# Examining the Limits of Predictability of Human Mobility

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## Abstract

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## 1. Introduction

#### 1.1. Benchmarking Limits of Mobility Prediction

**Problem Definition.**Before going into further details, we can state the central objective of this work as follows: knowing that we observe a discrepancy between the predictability upper bound and the empirical prediction accuracy, we aim at investigating the methodology behind the upper bound estimation and at understanding the primary reasons for this discrepancy. To this end, we adopt the approach consisting in the three steps listed hereafter, where each step acts as a causal verification for the next.

- Confirm the discrepancy between the upper limit of predictability and prediction accuracy through extensive experimentation using widely contrasting prediction models on contrasting datasets.
- Following the discrepancy confirmation, revisit the assumptions underlying the upper bound computation methodology.
- Scrutinize the assumptions, analyze the reasons contributing to the failure of the methodology.

#### 1.2. Discrepancies and Inconsistencies

#### 1.3. Questioning the Predictability Upper Bound

- Substantiate the observed discrepancy between ${\pi}_{acc}$ and ${\pi}^{max}$. To this end, we build prediction models using seven distinct approaches and conduct a comprehensive accuracy analysis based on three real-world mobility datasets.
- Revisit the assumptions hereafter, which might have lead to this discrepancy.
- (a)
- Human mobility is Markovian and thus possesses a memoryless structure.
- (b)
- The mobility entropy estimating technique achieves an asymptotic convergence.
- (c)
- The predictability upper bound accounts for (all) the long-distance dependencies in a mobility trajectory.

#### 1.4. Roadmap and Main Findings

- We discuss all the relevant concepts used in this work in Section 2 and illustrate how diverse concepts such as entropy, mutual information and predictive information interact with each other in the light of the predictability upper bound.
- In Section 3, we describe the mobility datasets used in this work and confirm the discrepancy between the maximum upper bound of mobility prediction derived by the previous works and the empirical prediction accuracy derived using recurrent-neural network variants. In order to minimize any bias, we construct seven different prediction models and compute the accuracy across three datasets differing with respect to their collection timespans, region, demographics, sampling frequency and several other parameters.
- In Section 4, we audit three underlying assumptions in the currently used methodology for ${\pi}^{max}$ computation.
- (a)
- In Section 4.1, we demonstrate the non-Markovian character of human mobility dynamics contrary to the previously held assumption. Our statistical tests to confirm the nature of human mobility include (i) rank-order distribution, (ii) inter-event and dwell time distribution, and (iii) mutual information decay.
- (b)
- In Section 4.2, we analyze the entropy convergence by comparing entropies derived by using Lempel–Ziv 78 and Lempel–Ziv 77 encoding schemes on mobility trajectories. Based on this result, we show that there does not exist an ideal entropy estimation scheme for mobility trajectories that achieves an asymptotic convergence.
- (c)
- In Section 4.3, we assert that the current methodology used to estimate ${S}^{real}$ does not represent an accurate entropy estimate of mobility trajectory. To this end, we demonstrate that the individual elements present in a mobility subsequence derived by the currently used encoding schemes have non-zero dependencies un-accounted for, when deriving the mobility entropy. We validate such a manifestation by computing the pointwise mutual information associated with mobility trajectories which indicate an on average positive pointwise mutual information (PMI).

- In Section 5 we discuss the likely causes behind this discrepancy being overlooked. We also present the potential reasons as to why recurrent neural networks (RNN) extensions exceed the theoretical upper bound and discuss the applicability of the prediction models in different contexts. We conclude the paper in Section 6.

## 2. Relevant Concepts

#### 2.1. Mobility Modeling

#### 2.2. Markov Processes

#### 2.3. Long-Distance Dependencies

#### 2.4. Recurrent Neural Networks and Extensions

#### 2.5. Mutual Information

#### 2.6. Entropy, Encoding and Compression

#### 2.7. Predictive Information

## 3. Confirming ${\pi}^{max}$ Discrepancy with Real-World Datasets

#### 3.1. Experimental Setup

#### 3.2. Confirming the Predictability Upper Bound Discrepancy

## 4. Revisiting the Underlying Assumptions

#### 4.1. Questioning the Markovian Nature of Human Mobility

#### 4.1.1. Location Rank-Order Distribution

#### 4.1.2. Inter-Event Time Distribution

#### 4.1.3. Mutual Information Decay

#### 4.2. Questioning the Asymptotic Convergence of the Entropy Estimate

#### 4.3. Questioning ${S}^{real}$ as a Relative Entropy Estimate for Human Mobility

## 5. Discussion

- number of unique locations present in the trajectory,
- length of the trajectory and the size of the dataset,
- number of interacting locations within a long-distance dependency,
- distance between the interacting locations.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Data Availability

## References

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**Figure 1.**Prediction accuracy for Markov models (order 1–5). The x-axis signifies the proportion of trajectory length considered for the train-test split and y-axis signifies the precision of the prediction model.

**Figure 2.**Prediction accuracy for recurrent-neural architectures. The x-axis signifies the proportion of trajectory length considered for the train-test split and y-axis signifies the precision of the prediction model.

**Figure 3.**Comparison of ${\pi}^{max}$ with the maximum predictability achieved using models from each category. The dotted lines indicate the predictability by each approach (indicated with the same colour). x-axis signifies the proportion of trajectory length considered for the train-test split and y-axis signifies the precision of the prediction model.

**Figure 4.**Rank distribution of location visits at the collective level for aggregated dataset. The data is binned into exponentially wider bins and normalised by the bin width. The straight line represents the fitting through least squares regression ($\alpha $ and ${x}_{min}$, computed through maximum likelihood estimation).

**Figure 10.**Location pair occurrences across all the sampling rates of the true sample. The x-axis represents the unique pair ID in the descending order of their frequency of occurrence. The y-axis is the ratio between the unique pairs and the total number of pairs contained in the an individual trajectory.

**Figure 11.**Mutual information decay and joint entropy estimated for all the datasets. The dataset consists of stacked sequences of temporally arranged individual points of interest.

**Figure 12.**Comparison of entropy derived using LZ78 and LZ77 encoding algorithms. The red curve is the maximum entropy.

**Figure 13.**Pointwise mutual information across longer substrings in a user trajectory. The x-axis denotes the index’s of element pairs in a substring derived from a user trajectory using LZ78 encoding algorithm. The y-axis denote the pointwise mutual information between the element pairs.

**Figure 14.**Pointwise mutual information across short substrings in a user trajectory. The x-axis denote the index’s of element pairs in a substring derived from a user trajectory using LZ78 encoding algorithm. The y-axis denote the pointwise mutual information between the element pairs.

**Table 1.**Comparison of ${\pi}^{max}$ and ${\pi}_{acc}$ at varying granularities of $\Delta s$ (spatial granularity) and $\Delta t$ (temporal granularity) reported by existing literature.

Authors (year) | ${\mathit{\pi}}^{\mathit{max}}$ ($\Delta \mathit{s},\Delta \mathit{t}$) | ${\mathit{\pi}}_{\mathit{acc}}$ | Prediction Model | Dataset Duration | Dataset Type |
---|---|---|---|---|---|

Song et al. [5] (2010) | 93% (3–4 km) | – | – | 3 months | CDR |

Lu et al. [11] (2013) | 88% (3–4 km) | 91% | Markov (first-order) | five months | CDR |

Smith et al. [12] (2014) | 93.05–94.7% (350 m, 5 min) 81.45–85.57% (100 m, 5 min) 74.23–78.20% (350 m, 60 min) | – | – | 36 months | GPS |

Ikanovic and Mollgaard [17] (2017) | 95.5 ± 1.8% (1.7 km) 71.1% (25 m) | 88.3 ± 3.8% 75.8% | Markov (first-order) | 36 months (same as previous) | GPS |

**Table 2.**Recurrent neural network variants with their respective architectural differences and features.

Extension | Architecture | Features |
---|---|---|

Vanilla-RNN [34] | • no cell state/gating mechanism • recurrent connections | • faster and stable training • simple architecture |

RNN-LSTM [31] | • similar connections as Vanilla-RNN • diff. cell state with gating mechanism | • actively maintain self-connecting loops • prevents memory degradation |

Dilated-RNN [35] | • similar cell structure as LSTM • dilated skip connections | • increased parallelism in the computation • improves long-term memorization capabilities |

RHN [32] | • diff. cell design • long credit assignment paths | • handles short-term patterns • reduces data-dependent parameters for LDD memorization |

PSMM [33] | • diff. gating function, shortcut connections • variable dimensionality hidden state | • improves handling of rare symbols • allows for better long-distance gradients |

**Table 3.**Mobility dataset specifications and their respective ${S}^{real}$ and ${\pi}^{max}$ values.

Datasets | Num. Users | Duration (months) | Avg. Trajectory Length | Distinct Locations | Avg. Spatio-Temporal Granularity | ${\mathit{S}}^{\mathit{real}}$ | ${\mathit{\pi}}^{\mathit{max}}$ |
---|---|---|---|---|---|---|---|

PrivaMov | 100 | 15 | 1,560,000 | 2651 | 246 m 24 s | 6.63 | 0.5049 |

NMDC | 191 | 24 | 685,510 | 2087 | 1874 m 1304 s | 5.08 | 0.6522 |

GeoLife | 182 | 36 | 8,227,800 | 3892 | 7.5 m 5 s | 7.77 | 0.4319 |

**Table 4.**Hyperparameters selected for each recurrent neural networks (RNN) variant for the prediction accuracy measurement experiments.

RNN Variant | Hidden-Layer Size | Unroll Steps | Learning Rate | Activation Function | Optimizer | Dropout Rate |
---|---|---|---|---|---|---|

Vanilla-RNN | 100 | 25 | 0.1 | tanh | Adam | 0.2 |

RNN-LSTM | 100 | 50 | 1.0 × ${}^{-8}$ | ReLU | Adam | 0.2 |

Dilated-RNN | 100 | 32 | 1.0 × ${}^{-6}$ | ReLU | Adam | 0.2 |

RHN | 100 | 50 | 1.0 × ${}^{-8}$ | ReLU | Adam | 0.2 |

PSMM | 100 | 50 | 1.0 × ${}^{-8}$ | ReLU | Adam | 0.2 |

Datasets | ${\mathit{\pi}}^{\mathit{max}}$ | ${\mathit{\pi}}_{\mathit{acc}}\left(\mathit{MC}\left(2\right)\right)$ | ${\mathit{\pi}}_{\mathit{acc}}\left(\mathit{MC}\left(3\right)\right)$ | ${\mathit{\pi}}_{\mathit{acc}}\left(\mathit{HMM}\left(2\right)\right)$ | ${\mathit{\pi}}_{\mathit{acc}}\left(\mathit{RHN}\right)$ | ${\mathit{\pi}}_{\mathit{acc}}\left(\mathit{RNN}\right)$ |
---|---|---|---|---|---|---|

PrivaMov | 0.50 | 0.47 | 0.46 | 0.60 | 0.76 | 0.72 (Dilated-RNN) |

NMDC | 0.65 | 0.70 | 0.68 | 0.66 | 0.78 | 0.72 (RNN-LSTM) |

GeoLife | 0.43 | 0.40 | 0.36 | 0.43 | 0.70 | 0.66 (PSMM) |

Name | Density $\mathit{p}\left(\mathit{x}\right)=\mathit{Cf}\left(\mathit{x}\right)$ | |
---|---|---|

$\mathit{f}\left(\mathit{x}\right)$ | $\mathit{C}$ | |

Power law with cutoff | ${x}^{-\alpha}{e}^{-\lambda x}$ | $\frac{{\lambda}^{1-\alpha}}{\tau (1-\alpha ,\lambda {x}_{min})}$ |

Exponential | ${e}^{-\lambda x}$ | $\lambda {e}^{\lambda {x}_{min}}$ |

Stretched exponential | ${x}^{\beta -1}{e}^{-\lambda {x}^{\beta}}$ | $\beta \lambda {e}^{\lambda {x}_{min}^{\beta}}$ |

Log-normal | $\frac{1}{x}exp[-\frac{{(lnx-\mu )}^{2}}{2{\sigma}^{2}}]$ | $\sqrt{\frac{2}{\pi {\sigma}^{2}}}{\left[erfc\left(\frac{ln{x}_{min}-\mu}{\sqrt{2}\sigma}\right)\right]}^{-1}$ |

Rank Order | Power Law p | Log-Normal | Exponential | Stretched Exp. | Power Law + Cutoff | Support for Power-Law | ||||
---|---|---|---|---|---|---|---|---|---|---|

LR | p | LR | p | LR | p | LR | p | |||

Privamov | 0.00 | −12.72 | 0.00 | −30.12 | 0.00 | −11.42 | 0.00 | −113.1 | 0.00 | with Cutoff |

NMDC | 0.00 | −11.28 | 0.00 | −27.23 | 0.00 | −13.95 | 0.00 | −320 | 0.00 | with Cutoff |

Geolife | 0.006 | −17.04 | 0.00 | −19.21 | 0.00 | −18.21 | 0.08 | −560.78 | 0.00 | with Cutoff |

**Table 8.**Maximum likelihood and K-S test for the cumulative distributions (lower value in boldface indicates a better fit). We clearly observe that high granularity points of interest depict a power-law unlike the CDR logs which are a rough approximation of human mobility.

MSE | ||||
---|---|---|---|---|

Measure | Log-Normal | Exponential | Stretched Exp. | Power Law + Cutoff |

NMDC-POIs | 0.04501 | 0.05648 | 0.02348 | 0.00616 |

GeoLife-POIs | 0.00324 | 0.07306 | 0.00378 | 0.00087 |

PrivaMov-POIs | 0.05824 | 0.09386 | 0.00739 | 0.00114 |

NMDC-GSM | 0.25584 | 0.00224 | 0.00584 | 0.07268 |

PrivaMov-GSM | 0.03655 | 0.00895 | 0.00098 | 0.00783 |

K-S Test | ||||

NMDC-POIs | 0.65843 | 0.75615 | 0.07456 | 0.00825 |

GeoLife-POIs | 0.63288 | 0.93644 | 0.04289 | 0.00046 |

PrivaMov-POIs | 0.96752 | 0.69748 | 0.27896 | 0.00116 |

NMDC-GSM | 0.56825 | 0.00987 | 0.00967 | 0.04568 |

PrivaMov-GSM | 0.85621 | 0.00567 | 0.00165 | 0.00927 |

Inter-Event Times | Power Law p | Log-Normal | Exponential | Stretched Exp. | Power Law + Cutoff | Support for Power-Law | ||||
---|---|---|---|---|---|---|---|---|---|---|

LR | p | LR | p | LR | p | LR | p | |||

Privamov | 0.12 | −1.13 | 0.28 | 5.69 | 0.00 | 0.09 | 0.00 | −0.34 | 0.74 | with Cutoff |

NMDC | 0.08 | −0.11 | 0.02 | 2.98 | 0.00 | 3.78 | 0.54 | −2.87 | 0.31 | weak |

Geolife | 0.86 | −7.76 | 0.00 | −20.43 | 0.00 | 17.87 | 0.08 | −0.30 | 0.59 | good |

**Table 10.**Kolmogorov–Smirnov goodness-of-fit test for mutual information decay of GeoLife dataset at varying sampling rates.

Sampling Rate | Power Law p | Power Law + Cutoff | Log-Normal | Exponential | Stretched Exp. | Support for Power Law | ||||
---|---|---|---|---|---|---|---|---|---|---|

LR | p | LR | p | LR | p | LR | p | |||

1X | 0.51 | 5.43 | 0.19 | 0.278 | 0.47 | 9.89 | 0.96 | 4.32 | 0.12 | good |

2X | 0.06 | 0.00 | 0.07 | −1.25 | 0.08 | 2.89 | 0.11 | 10.08 | 0.00 | with Cutoff |

4X | 0.46 | −0.065 | 0.67 | −0.072 | 0.87 | 1.89 | 0.87 | 1.78 | 0.07 | moderate |

0.5X | 0.00 | 0.00 | 0.00 | −5.54 | 0.01 | 8.66 | 0.38 | 11.88 | 0.00 | with Cutoff |

0.25X | 0.00 | 0.00 | 0.02 | −1.78 | 0.03 | 9.94 | 0.04 | 13.56 | 0.00 | with Cutoff |

**Table 11.**Kolmogorov–Smirnov goodness-of-fit test for mutual information decay across all the datasets.

Dataset | Power Law p | Power Law + Cutoff | Log-Normal | Exponential | Stretched Exp. | Support for Power Law | ||||
---|---|---|---|---|---|---|---|---|---|---|

LR | p | LR | p | LR | p | LR | p | |||

Privamov | 0.43 | 3.25 | 0.69 | 1.78 | 0.28 | 6.28 | 0.83 | 4.89 | 0.34 | good |

NMDC | 0.27 | 1.82 | 0.11 | −0.27 | 0.10 | 2.47 | 0.65 | 2.21 | 0.16 | moderate |

Geolife | 0.51 | 5.43 | 0.19 | 0.278 | 0.47 | 9.89 | 0.96 | 4.32 | 0.12 | good |

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Kulkarni, V.; Mahalunkar, A.; Garbinato, B.; Kelleher, J.D.
Examining the Limits of Predictability of Human Mobility. *Entropy* **2019**, *21*, 432.
https://doi.org/10.3390/e21040432

**AMA Style**

Kulkarni V, Mahalunkar A, Garbinato B, Kelleher JD.
Examining the Limits of Predictability of Human Mobility. *Entropy*. 2019; 21(4):432.
https://doi.org/10.3390/e21040432

**Chicago/Turabian Style**

Kulkarni, Vaibhav, Abhijit Mahalunkar, Benoit Garbinato, and John D. Kelleher.
2019. "Examining the Limits of Predictability of Human Mobility" *Entropy* 21, no. 4: 432.
https://doi.org/10.3390/e21040432