- Introduction
- Loading the packages and the data
- Classical Kmeans
- Classical c-means and generalized c-means
- Spatial c-means and generalized c-means
- Comparing the spatial consistency of FCM, GFCM, SFCM and SGFCM
- Interpreting the results of the final classification
- Ensuring clustering stability
- Advanced example
- Final words
- References

This document is a short introduction to the package **geocmeans**. It implements fuzzy classification methods bringing spatial information and neighbouring in its calculation

In their article, Cai, Chen, and Zhang (2007) described the method, originally applied in the analysis of brain imagery. The generalized version of the spatial fuzzy c-means is presented by Zhao, Jiao, and Liu (2013).

Gelb and Apparicio (2021) applied the method to socio-residential and environmental data comparing the results with other unsupervised classification algorithms (in French).

There are actually numerous packages and functions to perform unsupervised classification in R (*hclust*, *kmeans*, *cmeans*, *factoextra*, etc.). However, these methods are not always well-suited to analyze spatial data. Indeed, they do not account for spatial information such as proximity or contiguity between observations. This may lead to solutions for which close observations end up in different groups event though they are very similar.

To our knowledge, the package **ClustGeo** is the only package proposing an unsupervised classification method which directly consider spatial proximity between observations. The proposed approach is appealing because the user can select a parameter (*alpha*) that controls the weight of the spatial distance matrix (calculated between observations with their locations) versus the semantic distance matrix (calculated between observations with their variables).

However, this method belongs to the category of “hard-clustering” algorithms. Each observation ends up in one cluster/group. The main draw-back here is the difficulty to identify observations that are undecided, at the frontier of two clusters/groups. The soft or fuzzy clustering algorithms provide more information because they calculate the “probability” of each observation to belong to each group.

The algorithms SFCM (spatial fuzzy c-means) and SGFCM (spatial generalized fuzzy c-means) propose to combine the best of both worlds.

The package **geocmeans** is an implementation in R of these methods. It comes with a set of functions to facilitate the analysis of the final membership matrices:

- calculating many quality indices (coming mainly from the package
*fclust*) - mapping the results
- giving summary statistics for each cluster/group
- a shiny app to investigate dynamically the results.

The general workflow for applying a spatial fuzzy clustering algorithm with **geocmeans** is described with the following figure.

**geocmeans** comes with a toy dataset *LyonIris*, combining many demographic and environmental variables aggregated at the scale of the Iris (aggregated units for statistical information) in Lyon (France).

Note that the dataset used for a fuzzy classification can only contain continuous variables. For dataset with categorical variables, one could start with a PCA (Principal Component Analysis), MCA (multiple Correspondence Analysis) or a MDFA (Mixed Data Factor Analysis) and uses the obtained factors as inputs for the clustering.

Before starting the analysis, the data must be standardized because most of the calculus is based on Euclidean distance (we plan to also include Manhattan distance in a future release).

```
#charging packages and data
library(geocmeans)
library(ggplot2)
library(ggpubr)
library(dplyr)
library(viridis)
library(spdep)
data(LyonIris)
# selecting the columns for the analysis
<-c("Lden","NO2","PM25","VegHautPrt","Pct0_14",
AnalysisFields "Pct_65","Pct_Img","TxChom1564","Pct_brevet","NivVieMed")
# rescaling the columns
<- LyonIris@data[AnalysisFields]
Data for (Col in names(Data)){
<- scale(Data[[Col]])
Data[[Col]]
}
# preparing some elements for further mapping
$OID <- as.character(1:nrow(LyonIris))
LyonIris<- ggplot2::fortify(LyonIris,region="OID") FortiData
```

To explore the dataset and choose the right number of cluster/groups (k) we propose to start with a classical kmeans.

```
# finding the best k by using the r2 of the classification
# trying for k from 2 to 10
<- sapply(2:10,function(k){
R2s <- kmeans(Data,centers=k,iter.max = 150)
Clust <- Clust$betweenss / Clust$totss
R2 return(R2)
})
<- data.frame(K=2:10,
Df R2 = R2s)
ggplot(Df)+
geom_line(aes(x=K,y=R2s))+
geom_point(aes(x=K,y=R2s),color="red")+
xlab("Number of groups")+
ylab("R2 of classification")
```

By plotting the R-squared of the kmeans classification for each k between 2 and 10, we can see a first elbow at k=3. But this small number of groups leads to a classification explaining only 43% of the original data variance. We decide to keep k=4 to have one more group in the analysis.

Let us map the obtained groups.

```
<- kmeans(Data,centers=4,iter.max = 150)
KMeanClust $Cluster <-paste("cluster",KMeanClust$cluster,sep="_")
LyonIris
# mapping the groups
<- merge(FortiData,LyonIris,by.x="id",by.y="OID")
DFmapping
ggplot(data=DFmapping)+
geom_polygon(aes(x=long,y=lat,group=group,fill=Cluster),color=rgb(0,0,0,0))+
coord_fixed(ratio = 1)+
scale_fill_manual(name="Cluster",values = c("cluster_1"="palegreen3",
"cluster_2"="firebrick",
"cluster_3"="lightyellow2",
"cluster_4"="steelblue"))+
theme( axis.title = ggplot2::element_blank(),
axis.text = ggplot2::element_blank(),
axis.ticks = ggplot2::element_blank()
)
```

We can clearly distinguish 4 strong spatial structures, but with some mixing between the clusters.

We could now compare this solution with a classical c-means algorithm.

The classical c-means is a simple method to perform fuzzy unsupervised classification. The package **geocmeans** proposes the function *CMeans*. We set the fuzziness degree (m) to 1.5.

We do not present the algorithm here, but only the two main formulas.

The first one is used to update the values of the membership matrix at each iteration \(u_ik\)

\[u_{ik} = \frac{(||x_{k} - v{_i}||^{2}) ^{(-1/(m-1))}}{\sum_{j=1}^c(||x_{k} - v{_j}||^2 )^{(-1/(m-1))}}\] And the second one to update the centers of the clusters

\[v_{i} = \frac{\sum_{k=1}^N u_{ik}^m(x_{k})}{\sum_{k=1}^N u_{ik}^m}\]

With :

- \(x_k\) the values of the observation \(k\)
- \(v_i\) the values of the center of the cluster \(i\)
- \(c\) the number of clusters
- \(m\) the fuzziness index

`<- CMeans(Data,4,1.5,500,standardize = FALSE, seed = 456, tol = 0.00001, verbose = FALSE) Cmean `

We can now use the function *calcqualityIndexes* which combines many indices from the package **fclust** to analyze the quality of the classification. We will use these values later for the purpose of comparisons among the different algorithms.

`calcqualityIndexes(Data, Cmean$Belongings, m = 1.5)`

```
## $Silhouette.index
## [1] 0.362851
##
## $Partition.entropy
## [1] 0.7320161
##
## $Partition.coeff
## [1] 0.6070524
##
## $XieBeni.index
## [1] 1.352347
##
## $FukuyamaSugeno.index
## [1] 910.6537
##
## $Explained.inertia
## [1] 0.3018179
```

Note, **geocmeans** also proposes a so-called generalized version of the c-means algorithm. It is known to accelerate convergence and yield less fuzzy results by adjusting the membership matrix at each iteration. It requires an extra *beta* parameter controlling the strength of the modification. The modification only affects the formula updating the membership matrix.

\[u_{ik} = \frac{(||x_{k} - v{_i}||^{2} - \beta_k) ^{(-1/(m-1))}}{\sum_{j=1}^c(||x_{k} - v{_j}||^2 - \beta_k)^{(-1/(m-1))}}\]

with \(\beta_k = min(||x_{k} - v||^2)\) and \(0 \leq \beta \leq 1\).

To select an appropriate value for this parameter, we will try all the possible values between 0 and 1 with a step of 0.05.

```
<- selectParameters("GFCM",data = Data, k = 4, m = 1.5,
beta_values beta = seq(0,1,0.05), spconsist = FALSE,
tol = 0.00001, seed = 456)
```

```
::kable(beta_values[c("beta","Silhouette.index","XieBeni.index","Explained.inertia")],
knitrcol.names = c("beta", "silhouette index",
"Xie and Beni index", "explained inertia"),digits = 3)
```

beta | silhouette index | Xie and Beni index | explained inertia |
---|---|---|---|

0.00 | 0.363 | 1.352 | 0.302 |

0.05 | 0.365 | 1.268 | 0.310 |

0.10 | 0.366 | 1.193 | 0.319 |

0.15 | 0.365 | 1.128 | 0.327 |

0.20 | 0.361 | 1.071 | 0.336 |

0.25 | 0.360 | 1.020 | 0.345 |

0.30 | 0.363 | 0.984 | 0.354 |

0.35 | 0.361 | 0.963 | 0.363 |

0.40 | 0.370 | 0.943 | 0.372 |

0.45 | 0.369 | 0.924 | 0.381 |

0.50 | 0.366 | 0.907 | 0.390 |

0.55 | 0.365 | 0.891 | 0.399 |

0.60 | 0.360 | 0.876 | 0.409 |

0.65 | 0.358 | 0.863 | 0.418 |

0.70 | 0.358 | 0.852 | 0.428 |

0.75 | 0.353 | 0.843 | 0.437 |

0.80 | 0.347 | 0.838 | 0.447 |

0.85 | 0.340 | 0.837 | 0.456 |

0.90 | 0.332 | 0.844 | 0.464 |

0.95 | 0.323 | 0.861 | 0.471 |

1.00 | 0.317 | 0.919 | 0.475 |

Considering the table above, we select *beta* = 0.7, it maintains a good silhouette index, and increases Xie and Beni index and explained inertia. Let us compare the results of GFCM and FCM.

```
<- GCMeans(Data,k = 4,m = 1.5, beta = 0.7,500,standardize = FALSE, seed=456,
GCmean tol = 0.00001, verbose = FALSE)
<- calcqualityIndexes(Data,GCmean$Belongings,m=1.5)
r1 <- calcqualityIndexes(Data,Cmean$Belongings,m=1.5)
r2 <- cbind(unlist(r1), unlist(r2))
df
::kable(df,
knitrdigits = 3,col.names = c("GFCM", "FCM"))
```

GFCM | FCM | |
---|---|---|

Silhouette.index | 0.358 | 0.363 |

Partition.entropy | 0.323 | 0.732 |

Partition.coeff | 0.833 | 0.607 |

XieBeni.index | 0.852 | 1.352 |

FukuyamaSugeno.index | 219.640 | 910.654 |

Explained.inertia | 0.428 | 0.302 |

The results indicate that the GFCM provides a solution that is less fuzzy (higher explained inertia and lower partition entropy) but keeps a good silhouette index and an even better Xie and Beni index.

We can now map the two membership matrices and the most likely group for each observation. To do so, we use the function *mapClusters* from **geocmeans**. We propose here to define a threshold of 0.45. If an observation only has values below this probability in a membership matrix, it will be labeled as “undecided” (represented with transparency on the map).

We can compare the maps of the classical c-means and the generalized version.

```
<- mapClusters(LyonIris,Cmean$Belongings,undecided = 0.45)
cmeansMaps<- mapClusters(LyonIris,GCmean$Belongings,undecided = 0.45)
GcmeansMaps
ggarrange(cmeansMaps$ProbaMaps[[1]],GcmeansMaps$ProbaMaps[[1]],
nrow = 1, ncol = 2, common.legend = TRUE, legend = "bottom")
```

```
ggarrange(cmeansMaps$ProbaMaps[[2]],GcmeansMaps$ProbaMaps[[2]],
nrow = 1, ncol = 2, common.legend = TRUE, legend = "bottom")
```

```
ggarrange(cmeansMaps$ProbaMaps[[3]],GcmeansMaps$ProbaMaps[[3]],
nrow = 1, ncol = 2, common.legend = TRUE, legend = "bottom")
```

```
ggarrange(cmeansMaps$ProbaMaps[[4]],GcmeansMaps$ProbaMaps[[4]],
nrow = 1, ncol = 2, common.legend = TRUE, legend = "bottom")
```

```
ggarrange(cmeansMaps$ClusterPlot,GcmeansMaps$ClusterPlot,
nrow = 1, ncol = 2, common.legend = TRUE, legend = "bottom")
```

As expected, the results are very similar, but the generalized version provides a more clear-cut classification.

Now we can use the *SFCM* function to perform a spatial c-means. The first step is to define a spatial weight matrix indicating which observations are neighbours and the strength of their relationship. We propose here to use a basic queen neighbour matrix (built with **spdep**). The matrix must be row-standardized to ensure that the interpretation of all the parameters remains clear.

```
library(spdep)
<- poly2nb(LyonIris,queen = TRUE)
Neighbours <- nb2listw(Neighbours,style="W",zero.policy = TRUE) WMat
```

The main challenge with the SFCM method is to select the parameter *alpha*. It represents the weight of the spatial dimension (lagged values) in the calculus of the membership matrix and the cluster centers.

- If alpha=0, then we end up with a classical c-means algorithm.
- If alpha=1, then the original and the lagged values have the same weight
- If alpha=2, then the lagged values are twice more important than the original values
- end so on…

The two following formulas show how the functions updating the membership matrix and the centers of the clusters are modified.

\[u_{ik} = \frac{(||x_{k} - v{_i}||^2 + \alpha||\bar{x_{k}} - v{_i}||^2)^{(-1/(m-1))}}{\sum_{j=1}^c(||x_{k} - v{_j}||^2 + \alpha||\bar{x_{k}} - v{_j}||^2)^{(-1/(m-1))}}\]

\[v_{i} = \frac{\sum_{k=1}^N u_{ik}^m(x_{k} + \alpha\bar{x_{k}})}{(1 + \alpha)\sum_{k=1}^N u_{ik}^m}\] with \(\bar{x}\) the lagged version of x and \(\alpha \geq 0\)

As the formula suggests, the *SFCM* can be seen as a spatially smoothed version of the classical c-means and *alpha* controls the degree of spatial smoothness. This smoothing can be interpreted as an attempt to reduce spatial overfitting of the classical c-means.

To select *alpha*, we propose to check all possible values between 0 and 2 with a step of 0.05.

```
<- selectParameters(algo = "SFCM", data = Data,
DFindices_SFCM k = 4, m = 1.5, alpha = seq(0,2,0.05),
nblistw = WMat, standardize = FALSE,
tol = 0.0001, verbose = FALSE, seed = 456)
```

Now we are able to check the indices to select the best *alpha*. The goal is to reduce spatial inconsistency as much as possible and to maintain a good classification quality.

Let us start with the spatial inconsistency. This indicator (developed for this package) calculates the sum of the squared differences between each observation and its neighbours on the membership matrix. Thus, the maximum for each observation is \(k*j\) with *j* the number of neighbours for the observation and *k* the number of groups. A maximum is reached if each observation has 100% chance belonging to a cluster that is different from all its neighbours. So, when we sum up the values obtained for all the observations, we obtain a quantity of spatial inconsistency. This quantity is divided by the quantity obtained when randomly permuting the rows of the membership matrix. This second quantity represents the spatial inconsistency that we might expect if the observations were randomly scattered in space. We can repeat the permutation step (Monte Carlo approach) and keep the mean of the ratios to have a more robust indicator (see help(spConsistency) for details).

A smaller value indicates a smaller spatial inconsistency and thus a greater spatial consistency. 0 meaning that all observations have exactly the same values in the membership matrix as their neighbours (perfect spatial consistency).

```
ggplot(DFindices_SFCM)+
geom_smooth(aes(x=alpha,y=spConsistency), color = "black")+
geom_point(aes(x=alpha,y=spConsistency), color = "red")
```

`## `geom_smooth()` using method = 'loess' and formula 'y ~ x'`

Not surprisingly, increasing *alpha* leads to a decrease of the spatial inconsistency. This gain follows an inverse function.

Let us now check the explained inertia

```
ggplot(DFindices_SFCM)+
geom_smooth(aes(x=alpha,y=Explained.inertia), color = "black")+
geom_point(aes(x=alpha,y=Explained.inertia), color = "red")
```

`## `geom_smooth()` using method = 'loess' and formula 'y ~ x'`

As expected, the explained inertia decreases when alpha increases and again follows an inverse function. The classification has to find a compromise between the original values and the lagged values. However, the loss is very small here: only 3% between alpha = 0 and alpha = 2.

To finish here, we can observe the silhouette and Xie and Beni indicators.

```
ggplot(DFindices_SFCM)+
geom_smooth(aes(x=alpha,y=Silhouette.index), color = "black")+
geom_point(aes(x=alpha,y=Silhouette.index), color = "red")
```

```
ggplot(DFindices_SFCM)+
geom_smooth(aes(x=alpha,y=XieBeni.index), color = "black")+
geom_point(aes(x=alpha,y=XieBeni.index), color = "red")
```

The detail of the meaning of these indicators is beyond the scope of this vignette. Let us just stress that a larger silhouette index indicates a better classification, and a smaller Xie and Beni index indicates a better classification.

After considering all the previous charts, we decide to keep alpha = 0.7 as it seems to provide a good balance between spatial consistency and classification quality in this case.

```
<- SFCMeans(Data, WMat, k = 4, m = 1.5, alpha = 0.7,
SFCM tol = 0.0001, standardize = FALSE,
verbose = FALSE, seed = 456)
```

It is also possible to use the so-called generalized version of the spatial c-means. In that case, we must define both *alpha* and *beta*.

The next formula shows how the membership matrix is updated at each iteration. Note that the centres of the clusters are updated with the same formula as SFCM.

\[u_{ik} = \frac{(||x_{k} - v{_i}||^2 -\beta_k + \alpha||\bar{x_{k}} - v{_i}||^2)^{(-1/(m-1))}}{\sum_{j=1}^c(||x_{k} - v{_j}||^2 -\beta_k + \alpha||\bar{x_{k}} - v{_j}||^2)^{(-1/(m-1))}}\]

Because we select a high resolution for our grid search of *alpha* and *beta*, we will use a multiprocessing approach.

```
::plan(future::multiprocess(workers=4))
future<- selectParameters.mc(algo = "SGFCM", data = Data,
DFindices_SFGCM k = 4, m = 1.5, alpha = seq(0,2,0.05),
beta = seq(0,0.85,0.05),
nblistw = WMat, standardize = FALSE, chunk_size = 50,
tol = 0.0001, verbose = FALSE, seed = 456)
```

```
ggplot(DFindices_SFGCM) +
geom_raster(aes(x = alpha, y = beta, fill = Silhouette.index), size = 5) +
scale_fill_viridis() +
coord_fixed(ratio=1)
```

```
ggplot(DFindices_SFGCM) +
geom_raster(aes(x = alpha, y = beta, fill = XieBeni.index), size = 5) +
scale_fill_viridis() +
coord_fixed(ratio=1)
```

```
ggplot(DFindices_SFGCM) +
geom_raster(aes(x = alpha, y = beta, fill = spConsistency), size = 5) +
scale_fill_viridis() +
coord_fixed(ratio=1)
```

The first two plots indicate that some specific pairs of *alpha* and *beta* yield good results in the range 0.8 < *alpha* < 1.2 and 0.6 < *beta* < 0.8. The last plot shows that the selection of *beta* has no impact on the spatial consistency.

Considering the previous plots, we decide to retain the solution with *beta* = 0.65 and *alpha* = 0.95 which yield very good results for all the indices considered.

```
<- SGFCMeans(Data,WMat,k = 4,m=1.5, alpha=0.95, beta = 0.65,
SGFCM tol=0.0001, standardize = FALSE, verbose = FALSE, seed = 456)
```

Again, we compare here the generalized and the classical version of the spatial c-means algorithm.

```
<- calcqualityIndexes(Data, SFCM$Belongings,m = 1.5)
r1 <- calcqualityIndexes(Data, SGFCM$Belongings,m = 1.5)
r2
<- spatialDiag(SFCM$Belongings, nblistw = WMat,
diagSFCM undecided = 0.45,nrep = 500)
<- spatialDiag(SGFCM$Belongings, nblistw = WMat,
diagSGFCM undecided = 0.45,nrep = 500)
<- cbind(
df c(unlist(r1),diagSFCM$SpConsist),
c(unlist(r2),diagSGFCM$SpConsist)
)row.names(df)[length(row.names(df))] <- "sp.consistency"
::kable(df,digits = 3,col.names = c("SFCM","SGFCM")) knitr
```

SFCM | SGFCM | |
---|---|---|

Silhouette.index | 0.370 | 0.391 |

Partition.entropy | 0.765 | 0.544 |

Partition.coeff | 0.592 | 0.715 |

XieBeni.index | 1.976 | 0.971 |

FukuyamaSugeno.index | 1100.096 | 739.201 |

Explained.inertia | 0.274 | 0.347 |

sp.consistency | 0.183 | 0.171 |

The solution of the SGFCM is better on the semantic and the spatial aspects.

We can compare the maps

```
<- mapClusters(geodata = LyonIris, object = SFCM$Belongings,undecided = 0.45)
SFCMMaps <- mapClusters(geodata = LyonIris, object = SGFCM$Belongings,undecided = 0.45)
SGFCMMaps
ggarrange(SFCMMaps$ProbaMaps[[1]],SGFCMMaps$ProbaMaps[[1]], nrow = 1, ncol = 2,
common.legend = TRUE, legend = "bottom")
```

```
ggarrange(SFCMMaps$ProbaMaps[[2]],SGFCMMaps$ProbaMaps[[2]], nrow = 1, ncol = 2,
common.legend = TRUE, legend = "bottom")
```

```
ggarrange(SFCMMaps$ProbaMaps[[3]],SGFCMMaps$ProbaMaps[[3]], nrow = 1, ncol = 2,
common.legend = TRUE, legend = "bottom")
```

```
ggarrange(SFCMMaps$ProbaMaps[[4]],SGFCMMaps$ProbaMaps[[4]], nrow = 1, ncol = 2,
common.legend = TRUE, legend = "bottom")
```

```
ggarrange(SFCMMaps$ClusterPlot,SGFCMMaps$ClusterPlot, nrow = 1, ncol = 2,
common.legend = TRUE, legend = "bottom")
```

Now, we can do a deeper spatial analysis and compare the spatial consistency of the four classifications realized (FCM, GFCM, SFCM, SGFCM).

```
<- spatialDiag(Cmean$Belongings, nblistw = WMat, nrep=250)
spdiag_1 <- spatialDiag(GCmean$Belongings, nblistw = WMat, nrep=250)
spdiag_2 <- spatialDiag(SFCM$Belongings, nblistw = WMat, nrep=250)
spdiag_3 <- spatialDiag(SGFCM$Belongings, nblistw = WMat, nrep=250)
spdiag_4
#looking at the moran I values for each group
<- data.frame(cbind(spdiag_1$MoranValues$MoranI,
moran_table $MoranValues$MoranI,
spdiag_2$MoranValues$MoranI,
spdiag_3$MoranValues$MoranI
spdiag_4
))row.names(moran_table) <- paste("cluster ",1:4,sep="")
::kable(moran_table, digits = 3,
knitrcol.names = c("FCM","GFCM","SFCM","SGFCM"),
caption = "Moran I index for the columns of the membership matrix"
)
```

FCM | GFCM | SFCM | SGFCM | |
---|---|---|---|---|

cluster 1 | 0.835 | 0.789 | 0.908 | 0.913 |

cluster 2 | 0.723 | 0.624 | 0.860 | 0.865 |

cluster 3 | 0.389 | 0.408 | 0.645 | 0.703 |

cluster 4 | 0.547 | 0.489 | 0.749 | 0.741 |

Not surprisingly, the Moran I values calculated on the membership matrices are higher for SFCM and SGFCM, indicating stronger spatial structures in the classifications.

`print(c(spdiag_1$SpConsist, spdiag_2$SpConsist,spdiag_3$SpConsist,spdiag_4$SpConsist))`

`## [1] 0.3344399 0.3933804 0.1837933 0.1713737`

Considering the values of spatial inconsistency, we could check if the value obtained for SGFCM is significantly lower than the one of SFCM. Considering the previous 250 values obtained by permutations, we can calculate a pseudo p-value:

`sum(spdiag_4$SpConsist > spdiag_3$SpConsistSamples) / length(spdiag_3$SpConsistSamples)`

`## [1] 0`

It appears that out of 250 permutations, the observed values of spatial inconsistency of SGFCM are always lower than that of SFCM. The difference is significant at the threshold 0.004 (=1/250)

We can map the undecided observations of the final solution. These entities should be analyzed more precisely. Selecting them is easy with the function *undecidedUnits* of the **geocmeans** package.

```
<- undecidedUnits(SGFCM$Belongings,0.45)
Undecided $FinalCluster <- ifelse(Undecided=="Undecided",
LyonIris"Undecided",paste("cluster",Undecided,sep="_"))
# mapping the groups
<- merge(FortiData,LyonIris,by.x="id",by.y="OID")
DFmapping
ggplot(data=DFmapping)+
geom_polygon(aes(x=long,y=lat,group=group,fill=FinalCluster),color=rgb(0,0,0,0))+
coord_fixed(ratio = 1)+
scale_fill_manual(name="FinalCluster",values = c("cluster_V1"="palegreen3",
"cluster_V2"="firebrick",
"cluster_V3"="lightyellow2",
"cluster_V4"="steelblue",
"cluster_V5"="pink",
"Undecided"=rgb(0,0,0,0.4)))+
theme( axis.title = element_blank(),
axis.text = element_blank(),
axis.ticks = element_blank()
)
```

It is also possible to visualize the uncertainty by using jittered points. The idea is to draw random points in polygons and attribute them to a group proportionally to the membership values of the polygon the fall in.

```
<- c("palegreen3","firebrick","lightyellow2","steelblue","pink")
colors uncertaintyMap(LyonIris, SGFCM$Belongings, color = colors)
```

To quantify and map uncertainty, one could Calculate the Shanon entropy index and use it as a measure of uncertainty with the function `calcUncertaintyIndex.`

A value of 1 indicates that an observation has equal membership values among groups and a value of 0 indicates that an observation belongs to only one group.

```
$entropyidx <- calcUncertaintyIndex(SGFCM$Belongings)
LyonIris
# mapping the uncertainty
<- merge(FortiData,LyonIris,by.x="id",by.y="OID")
DFmapping
ggplot(data=DFmapping)+
geom_polygon(aes(x=long,y=lat,group=group,fill=entropyidx),color=rgb(0,0,0,0))+
coord_fixed(ratio = 1)+
labs(title = "Uncertainty evaluation", fill = "entropy index") +
theme(axis.title = element_blank(),
axis.text = element_blank(),
axis.ticks = element_blank()
)
```

One can obtain a lot of descriptive information about the final groups with four functions of **geocmeans** :

`summarizeClusters`

: calculate summary statistics for each group for a given dataset by using the membership matrix as weights (sticking with the fuzzy spirit). One can also use the generic function`summary`

`spiderPlots`

: display a spider plot allowing to compare quickly the differences between groups.`violinPlots`

: display a violin plot for each variable in a given dataset. Observations must be grouped before.`sp_clust_explorer`

: launch a shiny app to investigate the results dynamically.

```
summarizeClusters(LyonIris@data[AnalysisFields],belongmatrix = SGFCM$Belongings,
weighted = TRUE, dec = 3)
```

```
## $Cluster_1
## Lden NO2 PM25 VegHautPrt Pct0_14 Pct_65 Pct_Img TxChom1564
## Q5 50.589 27.985 16.784 6.338 8.32 6.77 6.046 7.785
## Q10 51.585 29.228 17.273 7.78 10.331 8.408 6.902 8.577
## Q25 53.776 30.867 18.333 10.334 12.47 10.736 8.825 11.114
## Q50 56.383 35.037 18.952 13.855 15.023 13.891 11.98 13.119
## Q75 59.117 38.881 19.582 18.091 17.456 17.495 15.248 15.495
## Q90 62.474 41.146 20.004 26.082 19.673 20.509 18.629 18.429
## Q95 64.175 44.365 20.206 31.659 21.192 22.987 20.193 20.58
## Mean 56.627 35.058 18.771 15.294 14.91 14.085 12.449 13.567
## Std 4.37 5.766 1.208 7.716 4.416 5.184 5.642 5.298
## Pct_brevet NivVieMed
## Q5 7.221 17419.89
## Q10 8.474 18717.44
## Q25 10.738 20126.27
## Q50 13.995 22572.24
## Q75 19.769 24717.11
## Q90 25.535 28046.09
## Q95 30.333 31046.21
## Mean 15.935 22869.05
## Std 8.182 4004.629
##
## $Cluster_2
## Lden NO2 PM25 VegHautPrt Pct0_14 Pct_65 Pct_Img TxChom1564
## Q5 44.821 14.772 12.649 12.107 12.776 12.691 3.787 6.723
## Q10 45.822 15.692 12.92 14.655 14.297 13.379 4.259 6.999
## Q25 49.654 18.685 13.741 21.742 16.697 16.145 5.738 7.934
## Q50 52.309 22.074 14.555 28.635 18.605 19.207 7.466 9.603
## Q75 54.808 25.971 15.695 36.48 20.816 22.612 10.08 12
## Q90 58.116 29 16.534 42.041 22.335 27.455 14.174 15.045
## Q95 60.163 32.396 17.279 44.911 22.837 29.579 17.32 17.7
## Mean 52.16 22.436 14.748 28.341 18.402 19.835 8.51 10.619
## Std 4.802 5.595 1.514 10.238 3.598 5.877 4.842 5.629
## Pct_brevet NivVieMed
## Q5 10.061 19832.54
## Q10 11.654 20783.62
## Q25 14.719 22402
## Q50 18.432 24529.81
## Q75 23.231 28399.48
## Q90 29.32 31031.92
## Q95 32.451 34184.69
## Mean 19.709 25380.5
## Std 8.288 4369.744
##
## $Cluster_3
## Lden NO2 PM25 VegHautPrt Pct0_14 Pct_65 Pct_Img TxChom1564
## Q5 49.798 17.78 13.828 6.066 12.75 9.204 5.735 7.014
## Q10 51.342 19.266 14.049 7.239 14.851 11.218 7.021 8.042
## Q25 53.119 21.734 15.145 10.739 17.284 13.915 9.308 10.341
## Q50 55.599 25.39 15.653 14.948 19.483 17.316 12.747 13.1
## Q75 58.777 30.382 17.072 20.608 21.744 20.907 18.397 16.651
## Q90 62.58 35.164 18.582 29.932 24.515 23.814 22.821 20.836
## Q95 63.937 38.523 18.883 34.84 26.302 25.504 28.424 24.37
## Mean 56.114 26.145 16.019 16.576 19.358 17.264 14.365 14.031
## Std 4.53 6.411 1.614 8.737 4.516 5.483 7.41 7.063
## Pct_brevet NivVieMed
## Q5 13.488 15635.58
## Q10 17.02 16542.05
## Q25 20.781 18807.47
## Q50 25.835 20432.88
## Q75 32.021 23025.93
## Q90 38.279 25212.17
## Q95 43.764 26277.2
## Mean 27.216 20877.06
## Std 10.51 3534.834
##
## $Cluster_4
## Lden NO2 PM25 VegHautPrt Pct0_14 Pct_65 Pct_Img TxChom1564
## Q5 51.576 20.041 14.136 6.455 16.48 7.549 10.481 11.133
## Q10 52.383 21.383 14.56 7.648 18.483 8.95 16.123 13.787
## Q25 55.067 23.266 15.907 10.844 21.452 11.048 21.007 17.872
## Q50 57.243 26.534 16.538 14.218 24.398 13.815 27.056 23.003
## Q75 59.53 31.85 17.733 18.387 27.6 17.125 32.865 31.83
## Q90 63.441 37.186 18.757 24.835 30.974 20.535 38.25 34.394
## Q95 64.732 39.63 19.068 28.607 32.41 23.228 41.005 37.971
## Mean 57.515 27.99 16.693 15.476 24.403 14.223 26.758 24.189
## Std 4.253 6.622 1.518 7.25 6.315 5.024 9.457 9.479
## Pct_brevet NivVieMed
## Q5 21.535 12404.91
## Q10 26.811 12891.14
## Q25 32.702 13852.68
## Q50 38.41 15648.12
## Q75 45.03 18117.52
## Q90 49.569 19545.97
## Q95 53.699 21898.08
## Mean 38.608 16111.03
## Std 11.927 3117.687
```

```
# equivalent to :
# summary(SGFCM, LyonIris@data[AnalysisFields])
```

```
spiderPlots(LyonIris@data[AnalysisFields], SGFCM$Belongings,
chartcolors = c("darkorange3","grey4","darkgreen","royalblue"))
violinPlots(LyonIris@data[AnalysisFields], SGFCM$Groups)
```

All the clustering algorithms of this package depends on a initial state that might influence the results. A good practice is to check the results to ensure that the groups obtained are stable if we replicate the analysis. The function `boost_group_validation`

proposes an approach based on bootstrapping.

```
<- boot_group_validation(SGFCM, nsim = 1000, maxiter = 1000,
bootvalues tol = 0.0001, verbose = FALSE)
```

We can now check if the groups are stable across simulations.

`<- reshape2::melt(bootvalues$group_consistency) melted_df `

`## No id variables; using all as measure variables`

```
$variable <- as.factor(melted_df$variable)
melted_df
ggplot() +
geom_histogram(mapping = aes(x = value), data = melted_df, bins = 30) +
labs(title = "stability of clusters", subtitle = "for 1000 iterations",
x = "Jaccard index") +
facet_wrap(vars(variable), ncol=2)
```

The Jaccard index is calculated at each replication and for each cluster in comparison with the original result. A greater value means a more stable cluster. Values above 0.75 indicate a stable cluster. Values between 0.5 and 0.75 indicate a cluster with some uncertainty. Under 0.5, the cluster tends to dissolve during the iterations. In the presented case, the group 3 is more worisome with many Jacard values under 0.5.

We can check the values of the centers of the group 3 across the 1000 iterations.

```
<- bootvalues$group_centers[["group3"]]
df_gp3
<- reshape2::melt(df_gp3) melted_df
```

`## No id variables; using all as measure variables`

```
$variable <- as.factor(melted_df$variable)
melted_df
ggplot() +
geom_histogram(mapping = aes(x = value), data = melted_df, bins = 30) +
labs(title = "stability of group 3 centers", subtitle = "for 1000 iterations") +
xlim(-3,3)+
facet_wrap(vars(variable), ncol=3)
```

It appears that the center values for NO_{2} and PM_{2.5} are bimodal, that might explain why this group tends to dissolve. This could mean that group three is the combination of two very similar sub-groups.

It has been proposed to modify the SFCM algorithm to include “non-local spatial information” instead of the information provided by close observations (Zhao, Liu, and Fan 2015). Basically, this implies a modified spatial weighting scheme combining neighbouring and semantic distance. This way, more similar neighbours get more weights in the calculation of the lagged dataset which can be desirable in dataset characterized by a lot of noise. When working with a classical vector dataset, one can use the function `adjustSpatialWeights`

and create a row-standardized weight matrix using the inverse of the euclidean distance between observation and their neighbours as weights.

```
# create the modified weight matrix
<- adjustSpatialWeights(Data, WMat$neighbours, style = "W")
WMat_adj
# calculate the modified version of FCM with non-local information
<- SFCMeans(Data, WMat_adj, k = 4, m = 1.5, alpha = 0.7,
nl_SFCM tol = 0.0001, standardize = FALSE,
verbose = FALSE, seed = 456)
```

When working with rasters, the same method can be apllied by giving `"nl"`

as value for the parameter `lag_method`

.

That’s all, folks ! Following are the enhancements for the next version

- introduce other methods of spatial c-means
- open some other parameters to the user (such as the function defining the convergence criterion)
- work on documentation

Cai, Weiling, Songcan Chen, and Daoqiang Zhang. 2007. “Fast and Robust Fuzzy c-Means Clustering Algorithms Incorporating Local Information for Image Segmentation.” *Pattern Recognition* 40 (3): 825–38.

Gelb, Jérémy, and Philippe Apparicio. 2021. “Apport de La Classification Floue c-Means Spatiale En géographie: Essai de Taxinomie Socio-résidentielle Et Environnementale à Lyon.” *Cybergeo: European Journal of Geography*.

Zhao, Feng, Licheng Jiao, and Hanqiang Liu. 2013. “Kernel Generalized Fuzzy c-Means Clustering with Spatial Information for Image Segmentation.” *Digital Signal Processing* 23 (1): 184–99.

Zhao, Feng, Hanqiang Liu, and Jiulun Fan. 2015. “A Multiobjective Spatial Fuzzy Clustering Algorithm for Image Segmentation.” *Applied Soft Computing* 30: 48–57.