# A Spatio-Temporal Analysis of the Health Situation in Poland Based on Functional Discriminant Coordinates

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## Abstract

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## 1. Introduction

- Firstly, functional data are normally used to cope with the problem of missing observations, which is inevitable in many areas of applied research. Unfortunately, most methods concerning data analysis require complete time series. The removal of a time series with missing observations from a data set is one of popular solutions, but this can lead, and in most cases does lead, to serious data loss. Another possibility is to use one of the many methods of missing data prediction, but, in that case, the results will depend on the interpolation method. Contrary to these approaches, in the case of functional data, the problem of missing observations is resolved by expressing a given time series in the form of a continuous function set.
- Secondly, in the statistical development of MFDCA, the structure of observations is naturally retained when using functional data, i.e., the temporal link is maintained and the information regarding any measurement is taken into account. Consequently, results are assumed to be robust.
- Thirdly, moments of observation do not have to be equally spaced in a particular time series, which can be a major advantage in online applications.
- Fourthly, when using functional data, one avoids the problem of dimensionality. When the total number of time points in which observations are made exceeds the number of time series under analysis, most statistical methods do not provide satisfactory results because of misleading false estimates. In the case of functional data, this problem can be avoided because the time series are replaced by a set of continuous representative functions, which are independent of the time points in which observations are made.

## 2. The Data

## 3. Statistical Methodology

#### 3.1. Functional Discriminant Coordinates

**Remark**

**1.**

#### 3.2. Cluster Analysis

#### 3.3. Functional Multivariate Coefficient of Variation

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Average values of 8 variables calculated from functional data for districts included in each of the 16 provinces.

**Note:**The ordinate axis shows the unitized values of a given variable.

**Figure 4.**Weight functions for the first (left) and the second (right) functional discriminant coordinate.

Variable | Description | Type of Variable |
---|---|---|

1 | Nurses and midwives per 10,000 population | S |

2 | Doctors per 10,000 population | S |

3 | Population per generally available pharmacy | D |

4 | Deaths of people due to cardiovascular disease per 100,000 population | D |

5 | Total deaths due to cancer per 100,000 population | D |

6 | Health out-patient departments per 10,000 population | S |

7 | Number of doctors consultations per 10,000 population | S |

8 | Infant deaths per 1000 live births | D |

Number | Province Name | Number of Districts |
---|---|---|

1 | dolnośląskie | 30 |

2 | kujawsko-pomorskie | 23 |

3 | lubelskie | 24 |

4 | lubuskie | 14 |

5 | łódzkie | 24 |

6 | małopolskie | 22 |

7 | mazowieckie | 42 |

8 | opolskie | 12 |

9 | podkarpackie | 25 |

10 | podlaskie | 17 |

11 | pomorskie | 20 |

12 | śląskie | 36 |

13 | świętokrzyskie | 14 |

14 | warmińsko-mazurskie | 21 |

15 | wielkopolskie | 35 |

16 | zachodniopomorskie | 21 |

Total | 380 |

Number | Eigenvalue | % Total Variance | % Cumulative Variance |
---|---|---|---|

1 | 48.2682 | 27.6745 | 27.6745 |

2 | 28.6024 | 16.3991 | 44.0735 |

3 | 26.3461 | 15.1055 | 59.1790 |

4 | 15.1611 | 8.6926 | 67.8716 |

5 | 11.6214 | 6.6631 | 74.5347 |

6 | 9.4012 | 5.3901 | 79.9248 |

7 | 8.5436 | 4.8984 | 84.8232 |

8 | 5.6155 | 3.2197 | 88.0429 |

9 | 4.9859 | 2.8587 | 90.9016 |

10 | 4.8982 | 2.8083 | 93.7099 |

11 | 3.4169 | 1.9591 | 95.6690 |

12 | 2.7111 | 1.5544 | 97.2234 |

13 | 2.0385 | 1.1687 | 98.3921 |

14 | 1.4576 | 0.8357 | 99.2279 |

15 | 1.3467 | 0.7721 | 100.0000 |

Number | Variable 1 | Variable 2 |
---|---|---|

1 | −1.0418 | −0.1272 |

2 | −0.9538 | −1.8010 |

3 | 2.1856 | 0.8277 |

4 | −0.2247 | −0.0520 |

5 | −0.3005 | 0.4795 |

6 | 1.0408 | 0.2084 |

7 | 0.4584 | −1.0042 |

8 | 0.9223 | 1.5262 |

9 | 2.5934 | −1.5963 |

10 | 0.5307 | 2.2876 |

11 | −1.0424 | 0.8810 |

12 | −0.7883 | 0.7896 |

13 | 3.0622 | 0.1767 |

14 | −1.6026 | 0.6644 |

15 | −1.5847 | −1.2004 |

16 | −0.9763 | 0.6872 |

First functional discriminant coordinate | |||||||

Variable | ${\widehat{\gamma}}_{10}$ | ${\widehat{\gamma}}_{11}$ | ${\widehat{\gamma}}_{12}$ | ${\widehat{\gamma}}_{13}$ | ${\widehat{\gamma}}_{14}$ | Area | Area (%) |

1 | 2.5407 | −3.2389 | 2.0978 | -0.3165 | −0.9303 | 9.0039 | 8.2381 |

2 | 0.5792 | 7.8019 | −0.7947 | −7.3340 | 12.8083 | 34.9609 | 31.9876 |

3 | 0.2994 | −0.5582 | 2.4586 | 6.3065 | 0.1062 | 14.4837 | 13.2519 |

4 | −3.0143 | 0.3007 | 6.7926 | −1.9114 | −1.2470 | 15.1264 | 13.8399 |

5 | 3.8320 | 0.7027 | 0.1091 | −0.9202 | −0.7656 | 9.3864 | 8.5882 |

6 | −0.1795 | −3.5460 | −0.4691 | 1.3195 | −1.3298 | 8.7501 | 8.0059 |

7 | 0.9525 | 0.6116 | −6.6553 | 4.7278 | 1.7190 | 16.0264 | 14.6634 |

8 | −0.1802 | 0.6553 | 0.0063 | −0.1974 | 0.3950 | 1.5574 | 1.4249 |

Second functional discriminant coordinate | |||||||

Variable | ${\widehat{\gamma}}_{20}$ | ${\widehat{\gamma}}_{21}$ | ${\widehat{\gamma}}_{22}$ | ${\widehat{\gamma}}_{23}$ | ${\widehat{\gamma}}_{24}$ | Area | Area (%) |

1 | −0.8113 | 2.1021 | 1.9010 | 7.7922 | 6.7172 | 23.1732 | 17.3879 |

2 | 1.3009 | −9.6903 | −5.7988 | 6.8425 | 3.5944 | 26.8581 | 20.1529 |

3 | 0.2159 | 5.0384 | 0.8509 | −7.6787 | 0.3243 | 18.8570 | 14.1493 |

4 | −1.4582 | 10.6698 | 0.1764 | −1.8355 | 8.9551 | 28.0178 | 21.0231 |

5 | −0.0163 | −2.8675 | −0.7596 | −0.2515 | −0.4281 | 6.5687 | 4.9288 |

6 | 0.7540 | 2.9261 | −1.3371 | 0.7459 | −2.1031 | 7.4906 | 5.6206 |

7 | 1.2655 | −0.1701 | 4.4019 | −4.0271 | −7.7531 | 20.7457 | 15.5665 |

8 | −0.5597 | −0.0570 | −0.2735 | −0.0484 | 0.3602 | 1.5604 | 1.1709 |

Number | Province | Cluster |
---|---|---|

1 | dolnośląskie | I |

2 | kujawsko-pomorskie | II |

3 | lubelskie | III |

4 | lubuskie | II |

5 | łódzkie | I |

6 | małopolskie | III |

7 | mazowieckie | I |

8 | opolskie | IV |

9 | podkarpackie | III |

10 | podlaskie | IV |

11 | pomorskie | II |

12 | śląskie | II |

13 | świętokrzyskie | III |

14 | warmińsko-mazurskie | II |

15 | wielkopolskie | II |

16 | zachodniopomorskie | II |

Provinces | MFCV |
---|---|

All | 0.3705 |

Cluster I | 0.2975 |

Cluster II | 0.2890 |

Cluster III | 0.1728 |

Cluster IV | 0.2047 |

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**MDPI and ACS Style**

Krzyśko, M.; Wołyńki, W.; Szymkowiak, M.; Wojtyła, A.
A Spatio-Temporal Analysis of the Health Situation in Poland Based on Functional Discriminant Coordinates. *Int. J. Environ. Res. Public Health* **2021**, *18*, 1109.
https://doi.org/10.3390/ijerph18031109

**AMA Style**

Krzyśko M, Wołyńki W, Szymkowiak M, Wojtyła A.
A Spatio-Temporal Analysis of the Health Situation in Poland Based on Functional Discriminant Coordinates. *International Journal of Environmental Research and Public Health*. 2021; 18(3):1109.
https://doi.org/10.3390/ijerph18031109

**Chicago/Turabian Style**

Krzyśko, Mirosław, Waldemar Wołyńki, Marcin Szymkowiak, and Andrzej Wojtyła.
2021. "A Spatio-Temporal Analysis of the Health Situation in Poland Based on Functional Discriminant Coordinates" *International Journal of Environmental Research and Public Health* 18, no. 3: 1109.
https://doi.org/10.3390/ijerph18031109