# Computing Spatiotemporal Accessibility to Urban Opportunities: A Reliable Space-Time Prism Approach in Uncertain Urban Networks

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. Non-Normal Reliable Network-Based Space-Time Prism Model

#### 2.2. Solution Algorithm

- Step 1: Adopt a large number (i.e., $N=1000or10000$).
- Step 2: Determine the confidence interval ($\alpha $).
- Step 3: Generate $N$ random samples from the probability distribution of each edge in the path.
- Step 4: Sum up values generated in Step 3.
- Step 5: Sort values obtained in Step 4.
- Step 6: Select element $\alpha \times N$ from the matrix achieved in Step 5. This element is inverse value of CDF of the path at confidence interval $\alpha $.

Algorithm 1. Inspired from [58] |

Inputs:$r$, $b$, and $\alpha $ |

Outputs:$NNRFC$ |

Step 1. Initialization |

Set $NNRFC=\varnothing $ |

For each edge ${e}_{ij}$ in $E$: |

Create forward path ${l}_{u}^{rj}=\varnothing $ |

End for |

For each link ${e}_{rj}$ emanating from $r$ |

Create forward path ${l}_{R}^{rj}={e}_{rj}$ |

Calculate ${F}_{{l}_{R}^{rj}}^{-1}\left(\alpha \right)$ for the constructed path |

Set $SE=SE{\displaystyle \cup}\left\{{l}_{R}^{rj}\right\}$ |

Set ${L}_{R}^{rj}={L}_{R}^{rj}{\displaystyle \cup}\left\{{l}_{R}^{rj}\right\}$ |

End for |

Step 2. Path selection |

If $SE=\varnothing $ |

Stop |

Else |

Select ${l}_{R}^{rj}$ at the top of $SE$ and set $SE-\left\{{l}_{R}^{rj}\right\}$ |

End if |

Step 3. Path extension |

For each allowed movement in ${\psi}_{jk}\in \Psi $ from $j$ in ${l}_{R}^{rj}$ |

Create new path ${l}_{R}^{r\u201ajk}={l}_{R}^{rj}\oplus {e}_{jk}$ |

Calculate ${F}_{{l}_{R}^{r\u201ajk}}^{-1}\left(\alpha \right)$ by MCS |

If ${F}_{{l}_{R}^{r\u201ajk}}^{-1}\left(\alpha \right)\le b$ |

If ${l}_{R}^{rjk}\notin FC$ |

Set $SE=SE{\displaystyle \cup}\left\{{l}_{R}^{r\u201ajk}\right\}$ |

Set ${L}_{R}^{rj}={L}_{R}^{rj}{\displaystyle \cup}\left\{{l}_{R}^{r\u201ajk}\right\}$ |

Else |

Check dominance condition for ${l}_{R}^{r\u201ajk}$ |

If ${l}_{R}^{ks}$ is non-dominated path |

Set $SE=SE{\displaystyle \cup}\left\{{l}_{R}^{r\u201ajk}\right\}$ |

Set ${L}_{R}^{rj}={L}_{R}^{rj}{\displaystyle \cup}\left\{{l}_{R}^{r\u201ajk}\right\}$ |

Remove dominated paths in $SE$ and ${L}_{R}^{rj}$ |

End if |

End if |

End if |

End for |

Set $NNRFC={L}_{R}^{rj}$ |

Algorithm 2. Backward cone calculator. |

Inputs:$s$, $b$, $\alpha $, and $NNRFC$ |

Outputs:$NNRBC$ |

Step 1. Initialization |

Set $NNRBC=\varnothing $ |

For each link ${e}_{ij}$ in $E$: |

Create ${l}_{u}^{is}:=\varnothing $ |

End for |

For each link ${e}_{is}$ emerging into $s$ |

If ${L}_{R}^{ri}\ne \varnothing $ |

For each ${l}_{R}^{ri}\in NNRFC$ |

Integrate ${l}_{R}^{ri}$ with ${e}_{is}$, ${l}_{R}^{r\u201ais}={l}_{R}^{ri}\oplus {e}_{is}$ |

Calculate ${F}_{{l}_{R}^{r\u201ais}}^{-1}\left(\alpha \right)$ by MCS |

End for |

If $\mathrm{min}({F}_{{l}_{R}^{r\u201ais}}^{-1}\left(\alpha \right))\le b$ |

Set $SE:=SE{\displaystyle \cup}\left\{{l}_{R}^{is}\right\}$ |

Set ${L}_{R}^{is}:={L}_{R}^{is}{\displaystyle \cup}\left\{{l}_{R}^{is}\right\}$ |

End if |

End if |

End for |

Step 2. Path selection |

If $SE=\varnothing $ |

Stop |

Else |

Select ${l}_{R}^{is}$ at the top of $SE$ and set $SE-\left\{{l}_{R}^{is}\right\}$ |

End if |

Step 3. Path extension |

For each allowed movement in ${\psi}_{ki}\in \Psi $ create ${l}_{R}^{ks}={e}_{ki}\oplus {l}_{R}^{is}$ |

If ${l}_{R}^{ri}={l}_{R}^{rk}\oplus {e}_{ki}\in NNRFC\ne \varnothing $ |

For each ${l}_{R}^{ri}\in NNRFC$ |

Integrate ${l}_{R}^{ri}$ with ${l}_{R}^{ks}$, ${l}_{R}^{rs}={l}_{R}^{ri}{\displaystyle \cup}{l}_{R}^{ks}={l}_{R}^{rk}\oplus {e}_{ki}\oplus {l}_{R}^{is}$ |

Calculate ${F}_{{l}_{R}^{r\u201ais}}^{-1}\left(\alpha \right)$ by MCS |

End for |

If $\mathrm{min}({F}_{{l}_{R}^{rs}}^{-1}\left(\alpha \right))\le b$ |

If ${l}_{R}^{ks}\notin {L}_{R}^{s}$ |

Set $SE=SE{\displaystyle \cup}\left\{{l}_{R}^{ks}\right\}$ |

Set ${L}_{R}^{is}={L}_{R}^{is}{\displaystyle \cup}\left\{{l}_{R}^{ks}\right\}$ |

Else |

Check dominance condition for ${l}_{R}^{ks}$ |

If ${l}_{R}^{ks}$ is non-dominated path |

Set $SE=SE{\displaystyle \cup}\left\{{l}_{R}^{ks}\right\}$ |

Set ${L}_{R}^{is}={L}_{R}^{is}{\displaystyle \cup}\left\{{l}_{R}^{ks}\right\}$ |

Remove dominated paths in $SE$ and ${L}_{R}^{is}$ |

End if |

End if |

End if |

End if |

End for |

Set $NNRBC={L}_{R}^{is}$ |

#### 2.3. Accessibility Model

## 3. Numerical Computation and Discussion

- Trip chain $Home\to \mathrm{Kowsar}\text{}market\to Home$ was defined due to the portion of home-based trips among the other trips (about 90% of all trips).
- Modeling period was considered the time window between 5:00 p.m. and 8:00 p.m., including the afternoon peak hour.
- Time budget was set to 60 minutes, following previous studies [36].
- Area of markets was considered as the indicator of opportunity attractiveness (Table 4).
- On-time arrival probability was defined 99% for simulating extreme conditions, which gives minimum value or lower bound of accessibility.
- Travel time distributions were set to normal, exponential, and log-normal distributions as travel time uncertainty in urban street networks is commonly defined by pre-allocated distributions [62]. Normal distribution was selected to provide a ground for comparing the results of the non-normal model with the conventional outputs. Additionally, log-normal distribution was taken into account because this distribution could model travel time in a more realistic manner [77].

- Null hypothesis (${H}_{0}$): There is no statistically meaningful difference between the average of TAZ accessibility values in normally and non-normally distributed networks.
- Alternative hypothesis (${H}_{1}$): There is a statistically meaningful difference between the average of TAZ accessibility values in normally and non-normally distributed networks.

## 4. Summary and Conclusion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Classical Network-Based Space-Time Prisms

## References

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**Figure 2.**Accessibility of markets and traffic analysis zones (TAZs) (normalized values). (

**a**) Accessibility with normal reliable space-time prism (NRNTP). (

**b**) Accessibility with exponential non-normal reliable space-time prism (NNRNTP). (

**c**) Accessibility with log-normal NNRNTP.

Symbol | Definition |
---|---|

$r$ | Index of trip origin, $r\in V$ |

$s$ | Index of trip destination, $s\in V$ |

${t}_{r}$ | Departure time from origin ($r$) |

${t}_{s}$ | Arrival time to destination ($s$) |

$b$ | Travel time budget ($b={t}_{s}-{t}_{r})$ |

$Q$ | Set of individuals or set of population groups |

$q$ | Index of an individual or a population group |

$\alpha $ | On-time arrival probability depicting risk-taking behavior of a traveler |

${\alpha}_{q}$ | On-time arrival probability of individual or population group $q\in Q$ |

${t}_{i}^{a}$ | The earliest arrival time to vertex $i\in V$ |

${t}_{i}^{b}$ | The latest departure time from vertex $i\in V$ |

${x}_{i}$ | $x$ co-ordinate of vertex $i\in V$ |

${y}_{i}$ | $y$ co-ordinate of vertex $i\in V$ |

$K$ | Set of all intermediate opportunities or points of interest $k$ |

$Z$ | Set of traffic analysis zones (TAZs) in a geographic area |

$z$, $\stackrel{\xb4}{z}$ | Indices of TAZs |

$CI$ | Set of confidence interval including ${\alpha}_{n}$ |

${P}_{z}$ | Population of TAZ $z\in Z$ |

${\stackrel{\xb4}{P}}_{z}^{{\alpha}_{q}}$ | Portion of population group $q$ in TAZ $z$ conducting their trip at confidence interval ${\alpha}_{q}$ |

${O}_{k}$ | Attractiveness of opportunity $k\in K$. Area could represent attractiveness of an opportunity [65,66,67,68,69]. |

${O}_{max}$ | Maximum attractiveness of opportunities, $\underset{k\in K}{\mathrm{max}}({O}_{k})$ |

$B\left(k\right)$ | A binary variable with value of 1 if activity $k$ be in space-time prism, and 0 otherwise. |

${L}_{}^{ij}$ | Set of paths start from vertex $i\in V$ and end to vertex $j\in V$ |

${L}_{}^{w\u201aij}$ | Set of paths from vertex $w\in V$ to vertex $j\in V$ passing through edge ${e}_{ij}$ |

$u$ | Index of a path in $L$ or ${L}_{}^{w\u201aij}$ |

${l}_{u}^{ij}$ | Path $u\in {L}^{ij}$ from vertex $i\in V$ to vertex $j\in V$ |

${l}_{u}^{w\u201aij}$ | Path $u\in {L}_{}^{w\u201aij}$ from vertex $w\in V$ to vertex $j\in V$ passing through edge ${e}_{ij}$ |

${l}_{NDM}^{ij}$ | Non-dominated path from vertex $i\in V$ to vertex $j\in V$ |

${l}_{NDM}^{r\u201aij}$ | Non-dominated path from vertex $w\in V$ to vertex $j\in V$ passing through edge ${e}_{ij}$ |

${F}_{{l}_{u}^{ij}}^{-1}\left(\alpha \right)$ | Value of inverse cumulative density function of ${l}_{u}^{ij}$ at confidence interval $\alpha $ |

Edge | First Parameter | Second Parameter |
---|---|---|

U1 | 5 | 7 |

U2 | 10 | 11 |

U3 | 15 | 13 |

U4 | 18 | 20 |

U5 | 4 | 6 |

**Table 3.**Result of Monte Carlo Simulation for the given example (The highlighted cell is travel time of the path at confidence interval 80%).

Random Samples Taken from Distributions in Table 2 | Summation | Sorting | ||||

U1 | U2 | U3 | U4 | U5 | ||

6.284 | 10.620 | 13.847 | 19.058 | 5.049 | 54.860 | 52.477 |

5.442 | 10.600 | 13.378 | 19.659 | 5.945 | 55.026 | 53.748 |

6.674 | 10.172 | 14.192 | 19.717 | 5.420 | 56.177 | 54.257 |

6.942 | 10.090 | 13.023 | 19.578 | 4.623 | 54.257 | 54.860 |

6.692 | 10.255 | 14.820 | 18.635 | 4.582 | 54.986 | 54.986 |

6.011 | 10.858 | 14.358 | 18.904 | 5.700 | 55.833 | 55.026 |

5.557 | 10.911 | 13.977 | 19.504 | 5.823 | 55.773 | 55.569 |

6.493 | 10.699 | 14.878 | 18.219 | 5.278 | 55.569 | 55.773 |

5.473 | 10.725 | 13.548 | 18.219 | 4.510 | 52.477 | 55.833 |

6.914 | 10.229 | 13.886 | 18.539 | 4.177 | 53.748 | 56.177 |

Opportunity_ID | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Attractiveness | 0.37 | 0.09 | 0.18 | 1 | 0.33 | 0.02 | 0.54 | 0.65 | 0.26 | 0.28 | 0.20 | 0.48 |

**Table 5.**t-test for analyzing the difference between the accessibility of TAZs generated by the models.

t-test in 95% of Confidence Interval for Accessibility Value of TAZs | ||||
---|---|---|---|---|

Distribution type | Accessibility with exponential NNRNTP | Accessibility with log-normal NNRNTP | ||

t value | Sig. (2-tailed) | t value | Sig. (2-tailed) | |

Accessibility with NRNTP | 8.354 | 0.000 | 6.254 | 0.000 |

Accessibility with log-normal NNRNTP | −7.938 | 0.000 | - | - |

t-test in 95% of confidence interval | ||||
---|---|---|---|---|

Size of NNRNTP with exponential distribution | Size of NNRNTP with log-normal distribution | |||

t value | Sig. (2-tailed) | t value | Sig. (2-tailed) | |

Size of NRNTP | 3.233 | 0.003 | 7.085 | 0.000 |

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**MDPI and ACS Style**

Sahebgharani, A.; Mohammadi, M.; Haghshenas, H.
Computing Spatiotemporal Accessibility to Urban Opportunities: A Reliable Space-Time Prism Approach in Uncertain Urban Networks. *Computation* **2019**, *7*, 51.
https://doi.org/10.3390/computation7030051

**AMA Style**

Sahebgharani A, Mohammadi M, Haghshenas H.
Computing Spatiotemporal Accessibility to Urban Opportunities: A Reliable Space-Time Prism Approach in Uncertain Urban Networks. *Computation*. 2019; 7(3):51.
https://doi.org/10.3390/computation7030051

**Chicago/Turabian Style**

Sahebgharani, Alireza, Mahmoud Mohammadi, and Hossein Haghshenas.
2019. "Computing Spatiotemporal Accessibility to Urban Opportunities: A Reliable Space-Time Prism Approach in Uncertain Urban Networks" *Computation* 7, no. 3: 51.
https://doi.org/10.3390/computation7030051