel, the search radius, and the total sample size. For traditional kernel density estimators, these parameters are invariant over space. Therefore, the standard error is a constant defined as

Quantifying Wildlife Home Range Changes 277

As an example, when home ranges are binary, no change will be detected at a location associated with one telemetry point in the first time period and twelve in the second time period. The kernel approach, in contrast, has the potential to identify these as significant changes. As with STAMP, output of the kernel density approach is mappable, accounts for

In the final method, STAMP and kernel density estimation change detection are integrated to characterize how wildlife intensity varies for different spatial-temporal home range patterns. The amount of statistically significant change in intensity can be determined for each spatial-temporal home range pattern. For disappearance and contraction, only negative change or decreasing use will occur; for generation and expansion, only positive change or increasing use will occur; and for stable patterns, both types of change may be present. With the caribou data, stable patterns have both positive and negative change, and differentiating the types of change identifies how intensity of use is varying within areas used consistently through time (Table 2). In the caribou example, 3.73% and 12.99% of stable pattern areas had statistically significant positive and negative change, respectively. Given the general decline in home range areas, stable areas that have had an increased use, or no

spatial variation in home range patterns, and integrates with other GIS data.

**3.3 Integrating STAMP and kernel change approaches (method 3)** 

decline in use, may be the most important habitat for conservation.

from 2006 to 2007.

**4. Conclusion** 

contraction in spatial-temporal patterns*.* 

**Positive Change Negative Change No Change Total** 

Table 2. Area of statistically significant changes in kernel density estimates and the proportion of significant change for each spatial-temporal pattern of home range change

Integrating results of STAMP and kernel density change demonstrate that 27.92% of the area of disappearance patterns had statistically significant negative change, while 49.42% of contraction area was associated with negative change (Table 2). One might anticipate that the spatially isolated disappearance events would have more negative change, as these are locations within regions where habitat use has ceased. However, in the caribou example, disappearance is often associated with a single telemetry point in *t* followed by no use in *t*+1. Greater magnitude changes in intensity are occurring in central portions of the home range where many telemetry points are identified in *t* and in an area associated with

Data on wildlife locations are increasingly detailed in both space and time. Conversion to binary home range maps has been useful. However, the methods presented here take

 km2 % km2 % km2 % km2 % **Stable** 29.17 3.73 101.52 12.99 650.79 83.28 781.48 100.00 **Disappearance** 0.00 0.00 83.58 27.92 215.80 72.08 299.38 100.00 **Contraction** 0.00 0.00 453.90 49.42 464.55 50.58 918.45 100.00 **Generation** 50.08 35.11 0.00 0.00 92.55 64.89 142.63 100.00 **Expansion** 52.75 17.17 0.00 0.00 254.42 82.83 307.17 100.00

$$se = \sqrt{\frac{\left[\int \left(k \left(\mathbf{z}\right)\right)^2 d\mathbf{z}\right]^2}{4m\tau^2}}\tag{4}$$

where *k*(*z*) is the Gaussian kernel with a mean of zero and a standard deviation of 2 . Significant positive change (α = 0.05) occurs at location *i* when changeiΔt > 1.96 and significant negative change occurs when changeiΔt < -1.96. Otherwise, no significant change is assumed to have occurred. Calculating changeiΔt in this way does not produce an exact measure of statistically significant change. Kernel density estimators produce estimates, and the standard error or variance can be thought of as a confidence envelope around that value. Therefore, significant change is also estimated (Bowman & Azzalini, 1997, pg. 116; Fotheringham et al., 2002, pg. 205).

When change in the caribou home range is quantified as the intersection of two binary home ranges, 31.81% of the area is found to overlap, suggesting that change has occurred in 68.09% of the home range. However, when change is defined by statistical significance, only 31.39% of the area has changed and change can be categorized as increasing use (positive change, 5.39%) or decreasing use (negative, change 26.00%) (Fig. 7). By assessing change with statistical significance, and enabling the rate of significant change to vary over space, no change is identified near edges of the home range, where intensity values are small and/or zero. Changes in intensity, not captured with binary approaches, are emphasized.

Fig. 7. Spatial-temporal change in 2006 to 2007 caribou home ranges. Change is defined as statistically significant variation in kernel density estimates generated from telemetry data. (The semicircles of positive and negative change are edge effects created by the software used for detecting change.)

( ) ( )

<sup>=</sup> **z z**

*k*

where *k*(*z*) is the Gaussian kernel with a mean of zero and a standard deviation of 2 . Significant positive change (α = 0.05) occurs at location *i* when changeiΔt > 1.96 and significant negative change occurs when changeiΔt < -1.96. Otherwise, no significant change is assumed to have occurred. Calculating changeiΔt in this way does not produce an exact measure of statistically significant change. Kernel density estimators produce estimates, and the standard error or variance can be thought of as a confidence envelope around that value. Therefore, significant change is also estimated (Bowman & Azzalini, 1997, pg. 116;

*se*

Fotheringham et al., 2002, pg. 205).

used for detecting change.)

4

When change in the caribou home range is quantified as the intersection of two binary home ranges, 31.81% of the area is found to overlap, suggesting that change has occurred in 68.09% of the home range. However, when change is defined by statistical significance, only 31.39% of the area has changed and change can be categorized as increasing use (positive change, 5.39%) or decreasing use (negative, change 26.00%) (Fig. 7). By assessing change with statistical significance, and enabling the rate of significant change to vary over space, no change is identified near edges of the home range, where intensity values are small and/or zero. Changes in intensity, not captured with binary approaches, are emphasized.

Fig. 7. Spatial-temporal change in 2006 to 2007 caribou home ranges. Change is defined as statistically significant variation in kernel density estimates generated from telemetry data. (The semicircles of positive and negative change are edge effects created by the software

*n*τ

<sup>2</sup> <sup>2</sup>

(4)

2 d As an example, when home ranges are binary, no change will be detected at a location associated with one telemetry point in the first time period and twelve in the second time period. The kernel approach, in contrast, has the potential to identify these as significant changes. As with STAMP, output of the kernel density approach is mappable, accounts for spatial variation in home range patterns, and integrates with other GIS data.

#### **3.3 Integrating STAMP and kernel change approaches (method 3)**

In the final method, STAMP and kernel density estimation change detection are integrated to characterize how wildlife intensity varies for different spatial-temporal home range patterns. The amount of statistically significant change in intensity can be determined for each spatial-temporal home range pattern. For disappearance and contraction, only negative change or decreasing use will occur; for generation and expansion, only positive change or increasing use will occur; and for stable patterns, both types of change may be present.

With the caribou data, stable patterns have both positive and negative change, and differentiating the types of change identifies how intensity of use is varying within areas used consistently through time (Table 2). In the caribou example, 3.73% and 12.99% of stable pattern areas had statistically significant positive and negative change, respectively. Given the general decline in home range areas, stable areas that have had an increased use, or no decline in use, may be the most important habitat for conservation.


Table 2. Area of statistically significant changes in kernel density estimates and the proportion of significant change for each spatial-temporal pattern of home range change from 2006 to 2007.

Integrating results of STAMP and kernel density change demonstrate that 27.92% of the area of disappearance patterns had statistically significant negative change, while 49.42% of contraction area was associated with negative change (Table 2). One might anticipate that the spatially isolated disappearance events would have more negative change, as these are locations within regions where habitat use has ceased. However, in the caribou example, disappearance is often associated with a single telemetry point in *t* followed by no use in *t*+1. Greater magnitude changes in intensity are occurring in central portions of the home range where many telemetry points are identified in *t* and in an area associated with contraction in spatial-temporal patterns*.* 
