3.1.2.11 Dureza Buchholz

Intensity Analysis (Aldwaik & Pontius 2012; Pontius, Shusas & McEachern 2004) is a method that considers maps at multiple instances for the same set of land cover classes, described as categories. The goal of intensity analysis is to account for transitions at interval, category and transition level (Aldwaik & Pontius 2012). Using a transition matrix for each time interval, changes are computed relative to hypothetical uniform change at the interval, category, as well as at the transition level. Interval level can show if the change is large merely because the duration of the interval is long or if the change over time is faster or slower than expected. Category level indicates whether the loss or gain of a particular land cover class is large, because the category is large, thereby identifying classes that are actively losing or gaining. The transition level shows if some transitions are targeting or are targeted by particular land cover classes. By analysing the off-diagonal entries of the confusion matrix, systematic transitions of land cover change can be identified. The method also tests for stationarity of changes across time intervals. Table 2-1 introduces common symbols used in the mathematical equations formulating the intensity analysis framework to reduce duplication.

Table 2-1 Common symbols used in mathematical notation in Equation 2-11–Equation 2-18 Symbol Meaning

𝐽 Number of categories

𝑖, 𝑗 Index for a category, i = initial time point, j = final time point 𝑚, 𝑛 Index for the losing (m) and gaining (n) category during transition

𝐶𝑡𝑖𝑗 This is the number of pixels that transition from category i to category j during interval t

𝑌𝑡, 𝑌𝑡+1 Year at start (t) and end (t+1) of time interval

2.4.5.1 Interval level

Equation 2-11 calculates the overall rate of change during an interval as the size of the change divided by the duration of the time interval expressed as a percentage of the spatial extent at one rate per time interval.

𝑆𝑡= change during [𝑌𝑡, 𝑌𝑡+1] (duration of [𝑌𝑡, 𝑌𝑡+1])(extent)100% = ∑𝐽𝑗=1[(∑𝐽𝑖=1𝐶𝑡𝑖𝑗) − 𝐶𝑡𝑗𝑗] (𝑌𝑡+1− 𝑌𝑡)(∑ ∑𝐽𝑖=1𝐶𝑡𝑖𝑗) 𝐽 𝑗=1 100% Equation 2-11

where 𝑆_𝑡 is the annual intensity of change for time interval [𝑌𝑡, 𝑌𝑡+1]; and

Equation 2-12 calculates one uniform rate based on rate of overall change for the entire study period given that the pattern of change were stationary in terms of rate of overall change.

𝑈 =area of change during all intervals_{(duration of all intervals)(extent)}100% =∑ {∑ [(∑ 𝐶𝑡𝑖𝑗

𝐽 𝑖=1 ) − 𝐶𝑡𝑗𝑗] 𝐽 𝑗=1 } 𝑇−1 𝑡=1 (𝑌𝑇− 𝑌1)(∑𝐽𝑗=1∑𝐽𝑖=1𝐶𝑡𝑖𝑗) 100% Equation 2-12

where 𝑈 is value of uniform line for time intensity analysis;

𝐶_𝑡𝑗𝑗 is the number of pixels that persist in category j during interval t; 𝑇 is the number of time points; and

𝑡 is the index for the initial time point of interval [𝑌𝑡, 𝑌𝑡+1], where t

ranges from 1 to T-1. 2.4.5.2 Category level

Category level analysis compares observed intensities of loss and gain for each category with uniform intensity of change during each time interval. It identifies the categories for which change is more intensive than the overall change intensity in the spatial extent. At category level, Equation 2-13 computes the annual gross per-category loss intensity during an interval, by dividing the size of the annual gross loss by the size of the category at the start of the interval. Similarly, Equation 2-14 gives the annual gross per-category gain intensity during an interval, this time calculated from the size of the annual gross gain divided by the category size at the end of the interval.

𝐿𝑡𝑖=annual loss of 𝑖 during [𝑌_{size of 𝑖 at 𝑌} 𝑡, 𝑌𝑡+1]

𝑡 100% =

[(∑𝐽𝑗=1𝐶𝑡𝑖𝑗) − 𝐶𝑡𝑖𝑖]

(𝑌𝑡+1− 𝑌𝑡) ∑𝐽𝑗=1𝐶𝑡𝑖𝑗

100% Equation 2-13

𝐺𝑡𝑖=

annual gain of 𝑗 during [𝑌𝑡, 𝑌𝑡+1]

size of 𝑗 at 𝑌𝑡+1 100% =

[(∑𝐽𝑖=1𝐶𝑡𝑖𝑗) − 𝐶𝑡𝑗𝑗]

(𝑌𝑡+1− 𝑌𝑡) ∑𝐽𝑖=1𝐶𝑡𝑖𝑗

100% Equation 2-14

where

𝐿𝑡𝑖 is the intensity of annual loss of category i during interval t _{relative to size of category i at 𝑌}

𝑡;

𝐺_𝑡𝑖 is the intensity of annual gain of category i during interval t relative to size of category i at 𝑌𝑡+1;

𝐶_𝑡𝑗𝑗 is the number of pixels that persist in category j during interval t; and

𝐶𝑡𝑖𝑖 is the number of pixels that persist in category i during interval t. For category level intensity analysis, the uniform hypothesis states that gross loss and gross gain will occur at the same annual intensity for all categories, which is equal to the speed of change during the interval (U from Equation 2-12). The loss of i is dormant during interval t when annual loss (Lti) < U. If Gtj < U, then the gain of j is dormant during interval t. Conversely,

if Lti > U, then the loss of i is active during interval t; and gain (Gtj) of j is active during interval t if Gtj > U.

2.4.5.3 Transition level

For each time interval, transition level of analysis produces two sets of outputs: one for gains of category n, and the other for losses of category m. The uniform hypothesis at transition level (Equation 2-15) is that during a particular interval, category n transitions to all other categories (not-n) with the same annual intensity. Equation 2-16 gives the annual transition intensity of the gain of category n from another category i.

𝑊𝑡𝑛=

annual gain of 𝑛 during [𝑌𝑡, 𝑌𝑡+1]

size of non − 𝑛 at 𝑌𝑡 100% =

[(∑𝐽𝑖=1𝐶𝑡𝑖𝑛) − 𝐶𝑡𝑛𝑛]100%

(𝑌𝑡+1− 𝑌𝑡) ∑𝐽𝑗=1[(∑𝐽𝑖=1𝐶𝑡𝑖𝑗) − 𝐶𝑡𝑛𝑗]

Equation 2-15

𝑅𝑡𝑖𝑛=

annual transition from 𝑖 to 𝑛 during [𝑌𝑡, 𝑌𝑡+1]

size of 𝑖 at 𝑌𝑡 100% =

𝐶𝑡𝑖𝑛100%

(𝑌𝑡+1− 𝑌𝑡) ∑𝐽𝑗=1𝐶𝑡𝑖𝑗

Equation 2-16

where _𝑊

𝑡𝑛 is the value of uniform intensity of transition to category n from _{all non-n categories at time Y}

t during time interval [Yt, Yt+1];

𝑅_𝑡𝑖𝑛 is the annual intensity of transition from category i to category n during time interval [Yt, Yt+1] where i ≠ n;

𝐶𝑡𝑖𝑛 is the number of pixels that transition from category i to category

n during interval t;

𝐶_𝑡𝑛𝑛 is the number of pixels that persist in category n during interval t; and

𝐶_𝑡𝑛𝑗 is the number of pixels that transition from category n to category

j during interval t.

If the observed transition intensity from i (Equation 2-16) is less than the uniform transition intensity (Equation 2-15), then the gain of n avoids category i, but if the observed transition intensity (Rtin) from i is greater than the uniform transition intensity (Wtn), then the gain of n

targets category i. The transition from m to n is stationary, given the gain of n, if the gain of category n either targets category m for all time intervals or avoids category m for all time intervals.

Equation 2-17 and Equation 2-18 similarly analyse the loss of category m.

𝑉𝑡𝑚=

annual loss of 𝑚 during [𝑌𝑡, 𝑌𝑡+1]

size of non − 𝑚 at 𝑌𝑡 100% =

[(∑𝐽𝑖=1𝐶𝑡𝑚𝑗) − 𝐶𝑡𝑚𝑚]100%

(𝑌𝑡+1− 𝑌𝑡) ∑𝐽𝑗=1[(∑𝐽𝑗=1𝐶𝑡𝑖𝑗) − 𝐶𝑡𝑖𝑚]

𝑄𝑡𝑚𝑗 =

annual transition from 𝑗 to 𝑚 during [𝑌𝑡, 𝑌𝑡+1]

size of 𝑗 at 𝑌𝑡 100% =

𝐶𝑡𝑚𝑗100%

(𝑌𝑡+1− 𝑌𝑡) ∑𝐽𝑗=1𝐶𝑡𝑖𝑗

Equation 2-18

where _𝑉

𝑡𝑚 is the value of uniform intensity of transition to category m from _{all non-m categories at time Y}

t during time interval [Yt, Yt+1];

𝑄𝑡𝑚𝑗 is the annual intensity of transition from category m to category j during time interval [Yt, Yt+1] where j ≠ m;

𝐶𝑡𝑚𝑗 is the number of pixels that transition from category m to category

j during interval t;

𝐶_𝑡𝑚𝑚 is the number of pixels that persist in category m during interval t; and

𝐶𝑡𝑖𝑚 is the number of pixels that transition from category i to category

m during interval t.

If category m loses to all other categories in a uniform manner, then Qtmj = Vtm for all j, and the

transition from m to n is defined as stationary, given the loss of m, if the loss of category m is either avoided by category n or targeted by category n of all time steps. Stationary processes will be defined in the next section.

2.4.5.4 Stationarity defined

In statistics, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space. For a stationary process, mean and variance do not change over time or position. In contrast, spatial non-stationarity describes modelled relationships that are not constant across space but are dependent on the absolute location in space (Haining 1993; Jones & Hanham 1995). Miller & Hanham (2011) warn that when underlying processes are non-stationary, global statistics, pattern measurements and model parameters will be inaccurate making any subsequent spatial inferences incorrect. The next section focuses on land change modelling used to simulate future land change.