# Re-Arranging Space, Time and Scales in GIS: Alternative Models for Multi-Scale Spatio-Temporal Modeling and Analyses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Triangular Model

#### 2.1. Concept

_{1}, all intervals (e.g. I

_{2}, I

_{3}, and I

_{4}) before I

_{1}are located in a triangular zone in the left corner of the study area (Figure 3). This zone is then defined as the before zone of I

_{1}which encloses all intervals before I

_{1}. Analogously, other temporal relations can be represented by unique zones in the 2D space in different spatial relations to the reference interval (Figure 2b). Based on this unique feature of the TM, Qiang et al. [14] developed a set of graphic query tools that enable users to define temporal queries by creating geometric zones in the 2D space of the TM (see Figure 4). All interval points that are spatially contained by the zone meet the temporal constraints (e.g., during, before, and overlaps) described by the zone. Composite queries connected by logical operators can be represented by the intersection and union of query zones, which is more intuitive than mathematic equations.

#### 2.2. Analysis of Crisp Time Intervals

#### 2.3. Imperfect Time Interval

#### 2.3.1. Rough Time Interval

#### 2.3.2. Fuzzy Time Interval

#### 2.4. Time Series

#### 2.4.1. Visualization

_{1}, I

_{2}, and I

_{3}) and long-term trends can be observed at higher levels (e.g., high speed in I

_{4}, low speed in I

_{5}). The hierarchical structure of variations at different scales is presented as well [17,25]. In addition to time series data, the TM can be used to represent other types of linear data such as traffic queues along a road and DNA sequences.

#### 2.4.2. Map Algebra for CTM

## 3. Pyramid Model

#### 3.1. The Concept

^{2}pixels. The position of a point in the 3D space is defined as coordinates of (x, y, z), where (x, y) represent the centroid of the square area and z indicates the spatial extent of the area. In this case, z is proportional to the size of the square. Thereby, the raster data can be represented by a uniform lattice of points in a 3D space (Figure 15)

#### 3.2. Multi-Scale Spatial Analysis

## 4. A Multi-Scale Analytical Framework

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The transformation from the linear model to the triangular model (TM). (

**a**) Time intervals in the linear model. (

**b**) Projecting a time interval into a point in the TM. (

**c**) Time intervals in (

**a**) represented in the TM (adapted from [13]).

**Figure 2.**Representing temporal relations in the TM. (

**a**) Thirteen topological relations between time intervals. (

**b**) The representation of the temporal relations in the TM. (adapted from [15]).

**Figure 3.**Representing temporal relations as spatial relations in the TM. (

**a**) Time intervals (I

_{1–4}) in the linear model. (

**b**) Time intervals (I

_{1–4}) in the TM. (

**c**) the before zone of I

_{1}.

**Figure 4.**Temporal queries in the TM by creating 2D zones. (

**a**) Selecting intervals during I

_{1}. (

**b**) Selecting intervals contained by I

_{1}. (

**c**) Selecting intervals containing I

_{1}and in-between (I

_{2}, I

_{3}) (adapted from [14]).

**Figure 5.**The interface of the GeoTM with a map view in the left and TM view in the right, which are dynamically linked. Temporal queries can be performed by creating geometric zones in the TM (adapted from [14]).

**Figure 7.**Representation of rough time intervals in the TM. (

**a**) The linear representation. (

**b**) Constructing a rough time interval in the TM. (

**c**) Intervals in (a) in the TM. (

**d**). Overlap between a query zone

**A**and a rough time interval

**R(I)**(adpated from [17]).

**Figure 8.**(

**a**) Visualizing rough time intervals of the military features in the WWI aerial photos. The dark areas are clusters of the intervals and I

_{1–4}indicate difference phases (time intervals) of the war. (

**b**) Selecting features of Cluster 2 (artillery attack from the Allies army) in the GeoTM (modified from [17]).

**Figure 9.**Representing the during relation to a fuzzy time interval $\tilde{I}(t)$ in TM (adapted from [15]).

**Figure 10.**Representing temporal relations of a fuzzy time interval $\tilde{I}(t)$ in TM (adapted from [15]).

**Figure 11.**Representing time series in the TM. (

**a**) Time series represented in a line chart and color-coded linear raster. (

**b**) The TM representation of the base intervals in a time series. (

**c**) The TM representation of all intervals in a time series. (

**d**) Rasterized TM with grey-coded attributes.

**Figure 12.**A comparison of the line chart (

**a**) and TM representation (

**b**) of the moving speed of a soccer player in a game (adapted from [18]).

**Figure 13.**Using map algebra to compare air quality in Beijing in 2007 and 2008. (

**a**) Time series of PM10 AQI in Beijing in 2007 and 2008. (

**b**) The TM of the 2007 AQI. (

**c**) The TM of 2008 AQI. (

**d**) The binary result of subtracting the 2007 TM from the 2008 TM (TM

_{2008}–TM

_{2007}).

**Figure 14.**Illustration of an image pyramid (

**left**) and the configuration of the pyramid model for a raster (

**right**).

**Figure 16.**Calculating local fractal dimensions of a land cover raster. (

**a**) A binary land cover raster in the wetland in the Mississippi Delta. (

**b**) Local fractal dimension of the land cover data in a 11 × 11 cells moving window. (

**c**) Local fractal dimension in a 21 × 21 cells moving window. (

**d**) Local fractal dimension in a 31 × 31 cells moving window.

**Figure 18.**The isosurface of voxels with a 0.99 fractal dimension in the PM. (

**a**) An oblique view. (

**b**) A top-down view. (

**c**) A horizontal view along the x axis (east). (

**d**) A horizontal view along the y axis (north).

**Figure 19.**The representation of multi-scale spatio-temporal data in the continuous spatio-temporal model (CSTM). (

**a**): Time series of spatial data. (

**b**): Time series of PMs. (

**c**): PMs of different time intervals in a TM.

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**MDPI and ACS Style**

Qiang, Y.; Van de Weghe, N.
Re-Arranging Space, Time and Scales in GIS: Alternative Models for Multi-Scale Spatio-Temporal Modeling and Analyses. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 72.
https://doi.org/10.3390/ijgi8020072

**AMA Style**

Qiang Y, Van de Weghe N.
Re-Arranging Space, Time and Scales in GIS: Alternative Models for Multi-Scale Spatio-Temporal Modeling and Analyses. *ISPRS International Journal of Geo-Information*. 2019; 8(2):72.
https://doi.org/10.3390/ijgi8020072

**Chicago/Turabian Style**

Qiang, Yi, and Nico Van de Weghe.
2019. "Re-Arranging Space, Time and Scales in GIS: Alternative Models for Multi-Scale Spatio-Temporal Modeling and Analyses" *ISPRS International Journal of Geo-Information* 8, no. 2: 72.
https://doi.org/10.3390/ijgi8020072