# Comparing Equation-Based and Agent-Based Data Generation Methods for Early Warning Signal Analysis

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## Abstract

**:**

## 1. Introduction

## 2. System Used for Data Generation

#### 2.1. Pitchfork System

#### 2.2. Hysteresis System

## 3. Data Generation and Comparison

## 4. Early Warning Signal Analysis

## 5. Results

#### 5.1. Pitchfork System

#### 5.2. Hysteresis System

## 6. Discussion and Limitations

## 7. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Pseudocode: 2D Ising model using Metropolis algorithm |

Create an initial lattice configuration of $N$ × $N$ spins for the total number of spins do: Select a random spin ${S}_{i}$ Compute the change in energy $\mathsf{\Delta}E$ associated with flipping the spin: (A) Pitchfork Model: $\mathsf{\Delta}E=2\cdot {S}_{i}\cdot {\sum}_{\langle i,j\rangle}{S}_{j}$ (B) Hysteresis Model: $\mathsf{\Delta}E=2\cdot {S}_{i}\cdot \left({\sum}_{\langle i,j\rangle}{S}_{i,j}-H\right)$ Generate a uniformly distributed random number $R$ between 0 and 1 if $R<{e}^{-\mathsf{\Delta}E\cdot T}$ then: Accept the flip: ${S}_{i}\leftarrow -{S}_{i}$ else: Do not accept the flip: ${S}_{i}\leftarrow {S}_{i}$ end ifend for |

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**Figure 1.**Top row plots: mean (bold), min and max (brighter) of 100 time series each as used for early warning signal (EWS) analysis from the pitchfork (left column) and the hysteresis (right column) system, generated from simulations with agent-based model (ABM) (blue) and equation-based model (EBM) (green). In red are the theoretical (i.e., mathematically calculated) equilibria (or tipping) of the systems. Second row plots: results of dynamic time warping (DTW) comparison of ABM and EBM data, indicating a distinctively closer match in the pitchfork system. Bottom row plots: phase portrait of ABM and EBM data, confirming the closer match in the pitchfork system with additional indicators.

**Figure 2.**EWS analysis of Ising time-series (TS) in pitchfork mode. Left column shows analysis of EBM-generated data; the right, of ABM-generated data (blue: original data; green: detrended). Solid lines indicate mean curves; brighter areas mark upper and lower bounds of samples. The bold solid line segment (time steps 100 to 1700) shows the range to which Kendall’s τ calculation was applied. The red dashed line in the first-row plots shows the mathematically calculated (i.e., theoretical) upper equilibrium branch of the pitchfork bifurcation. Note that the coefficient of variation (CV), detrended fluctuation analysis (DFA) and Reddening are indicators that by themselves apply detrending to data. That is why no separate (green) signals are shown in these cases.

**Figure 3.**EWS analysis of Ising time-series (TS) in hysteresis mode. All plots and indications in analogy to Figure 2. In this case, the red dashed line in the first-row plots shows the mathematically calculated (i.e., theoretical) sigmoid curve of a combined saddle-node bifurcation.

Agent-Based Model | Equation-Based Model | |
---|---|---|

Pitchfork system | 2D Ising model without external field | Mean-field approximation of 2D Ising model without external field |

Hysteresis system | 2D Ising model with external field | Mean-field approximation of 2D Ising model with external field |

Agent-Based Pitchfork Model | Equation-Based Pitchfork Model | ||
---|---|---|---|

Temperature T (critical parameter) | Over [1.42, 3.12] | Inverse temp. B (critical parameter) | Over [2.7, −0.3] |

Lattice size N | 50 < N < 100 | Time step ∆t | 0.1 |

Weight of normal distribution p | 0.01 < p < 0.02 |

Agent-Based Pitchfork Model | Equation-Based Pitchfork Model | ||
---|---|---|---|

External field H (critical parameter) | Over [−0.2, 0.2] | External field H (critical parameter) | Over [0.6, −0.5] |

Lattice size N | 50 < N < 100 | Time step ∆t | 0.1 |

Temperature T | 2.12 | Inverse temp. B | 1.3 |

Weight of normal distribution p | 0.004 < p < 0.008 |

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**MDPI and ACS Style**

Reisinger, D.; Füllsack, M. Comparing Equation-Based and Agent-Based Data Generation Methods for Early Warning Signal Analysis. *Systems* **2020**, *8*, 54.
https://doi.org/10.3390/systems8040054

**AMA Style**

Reisinger D, Füllsack M. Comparing Equation-Based and Agent-Based Data Generation Methods for Early Warning Signal Analysis. *Systems*. 2020; 8(4):54.
https://doi.org/10.3390/systems8040054

**Chicago/Turabian Style**

Reisinger, Daniel, and Manfred Füllsack. 2020. "Comparing Equation-Based and Agent-Based Data Generation Methods for Early Warning Signal Analysis" *Systems* 8, no. 4: 54.
https://doi.org/10.3390/systems8040054