# Timescape: A Novel Spatiotemporal Modeling Tool

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Space–Time Distances

#### 2.2. Causal Structure: Topology of the Events Space

#### Accommodating Seasonal Variability

#### 2.3. The Algorithm

#### 2.4. Model Tuning

#### 2.4.1. Spatial Interpolator

#### 2.4.2. Metric Parameters

#### 2.4.3. Form Factor

#### 2.5. Timescape Implementations

#### 2.5.1. Python

#### 2.5.2. Java

#### 2.6. Coping with Binary and Count Data

## 3. Results

#### 3.1. Fungi ${\delta}^{15}$N

#### 3.2. Lowest Temperatures

#### 3.3. Extra Virgin Olive Oil ${\delta}^{18}$O

#### 3.4. Precipitation ${\delta}^{2}$H

#### 3.5. Performance Analysis

- -
- configuration 1: laptop–Intel i7 @ 2.60 GHz, 4 cores, 16 GByte RAM, SSD storage, Python 3.8 on Windows 10 OS.
- -
- configuration 2: laptop–Intel i7 @ 2.30 GHz, 4 cores, 8 GByte RAM, SSD storage, Python 3.7 on Ubuntu Linux 19.10 OS.
- -
- configuration 3: virtual machine–Oracle Virtualbox on MacPro host, Intel Xeon-E5 @ 3.50 GHz–4 dedicated cores, 16 GByte reserved RAM out of 64, SSD storage, Python 3.8 on Xubuntu Linux 20.04-LTS OS.

## 4. Discussion

#### 4.1. Measurements, Stationarity, Accuracy

#### 4.2. Timescape Topology Modification

#### 4.3. Timescape vs. Minkowskian Geometry

#### 4.4. A Universal Tool?

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The whole space–time centered about the event $\mathit{x}$ of coordinates $({x}_{0},{x}_{1},{x}_{2})$: $\mathit{x}$ is located at $({x}_{1},{x}_{2})$ in a bi-dimensional space at time ${x}_{0}$: $\mathit{x}=({x}_{0},{x}_{1},{x}_{2})$.

**Figure 2.**A limited portion of the space–time centered about the event $\mathit{x}=({x}_{0},{x}_{1},{x}_{2}\dots {x}_{n})$.

**Figure 3.**A geometrical interpretation of $\Delta $: $\Delta (\mathit{x},\mathit{y})$ is the hypotenuse of the triangle formed by ${D}_{s}(\mathit{x},\mathit{y})$ and ${D}_{t}(\mathit{x},\mathit{y})$ if $\mathit{y}\in {\mathcal{C}}_{x}^{+}$. For all the other events $\notin {\mathcal{C}}_{x}^{+}$ the distance is infinte. For example: $\Delta (\mathit{x},\mathit{u})=\infty $ because ${u}_{0}<{x}_{0}$, $\Delta (\mathit{x},\mathit{v})=\infty $ because ${v}_{0}={x}_{0}$ and $\Delta (\mathit{x},\mathit{w})=\infty $ because ${D}_{s}(\mathit{x},\mathit{w})>k{D}_{t}(\mathit{x},\mathit{w})$ even if ${w}_{0}>{x}_{0}$.

**Figure 4.**Sheaves of future and past causal cones. $\mathit{x}$ sits at the common tip of the cones. The cones widen as k increases. For $k=0$ the cones shrink to segments, and for $k\to \infty $ the cones coincide to the whole past (${\mathit{E}}_{x}^{-}$) and future(${\mathit{E}}_{x}^{+}$) of $\mathit{x}$.

**Figure 5.**(

**Left**): the causal cones ${\mathcal{C}}_{x}^{-}$ and ${\tilde{\mathcal{C}}}_{x}^{-}$ of $\mathit{x}$; note that $\tilde{\Delta}(\mathit{y},\mathit{x})<\tilde{\Delta}(\mathbf{z},\mathit{x})=\infty $ even if $\Delta (\mathit{y},\mathit{x})>\Delta (\mathbf{z},\mathit{x})$. (

**Right**): the cross section of ${\tilde{\mathcal{C}}}_{x}^{-}$, backwards in time, in units of T. The slope of the dashed line is 1 and that of the dotted line is $\alpha $.

**Figure 6.**Source (•) and target (•) events. The target events are the centers of a regular lattice of equal cells arranged as an $(n\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}1)$-dimensional box.

**Figure 7.**Two target events $\mathit{x}$ and $\mathit{y}$ with their input sets ${\mathit{S}}_{x}$ and ${\mathit{S}}_{y}$.

**Figure 8.**Timescape flowchart. Users must choose a target set T, a spatial interpolator, the distance parameters c and k, and form factor ψ (thick boxes). The accuracy can be evaluated by the spatial interpolator itself or a posteriori, depending on the chosen spatial interpolator.

**Figure 9.**(

**Left**): An example of ${R}^{2}(c,k)$. (

**Right**): $N(c,k)$ in percent units. Plots from a dedicated function of the Python package [31].

**Figure 11.**(

**Left**): three periodic form factors, solid line $\psi \left(t\right)={cos}^{2}\omega t$, dashed line (stricter) $\psi \left(t\right)={cos}^{20}\omega t$, dotted line (looser) $\psi \left(t\right)=\sqrt{|cos\omega t|}$. (

**Right**): the corresponding cross-sections of the causal cones.

**Figure 12.**(

**Top left**): source events $\mathit{S}$. (

**Top right**): target field values $\widehat{\mathrm{\Phi}}\left(\mathit{x}\right)$. (

**Bottom left**): $\eta \left(\mathit{x}\right)$ i.e., events in the causal neighborhood ${\mathit{S}}_{x}$. (

**Bottom right**): accuracy $\widehat{\sigma}\left(\mathit{x}\right)$. The source coordinates are the actual ones, the target coordinates are cell indexes.

**Figure 13.**The topmost sheet of three models. (

**Left**): fine-tuned Timescape, i.e., the topmost horizontal sheet of predicted values $\widehat{\mathrm{\Phi}}\left(\mathit{x}\right)$ (top right of Figure 12). (

**Center**): the loosest possible Timescape. (

**Right**): three-dimensional Kriging.

**Figure 14.**(

**Left**): The weather stations network (red dots) in Umbria; A, B and C are referred to the actual sites of the time series of Figure 15. (

**Right**): spatio-temporal distribution of the source events. Base map and boundaries from NaturalEarthData [70] and Geoportale Nazionale [71], map prepared with Qgis [72].

**Figure 15.**Time series extracted at the sites A (

**top**), B (

**middle**) and C (

**botom**) marked in Figure 14. Time is in years and field values in °C. The shadowed area marks the extrapolated values. Time series A starts in 1988, as there were no recording stations nearby before.

**Figure 16.**The same time series of Figure 15 at sites (

**A**)–(

**C**) compared with different interpolation techniques. Black: periodic Timescape, the actual case. Red: Straight cone (i.e., non-seasonal) Timescape. Blue: three-dimensional spatial Kriging. Time is in years and field values in °C, error bars not shown.

**Figure 17.**(

**Left**): source events spatial distribution. (

**Center**): source events spatiotemporal distribution, notice the layered time structure, time is expressed in years, space distances in thousands of kilometers. (

**Right**): the spatiotemporal variogram (13), distances in kilometers.

**Figure 18.**2009, 2010, and 2011 ${\delta}^{18}$O isoscapes as sheets of the model. Plots prepared with DataGraph [81] from exported geotiff images.

**Figure 19.**Three sheets of the Timescape model, time increasing top to bottom (year 1981, 1985, and 1989). (

**Left**): Causal neighborhood $\eta \left(\mathit{x}\right)$, source events’ sites are shown as red dots. (

**Right**): field values $\widehat{\mathrm{\Phi}}\left(\mathit{x}\right)$, note the typical bull’s eye shapes due to the IDW algorithm.

**Figure 20.**Symmetric double causal cones. Note that $\Delta (\mathit{x},\mathit{y})<\infty $ and $\Delta (\mathit{y},\mathbf{z})<\infty $, nonetheless $\Delta (\mathit{x},\mathbf{z})=\infty $ because ${x}_{0}={z}_{0}\wedge \overrightarrow{x}\ne \overrightarrow{z}\phantom{\rule{0.277778em}{0ex}}\u27f9\mathit{x}\notin {\mathcal{C}}_{z}$.

**Figure 21.**An input-driven decision tree. The extreme cases (purely spatial and purely temporal dependencies, see Table 5) suggest that the Timescape algorithm is either unsuitable or far too involved, and the entangled cases can be advantageously treated.

**Table 1.**Target events’ loop (dashed box of Figure 8).

1 | ${\mathcal{C}}_{\mathit{x}}^{-}\leftarrow \mathit{u}\in \mathit{S}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\mathrm{\Delta}(\mathit{u},\mathit{x})<\mathit{\infty}$ | Equation (7) Equation (9) Figure 3 |

2 | ${\mathit{S}}_{x}\leftarrow \mathit{S}\cap {\mathcal{C}}_{x}^{-}$ | filtering–Equation (11) Figure 7 |

3 | $\eta \left(\mathit{x}\right)\leftarrow |{\mathit{S}}_{x}|$ | Equation (15) |

4 | if ${\mathit{S}}_{x}=\u2300\u27f9\widehat{\mathrm{\Phi}}\left(\mathit{x}\right)\leftarrow \mathsf{null}$ | |

else $\widehat{\mathrm{\Phi}}\left(\mathit{x}\right)\leftarrow {\sum}_{{\mathit{S}}_{x}}W(\mathit{u}...)\phantom{\rule{0.166667em}{0ex}}\mathrm{\Phi}\left(\mathit{u}\right)$ | estimate–Equation (14) | |

if interp. allows $\widehat{\sigma}\left(\mathit{x}\right)\leftarrow \sqrt{\mathrm{var}}$ | Kriging… | |

else $\widehat{\sigma}\left(\mathit{x}\right)\leftarrow \mathsf{null}$ | IDW… | |

5 | update $\mathit{x}$: $\left[\phantom{\rule{0.277778em}{0ex}}\widehat{\mathrm{\Phi}}\left(\mathit{x}\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\sigma}\left(\mathit{x}\right),\phantom{\rule{0.277778em}{0ex}}\eta \left(\mathit{x}\right)\phantom{\rule{0.277778em}{0ex}}\right]$ |

Model | $\left|\mathit{S}\right|$ | Coordinates | ∼Scale | Time Span-Resol. |
---|---|---|---|---|

Fungi ${\delta}^{15}$N | 62 | Local Euclidean $(x,y)$ | 100 m | few days– 1 day |

Min temperature | 2521 | WGS84 UTM33 N | 100 km | 20 years–1 month |

Olive oil ${\delta}^{18}$O | 275 | European Lambert | 1000 km | 3 years–1 year |

Precipitation ${\delta}^{2}$H | 1152 | Geographical $(\lambda $, $\varphi )$ | ${10}^{4}$ km | 10 years–1 year |

**Table 3.**Model parameters. The target cells are ordered by time $\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}$ horizontal $\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}$ vertical.

Model | Method | Distance | Form | Target Cells $\phantom{\rule{0.277778em}{0ex}}\left|\mathit{T}\right|$ |
---|---|---|---|---|

Fungi ${\delta}^{15}$N | Kriging | Euclidean | $\psi \equiv 1$ | $64\times 128\times 128=\mathbf{1},\mathbf{048},\mathbf{576}$ |

Min temperature | Kriging | Euclidean | Equation (10) $\alpha =0$ | $240\times 25\times 32=\mathbf{192},\mathbf{000}$ |

Olive oil ${\delta}^{18}$O | IDW | Euclidean | $\psi \equiv 1$ | $7\times 74\times 99=\mathbf{51},\mathbf{282}$ |

Precipitation ${\delta}^{2}$H | IDW | Geodesic | $\psi \equiv 1$ | $20\times 120\times 50=\mathbf{120},\mathbf{000}$ |

**Table 4.**Model performances, measured in target events per second. ${\eta}_{\mathsf{model}}$ (17) is expressed in percent, running times in minute seconds.

Model | ${\mathit{\eta}}_{\mathsf{model}}$ | Configuration 1 | Configuration 2 | Configuration 3 | |||
---|---|---|---|---|---|---|---|

Time | evt/s | Time | evt/s | Time | evt/s | ||

Fungi ${\delta}^{15}$N | 55% | 87${}^{\prime}$32${}^{\prime \prime}$ | 200 | 125${}^{\prime}$2${}^{\prime \prime}$ | 140 | 75${}^{\prime}$33${}^{\prime \prime}$ | 231 |

Min temperature | 73% | 115${}^{\prime}$49${}^{\prime \prime}$ | 28 | 118${}^{\prime}$6${}^{\prime \prime}$ | 27 | 78${}^{\prime}$20${}^{\prime \prime}$ | 41 |

Olive oil ${\delta}^{18}$O | 81% | 39${}^{\prime \prime}$ | 1315 | 47${}^{\prime \prime}$ | 1091 | 35${}^{\prime \prime}$ | 1478 |

Precipitation ${\delta}^{2}$H | 62% | 15${}^{\prime}$42${}^{\prime \prime}$ | 157 | 16${}^{\prime}$4${}^{\prime \prime}$ | 124 | 11${}^{\prime}$21${}^{\prime \prime}$ | 176 |

Field Dependence | Timescape | Alternatives | References |
---|---|---|---|

temporal only | useless | time series analysis | [2,3] |

ODE and PDE | |||

prev. temporal | sharp cones $k\to 0$ | dynamical systems | [16,17,88] |

machine learning | [89] | ||

entangled S + T | optimal suitability | stochastic PDE | [17,18] |

covariance-based | [12] | ||

Bayesian modeling | [21] | ||

neural networks | [19,20] | ||

prev. spatial | broad cones $k\to \infty $ | gen. lin. models | [22] |

regression trees | [23] | ||

spatial only | useless | geostatistics | [4,5] |

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## Share and Cite

**MDPI and ACS Style**

Ciolfi, M.; Chiocchini, F.; Pace, R.; Russo, G.; Lauteri, M.
Timescape: A Novel Spatiotemporal Modeling Tool. *Earth* **2022**, *3*, 259-286.
https://doi.org/10.3390/earth3010017

**AMA Style**

Ciolfi M, Chiocchini F, Pace R, Russo G, Lauteri M.
Timescape: A Novel Spatiotemporal Modeling Tool. *Earth*. 2022; 3(1):259-286.
https://doi.org/10.3390/earth3010017

**Chicago/Turabian Style**

Ciolfi, Marco, Francesca Chiocchini, Rocco Pace, Giuseppe Russo, and Marco Lauteri.
2022. "Timescape: A Novel Spatiotemporal Modeling Tool" *Earth* 3, no. 1: 259-286.
https://doi.org/10.3390/earth3010017