# Grey Systems Theory as an Effective Method for Analyzing Scarce, Incomplete and Uncertain Data on the Example of a Survey of Public Perceptions of Safety in Urban Spaces

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## Abstract

**:**

## 1. Introduction

#### 1.1. Heuristic Methods in Spatial Planning

#### 1.2. Strategies Promoting the Creation of Safe Urban Spaces

#### 1.3. Methods for Analyzing Spatial Data

#### 1.4. Grey Systems vs. Statistical Analyses

## 2. Materials and Methods

#### 2.1. Characteristics of the Grey Systems Theory

- –
- white (white box) where information is completely known,
- –
- black (black box) where information is unknown and uncertain, and the user can only observe the system based on a sequence of statements that start at an entry and end at an exit,
- –
- grey (grey box) where information is limited: Part of the information is known, and part of the information is unknown [38].

- –
- the definition of observation vectors,
- –
- calculation of the reflection of observation vectors,
- –
- calculation of behavior measures,
- –
- calculation of the values of absolute degree of similarity, i.e., the similarity coefficient,
- –
- determination of the order of impact of the analyzed system factors on the characteristics of the system.

_{0}) and behaviors (X

_{1}, X

_{2}, ..., X

_{k}) are defined in the first step. The number of behavior vectors is determined by the number of the observed variables. Each vector contains information about a given variable that has been received from a given number of respondents. In a grey model, the system’s real-world behavior is described by the predicted/endogenous variable X

^{(0)}(k), where k = 1, 2, ..., n is a set of explanatory variables that determine the state of the predicted variable. Therefore, an endogenous real-world process (X

^{(0)}(k)) is explained in time by N number of independent (explanatory) variables [40,41].

_{0}and X

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5}, [33]:

- (1)
- 0 < ε ≤ 1;
- (2)
- ε is related only to the geometric shape of vectors X
_{0}and X_{k}, and it is unrelated to their spatial arrangement; - (3)
- ε does not equal zero if every two vectors are even minimally related;
- (4)
- the higher the degree of relatedness (similarity) between two observation vectors, the higher the value of ε;
- (5)
- ε is equal to or close to 1 if observation vectors are parallel or nearly parallel to each other [23].

#### 2.2. Identification of Factors that Compromise Safety in Urban Space

## 3. Results

#### 3.1. Analysis Based on Input Data

- –
- unlit streets (X
_{1}), - –
- clubs, pubs, 24/7 shops, alcohol shops (X
_{2}), - –
- unmanaged green areas (X
_{3}), - –
- neglected buildings (damaged exterior, window and door frames) (X
_{4}), - –
- vacant buildings, ruins (X
_{5}). - –
- trampled paths (X
_{6}), - –
- walled structures for waste containers (X
_{7}), - –
- narrow passages between buildings (X
_{8}), - –
- proximity of a cemetery (X
_{9}), - –
- unguarded car parks (X
_{10}). - –
- residential buildings without intercoms, gates or inner courts (X
_{11}), - –
- embankments (X
_{12}), - –
- other (underpass, bridge, etc.) (X
_{13}), - –
- forests, parks (X
_{14}), - –
- graffiti (X
_{15}). - –
- bus stop shelters (X
_{16}), - –
- illegal waste dumping sites (X
_{17}).

_{0}) and the adopted geospatial attributes (X

_{1}, X

_{2},… X

_{17}) was calculated in the last step of the procedure (Table 1).

_{1}, X

_{2}, X

_{3}, … X

_{17}and the sense of insecurity X

_{0}was determined in the following order: Ɛ

_{01}> Ɛ

_{02}> Ɛ

_{05}> Ɛ

_{14}> Ɛ

_{06}> Ɛ

_{13}> Ɛ

_{04}> Ɛ

_{11}> Ɛ

_{08}> Ɛ

_{10}> Ɛ

_{09}> Ɛ

_{03}> Ɛ

_{07}> Ɛ

_{12}> Ɛ

_{17}> Ɛ

_{16}> Ɛ

_{15}.

_{1}); clubs, pubs, 24/7 shops, alcohol shops (X

_{2}); and vacant buildings, ruins (X

_{5}), for which the value of the similarity coefficient Ɛ reached 0.7805, 0.6778 and 0.6312, respectively. Perceptions of insecurity were least likely to be influenced by: Graffiti (X

_{15}), bus stop shelters (X

_{16}) and illegal waste dumping sites (X

_{17}), for which the value of Ɛ was determined at X

_{15}= 0.5600, X

_{16}=0.5646 and X

_{17}= 0.5708. Forests, parks (X

_{14}, Ɛ = 0.6177) and trampled paths (X

_{6}, z Ɛ = 0.6109) also exerted a considerable influence on the respondents’ sense of insecurity.

#### 3.2. Analysis Based on the Number of Observations

## 4. Discussion

_{01}> Ɛ

_{02}> Ɛ

_{05}> Ɛ

_{14}). In turn, the last three values of ε were identical in models with 25 and more observations (Ɛ

_{17}> Ɛ

_{16}> Ɛ

_{15}).

_{01}> Ɛ

_{02}> Ɛ

_{05}> Ɛ

_{14}> Ɛ

_{06}> Ɛ

_{13}> Ɛ

_{11}> Ɛ

_{04}> Ɛ

_{08}> Ɛ

_{10}). The three least important attributes in the model were also characterized by a stable sequence (Ɛ

_{17}> Ɛ

_{16}> Ɛ

_{15}) (Table 3).

_{2}(club, pub, 24/7 shop, alcohol shop), where Ɛ ranged from 0.6706 to 0.7024. These attributes were followed by: X

_{5}(vacant buildings, ruins) with Ɛ values of 0.6312–0.6491; X

_{14}(forests, parks) with Ɛ values of 0.6160–0.6328; X

_{6}(trampled paths) with Ɛ values of 0.6066 to 0.62781; and X

_{13}(other, such as underpasses and bridges) with Ɛ values of 0.6052–0.6241. The values of Ɛ ranged from 0.5835 to 0.6011 for X

_{11}(residential buildings without intercoms, gates or inner courts), X

_{4}(neglected buildings with damaged exterior, window and door frames), X

_{8}(narrow passages between buildings) and X

_{10}(unguarded car parks). The above attributes contributed to a sense of insecurity in space. The following attributes were least associated with perceptions of insecurity: X

_{17}(illegal waste dumping sites) with Ɛ values of 0.5708–0.5872; X

_{16}(bus stop shelters) with Ɛ values of 0.5646–0.57491; and X

_{15}(graffiti) with Ɛ values of 0.5600–0.5694.

## 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.**A diagram of the process of information whitening in the grey system theory (the colors of grey system models: White, grey and black). Source: Own elaboration based on Dytczak Ginta 2014.

**Figure 2.**Diagram of the values of the similarity coefficient ε representing the strength of the analyzed relationships in terms of the number of observations included in the model.

Attributes That Influence the Sense of Insecurity in Urban Space | Value of Coefficient ε Describing the Influence of System Factors on X_{0} |
---|---|

X_{1}-unlit streets | Ɛ_{01} = 0.7805 |

X_{2}-clubs, pubs, 24/7 shops, alcohol shops | Ɛ_{02} = 0.6778 |

X_{3}-unmanaged green areas | Ɛ_{03} = 0.5795 |

X_{4}-neglected buildings (damaged exterior, window and door frames) | Ɛ_{04} = 0.5948 |

X_{5}-vacant buildings, ruins | Ɛ_{05} = 0.6312 |

X_{6}-trampled paths | Ɛ_{06} = 0.6109 |

X_{7}-walled structures for waste containers | Ɛ_{07} = 0.5786 |

X_{8}-narrow passages between buildings | Ɛ_{08} = 0.5871 |

X_{9}-proximity of a cemetery | Ɛ_{09} = 0.5797 |

X_{10}-unguarded car parks | Ɛ_{10} = 0.5835 |

X_{11}-residential buildings without intercoms, gates or inner courts | Ɛ_{11} = 0.5932 |

X_{12}-embankments | Ɛ_{12} = 0.5748 |

X_{13}-other (underpass, bridge, etc.) | Ɛ_{13} = 0.6052 |

X_{14}-forests, parks | Ɛ_{14} = 0.6177 |

X_{15}-graffiti | Ɛ_{15} = 0.5600 |

X_{16}-bus stop shelters | Ɛ_{16} = 0.5646 |

X_{17}-illegal waste dumping sites | Ɛ_{17} = 0.5708 |

Similarity Factor | Number of Observations | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

4 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | |

Ɛ_{01} | 0.9444 | 0.8462 | 0.7600 | 0.9138 | 0.8875 | 0.8333 | 0.8083 | 0.8309 | 0.8049 | 0.7835 | 0.7762 | 0.8028 | 0.7805 |

Ɛ_{02} | 0.8462 | 0.8824 | 0.9464 | 0.7636 | 0.7183 | 0.7024 | 0.6888 | 0.6800 | 0.6852 | 0.6730 | 0.6706 | 0.6844 | 0.6778 |

Ɛ_{03} | 0.7143 | 0.7167 | 0.7049 | 0.7049 | 0.6394 | 0.6099 | 0.5885 | 0.5849 | 0.5856 | 0.5836 | 0.5790 | 0.5842 | 0.5795 |

Ɛ_{04} | 0.7250 | 0.7241 | 0.7500 | 0.6667 | 0.6360 | 0.6083 | 0.5989 | 0.6061 | 0.6055 | 0.6058 | 0.5990 | 0.6019 | 0.5948 |

Ɛ_{05} | 0.7368 | 0.7407 | 0.8125 | 0.7302 | 0.6632 | 0.6491 | 0.6370 | 0.6398 | 0.6404 | 0.6375 | 0.6318 | 0.6375 | 0.6312 |

Ɛ_{06} | 0.7045 | 0.7167 | 0.7551 | 0.6986 | 0.6462 | 0.6278 | 0.6171 | 0.6160 | 0.6168 | 0.6118 | 0.6066 | 0.6146 | 0.6109 |

Ɛ_{07} | 0.7143 | 0.7097 | 0.7551 | 0.6648 | 0.6092 | 0.5885 | 0.5853 | 0.5859 | 0.5874 | 0.5828 | 0.5795 | 0.5835 | 0.5786 |

Ɛ_{08} | 0.6875 | 0.7031 | 0.7232 | 0.6648 | 0.6183 | 0.6000 | 0.5949 | 0.6000 | 0.5984 | 0.5972 | 0.5909 | 0.5932 | 0.5871 |

Ɛ_{09} | 0.6324 | 0.6512 | 0.6603 | 0.6239 | 0.5994 | 0.5924 | 0.5860 | 0.5852 | 0.5847 | 0.5841 | 0.5795 | 0.5815 | 0.5797 |

Ɛ_{10} | 0.6500 | 0.6757 | 0.6761 | 0.6295 | 0.6054 | 0.5971 | 0.5885 | 0.5918 | 0.5903 | 0.5890 | 0.5858 | 0.5866 | 0.5835 |

Ɛ_{11} | 0.7143 | 0.7321 | 0.6923 | 0.6510 | 0.6211 | 0.6111 | 0.6016 | 0.6023 | 0.6004 | 0.6004 | 0.5948 | 0.5959 | 0.5932 |

Ɛ_{12} | 0.6731 | 0.6970 | 0.6623 | 0.6272 | 0.6069 | 0.5914 | 0.5808 | 0.5875 | 0.5856 | 0.5851 | 0.5792 | 0.5791 | 0.5748 |

Ɛ_{13} | 0.7368 | 0.7708 | 0.6923 | 0.6706 | 0.6462 | 0.6241 | 0.6101 | 0.6148 | 0.6116 | 0.6146 | 0.6098 | 0.6107 | 0.6052 |

Ɛ_{14} | 0.7647 | 0.8421 | 0.7451 | 0.7101 | 0.6535 | 0.6328 | 0.6225 | 0.6243 | 0.6190 | 0.6180 | 0.6160 | 0.6209 | 0.6177 |

Ɛ_{15} | 0.6250 | 0.6383 | 0.6302 | 0.6058 | 0.5824 | 0.5694 | 0.5636 | 0.5678 | 0.5667 | 0.5655 | 0.5626 | 0.5633 | 0.5600 |

Ɛ_{16} | 0.6875 | 0.6970 | 0.6603 | 0.6151 | 0.5886 | 0.5749 | 0.5693 | 0.5719 | 0.5708 | 0.5681 | 0.5658 | 0.5676 | 0.5646 |

Ɛ_{17} | 0.6957 | 0.7241 | 0.6563 | 0.6272 | 0.6026 | 0.5872 | 0.5771 | 0.5843 | 0.5796 | 0.5768 | 0.5730 | 0.5747 | 0.5708 |

**Table 3.**The sequence of the calculated values of the similarity coefficient Ɛ in terms of the number of observations included in the model.

Nr. of Obs. | Sequence of the Values of ε Representing the Streng thof the Analyzed Relationships |
---|---|

4 | Ɛ_{01} > Ɛ_{02} > Ɛ_{14} > Ɛ_{05} = Ɛ_{13} > Ɛ_{04} > Ɛ_{03} = Ɛ_{07} = Ɛ_{11} > Ɛ_{06} > Ɛ_{17} > Ɛ_{08} = Ɛ_{16} > Ɛ_{12} > Ɛ_{10} Ɛ_{09} > Ɛ_{15} |

5 | Ɛ_{02} > Ɛ_{01} > Ɛ_{14} > Ɛ_{13} > Ɛ_{05} > Ɛ_{11} > Ɛ_{04} = Ɛ_{17} > Ɛ_{03} = Ɛ_{06} > Ɛ_{07} > Ɛ_{08} > Ɛ_{12} = Ɛ_{16} > Ɛ_{10} > Ɛ_{09} > Ɛ_{15} |

10 | Ɛ_{02} > Ɛ_{05} > Ɛ_{01} > Ɛ_{06} = Ɛ_{07} > Ɛ_{04} > Ɛ_{14} > Ɛ_{08} > Ɛ_{03} > Ɛ_{11} = Ɛ_{13} > Ɛ_{10} > Ɛ_{12} > Ɛ_{09} = Ɛ_{16} > Ɛ_{17} > Ɛ_{15} |

15 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{06} > Ɛ_{13} > Ɛ_{04} > Ɛ_{07} = Ɛ_{08} > Ɛ_{11} > Ɛ_{03} > Ɛ_{10} > Ɛ_{12} = Ɛ_{17} > Ɛ_{09} >Ɛ_{16} > Ɛ_{15} |

20 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{06} = Ɛ_{13} > Ɛ_{04} > Ɛ_{11} > Ɛ_{08} > Ɛ_{03} > Ɛ_{07} > Ɛ_{12} >Ɛ_{10} >Ɛ_{17} >Ɛ_{09} >Ɛ_{16} > Ɛ_{15} |

25 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{06} > Ɛ_{13} > Ɛ_{11} >Ɛ_{04} > Ɛ_{08} > Ɛ_{10} > Ɛ_{09} > Ɛ_{12} > Ɛ_{03} = Ɛ_{07} >Ɛ_{17} > Ɛ_{16} > Ɛ_{15} |

30 | Ɛ_{01} > Ɛ_{02} >Ɛ_{05} > Ɛ_{14} > Ɛ_{06} > Ɛ_{13} > Ɛ_{11} > Ɛ_{04} > Ɛ_{08} > Ɛ_{10} > Ɛ_{09} > Ɛ_{07} > Ɛ_{03} > Ɛ_{12} > Ɛ_{17} > Ɛ_{16} > Ɛ_{15} |

35 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{06} > Ɛ_{13} > Ɛ_{04} > Ɛ_{11} > Ɛ_{08} > Ɛ_{10} > Ɛ_{12} > Ɛ_{07} > Ɛ_{03} > Ɛ_{09} > Ɛ_{17} > Ɛ_{16} > Ɛ_{15} |

40 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{06} > Ɛ_{13} > Ɛ_{04} > Ɛ_{11} > Ɛ_{08} > Ɛ_{10} > Ɛ_{07} > Ɛ_{03} = Ɛ_{12} > Ɛ_{09} > Ɛ_{17} > Ɛ_{16} > Ɛ_{15} |

45 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{13} > Ɛ_{06} > Ɛ_{04} > Ɛ_{11} > Ɛ_{08} > Ɛ_{10} > Ɛ_{12} > Ɛ_{09} > Ɛ_{03} > Ɛ_{07} > Ɛ_{17} > Ɛ_{16} > Ɛ_{15} |

50 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{13} > Ɛ_{06} > Ɛ_{04}> Ɛ_{11} > Ɛ_{08} > Ɛ_{10} > Ɛ_{07} = Ɛ_{09} > Ɛ_{12} > Ɛ_{03} > Ɛ_{17}> Ɛ_{16} > Ɛ_{15} |

55 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{06} > Ɛ_{13} > Ɛ_{04} > Ɛ_{11} > Ɛ_{08} > Ɛ_{10} > Ɛ_{03} > Ɛ_{07} > Ɛ_{09} > Ɛ_{12} > Ɛ_{17} > Ɛ_{16} > Ɛ_{15} |

60 | Ɛ_{01} > Ɛ_{02} > Ɛ_{05} > Ɛ_{14} > Ɛ_{06} > Ɛ_{13} > Ɛ_{04} > Ɛ_{11} > Ɛ_{08} > Ɛ_{10} > Ɛ_{09} > Ɛ_{03} > Ɛ_{07} > Ɛ_{12} > Ɛ_{17} > Ɛ_{16} > Ɛ_{15} |

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## Share and Cite

**MDPI and ACS Style**

Gerus-Gościewska, M.; Gościewski, D.
Grey Systems Theory as an Effective Method for Analyzing Scarce, Incomplete and Uncertain Data on the Example of a Survey of Public Perceptions of Safety in Urban Spaces. *Land* **2021**, *10*, 73.
https://doi.org/10.3390/land10010073

**AMA Style**

Gerus-Gościewska M, Gościewski D.
Grey Systems Theory as an Effective Method for Analyzing Scarce, Incomplete and Uncertain Data on the Example of a Survey of Public Perceptions of Safety in Urban Spaces. *Land*. 2021; 10(1):73.
https://doi.org/10.3390/land10010073

**Chicago/Turabian Style**

Gerus-Gościewska, Małgorzata, and Dariusz Gościewski.
2021. "Grey Systems Theory as an Effective Method for Analyzing Scarce, Incomplete and Uncertain Data on the Example of a Survey of Public Perceptions of Safety in Urban Spaces" *Land* 10, no. 1: 73.
https://doi.org/10.3390/land10010073