# Comparison of Soft Indicator and Poisson Kriging for the Noise-Filtering and Downscaling of Areal Data: Application to Daily COVID-19 Incidence Rates

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## Abstract

**:**

## 1. Introduction

^{2}endemic area of Bangladesh using PK and data collected at the household level. Another study [8] applied PK to filter noise attached to lung and cervix cancer mortality rates recorded for white females in two contrasted county geographies: (1) state of Indiana that consists of 92 counties of fairly similar size and shape, and (2) four states in the western US (Arizona, California, Nevada and Utah) forming a set of 118 counties that are vastly different geographical units. In the western US and Utah, PK was used to smooth county-level incidence rates of drug poisoning deaths, populate data gaps and improve the reliability of rates recorded in sparsely populated counties [9].

## 2. Materials and Methods

#### 2.1. COVID-19 Incidence Data

^{2}, and finer details on the spatial distribution of the population was provided by a raster (31,557 cells of 1 km

^{2}each) of the 2016 population (Figure 1C). These high-resolution population data were derived through the geocoding of individual addresses from the National Register of Natural Persons (RNPP) and are publicly available [20]. Only very few individuals could not be geolocated (6827 individuals out of a population of 11,492,641 in 2020), leading to very accurate statistics.

#### 2.2. Poisson Kriging

_{α},t), α = 1, …, M} be the set of COVID-19 incidence rates (areal data) recorded at M = 581 municipalities v

_{α}on day t. Since the same analysis is undertaken independent-ly for every day t, the temporal reference is omitted from the notation hereafter for simplicity. Each rate is calculated as z(v

_{α}) = d(v

_{α})/n(v

_{α}), where d(v

_{α}) is the number of positive cases and n(v

_{α}) is the size of the population at risk (i.e., total population of the municipality). The objective of the analysis is twofold: (1) filter the noise attached to the observed rates z(v

_{α}), and (2) estimate incidence rates at L = 31,557 nodes ${\mathit{u}}_{l}$ of a 1 km spacing grid discretizing the entire country of Belgium (spatial disaggregation). The results will be two sets of rate estimates for any single day t: {${\widehat{z}}_{PK}$(v

_{α}), α = 1, …, M} and {${\widehat{z}}_{PK}$(${\mathit{u}}_{l}$), l = 1, …, L} with the following coherency constraint that is satisfied for each municipality v

_{α}:

_{α}and $n\left({\mathit{u}}_{l}\right)$ represents the population within the grid cell centered on the node ${\mathit{u}}_{l}$. In both cases, each rate estimate is calculated as the weighted sum of rates recorded in (B − 1) neighboring municipalities, besides the municipality v

_{α}where the estimation is taking place:

**h**) calculated between any two locations discretizing the geographical units corresponding to municipalities ${v}_{\beta}$ and ${v}_{\theta}$. Similarly, the area-to-point covariance ${\overline{C}}_{R}\left({\mathit{u}}_{l},{v}_{\theta}\right)$ is estimated by averaging the covariance C (

**h**) computed between grid node ${\mathit{u}}_{l}$ and a set of locations discretizing the geographical unit ${v}_{\theta}$.

**h**) is inferred in three steps. First, an area-based semivariogram is calculated from incidence rates using the Poisson estimator introduced in [5]:

**h**) is the number of pairs of municipalities (${v}_{\alpha},{v}_{\beta}$) whose population-weighted centroids are separated by the vector

**h**, and $\stackrel{\_}{\mathrm{z}}$ is the population-weighted mean of the M incidence rate. The squared spatial increments ${\left[z\left({v}_{\alpha}\right)-z\left({v}_{\beta}\right)\right]}^{2}$ are weighted by a function of their respective population sizes, $n\left({v}_{\alpha}\right)\times n\left({v}_{\beta}\right)$/[$n\left({v}_{\alpha}\right)+n\left({v}_{\beta}\right)$], a term which is inversely proportional to their standard deviations [6]. More importance is thus given to the more reliable data pairs (i.e., smaller standard deviations). Second, a model is fitted to the experimental semivariogram ${\widehat{\gamma}}_{v}\left(\mathit{h}\right)$ and deconvoluted using an iterative procedure [21] to derive the point-support semivariogram model $\gamma \left(\mathit{h}\right)$. Last, the covariance is calculated as C(h) = C (0) − $\gamma \left(\mathit{h}\right)$, where C (0) is the sill of the semivariogram $\gamma \left(\mathit{h}\right)$.

#### 2.3. Soft Indicator Kriging

- Compute K = 50 percentiles ${z}_{k}$ of the frequency distribution of M = 581 municipality rates, F(.), as: ${z}_{k}$=F
^{-1}(${p}_{k}$) with $\left\{{p}_{k}={p}_{min}+\left(k-1\right)\times \frac{\left(1-{p}_{min}\right)}{50},k=1,\cdots ,K\right\}$ where ${p}_{min}=F\left({z}_{min}\right)$ is the proportion of rates no greater than the minimum observed rate ${z}_{min}$. This formulation avoids obtaining a series of zero-valued thresholds for days where no cases were recorded in many municipalities. - For each municipality ${v}_{\alpha}$:
- Create a binomial distribution Bi ($n\left({v}_{\alpha}\right)$,$z\left({v}_{\alpha}\right)$) characterized by the daily incidence rate $z\left({v}_{\alpha}\right)$ and the population $n\left({v}_{\alpha}\right)$ within that geographical unit. This step allows one to capture the uncertainty attached to the observed rate $z\left({v}_{\alpha}\right)$, which can be substantial for municipalities that are sparsely populated (i.e., small population size $n\left({v}_{\alpha}\right)),$
- Discretize the probability distribution using the set of K thresholds ${z}_{k}$ calculated at step 1: $\left\{j\left({v}_{\alpha};{z}_{k}\right)={F}_{Bi}\left({v}_{\alpha};{z}_{k}\right),k=1,\cdots ,K\right\}$ where ${F}_{Bi}\left({v}_{\alpha}\right)$ is the cumulative binomial distribution for the α-th municipality. The quantity $j\left({v}_{\alpha};{z}_{k}\right)$ represents the probability that the underlying rate is no greater than the threshold ${z}_{k}$ for municipality ${v}_{\alpha}$.

- For each threshold ${z}_{k}$:
- Calculate and model the population-weighted indicator semivariogram as:$$\gamma \left(\mathit{h};{z}_{k}\right)=\frac{1}{{{\displaystyle \sum}}_{\alpha ,\beta}^{N\left(\mathit{h}\right)}n\left({v}_{\alpha}\right)\times n\left({v}_{\beta}\right)}{\displaystyle \sum}_{\alpha ,\beta}^{N\left(\mathit{h}\right)}n\left({v}_{\alpha}\right)\times n\left({v}_{\beta}\right){\left[j\left({v}_{\alpha};{z}_{k}\right)-j\left({v}_{\beta};{z}_{k}\right)\right]}^{2}$$
**h**) is the number of pairs of municipalities (${v}_{\alpha}$,${v}_{\beta}$) whose population-weighted centroids are separated by the vector**h**[19]. - Use this model and ATP ordinary kriging to disaggregate the probabilities $j\left({v}_{\alpha};{z}_{k}\right)$ (i.e., soft indicator data) at the nodes ${\mathit{u}}_{l}$ of a 1 km spacing grid discretizing the country (total number of nodes is L = 31,557).

- For each node ${\mathit{u}}_{l}$ of the discretization grid:
- Assemble the K estimated probabilities ${j}^{*}\left({\mathit{u}}_{l};{z}_{k}\right)$ into a probability distribution.
- Correct for order relation deviations [15] as: (1) each probability ${j}^{*}\left({\mathit{u}}_{l};{z}_{k}\right)$ can be negative or larger than 1 since it was estimated by ATP kriging (same potential issues as PK), and (2) the set of K probabilities ${j}^{*}\left({\mathit{u}}_{l};{z}_{k}\right)$ were estimated separately, with no guarantee that ${j}^{*}\left({\mathit{u}}_{l};{z}_{k}\right)\le {j}^{*}\left({\mathit{u}}_{l};{z}_{k+1}\right)\forall {z}_{k+1}{z}_{k}$.
- Create a continuous distribution using linear interpolation between thresholds, as well as between the first (last threshold) and the minimum (maximum) observed rate.
- Calculate the mean ${\widehat{z}}_{IK}\left({\mathit{u}}_{l}\right)$ and variance of the local probability distribution (ccdf).

- For each municipality ${v}_{\alpha}$:
- Estimate each of the K probabilities ${j}^{*}\left({v}_{\alpha};{z}_{k}\right)$ as the population-weighted average of the corresponding probabilities ${j}^{*}\left({\mathit{u}}_{l};{z}_{k}\right)$ for all nodes ${\mathit{u}}_{l}$ that fall within that geographical unit, i.e.,$${j}^{*}\left({v}_{\alpha};{z}_{k}\right)=\frac{1}{n\left({v}_{\alpha}\right)}{\displaystyle \sum}_{l=1}^{L}i({\mathit{u}}_{l};{v}_{\alpha})\times n\left({\mathit{u}}_{l}\right)\times {j}^{*}\left({\mathit{u}}_{l};{z}_{k}\right)$$
- Assemble the K estimated probabilities ${j}^{*}\left({v}_{\alpha};{z}_{k}\right)$ into a probability distribution.
- Correct for order relation deviations and create a continuous distribution using linear interpolation between thresholds, as well as between the first (last threshold) and the minimum (maximum) observed rate.
- Calculate the mean ${\widehat{z}}_{IK}\left({v}_{\alpha}\right)$ and variance of the local probability distribution (ccdf).

#### 2.4. Software

## 3. Results

#### 3.1. Temporal Trend and Spatial Patterns

#### 3.2. Kriging Estimates

#### 3.3. Kriging Variances

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

^{2}grid cells) are available at https://statbel.fgov.be/en/open-data/population-according-km2-grid-2020, , accessed on 9 May 2023. A free 1-yr license of SpaceStat can be downloaded at https://biomedware.com/products/spacestat/, , accessed on 9 May 2023.

## Conflicts of Interest

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**Figure 1.**(

**A**) Google Earth map overlaid with boundaries of 581 municipalities in Belgium. Population size recorded for: (

**B**) each municipality as of 1 January 2020 and (

**C**) 31,557 cells of 1 km

^{2}each in 2016, providing details about the spatial distribution of population with each administrative unit.

**Figure 2.**Time series of COVID-19 incidence rates (number of cases per 10,000 inhabitants), which were averaged over all 581 municipalities in Belgium. The time axis represents the number of days after 1 March 2020, while vertical red lines denote the four dates used for the comparison study.

**Figure 3.**Maps of municipality-level COVID-19 incidence rates (number of cases per 10,000 inhabitants) recorded at four different dates.

**Figure 4.**Experimental Poisson semivariogram calculated from municipality-level COVID-19 incidence rates (number of cases per 10,000 inhabitants) recorded at four different dates, with the model fitted.

**Figure 5.**Experimental population-weighted indicator semivariogram calculated for the first (second for 28 October 2020) threshold at four different dates, with the model fitted. The first threshold corresponds to a null COVID-19 incidence rate that was recorded among 1.9% (25 December 2020) to 64.4% (10 March 2020) of Belgian municipalities. COVID-19 cases were identified in all 581 Belgian municipalities on 28 October 2020 when the mean incidence rate peaked.

**Figure 6.**Experimental population-weighted indicator semivariogram calculated for three different thresholds (# 10, 25, 40) and two different dates (10 March 2020, 25 December 2020), with the model fitted.

**Figure 7.**Maps of smoothed COVID-19 incidence rates (number of cases per 10,000 inhabitants) calculated by PK and IK at the municipality level (ATA kriging,

**A**,

**B**) and at the nodes of a 1 km spacing grid (ATP kriging,

**C**,

**D**) for time period 1 (10 March 2020). Black pixels indicate negative kriging estimates. Bottom maps (

**E**,

**F**) show the kriging variance for ATP kriging.

**Figure 8.**Maps of smoothed COVID-19 incidence rates (number of cases per 10,000 inhabitants) calculated by PK and IK at the municipality level (ATA kriging,

**A**,

**B**) and at the nodes of a 1 km spacing grid (ATP kriging,

**C**,

**D**) for time period 1 (25 December 2020). Black pixels indicate negative kriging estimates. Bottom maps (

**E**,

**F**) show the kriging variance for ATP kriging.

**Figure 9.**Scatterplots of COVID-19 incidence rates smoothed by PK and IK at the municipality level (ATA kriging) versus rates recorded for time period 1 (10 March 2020). Notice the larger smoothing effect (i.e., smaller spread) caused by PK vs. IK, except for null incidence rates where the estimates range from 0 to 2 cases per 10,000 inhabitants vs. 0 to 1 for IK (green ellipse).

**Table 1.**Summary statistics of COVID-19 incidence rates (number of cases per 10,000 inhabitants) recorded at four different dates for all 581 Belgian municipalities.

Dates | ||||
---|---|---|---|---|

Statistics | 10 March 2020 (t = 10) | 16 September 2020 (t = 200) | 28 October 2020 (t = 242) | 25 December 2020 (t = 300) |

Mean | 0.444 | 6.049 | 105.6 | 11.66 |

Variance | 0.932 | 31.97 | 3573 | 55.65 |

Minimum | 0.0 | 0.0 | 12.98 | 0.0 |

Maximum | 6.981 | 45.59 | 303.8 | 50.84 |

% null values | 64.4 | 10.8 | 0.0 | 1.9 |

**Table 2.**Variance of Poisson (PK) and indicator kriging (IK) estimates calculated at the municipality level (ATA) and after spatial disaggregation (ATP) for four different dates.

Dates | ||||
---|---|---|---|---|

Variance | 10 March 2020 (t = 10) | 16 September 2020 (t = 200) | 28 October 2020 (t = 242) | 25 December 2020 (t = 300) |

ATA PK | 0.194 | 15.91 | 3219 | 25.10 |

ATA IK | 0.754 | 25.77 | 3575 | 49.07 |

ATP PK | 0.190 | 10.49 | 3031 | 26.95 |

ATP IK | 0.559 | 16.73 | 3225 | 45.81 |

**Table 3.**Rank correlation between Poisson (PK) and indicator kriging (IK) estimates calculated at the municipality level (ATA) and after spatial disaggregation (ATP) for four different dates. At the municipality level, the rank correlation with observed rates is also listed.

Dates | ||||
---|---|---|---|---|

Correlation | 10 March 2020 (t = 10) | 16 September 2020 (t = 200) | 28 October 2020 (t = 242) | 25 December 2020 (t = 300) |

ATA PK vs. rate | 0.645 | 0.846 | 0.989 | 0.851 |

ATA IK vs. rate | 0.883 | 0.976 | 0.998 | 0.997 |

PK vs. IK (ATA) | 0.645 | 0.889 | 0.989 | 0.862 |

PK vs. IK (ATP) | 0.849 | 0.875 | 0.980 | 0.907 |

**Table 4.**Frequency and magnitude of corrections applied to IK-based ccdfs that violate order relations at the grid node (ATP IK) and municipality (ATA IK) levels.

Dates | ||||
---|---|---|---|---|

Order Relations | 10 March 2020 (t = 10) | 16 September 2020 (t = 200) | 28 October 2020 (t = 242) | 25 December 2020 (t = 300) |

Freq. (ATP) | 0.798 | 0.791 | 0.427 | 0.708 |

Magn. (ATP) | 0.044 | 0.042 | 0.051 | 0.049 |

Freq. (ATA) | 0.0 | 0.0 | 0.0 | 0.0 |

Magn. (ATA) | 0.0 | 0.0 | 0.0 | 0.0 |

**Table 5.**Rank correlation between Poisson (PK) and indicator kriging (IK) outputs (i.e., estimated value, error standard deviation) calculated at the municipality level (ATA) and after spatial disaggregation (ATP) for four different dates. Correlation with municipality population size is also listed.

Dates | ||||
---|---|---|---|---|

Correlation | 10 March 2020 (t = 10) | 16 September 2020 (t = 200) | 28 October 2020 (t = 242) | 25 December 2020 (t = 300) |

ATA | ||||

PKstd vs. IKstd | −0.176 | 0.379 | 0.728 | 0.721 |

PKstd vs. PKest | −0.197 | −0.053 | 0.259 | 0.115 |

IKstd vs. IKest | 0.986 | 0.657 | 0.724 | 0.619 |

PKstd vs. Population | −0.908 | −0.933 | −0.980 | −0.919 |

IKstd vs. Population | 0.175 | −0.339 | −0.737 | −0.689 |

ATP | ||||

PKstd vs. IKstd | −0.106 | 0.414 | 0.499 | 0.508 |

PKstd vs. PKest | −0.208 | −0.040 | 0.257 | 0.116 |

IKstd vs. IKest | 0.952 | 0.571 | 0.631 | 0.585 |

PKstd vs. Population | −0.665 | −0.711 | −0.680 | −0.669 |

IKstd vs. Population | 0.040 | −0.302 | −0.384 | −0.390 |

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## Share and Cite

**MDPI and ACS Style**

Goovaerts, P.; Hermans, T.; Goossens, P.F.; Van De Vijver, E.
Comparison of Soft Indicator and Poisson Kriging for the Noise-Filtering and Downscaling of Areal Data: Application to Daily COVID-19 Incidence Rates. *ISPRS Int. J. Geo-Inf.* **2023**, *12*, 328.
https://doi.org/10.3390/ijgi12080328

**AMA Style**

Goovaerts P, Hermans T, Goossens PF, Van De Vijver E.
Comparison of Soft Indicator and Poisson Kriging for the Noise-Filtering and Downscaling of Areal Data: Application to Daily COVID-19 Incidence Rates. *ISPRS International Journal of Geo-Information*. 2023; 12(8):328.
https://doi.org/10.3390/ijgi12080328

**Chicago/Turabian Style**

Goovaerts, Pierre, Thomas Hermans, Peter F. Goossens, and Ellen Van De Vijver.
2023. "Comparison of Soft Indicator and Poisson Kriging for the Noise-Filtering and Downscaling of Areal Data: Application to Daily COVID-19 Incidence Rates" *ISPRS International Journal of Geo-Information* 12, no. 8: 328.
https://doi.org/10.3390/ijgi12080328