Formulas¶

This page provides the mathematical formulas used in the Spatiotemporal LULC Analysis plugin.

Notation¶

Symbol	Definition
\(\Omega_t\)	Set of valid pixels at time \(t\) (after NoData and AOI masking)
\(L_t(p)\)	Class label of pixel \(p\) at time \(t\)
\(A_{px}\)	Area per pixel in selected units
\(\mathbf{1}[\cdot]\)	Indicator function (1 if condition true, 0 otherwise)

Area by Class¶

Pixel Count¶

The number of pixels belonging to class \(k\) at time \(t\):

\[A_{k,t} = \sum_{p \in \Omega_t} \mathbf{1}[L_t(p) = k]\]

This counts all valid pixels where the class label equals \(k\).

Area¶

Convert pixel count to area using the pixel area factor:

\[\text{Area}_{k,t} = A_{k,t} \cdot A_{px}\]

Where \(A_{px}\) depends on output units:

Units	\(A_{px}\) Value
Pixels	1
Square meters	pixel width × pixel height (in meters)
Square kilometers	pixel area in m² ÷ 1,000,000

Percentage¶

Class percentage of total valid area:

\[\text{Percent}_{k,t} = \frac{A_{k,t}}{\sum_k A_{k,t}} \times 100\]

Interval Metrics¶

Valid Pixel Set¶

For interval \(t_0 \to t_1\), only pixels valid in both years are analyzed:

\[\Omega = \Omega_{t_0} \cap \Omega_{t_1}\]

Total Pixels¶

\[\text{Total} = |\Omega|\]

Changed Pixels¶

Count pixels where the class differs between years:

\[\text{Changed} = \sum_{p \in \Omega} \mathbf{1}[L_{t_0}(p) \ne L_{t_1}(p)]\]

Unchanged Pixels¶

\[\text{Unchanged} = \text{Total} - \text{Changed}\]

Gain and Loss¶

Gain¶

Pixels that transitioned into class \(k\):

\[\text{Gain}_k = \sum_{p \in \Omega} \mathbf{1}[L_{t_1}(p) = k \land L_{t_0}(p) \ne L_{t_1}(p)]\]

Only pixels that actually changed class are counted as gains.

Loss¶

Pixels that transitioned out of class \(k\):

\[\text{Loss}_k = \sum_{p \in \Omega} \mathbf{1}[L_{t_0}(p) = k \land L_{t_0}(p) \ne L_{t_1}(p)]\]

Net Change¶

\[\text{Net}_k = \text{Gain}_k - \text{Loss}_k\]

Positive: Class expanded
Negative: Class contracted
Zero: Balanced turnover

Gross Change¶

\[\text{Gross}_k = \text{Gain}_k + \text{Loss}_k\]

Total turnover for the class, regardless of direction.

Transition Matrix¶

Matrix Definition¶

The transition matrix \(M\) counts pixel movements between classes:

\[M_{i,j} = \sum_{p \in \Omega} \mathbf{1}[L_{t_0}(p) = i \land L_{t_1}(p) = j]\]

Where:

\(i\) = class at time \(t_0\) (row index)
\(j\) = class at time \(t_1\) (column index)
\(M_{i,j}\) = count of pixels moving from \(i\) to \(j\)

Matrix Properties¶

Diagonal elements (\(M_{i,i}\)): Pixels that stayed in class \(i\) (persistence)

Off-diagonal elements (\(M_{i,j}\) where \(i \ne j\)): Pixels that changed from \(i\) to \(j\)

Row sum: Total area of class \(i\) at \(t_0\)

\[\sum_j M_{i,j} = A_{i,t_0}\]

Column sum: Total area of class \(j\) at \(t_1\)

\[\sum_i M_{i,j} = A_{j,t_1}\]

Relationship to Gain/Loss¶

\[\text{Gain}_k = \sum_{i \ne k} M_{i,k}\]

\[\text{Loss}_k = \sum_{j \ne k} M_{k,j}\]

Top Transitions¶

Total Changed Area¶

Sum of all off-diagonal elements:

\[\text{Total}_{\text{change}} = \sum_{i \ne j} M_{i,j}\]

Transition Percentage¶

Share of total change for each transition:

\[\text{Percent}_{i,j} = \frac{M_{i,j}}{\text{Total}_{\text{change}}} \times 100\]

Transition Area¶

\[\text{Area}_{i,j} = M_{i,j} \cdot A_{px}\]

Change Intensity¶

Interval Intensity¶

Proportion of pixels that changed within the interval:

\[I_{\text{interval}} = \frac{\text{Changed}}{\text{Total}}\]

Range: 0 (no change) to 1 (complete change)

Annual Intensity¶

Average per-year change rate:

\[I_{\text{annual}} = \frac{I_{\text{interval}}}{t_1 - t_0}\]

Where \((t_1 - t_0)\) is the number of years in the interval.

Change Frequency Raster¶

Per-Pixel Frequency¶

For a time series with \(T\) observations, the change frequency at pixel \(p\):

\[F(p) = \sum_{t=1}^{T-1} \mathbf{1}[L_t(p) \ne L_{t+1}(p)]\]

Conditions¶

Computed only if \(p\) is valid in all years
NoData (-1) if \(p\) is invalid in any year
Range: 0 to \(T-1\)

Change Hotspots¶

Point Generation¶

Changed pixels are converted to point events with unit weight at their centroid locations.

Kernel Density Estimation¶

The density at location \((x, y)\) is:

\[\hat{f}(x, y) = \sum_{i=1}^{n} K\left(\frac{d_i}{h}\right)\]

Where:

\(n\) = number of change points (up to 50,000)
\(K\) = Gaussian kernel function
\(d_i\) = distance from \((x, y)\) to point \(i\)
\(h\) = bandwidth (1000 map units)

Gaussian Kernel¶

\[K(u) = \frac{1}{\sqrt{2\pi}} e^{-\frac{u^2}{2}}\]

Unit Conversions¶

Area Factor¶

The area factor \(A_{px}\) converts pixel counts to area:

\[A_{px} = |X_{\text{res}}| \times |Y_{\text{res}}| \times S\]

Where:

\(X_{\text{res}}\) = pixel width (from raster metadata)
\(Y_{\text{res}}\) = pixel height (from raster metadata)
\(S\) = scale factor based on CRS units and output units

Scale Factor¶

CRS Units	Output Units	Scale Factor
Meters	Pixels	0 (use count)
Meters	m²	1
Meters	km²	1/1,000,000
Degrees	m²	Requires projection

Summary Table¶

Metric	Formula	Output Type
Area by class	\(A_{k,t} \cdot A_{px}\)	CSV
Percentage	\(\frac{A_{k,t}}{\sum_k A_{k,t}} \times 100\)	CSV
Gain	\(\sum_{i \ne k} M_{i,k}\)	CSV
Loss	\(\sum_{j \ne k} M_{k,j}\)	CSV
Net change	\(\text{Gain}_k - \text{Loss}_k\)	CSV
Gross change	\(\text{Gain}_k + \text{Loss}_k\)	CSV
Transition count	\(M_{i,j}\)	CSV
Interval intensity	\(\frac{\text{Changed}}{\text{Total}}\)	CSV
Annual intensity	\(\frac{I_{\text{interval}}}{t_1 - t_0}\)	CSV
Change frequency	\(\sum_{t=1}^{T-1} \mathbf{1}[L_t(p) \ne L_{t+1}(p)]\)	Raster
Hotspot density	\(\sum_{i=1}^{n} K\left(\frac{d_i}{h}\right)\)	Raster