We use this vignette to reproduce the real world example for local
outlier detection analysed and described in Puchhammer and Filzmoser
(2023). All functions are included in the package
`ssMRCD`

.

```
library(ssMRCD)
library(ggplot2)
```

The original data from GeoSphere Austria (2022) is pre-cleaned and
saved in the data frame object `weatherAUt2021`

. Additional
information can be found on the helping page.

` ? weatherAUT2021`

```
data("weatherAUT2021")
head(weatherAUT2021)
#> p s vv t rsum rel name lon lat alt
#> 1 941.35 154.5 2.15 6.15 49.5 77.5 AIGEN/ENNSTAL 14.13833 47.53278 641
#> 2 945.80 172.5 2.95 6.25 39.5 76.0 ALLENTSTEIG 15.36694 48.69083 599
#> 3 985.50 152.0 2.55 8.20 46.5 78.0 AMSTETTEN 14.89500 48.10889 266
#> 4 893.20 123.5 1.85 5.10 66.0 74.5 BAD GASTEIN 13.13333 47.11055 1092
#> 5 984.50 176.0 1.50 8.45 40.5 75.0 BAD GLEICHENBERG 15.90361 46.87222 269
#> 6 992.05 188.5 2.20 8.95 60.5 77.0 BAD RADKERSBURG 15.99333 46.69222 207
```

To apply the local outlier detection function
`local_outliers_ssMRCD`

we need to specify a structure to
calculate the ssMRCD estimator, meaning we need a **neighborhood
assignment** and a weight matrix specifying the relative
influence of the neighborhoods on each.

Since a prominent part of Austria is Alpine landscape small sized neighborhoods might be appropriate. Thus, we construct a grid of the longitude and latitude values and classify the observations. We summarize neighborhoods for a very small number of observations.

```
= c(9:16, 18)
cut_lon = c(46, 47, 47.5, 48, 49)
cut_lat = ssMRCD::N_structure_gridbased(weatherAUT2021$lon, weatherAUT2021$lat, cut_lon, cut_lat)
N table(N)
#> N
#> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
#> 6 1 1 12 1 1 9 5 9 3 1 8 6 7 8 12 6 5 9 5 7 7 17 7 9 21
== 2] = 1
N[N == 3] = 4
N[N == 5] = 4
N[N == 6] = 7
N[N == 11] = 15
N[N = as.numeric(as.factor(N)) N
```

The final neighborhood structure can be seen in the following plot, where the neighborhoods are differently colorized.

```
= ggplot() +
g1 geom_text(data = weatherAUT2021, aes(x = lon, y = lat, col = as.factor(N), label = N)) +
geom_hline(aes(yintercept = cut_lat), lty = "dashed", col = "gray") +
geom_vline(aes(xintercept = cut_lon), lty = "dashed", col = "gray") +
labs(x = "Longitude", y = "Latitude", title = "Austrian Weather Stations: Neighborhood Structure") +
coord_fixed(1.4) +
theme_classic() +
theme(legend.position = "none")
g1
```

A natural choice for the **weight matrix** when using a
grid-based neighborhood structure is the geographical inverse-distance
weighting matrix. It is the default value for the
`local_outliers_ssMRCD`

function but can explicitly be
calculated using the function `geo_weights`

. This function
returns also the center of each neighborhood. E.g. for neighborhood 4 we
have the following weights, corresponding to the transparancy of the
arrows.

```
= geo_weights(coordinates = weatherAUT2021[, c("lon", "lat")],
GW N_assignments = N)
$W[4, ]
GW#> 1 2 3 4 5 6 7
#> 0.02554129 0.03265041 0.06004253 0.00000000 0.19041215 0.09170249 0.08164324
#> 8 9 10 11 12 13 14
#> 0.08849001 0.06215254 0.04767836 0.04370851 0.04285327 0.03754845 0.03340820
#> 15 16 17 18 19 20 21
#> 0.02547782 0.02752795 0.02472584 0.02375660 0.02127099 0.02047626 0.01893310
+
g1 labs(title = "Austrian Weather Stations: Weighting Matrix W") +
geom_segment(aes(x = GW$centersN[4, 1],
y = GW$centersN[4, 2],
xend = GW$centersN[-4, 1]-0.1,
yend = GW$centersN[-4, 2]-0.05,
alpha = GW$W[4, -4]),
arrow = arrow(length = unit(0.25, "cm")),
linewidth = 2,
col = "blue")+
geom_text(aes(x = GW$centersN[, 1], y = GW$centersN[, 2], label = 1:length(GW$centersN[, 2])))
```

Regarding the parameter settings we have many possibilities. When it
comes to the **number of neighbors** we want to compare
each observations with there are different possibilities. Either by
specifying a value for `k`

like `k = 10`

which
gives the number of observations to compare with or the value for
`dist`

(e.g.Â `dist = 1.5`

) as the radius of a
circle around the observation where all observations inside are compared
to the initial observation. A default value for `k`

that is
used among various local outlier detection methods is 10. However,
depending on the spatial structure of the data it makes sense to use
other values as well.

For the **spatial smoothing parameter
lambda** the restriction is it has to be in the
interval [0, 1). One can set the values according to some argumentation
based on the spatial distribution but also on underlying structures. A
value of

`0.5`

is the default value.A more sophisticated approach is to introduce outliers to the data
set and examine how well the parameter setting is able to find those.
The function `parameter_tuning`

can be used to analyse
different combinations of `k`

(or `dist`

) and
`lambda`

(see also help page).

` ? parameter_tuning`

Observations are switched and thus, this should introduce outliers that we know of. Nevertheless you should keep in mind that the observations are switched entirely randomly. If unlucky we might switch similar observations and thus, the measured performance of the detection method might suffer. However, since we want to compare the different parameter settings on the same contaminated data sets this should not affect the fine tuning itself.

Moreover, we implicitly assume that the original data set has no outliers. This is in general not the case. Thus, the false-negative rate (FNR) is not the true FNR regarding all outliers. Nevertheless, the parameter tuning is a way to see the effects of the parameter setting on the FNR and the false-positive rate (FPR) which is represented by the total number of found outliers.

```
set.seed(123)
parameter_tuning(data = weatherAUT2021[, 1:6],
coords = weatherAUT2021[, c("lon", "lat")],
N_assignments = N,
repetitions = 3,
k = c(5, 10, 15),
lambda = c(0, 0.25, 0.5, 0.75))$plot
```

It is apparent that some parameter settings are non-optimal compared
to others. In general the final choice of the best parameter setting
based on the output above depends on the overall goal. We choose
`lambda = 0.5`

and `k = 10`

. Although
`k = 10`

is not the optimal choice here, the difference in
performance is quite small and thus, we prefer to use the default
value.

If we are only interested in the covariance estimation we can use the
function `ssMRCD`

. A more convenient way if we are interested
in local outlier detection is to use the function
`local_outliers_ssMRCD`

, which already embeds the covariance
estimation.

` ? local_outliers_ssMRCD`

```
= local_outliers_ssMRCD(data = weatherAUT2021[, 1:6],
res coords = weatherAUT2021[, c("lon", "lat")],
N_assignments = N,
lambda = 0.5,
k = 10)
summary(res)
#> Local outliers: 10 of 183 observations (5.46%).
#> Used cut-off value: 14.20085.
#> Observations are compared to their k=10 neighbors.
```

The found outliers can be accessed by the list element
`outliers`

.

```
cat(weatherAUT2021$name[res$outliers], sep = ",\n")
#> OBERGURGL,
#> RUDOLFSHUETTE,
#> SCHOECKL,
#> PATSCHERKOFEL,
#> FEUERKOGEL,
#> LOIBL/TUNNEL,
#> SONNBLICK (TAWES),
#> RAX/SEILBAHNBERGSTATION,
#> GALZIG,
#> VILLACHER ALPE
```

For the covariance estimation there are two diagnostic plots
implemented for the S3-object `"ssMRCD"`

, which can be
specified by `type`

. For type `"convergence"`

the
convergence property for this specific data set is shown. For
`"ellipses"`

the estimated covariance matrices are shown in
the first two eigenvectors fo the overall data set. The
`colour_scheme`

parameter yields even more insight into the
covariances (see also `?plot.ssMRCD`

for more details).

```
= res$ssMRCD
covariance_res plot(covariance_res,
centersN = res$centersN,
manual_rescale = 0.5,
type = c("convergence", "ellipses"),
colour_scheme = "regularity",
legend = TRUE,
xlim = c(9, 19))
```

For completeness the global biplot based on the global covariance matrix is also plotted:

```
biplot(stats:: princomp(weatherAUT2021[, 1:6],
cor = TRUE,
covmat = robustbase::covMcd(weatherAUT2021[, 1:6])),
col = c("grey", "black"))
```

For the local outlier detection method there are several plots
regarding diagnostics available (see `?plot.locOuts`

). The
histogram shows all next distances and the cut-off value. The spatial
plot shows the outliers on the map, the 3D-plot as well, but with an
additional axis for the next-distance/next-distance divided by cut-off
value. The line plot shows the scaled values and the specified area in
the map. Orange-coloured lines are other outliers on the map, the
darkred colored line is the selected observation
(`focus`

).

` ? plot.locOuts`

`plot(res, type = "hist", pos = 4)`

`plot(res, type = "spatial", colour = "outScore", xlim = c(9.5, 19))`

```
# SCHOECKL
plot(res, type = "lines", focus = res$outliers[3])
```

`plot(res, type = "3D", colour = "outScore", theta = 0, phi = 10)`

If you want a fancy GIF you can also use the following code making
use of the package animation. You should have `ImageMagick`

installed.

```
library(animation)
# fineness of movement
= 90
n = rbind(cbind(rep(0, 2*n + n/2), c(seq(90, 0, length.out = n), seq(0, 90, length.out = n), seq(90, 30, length.out = n/2))),
param cbind(c(seq(0, 90, length.out = n), seq(90, 0, length.out = n)), rep(30, 2*n)),
cbind(rep(0, n/2), seq(30, 90, length.out = n/2)))
# use function saveGIF
saveGIF(interval = 0.2,
movie.name = "local_outliers_3D.gif",
expr = {
for (i in 1:dim(param)[1]) {
plot(outs, type = "3D", colour = "outScore", theta = param[i, 1], phi = param[i, 2])
}} )
```

GeoSphere Austria (2022): https://data.hub.zamg.ac.at.

Puchhammer P. and Filzmoser P. (2023): Spatially smoothed robust covariance estimation for local outlier detection. https://doi.org/10.48550/arXiv.2305.05371