# Towards an Integrated Methodology for Model and Variable Selection Using Count Data: An Application to Micro-Retail Distribution in Urban Studies

## Abstract

**:**

## 1. Introduction

- (i)
- the discrete, non-negative and highly skewed nature of micro-retail distribution, which is incompatible with the assumptions underlying traditional statistical approaches;
- (ii)
- the store absence, which characterizes urban spaces, is represented by a highly zero-inflated statistical distribution; store absence has both theoretically and methodologically been excluded from analysis;
- (iii)
- the role of contextual descriptors and their inclusion in traditional regression approaches;
- (iv)
- the presence of high multicollinearity when considering a large set of urban descriptors, which has been an issue in several methodological and theoretical approaches.

## 2. Methodological Literature Review

#### 2.1. Analysing the Relationship between Urban Form and Micro-retail Distribution

#### 2.2. Stores Absence and the Survivorship Bias

#### 2.3. The Contextual Effect

#### 2.4. Multicollinearity and Variable Selection Procedure

#### 2.5. Objective

## 3. Materials and Methods

#### 3.1. Case Study and Data Sources

#### 3.2. The Variables under Investigation

#### 3.3. Modelling Micro-retail Distribution: From Linear to Count Regression Approaches

#### 3.4. Modelling Selection: Goodness-of-fit Measures

#### 3.5. Feature Selection.

## 4. Results: Application to the French Riviera Case Study

#### 4.1. Model Selection

#### 4.2. Variable Selection

^{2}).

## 5. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AIC | Akaike Information Criteria |

BIC | Bayesian Information Criteria |

MCA | Multiple Centrality Assessment |

MFA | Multiple Fabric Assessment |

Enet | Elastic Net Penalized Regression |

GLM | Generalized Linear Model |

MLR | Multiple Linear Regression |

NB | Negative Binomial |

P | Poisson |

PR | Penalized Regression |

SSx | Space Syntax |

ZA | Zero Altered |

ZANB | Zero Altered Negative Binomial |

ZAP | Zero Altered Poisson |

ZI | Zero Inflated |

ZINB | Zero Inflated Negative Binomial |

ZIP | Zero Inflated Poisson |

## Appendix A. Urban Form and Retail Distribution Literature Review

**Table A1.**Literature review: papers investigating the relationship between urban form (mainly street-network configuration properties) and micro-retail distribution. E/NC-KDE, Euclidean/network-constrained Kernel Density Estimation; CDF, cumulative distribution function; MLR/BLR, muliple/bivariate linear regression; ExpR, exponential regression; P-corr, Pearson correlation; NBR, negative binomial regression; K-W H-test, Kruskal–Wallis H test.

Urban Form and Micro-retail Distribution | |||||
---|---|---|---|---|---|

Authors | Year | Dependent Variable | Phenomena | Space Study | Analytical Approach |

Hillier | 1999 | N° Stores/street | Micro-retail pattern | Camden London, UK | MLR |

Cutini | 2001 | N° Stores/25 m (100 streets) | Micro-retail pattern | 3 small-medim sized Italian towns | ExpR |

Van Nes | 2005 | - | Micro-retail aggl. | Amsterdam, Netherland | Visual |

Joosten and Van Nes | 2005 | - | Catering businesses | Berlin, Germany | Visual |

Sarma | 2006 | N° Stores/aggl. | Micro-retail aggl. | New Delhi, India | BLR |

Porta | 2006 | E-KDE: 100m-cells, bandwidth 100–300) | Micro-retail pattern | Bologna, Italy | P-Corr |

Ortiz-Chao | 2008 | N° Stores/street | Micro-retail (land use) | Mexico City, Mexico | CDF |

Porta | 2012 | E-KDE (300-mt BW) on a 10m-size cell raster | Micro-retail Pattern | Barcelona, Spain | P-Corr |

Tsou, Chen | 2013 | Micro-retail density within traffic zones | Micro-retail Pattern | Taipei city, taiwan | MlogLR |

Van Nes | 2014 | - | Micro-retail Pattern | Pompeii, Rome | Visual |

Wang et al. | 2014 | E-KDE (1.5-km BW on 100-m cell-side raster) | Micro-retail pattern | ChangChun, China | P-corr |

Sevtsuk | 2014 | Presence/absence micro-retail building level | Micro-retail pattern | Cambridge and Sommerville, USA | MLR-Spatial Lag and Error |

Sevtsuk | 2010 | ||||

Cui and Han | 2015 | E-KDE (1.5-km BW, 100-m size cell) | Micro-retail (Point of Interest) | Zhengzhou, China | P-corr |

Omer and Goldblatt | 2015 | N° Build. with micro-retail 50m street-buffer | Micro-retail pattern (comm.build.) | 8 Israeli Cities (3 types) | P-corr MLR |

Scoppa | 2013 | Micro-retail frontage/ street length | Micro-retail pattern (comm.parcels) | Buenos Aires, Argentina | BLR, PCA-MLR |

Peponis Scoppa | 2015 | ||||

Ye et al. | 2017 | N° Stores/street block | Catering businesses | Shenzhen, China | NBR |

Lin et al. | 2018 | E-KDE (3.5-km BW, 100-m size cells) | Micro-retail pattern (POI) | Guangzhou, China | |

Cutini et al. | 2018 | N° Stores/ street (30 streets) | Micro-retail pattern | Milan, Italy | Exp-Corr |

Saraiva et al. | 2019 | E-KDE (20-m size cells) | Micro-retail vacancy | 4 medium-sized Portuguese cities | P-corr |

Bobkova et al. | 2019 | N° Stores/ plot | Micro retail pattern | London, Amsterdam Stockholm | K-W H-test |

## Appendix B. Street-based Urban Form Measures

#### Appendix B.1. Street Network Configurational Indicators

- d[i, j] represents the distance of the shortest path between the reference midpoint i and each destination midpoint j within the sub-network identified by the radius r;
- δ[i, j] represents the relative Euclidean distance between each midpoint i and each destination midpoint j within the same distance;
- ${n}_{jk}\left[i\right]$ is the number of minimum paths from node j to node k on network G passing through point i, with j and k at a distance less than or equal to r.

**Table A2.**Summary table of the 40 street-network configurational indicators. r, radius; n, normalized.

Pedestrian r [meters] | Vehicle [minutes] | |||||
---|---|---|---|---|---|---|

300 | 600 | 1200 | 5 | 20 | ||

$Reach{}_{r}$ | ${R}_{300}$ | ${R}_{600}$ | ${R}_{1200}$ | ${R}_{5}$ | ${R}_{20}$ | |

$Reach{}_{r}^{N}$ | ${R}_{300}^{N}$ | ${R}_{600}^{N}$ | ${R}_{1200}^{N}$ | ${R}_{5}^{N}$ | ${R}_{20}^{N}$ | |

$Closenes{s}_{r}$ | ${C}_{300}$ | ${C}_{600}$ | ${C}_{1200}$ | ${C}_{5}$ | ${C}_{20}$ | |

$Closenes{s}_{r}^{N}$ | ${C}_{300}^{N}$ | ${C}_{600}^{N}$ | ${C}_{1200}^{N}$ | ${C}_{5}^{N}$ | ${C}_{20}^{N}$ | |

$Straightnes{s}_{r}$ | ${S}_{300}$ | ${S}_{600}$ | ${S}_{1200}$ | ${S}_{5}$ | ${S}_{20}$ | |

$Straightnes{s}_{r}^{N}$ | ${S}_{300}^{N}$ | ${S}_{600}^{N}$ | ${S}_{1200}^{N}$ | ${S}_{5}^{N}$ | ${S}_{20}^{N}$ | |

$Betweennes{s}_{r}$ | ${B}_{300}$ | ${B}_{600}$ | ${B}_{1200}$ | ${B}_{5}$ | ${B}_{20}$ | |

$Betweennes{s}_{r}^{N}$ | ${B}_{300}^{N}$ | ${B}_{600}^{N}$ | ${B}_{1200}^{N}$ | ${B}_{5}^{N}$ | ${B}_{20}^{N}$ |

**Table A3.**Summary table of the 36 directional centrality indicators. r, radius; S, squares; C, coastline, AS, anchor stores.

Towards Squares r [metres] | Towards Coastline r [meters] | Towards Anchor Stores r [meters] | |||||||
---|---|---|---|---|---|---|---|---|---|

300 | 600 | 1200 | 600 | 1200 | 2400 | 300 | 600 | 1200 | |

$Reac{h}_{r}$ | ${R}_{300}^{S}$ | ${R}_{600}^{S}$ | ${R}_{1200}^{S}$ | ${R}_{300}^{C}$ | ${R}_{600}^{C}$ | ${R}_{1200}^{C}$ | ${R}_{300}^{AS}$ | ${R}_{600}^{AS}$ | ${R}_{1200}^{AS}$ |

$Closenes{s}_{r}$ | ${C}_{300}^{S}$ | ${C}_{600}^{S}$ | ${C}_{1200}^{S}$ | ${C}_{300}^{C}$ | ${C}_{600}^{C}$ | ${C}_{1200}^{C}$ | ${C}_{300}^{AS}$ | ${C}_{600}^{AS}$ | ${C}_{1200}^{AS}$ |

$Straightnes{s}_{r}$ | ${S}_{300}^{S}$ | ${S}_{600}^{S}$ | ${S}_{1200}^{S}$ | ${S}_{300}^{C}$ | ${S}_{600}^{C}$ | ${S}_{1200}^{C}$ | ${S}_{300}^{AS}$ | ${S}_{600}^{AS}$ | ${S}_{1200}^{AS}$ |

$Betweennes{s}_{r}$ | ${B}_{300}^{S}$ | ${B}_{600}^{S}$ | ${B}_{1200}^{S}$ | ${B}_{300}^{C}$ | ${B}_{600}^{C}$ | ${B}_{1200}^{C}$ | ${B}_{300}^{AS}$ | ${B}_{600}^{AS}$ | ${B}_{1200}^{AS}$ |

#### Appendix B.2. Skeletal Streetscape Descriptors

**Figure A1.**Graphical representation of the two skeletal streetscape GIS protocols. On the left: building façade described through sightlines perpendicular to the street centerline, homogeneously distributed (3 m). On the right: building footprint and volumes captured by the proximity band approach (source: [89]).

**Table A4.**Summary table of the 36 directional centrality indicators. r, radius; S, squares; C, coastline; AS, anchor stores.

Streetscape Indicator from Street Sightlines | ||
---|---|---|

Urban Streetscape Component | Indicator | Implementation Formulae |

Open Space | Openness | $\frac{1}{N}{\displaystyle \sum _{j=1}^{N}}{S}_{r}\left(\mathrm{j}\right)+{S}_{l}\left(\mathrm{j}\right)$ |

Openness Roughness | $\sqrt{\frac{\left({{\displaystyle \sum}}_{j=1}^{N}\left({S}_{r}\left(\mathrm{j}\right)-\overline{{S}_{r}\left(\mathrm{j}\right)}\right)+{{\displaystyle \sum}}_{j=1}^{N}\left({S}_{l}\left(\mathrm{j}\right)-\overline{{S}_{l}\left(\mathrm{j}\right)}\right)\right)}{N-1}}$ | |

Facades-Street Network-Parcels Relationship | Building Setback * | $\frac{1}{n}{\displaystyle \sum}_{j=1}^{n}{W}_{r}\left(\mathrm{j}\right)+{W}_{l}\left(\mathrm{j}\right)$ |

Facades Misalignment | $\sqrt{\frac{\left({{\displaystyle \sum}}_{j=1}^{n}{\left({W}_{r}\left(\mathrm{j}\right)-\overline{{W}_{r}\left(\mathrm{j}\right)}\right)}^{2}\right)}{{n}_{r}-1}+\frac{\left({{\displaystyle \sum}}_{j=1}^{n}{\left({W}_{l}\left(\mathrm{j}\right)-\overline{{W}_{l}\left(\mathrm{j}\right)}\right)}^{2}\right)}{{n}_{l}-1}}$ | |

Average Building Height | $\frac{1}{n}{\displaystyle \sum _{j=1}^{n}}{H}_{r}\left(\mathrm{j}\right)+{H}_{l}\left(\mathrm{j}\right)$ | |

Building Height Misalignment | $\sqrt{\frac{\left({{\displaystyle \sum}}_{j=1}^{n}{\left({H}_{r}\left(\mathrm{j}\right)-\overline{{H}_{r}\left(\mathrm{j}\right)}\right)}^{2}\right)}{{n}_{r}-1}+\frac{\left({{\displaystyle \sum}}_{j=1}^{n}{\left({H}_{l}\left(\mathrm{j}\right)-\overline{{H}_{l}\left(\mathrm{j}\right)}\right)}^{2}\right)}{{n}_{l}-1}}$ | |

Facades Cross-sectional Ratio | Cross-sectional proportion | $\frac{1}{n}{\displaystyle \sum _{j=1}^{n}}H{W}_{r}\left(\mathrm{j}\right)+H{W}_{l}\left(\mathrm{j}\right)$ |

Variability of Cross-sectional proportion | $\sqrt{\frac{\left({{\displaystyle \sum}}_{j=1}^{n}{\left(H{W}_{r}\left(\mathrm{j}\right)-\overline{H{W}_{r}\left(\mathrm{j}\right)}\right)}^{2}\right)}{{n}_{r}-1}+\frac{\left({{\displaystyle \sum}}_{j=1}^{n}{\left(H{W}_{l}\left(\mathrm{j}\right)-\overline{H{W}_{l}\left(\mathrm{j}\right)}\right)}^{2}\right)}{{n}_{l}-1}}$ |

**Table A5.**Streetscape indicators implemented through the proximity band procedure (source: [56]).

Streetscape Indicator from Proximity Bands | ||||
---|---|---|---|---|

Urban Fabric Component | Indicator | Definition and Implementation Formulae | Proximity Band Width | |

Network Morphology | Street Length | Street segments length between two intersections | ${L}_{street}$ | / |

Windingness | 1−(Euclidean distance/network distance) between two intersections | $1-\raisebox{1ex}{${L}_{eucl.}$}\!\left/ \!\raisebox{-1ex}{${L}_{street}$}\right.$ | / | |

Local connectivity | Average of the presence nodes of degree 1 (ND1) | $\sum}{\mathrm{ND}}_{i}\left[0,1\right]/2$ | / | |

Average presence nodes of degree 4 (ND4) | / | |||

Average presence nodes of degree 3, 5 + (ND35+) | / | |||

Built-up Morphology | Prevalence of Building types | (0:125] m2 building surf./total built-up surf. | $\frac{{\displaystyle \sum}Sj}{{S}_{builted}}$ | 50 |

(125:250] m2 building surf./total built-up surf. | ||||

(250:1000] m2 building surf./total built-up surf. | ||||

(1000:4000] m2 building surf./total built-up surf. | ||||

(4000: max] m2 building surf./total built-up surf. | ||||

PB coverage ratio | Built-up Surface/PB Surf. | $\sum}{S}_{tot}/{\displaystyle \sum}{S}_{PB$ | ||

Building Contiguity | Weighted average of buildings frequency on built-up units | $\frac{{\displaystyle \sum}{S}_{b-u\left(i\right)}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${N}_{buildinb-u\left(i\right)}$}\right.\right)}{{\displaystyle \sum}{S}_{b-u\left(i\right)}}$ | ||

Specialization of Building Types | Specialized Building surf./PB surf. | $\frac{{\displaystyle \sum}{S}_{spec}}{{\displaystyle \sum}{S}_{PB}}$ | ||

Network-Building Relationship | Street corridor effect | Parallel façades length/street length | ${L}_{par.fac}/{L}_{street}$ | 10 |

PB building height H | Building volume/PB surface | $\sum}{V}_{builted}/{\displaystyle \sum}{S}_{builted$ | 20 | |

Open Space Width W | (PB surf.-built surf.)/street length | $({S}_{PB-}{S}_{built})/{L}_{street}$ | ||

Height/Width Ratio | PB Building Height/Open Space Width | $H/W$ | ||

Building frequency along SN | N. of Buildings/Str. length | ${N}_{build}/{L}_{street}$ | ||

Network-Plot Relationship | Parcel Frequency | N. of Plots/Street length | ${N}_{plot}/{L}_{street}$ | 50 |

Site Morphology | Surface slope | High sloped surf. (S > 30%)/PB Surface | $\sum}SlopedSur{f}_{i}/{S}_{PB$ | 50 |

Network-Site Relationship | Street acclivity | Avg. arct(slope) along the street centerline | E [arct(slope)_{i}] | / |

#### Appendix B.3. Urban Fabrics

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**Figure 2.**Rootgram of the global model (overall space study and variables) for: linear model G (top), Poisson regressions P-ZIP-ZAP (bottom-left) and negative binomial regressions NB-ZINB-ZANB (bottom-right). The last group of models showed a better fit between expected and observed values.

**Figure 3.**Projection in the geographical space of the observed, predicted and residual values for the global model, First/Second-Age City and urban fabric composite models.

**Table 1.**Micro-retail distribution, variance and zero-inflation values for the overall study region and within each morphological context.

Global | 1st-A.C. | 2nd-A.C. | Natural | UF1 | UF2 | UF3 | UF4 | UF5 | UF6 | UF7 | UF8 | UF9 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

N° streets with build. | 63071 | 14143 | 43969 | 4959 | 5523 | 8030 | 7210 | 10310 | 16528 | 9032 | 2018 | 2543 | 1877 |

% streets UF(i) with build. | 63.1 | 85.3 | 72.5 | 23.3 | 93.1 | 87.6 | 90.2 | 68.3 | 84.4 | 74.1 | 27.5 | 24.2 | 17.6 |

[%] Streets with Retail | 22.6 | 36.9 | 14.3 | 5.5 | 25 | 48 | 18 | 19 | 11 | 10 | 14 | 7 | 5 |

Avg. Retail Street Count | 0.66 | 1.81 | 0.35 | 0.08 | 0.69 | 2.8 | 0.37 | 0.63 | 0.17 | 0.14 | 0.64 | 0.13 | 0.06 |

Model Selection | ||||||||
---|---|---|---|---|---|---|---|---|

Global | G | P | NB | ZIP * | ZINB ** | ZAP * | ZANB * | |

AIC | 284,770.9 | 120,492 | 90,024.06 | 100,454.9 | 87,373.54 | 101,170.5 | 88,247.1 | |

−2log-likelihood | 284,558 | 120,282 | 89,812 | 100,034 | 86,956 | 100,750 | 87,824 | |

n | 63,071 | 63,071 | 63,071 | 63,071 | 63,071 | 63,071 | 63,071 | |

c | 105 + 2 | 105 + 1 | 105 + 2 | 210 + 2 | 210 + 3 | 210 + 2 | 210 + 3 | |

First | ** | ** | * | * | ||||

AIC | 77,041 | 47,955.88 | 36,477.71 | 41,594 | 35,451 | 41,729 | 35,869 | |

−2log-likelihood | 76,828 | 47,696 | 36,264 | 41,150 | 35,036 | 41,280 | 35,450 | |

n | 14,143 | 14,143 | 14,143 | 14,143 | 14,143 | 14,143 | 14,143 | |

c | 101 + 2 | 101 + 1 | 101 + 2 | 202 + 2 | 202 + 3 | 202 + 2 | 202 + 3 | |

Second | * | ** | * | * | ||||

AIC | 171,475 | 68,000 | 49,808 | 54,969 | 48,841 | 55,431 | 49,198 | |

−2log-likelihood | 171,260 | 67,862 | 49,616 | 54,522 | 48,416 | 54,976 | 48,746 | |

n | 43,969 | 43,969 | 43,969 | 43,969 | 43,969 | 43,969 | 43,969 | |

c | 102 + 2 | 102 + 1 | 102 + 2 | 204 + 2 | 204 + 3 | 204 + 2 | 204 + 2 | |

UF1 | ** | * | * | * | ||||

AIC | 22,254 | 10,816 | 9405 | 9883 | 9137 | 9968 | 9480 | |

−2log-likelihood | 22,078 | 10,774 | 9280 | 9580 | 8968 | 9662 | 9164 | |

n | 5506 | 5506 | 5506 | 5506 | 5506 | 5506 | 5506 | |

c | 93 + 2 | 93 + 1 | 93 + 2 | 186 + 2 | 186 + 3 | 186 + 2 | 186 + 3 | |

UF2 | ** | ** | * | * | ||||

AIC | 47,155 | 37,145 | 27,049 | 31,622 | 26,158 | 31,721 | 26,352 | |

−2log-likelihood | 46,948 | 36,914 | 26,866 | 31,218 | 25,786 | 31,308 | 25,976 | |

n | 7954 | 7954 | 7954 | 7954 | 7954 | 7954 | 7954 | |

c | 94 + 2 | 94 + | 94 + 2 | 188 + 2 | 188 + 3 | 188 + 2 | 188 + 3 | |

UF3 | * | * | * | * | ||||

AIC | 22,097 | 9976 | 9025 | 9161 | 8909 | 9267 | 9058 | |

−log-likelihood | 21,890 | 9766 | 8830 | 8760 | 8522 | 8890 | 8662 | |

n | 7108 | 7108 | 7108 | 7108 | 7108 | 7108 | 7108 | |

c | 93 + 2 | 93 + 1 | 93 + 2 | 186 + 2 | 186 + 3 | 186 + 2 | 186 + 3 | |

UF4 | * | ** | * | * | ||||

AIC | 48,306 | 25,602 | 16,408 | 19,036 | 16,023 | 19,154 | 16,135 | |

−2log-likelihood | 48,108 | 25,340 | 16,186 | 18,598 | 15,622 | 9357 | 15,730 | |

n | 10,259 | 10,259 | 10,259 | 10,259 | 10,259 | 10,259 | 10,259 | |

c | 94 + 2 | 94 + | 94 + 2 | 188 + 2 | 188 + 3 | 188 + 2 | 188 + 3 | |

UF5 | * | * | * | * | ||||

AIC | 36,232 | 15,636 | 13,765 | 14,156 | 13,691 | 14,267 | 13,778 | |

−2log-likelihood | 36,138 | 15,462 | 13,596 | 13,800 | 13,316 | 13,946 | 13,458 | |

n | 16,453 | 16,453 | 16,453 | 16,453 | 16,453 | 16,453 | 16,453 | |

c | 86 + 2 | 86 + 1 | 86 + 2 | 86 + 2 | 86 + 3 | 86 + 2 | 86 + 3 | |

UF6 | * | * | * | * | ||||

AIC | 16,590 | 7186 | 6721 | 6793 | 6638 | 6883 | 6800 | |

−2log-likelihood | 17,878 | 7358 | 6720 | 6714 | 6586 | 6792 | 6658 | |

n | 8889 | 8889 | 8889 | 8889 | 8889 | 8889 | 8889 | |

c | 86 + 2 | 86 + 1 | 86 + 2 | 86 + 2 | 86 + 3 | 86 + 2 | 86 + 3 |

**Table 3.**Results of the Vuong LR-test between our seven models for the overall study area and each morphological region.

Global | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | ZIP vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZINB |

Vuong test statistic | −89.546 | −21.39 | 13.692 | −16.887 | 17.724 | 9.313 | w2 = 0 | |

p | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | 0.5 | |

Best Model | P | NB | ZIP | ZINB | ZINB | ZINB | - | |

First | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | NB vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZINB |

Vuong test statistic | −37.51 | −19.416 | 11.817 | −18.483 | 15.491 | 7.93 | 0 | |

p | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.2 × 10^{−15} | <0.713 × 10^{12} | 0.5 | |

Best Model | P | NB | NB | ZINB | ZINB | ZINB | - | |

Second | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | NB vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZINB |

Vuong test statistic | −66.592 | −14.391 | 8.313 | −14.731 | 11.29 | 4.46 | −8.854 | |

p | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.74 × 10^{−8} | < 0.22 × 10^{−15} | <1.296 × 10^{−6} | < 0.22 × 10^{−15} | |

Best Model | P | NB | NB | ZINB | ZINB | ZINB | - | |

UF1 | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | NB vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZINB |

Vuong test statistic | −40.545 | −9.674 | 3.179 | −9.66 | 7.186 | 4.663 | 0 | |

p | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | 0.7 × 10^{−3} | <0.2 × 10^{−15} | 3.34 × 10^{−14} | 1.56 × 10^{−6} | 0.5 | |

Best Model | P | NB | NB | ZINB | ZINB | ZINB | / | |

UF2 | Tested Models | >G vs. P | >P vs. NB | >NB vs.ZIP | >NB vs. ZINB | >ZINB vs. ZAP | ZINB vs. ZANB | >ZINB vs. ZINB |

Vuong test statistic | −13.339 | −17.263 | 10.96 | 16.99 | 14.338 | 5.633 | 0 | |

p | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | 8.85 × 10^{−9} | 0.5 | |

Best Model | P | NB | NB | ZINB | ZINB | ZINB | / | |

UF3 | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | ZIP vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZINB |

Vuong test statistic | −49.992 | −9.183 | −1.04 | -5.59 | 6.9 | 4.357 | 0.385 | |

p | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | 0.0856 | <0.22 × 10^{−15} | 4.42 × 10^{−12} | 2.18 × 10^{−5} | 0.5 | |

Best Model | P | NB | ZIP/NB | ZINB | ZINB | ZINB | / | |

UF4 | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | NB vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZINB |

Vuong test statistic | −22092 | −10.486 | 6.835/7.436 | −8.246 | 8.558 | 2.103 | 0.001 | |

p | < 0.22 × 10^{−15} | < 0.22 ×10^{−15} | 0.4 × 10^{−12} | < 0.22 ×10^{−15} | < 0.22 × 10^{−15} | 0.0178 | 0.987 | |

Best Model | P | NB | NB | ZINB | ZINB | ZINB | / | |

UF5 | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | NB vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZANB |

Vuong test statistic | −52 | −8.314 | 2.19 | −9.16 | 6.505 | 1.97 | 0 | |

p | < 0.22 × 10^{−15} | < 0.22 × 10^{−15} | 0.0153 | <0.22 × 10^{−15} | 3.87 × 10^{−11} | 0.9756 | 0.5 | |

Best Model | P | NB | NB | ZINB | ZINB | ZINB/ZANB | / | |

UF6 | Tested Models | G vs. P | P vs. NB | NB vs.ZIP | NB vs. ZINB | ZINB vs. ZAP | ZINB vs. ZANB | ZINB vs. ZANB |

Vuong test statistic | −43.096 | −6.158 | −0.187 | −5.8/15.64 | 1.95/15.64 | 1.85 | 1.2 | |

p | < 0.22 × 10^{−15} | 3.69 × 10^{−10} | 0.425 | 2.06 × 10^{−9} | 2.3 × 10^{−2} | 0.107 | 0.107 | |

Best Model | P | NB | NB | ZINB | ZINB | ZINB/ZANB | / |

**Table 4.**Comparison of the results of ZINB models when global and sub-regions are evaluated for the same subgroup of features.

ZINB | C | Sz | Sc | Pz | Pc | F1 | E(T) | E(Tz) | E(Tc) | Sd(T) | Sd(Tz) | Sd(Tc) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | val | ±[%] | |

Global | 0.716 | 0.839 | 0.178 | 0.907 | 0.134 | 0.536 | 0.624 | 0.228 | 2.350 | 1.997 | 0.707 | 3.943 | ||||||||||||

Global * | 0.697 | 0.824 | 0.181 | 0.902 | 0.134 | 0.541 | 0.668 | 0.249 | 2.370 | 2.067 | 0.727 | 3.981 | ||||||||||||

Glob. (F + S) | 0.704 | 0.98 | 0.834 | 1.15 | 0.177 | −2.21 | 0.902 | −0.03 | 0.136 | 1.13 | 0.547 | 1.06 | 0.662 | −0.89 | 0.239 | −4.02 | 2.380 | 0.39 | 2.091 | 1.15 | 0.751 | 3.16 | 4.018 | 0.93 |

First * | 0.456 | 0.610 | 0.192 | 0.878 | 0.126 | 0.678 | 1.476 | 0.639 | 2.908 | 2.995 | 1.231 | 4.298 | ||||||||||||

First | 0.484 | 5.78 | 0.654 | 6.78 | 0.192 | -0.10 | 0.882 | 0.37 | 0.133 | 5.27 | 0.696 | 2.58 | 1.434 | −2.93 | 0.574 | −11.34 | 2.906 | −0.09 | 2.998 | 0.08 | 1.248 | 1.32 | 4.273 | −0.58 |

Second * | 0.775 | 0.875 | 0.172 | 0.906 | 0.143 | 0.411 | 0.408 | 0.156 | 1.922 | 1.575 | 0.501 | 3.637 | ||||||||||||

Second | 0.775 | 0.01 | 0.876 | 0.13 | 0.165 | −3.86 | 0.905 | −0.07 | 0.139 | -2.84 | 0.409 | −0.56 | 0.414 | 1.42 | 0.160 | 2.08 | 1.944 | 1.10 | 1.624 | 3.03 | 0.535 | 6.43 | 3.750 | 3.02 |

Global * | 0.703 | 0.831 | 0.180 | 0.902 | 0.137 | 0.547 | 0.652 | 0.237 | 2.332 | 1.985 | 0.706 | 3.792 | ||||||||||||

MFA(6) | 0.714 | 1.65 | 0.845 | 1.60 | 0.186 | 3.24 | 0.907 | 0.50 | 0.146 | 6.34 | 0.569 | 3.90 | 0.634 | −2.80 | 0.224 | −6.02 | 2.293 | −1.68 | 1.967 | −0.88 | 0.725 | 2.57 | 3.739 | −1.41 |

UF1 * | 0.628 | 0.756 | 0.246 | 0.888 | 0.171 | 0.584 | 0.685 | 0.301 | 1.833 | 1.512 | 0.615 | 2.496 | ||||||||||||

UF1 | 0.659 | 4.66 | 0.793 | 4.62 | 0.259 | 5.03 | 0.903 | 1.74 | 0.190 | 10.08 | 0.631 | 7.38 | 0.650 | −5.42 | 0.261 | −15.21 | 1.811 | −1.20 | 1.649 | 8.35 | 0.621 | 0.95 | 2.812 | 11.22 |

UF2 * | 0.288 | 0.400 | 0.167 | 0.876 | 0.106 | 0.728 | 2.209 | 1.102 | 3.395 | 3.747 | 1.563 | 4.874 | ||||||||||||

UF2 | 0.328 | 12.23 | 0.479 | 16.48 | 0.166 | −0.94 | 0.885 | 1.02 | 0.111 | 4.84 | 0.749 | 2.89 | 2.113 | −4.55 | 0.995 | −10.80 | 3.312 | −2.53 | 3.651 | −2.65 | 1.652 | 5.38 | 4.681 | −4.12 |

UF3 * | 0.743 | 0.858 | 0.225 | 0.888 | 0.195 | 0.475 | 0.394 | 0.151 | 1.490 | 1.013 | 0.389 | 1.873 | ||||||||||||

UF3 | 0.763 | 2.56 | 0.881 | 2.63 | 0.228 | 1.36 | 0.892 | 0.40 | 0.216 | 9.91 | 0.503 | 5.50 | 0.380 | −3.44 | 0.145 | −3.78 | 1.443 | −3.28 | 1.000 | −1.27 | 0.510 | 23.76 | 1.722 | −8.76 |

UF4 * | 0.611 | 0.702 | 0.230 | 0.900 | 0.120 | 0.463 | 0.787 | 0.389 | 2.448 | 2.319 | 0.778 | 4.677 | ||||||||||||

UF4 | 0.615 | 0.76 | 0.707 | 0.62 | 0.236 | 2.56 | 0.902 | 0.23 | 0.124 | 3.20 | 0.469 | 1.24 | 0.777 | −1.25 | 0.379 | −2.65 | 2.439 | −0.35 | 2.319 | −0.02 | 0.756 | −2.87 | 4.690 | 0.28 |

UF5 * | 0.867 | 0.958 | 0.108 | 0.907 | 0.204 | 0.239 | 0.191 | 0.043 | 1.427 | 0.686 | 0.206 | 1.527 | ||||||||||||

UF5 | 0.868 | 0.13 | 0.959 | 0.09 | 0.112 | 3.05 | 0.910 | 0.27 | 0.204 | 0.05 | 0.266 | 10.34 | 0.192 | 0.41 | 0.045 | 4.70 | 1.417 | −0.72 | 0.690 | 0.51 | 0.241 | 14.31 | 1.510 | −1.15 |

UF6 * | 0.886 | 0.971 | 0.100 | 0.916 | 0.225 | 0.248 | 0.156 | 0.029 | 1.326 | 0.546 | 0.169 | 1.119 | ||||||||||||

UF6 | 0.885 | −0.03 | 0.967 | −0.48 | 0.140 | 28.69 | 0.922 | 0.66 | 0.253 | 10.95 | 0.317 | 21.91 | 0.156 | −0.14 | 0.035 | 16.85 | 1.270 | −4.42 | 0.583 | 6.37 | 0.193 | 12.18 | 1.320 | 15.25 |

**Table 5.**Outcomes of feature selection procedures. Selection frequencies of the most recurrent descriptors of urban form in relation to micro-retail spatial distribution

Indicator Ranking by | |||
---|---|---|---|

N° Appearances | Overall Impact | ||

Betweenness 1200 | 9 | Buil. Coverage Ratio | 3.036 |

Street Acclivity | 9 | Betweenness 1200 | 2.087 |

Buil. Coverage Ratio | 8 | Street Acclivity | 1.732 |

Street Corridor Effect | 8 | Buil. Fragmentation | 1.364 |

Buil. Fragmentation | 7 | Street Corridor Effect | 1.173 |

Avg. Build. Height | 7 | Street Length | 1.121 |

Freq Parc | 7 | Avg Height | 0.973 |

Avg. Open Space | 6 | Betweenness N 5 | 0.943 |

Between AS 1200 | 5 | Avg. Street Wide | 0.911 |

Street Length | 4 | Parcel Frequency | 0.726 |

BetweennessN 5 | 4 | UF7 | 0.563 |

StraightnessN 5 | 4 | Avg SetBack | 0.549 |

UF7 | 3 | Std SetBack | 0.504 |

Avg SetBack | 3 | Std Buil.Height | 0.491 |

Std SetBack | 3 | UF4 | 0.483 |

UF4 | 3 | Small Buil. (<125 m^{2}) | 0.454 |

Small Buil. (<150 m^{2}) | 3 | Betw. Coast 600 | 0.445 |

Betw. Coast 600 | 3 | Reach 20 | 0.433 |

Reach 20 | 3 | Betweeness 600 | 0.381 |

Betweeness 600 | 3 | Straightness coast | 0.320 |

Straight. Coast 1200 | 3 | Specialisation | 0.317 |

Std. Open Space | 3 | Std. Open Space | 0.315 |

Straightness 20 | 3 | Straightness 20 | 0.300 |

StraightnessN 300 | 3 | StraightnessN 5 | 0.264 |

Betw. Coast 2400 | 3 | Reach 300 | 0.254 |

Straightness 1200 | 3 | Closeness N 600 | 0.237 |

StraightnessN 1200 m | 3 | StraightnessN 1200 m | 0.221 |

**Table 6.**Outcomes of the variable selection procedure (Enet-PR ZINB) implemented on the overall space of the French Riviera (global) and its contextual partitions (First-/Second-Age City, UF1–6).

COUNT-PART | Global | 1st A.C. | 2nd A.C. | UF1 | UF2 | UF3 | UF4 | UF5 | UF6 | ||
---|---|---|---|---|---|---|---|---|---|---|---|

Impact | N° select | 19 | 25 | 21 | 14 | 18 | 13 | 16 | 11 | 12 | |

Built-up Coverage Ratio | 3.036 | 8 | 2.005 | 1.234 | 1.753 | 1.152 | 1.308 | 1.213 | 1.368 | 1.002 | |

Betw. 1200 m | 2.023 | 7 | 1.160 | 1.169 | 1.031 | 1.486 | 1.704 | 1.361 | 1.112 | ||

Street Acclivity | 1.732 | 9 | 0.854 | 0.699 | 0.842 | 0.759 | 0.730 | 0.878 | 0.725 | 0.963 | 0.819 |

Built-up Fragmentation | 1.204 | 5 | 1.303 | 1.442 | 1.407 | 1.041 | 1.010 | ||||

Street Length | 1.121 | 4 | 1.147 | 1.329 | 1.252 | 1.392 | |||||

Corridor Effect | 1.061 | 4 | 1.212 | 1.281 | 1.399 | 1.169 | |||||

Avg. Height | 0.973 | 7 | 1.155 | 1.072 | 1.136 | 1.376 | 1.009 | 1.197 | 1.028 | ||

Betw. N 5 m | 0.943 | 4 | 1.232 | 1.126 | 1.393 | 1.193 | |||||

Avg. Open Space | 0.911 | 6 | 1.127 | 1.076 | 1.042 | 1.262 | 1.024 | 1.379 | |||

Parcel Frequency | 0.578 | 4 | 1.135 | 1.248 | 0.981 | 0.823 | |||||

UF7 | 0.563 | 3 | 1.234 | 0.881 | 1.210 | ||||||

Avg. Setback | 0.549 | 3 | 1.297 | 1.116 | 1.136 | ||||||

Std. Setback | 0.504 | 3 | 0.896 | 0.822 | 0.778 | ||||||

Std. Height | 0.491 | 2 | 1.132 | 1.359 | |||||||

UF4 | 0.483 | 3 | 1.169 | 0.915 | 1.229 | ||||||

Small Build. (<125 m^{2}) | 0.454 | 3 | 0.829 | 0.907 | 0.810 | ||||||

Betw. Coast 600 m | 0.445 | 3 | 1.096 | 1.075 | 1.275 | ||||||

Reach 20 min | 0.433 | 3 | 1.050 | 1.154 | 1.228 | ||||||

Betw. 600 m | 0.381 | 3 | 1.073 | 1.254 | 1.055 | ||||||

Straig. Coast 1200 m | 0.320 | 3 | 1.115 | 1.076 | 1.128 | ||||||

Build. Specialization | 0.317 | 1 | 1.317 | ||||||||

Std. Open Space | 0.315 | 3 | 0.994 | 0.755 | 0.936 | ||||||

Straig. 20 min | 0.300 | 3 | 1.146 | 1.036 | 1.117 | ||||||

Straig. N 5 min | 0.264 | 4 | 1.039 | 1.091 | 1.094 | 1.039 | |||||

Reach 300 | 0.254 | 1 | 1.254 | ||||||||

Clos. N 600 m | 0.237 | 2 | 0.896 | 0.868 | |||||||

Straig. N 300 m | 0.221 | 3 | 1.115 | 0.912 | 1.018 | ||||||

Reach N 5 min | 0.205 | 2 | 0.859 | 1.064 | |||||||

Betw. Coast 2400 m | 0.199 | 3 | 1.037 | 1.093 | 1.069 | ||||||

Betw. Coast 1200 m | 0.191 | 2 | 1.038 | 1.153 | |||||||

Betw. AS 1200 m | 0.170 | 5 | 1.005 | 1.079 | 1.024 | 1.059 | 1.004 | ||||

Reach N 600 m | 0.159 | 1 | 0.841 | ||||||||

Straig. Places 300 m | 0.153 | 2 | 1.050 | 0.897 | |||||||

Reach Coast 1200 m | 0.153 | 1 | 1.153 | ||||||||

Nodes 4 | 0.145 | 1 | 1.145 | ||||||||

Straig. 1200 m | 0.139 | 3 | 1.034 | 1.075 | 1.030 | ||||||

Reach 600 m | 0.123 | 2 | 1.008 | 0.884 | |||||||

UF5 | 0.121 | 1 | 0.879 | ||||||||

Betw. 300 m | 0.118 | 1 | 1.118 | ||||||||

Std. HW Ratio | 0.115 | 1 | 0.885 | ||||||||

Straig. 5 min | 0.115 | 1 | 1.115 | ||||||||

UF3 | 0.112 | 1 | 0.888 | ||||||||

Small Build. (125–250 m^{2}) | 0.105 | 2 | 0.937 | 0.958 | |||||||

Reach 1200 m | 0.086 | 1 | 1.086 | ||||||||

Betw. Places 1200 m | 0.080 | 1 | 1.080 | ||||||||

Straig. AS 600 m | 0.079 | 2 | 1.002 | 1.077 | |||||||

Straig.N 1200 m | 0.067 | 3 | 1.014 | 1.025 | 1.029 | ||||||

Straig. AS 1200 m | 0.041 | 2 | 1.033 | 1.008 | |||||||

AVG HW | 0.041 | 1 | 0.959 | ||||||||

Large Build. (250–1000 m^{2}) | 0.038 | 1 | 1.038 | ||||||||

Straig. N 600 m | 0.035 | 2 | 1.029 | 1.006 | |||||||

Betw. N 300 m | 0.029 | 1 | 1.029 | ||||||||

Betw. N 600 m | 0.009 | 1 | 1.009 | ||||||||

Clos. 5 m | 0.006 | 1 | 0.994 | ||||||||

ZERO-PART | Impact | N° select | 0 | 2 | 2 | 0 | 4 | 2 | 2 | 0 | 0 |

Built-up Fragmentation | 0.153 | 2 | 0.951 | 0.896 | |||||||

Parcel Frequency | 0.144 | 3 | 0.925 | 0.941 | 0.989 | ||||||

Reach 5 min | 0.13 | 1 | 0.870 | ||||||||

Corridor Effect | 0.11 | 4 | 0.976 | 0.987 | 0.954 | 0.973 | |||||

Betw. 1200 m | 0.062 | 2 | 0.994 | 0.943 |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Araldi, A.
Towards an Integrated Methodology for Model and Variable Selection Using Count Data: An Application to Micro-Retail Distribution in Urban Studies. *Urban Sci.* **2020**, *4*, 21.
https://doi.org/10.3390/urbansci4020021

**AMA Style**

Araldi A.
Towards an Integrated Methodology for Model and Variable Selection Using Count Data: An Application to Micro-Retail Distribution in Urban Studies. *Urban Science*. 2020; 4(2):21.
https://doi.org/10.3390/urbansci4020021

**Chicago/Turabian Style**

Araldi, Alessandro.
2020. "Towards an Integrated Methodology for Model and Variable Selection Using Count Data: An Application to Micro-Retail Distribution in Urban Studies" *Urban Science* 4, no. 2: 21.
https://doi.org/10.3390/urbansci4020021