*3.3. Geographic Information Mapping Trajectory and the Index Model of Comprehensive Land Use Intensity*

ArcGIS10.2 software was used to overlay five phases of land use maps and analyze the spatial change process of land use using change mapping with the following equation:

$$Y = G\_1 10^{\mathfrak{n}-1} + G\_2 10^{\mathfrak{n}-2} + \dots + G\_n 10^0 \tag{2}$$

where *Y* is the *n*–digit number calculated synthesis of the land use code; *n* is the number of periods of land use; *Gn* is the nth period of land use unit.

**Figure 2.** Typical settlement division and selection.

The index model of comprehensive land use intensity [56] was constructed to model the change in land use intensity around each settlement in the study area so that it could be implemented in the rural settlement spatial unit. The specific equation is as follows:

$$L = \sum\_{i=1}^{n} A\_i \mathbb{C}\_i = \sum\_{i=1}^{n} A\_i (S\_i / S) \tag{3}$$

where *L* is the land use intensity of a single sample; *Ai* and *Ci* are the graded indices of land use intensity at level *i* and the percentage of area occupied in the sample; *Si* is the area of land use type at level *i* in the sample; *S* is the total land area of the sample.

#### *3.4. Average Nearest Neighbor*

The Average Nearest Neighbor Index (ANN) is derived from the average distance between each rural settlement's center of mass and its nearest neighbor's center of mass and is one of the most common methods used to determine the spatial distribution pattern of rural settlements. The average Nearest Neighbor Index value is distributed between [−1, 1], and the closer the result is to 1, the more discrete the distribution is, and the opposite is the more clustered [31].

$$ANN = \gamma\_{\pi} \gamma\_{\beta} = \frac{\sum \frac{d\_{\text{min}}}{n}}{\frac{\sqrt{n/A}}{2}} = \frac{2\sqrt{\lambda}}{N} \sum d\_{\text{min}} \tag{4}$$

*ANN* is the average nearest neighbor index. *γα* is the average distance of nearest neighbors of village settlement points; *γβ* is the theoretical average under the random spatial distribution of village settlement points. *d*min is the distance between a village settlement point and the nearest neighboring village settlement; *n* is the number of village settlements; *A* is the total area of spatial units; *λ* is the spatial distribution density of village settlements.

#### *3.5. Standard Deviational Ellipse*

Standard deviational ellipse (SDE) can accurately reveal the spatial distribution center, dispersion, and directional trends of geographical elements and is a spatial statistical method to quantitatively analyze the overall characteristics of the spatial distribution of geographical elements [56,57]. The rotation angle is the angle formed by clockwise rotation from due north to the central axis, reflecting the main trend direction of its distribution, and the long axis characterizes the dispersion of rural settlement sites in the main trend direction, whose mathematical expression is [56,57]:

$$\mathcal{A}(A) \approx \tan \theta = \left\{ \left[ \left( \sum\_{i=1}^{n} w\_i^2 \mathbf{x}^{\prime 2}\_i - \sum\_{i=1}^{n} w\_i^2 \mathbf{y}^{\prime 2}\_j \right) \right] + \sqrt{\left( \left[ \sum\_{i=1}^{n} w\_i^2 \mathbf{x}^{\prime 2}\_i - \sum\_{i=1}^{n} w\_i^2 \mathbf{y}^{\prime 2}\_j \right]^2 + 4 \left( \sum\_{i=1}^{n} w\_i^2 \mathbf{x}^{\prime 2}\_i \mathbf{y}^{\prime 2}\_j \right)^2 \right)} \right\} / 2 \left( \sum\_{i=1}^{n} w\_i^2 \mathbf{x}^{\prime 2}\_i \mathbf{y}^{\prime 2}\_j \right) \tag{5}$$

$$\delta(B) \approx \delta\_{\mathbf{x}} = \sqrt{\left[\sum\_{i=1}^{n} \left(w\_i \mathbf{x}\_i^\prime \cos\theta - w\_i y\_i^\prime \sin\theta\right)^2 / \sum\_{i=1}^{n} w\_i^2\right]}\tag{6}$$

$$\delta(B) \approx \delta\_{\mathbf{\hat{y}}} = \sqrt{\left[\sum\_{i=1}^{n} \left(w\_i \mathbf{x'}\_i \sin \theta - w\_i y'\_i \cos \theta\right)^2 / \sum\_{i=1}^{n} w\_i^2\right]}\tag{7}$$

where (*A*) and (*B*), the azimuthal angle is derived from tan *θ*, *δx*, and *δ<sup>y</sup>* are the standard deviations along the *x* and *y* axes, respectively, and *xi* and *yi* represent the coordinate deviations from the mean center, *xi,* and *yi* indicate the deviation of coordinates from the mean center. The center (center of gravity) is the average distribution center of the rural settlement land space in the trough valley area. The center uses the main trend direction of rural settlement distribution as the azimuth, the standard deviation in the *x*–direction and *y*–direction as the ellipse axis, and the spatial distribution ellipse of rural settlement land is constructed to explain the characteristics of centrality, directionality, and spatial distribution pattern of the evolution of rural settlement type land in the trough valley area. Meanwhile, the direction, intensity, and spatial dispersion trends of the development changes of rural settlements in karst trough valleys are identified by the standard deviation ellipse eigenvalues in different years. This paper calculated the standard deviation ellipse parameters of rural settlement sites in karst valleys with the help of the ArcGIS software spatial statistics module and visualized the results.

## *3.6. Kernel Density Estimation*

Kernel density estimation (KDE) is a non–parametric density calculation method, which reveals the distribution characteristics of points through the spatial variation of the density of settlement points, and is suitable for measuring the spatial distribution density of rural settlement sites:

$$f(x,y) = \frac{1}{nh^2} \sum\_{i=1}^{n} k\left(\frac{d\_i}{n}\right) \tag{8}$$

where *f*(*x*,*y*) is the density estimate at point (*x*,*y*); *k*() is the kernel function; bandwidth *h* > 0; *n* is the number of observations; di is the distance of (*x*,*y*) location from the *i*th element. The higher the kernel density value, the higher the density of spatial distribution of rural settlement sites [58].

#### **4. Results**

#### *4.1. Spatial Pattern Analysis of the Evolution of Rural Settlement Types in the Karst Trough*

In order to clarify the spatial aggregation characteristics of the evolution of rural settlements in the LangXi trough valley and to classify the types of settlements, the regional rural settlements were analyzed using the nearest neighbor index. The nearest neighbor index analysis was conducted on the regional rural settlements, and the results of the analysis showed that, during the nearly 60 years from 1964 to 2021, the z–value of LangXi trough valley rural settlement was less than 1 in all four time periods, then the trough valley rural settlement showed a clustering trend. The significance level was less than 0.01, indicating that the spatial aggregation of rural settlement types within the trough valley territorial system rejects the null hypothesis of random distribution. From 1964 to 1999, the average observed distances in this period were all smaller than the expected average distances, and the nearest neighbor ratio was approximately 0.4 (0.37–0.39), with a significance level of *p* < 0.01, indicating that the karst valley rural settlements showed an overall clustering trend in this period. The number of rural settlement patches clusters significantly decreased, the average observed distance slightly increased, and the nearest neighbor ratio slightly decreased, from 0.396 to 0.372 (Table 2 and Figure 3). This shows that the spatial agglomeration of settlements tended to weaken with time evolution. From 2004 to 2021, the cluster z–value decreased sharply from −20.76 to −48.39, indicating that the spatial agglomeration of the clusters showed a sharp weakening trend over time.

**Table 2.** The nearest neighbor ratio of rural settlements in 1964 and 2021.


Drawing the standard deviation ellipse of the spatial distribution of rural settlement patches can explain the characteristics of centrality, direction, and spatial distribution patterns of rural settlement types in the karst trough valley area. Meanwhile, the direction and intensity of rural settlement development changes and their spatial dispersion trends can be identified by the standard deviation ellipse characteristic values in different periods. The average length of the x-axis from 1964 to 2021 was 1.4 km, the average length of the y-axis was 1.6 km, the rotation angle decreased from 25.59◦ to 25.31◦, and the deviation range of the main parameters of the standard deviation ellipse for each year was approximately 2%, and the basic spatial pattern of the settlement was relatively stable and maintained its distribution in the W–N direction (Figure 4). This shows that the basic spatial pattern of the settlement in the study area is controlled by the topography of the trough valley and trough dam, as well as the topography of the trough dam, which is surrounded by mountains on both sides, with east-west trough slopes and narrow north-south slopes. The center of the standard deviation ellipse is the center of gravity of rural settlements in the corresponding year, and its migration changes can reflect the overall spatial process of the evolution of rural settlement types in the study area. The center of gravity of the settlement in 1964 was used as the coordinate origin to measure the rate and direction of the settlement center of gravity migration in each period and visualize it. The calculation results show that the average annual rate of gravity migration was 32.12 m/a. In directional change, the gravity of the settlement shifted southeast from 1964 to 2021 and pointed to

the trough dam area. The spatial evolution of hotspot areas of rural settlement types in the trough valley differed in each period, and the spatial directionality was stronger from 2004 to 2021 than from 1964 to 2004. The main reason for this is the accelerated urbanization and industrialization of the trough valley area since 2004 and the significant changes in the spatial pattern of settlements.

**Figure 3.** Distribution of Nearest Neighbor Index of Rural Settlements, 1964–2021.

The study extracted the center of gravity of each settlement patch, used the center of gravity to represent the settlement, and calculated the spatial distribution density of the settlement. Using the ArcGIS nuclear density module, a spatial analysis of nuclear density was conducted to classify the settlement nuclear density in each period into the background, low density, medium density, and high-density zones (Figure 4). The settlement nucleation density of each period was also classified into the background, low density, medium density, and high-density zones (Figure 4). The spatial heterogeneity of settlement density distribution in the trough and valley area is prominent, and the high-value area of settlement density distribution from 1964 to 2021 tended to be the trough and dam areas. The background area is mainly located on the slope and top of the valley, part of the geological environment in this area is not suitable for forming settlements, and the distribution of settlements is small. The medium and high-density areas are mainly located in the karst valley trough and dam area with flat terrain, convenient transportation, and good farming conditions and are primarily distributed in a band. The high-density areas are distributed along the traffic arteries and the Yinjiang River, while the low-density areas are scattered in the two wings of the troughs and valleys.

#### *4.2. Analysis of the Buffer Zones in Land Use Change around Typical Rural Settlement Types*

From 1964 to 2021, buffer changes, the types, and amounts of land use around the settlements showed differential change characteristics within the buffer area. The land use changes in the settlement's 0 to 400 m buffer zone showed the evolution characteristics of three buffer interval dimensions. The land use mapping of 0 to 50 m, 50 to 200 m, and 200 to 400 m buffers in four troughs and valleys in typical settlement classes had varying characteristics (Figure 5).

**Figure 4.** (**a**) Kernel density analysis of rural settlements from1964 to 2021. (**b**) Gravity shift and standard deviational ellipse of rural settlement distribution from 1964 to 2021.

**Figure 5.** *Cont*.

**Figure 5.** Evolution of land use in the typical settlement type's 0–400 m buffer zone.

In general, the change in land type in the buffer zone of the ES of the dam of karst trough valley is mainly concentrated in the buffer range of 0 to 50 m, and the land use types around the settlement are mainly steep-slope arable land, gentle slope arable land, flat dam arable land, and rural settlement, and the number of land types accounts for 30.84%, 13.21%, 12.67%, and 8.23% of the buffer area, respectively. In the buffer zone, the overall trend of land use change shows the expansion of forest land and abandoned land, while rural residential areas and arable land maintain a balance. However, within the 0 to 50m buffer zone, the land use land types around the BS are mainly rural residential areas, orchards, flat dam arable land, and steep-slope arable land, with the numbers accounting for 12.13%, 10.36%, 28.31%, and 21.62%, respectively. The rate of change is mainly based on the expansion of rural settlements, flat dams of arable land over time, and the increase in abandoned land and orchards. The changes in the land use buffer zone around the AS of Xinchao and Ganlong are mainly manifested in four types of land: forest land, arable land, and abandoned land (Figure 5). Within the 0 to 50 m buffer zone, the overall land use types are mainly steep slope arable land, gently sloping arable land, rural residential areas, and abandoned land, accounting for 36.12%, 23.03%, 12.35%, and 9.16% of the buffer zone area, respectively. The land use around the DS Taiyangping has apparent differences in land use types, quantity, and structural changes in the buffer zone dimension; the number of land use types has increased, and the change in land use quantity is mainly in forestland, cropland, grassland, and abandoned land. In the 0 to 50 m buffer zone range, the buffer zone land types of rural settlements, low-cover grassland, and flat dam cropland show an increase and decrease with time.
