library(tidydfidx)
library(dplyr)February 6, 2026
In some situations, series from a data frame have a natural two-dimensional (tabular) representation because each observation can be uniquely characterized by a combination of two indexes. Two major cases of these situations in applied econometrics are:
The idea of dfidx is to keep in the same object the data and the information about its structure. A dfidx object is a data frame with an idx column, which is a data frame that contains the series that define the indexes.
From version 0.1-2, dfidx doesn’t depend anymore on some of tidyverse packages. If you want to use dfidx along with tidyverse in order to use tibbles instead of ordinary data frames and dplyr’s verbs, you should use the new tidydfidx package instead of dfidx.
This vignette supersede the preceding vignette of the dfidx package by showing the advantages of creating dfidx objects from a tibble and not from an ordinary data frame. It also introduces a new vector interface to define the indexes.
dfidx functionThe dfidx package is loaded using:
library(tidydfidx)
library(dplyr)We also attach the dplyr package (Wickham et al. 2023), which exports functions from the tibble (Müller and Wickham 2023) and the margrittr (Bache and Wickham 2022) packages because we’ll use throughout this vignette tibbles and not ordinary data frames and we’ll show how dplyr’s verbs can be used with dfidx objects thanks to appropriate methods.
To illustrate the features of dfidx, we’ll use the munnell data set (Munnell 1990) that is used in Baltagi (2013)’ and is part of the plm package as Produc. It contains several economic series for American states from 1970 to 1986. We’ve added to the initial data set a president series which indicates the name of the American president in power for the given year. We convert it to a tibble:
data("munnell", package = "dfidx")
munnell <- as_tibble(munnell)The two indexes are state and year and both are nested in another variable: state in region and year in president. A dfidx object is created with the dfidx function: the first argument should be a data frame (or a tibble) and the second argument idx is used to indicate the indexes. As, in the munnell data set, the first two columns contain the two indexes, the idx argument is not mandatory and a dfidx can be obtained from the munnell tibble simply by using:
munnell %>% dfidx# A tibble: 816 × 11
# Index: 48 (state) x 17 (year)
# Balanced: yes
region president publiccap highway water utilities privatecap gsp
<chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 East-S… Nixon 15033. 7326. 1656. 6051. 35794. 28418
2 East-S… Nixon 15502. 7526. 1721. 6255. 37300. 29375
3 East-S… Nixon 15972. 7765. 1765. 6442. 38670. 31303
# ℹ 813 more rows
# ℹ 3 more variables: labor <dbl>, unemp <dbl>, idx <idx>
The resulting object is of class dfidx and is a tibble with an idx column, which is a tibble containing the two indexes. Note that the two indexes are now longer standalone series in the resulting tibble, because the default value of the drop.index argument is TRUE. The header of the tibble indicates the names and the cardinal of the two indexes. It also indicated whether the data set is balanced ie, in this panel data context, whether all the states are observed for the same set of years (which is the case for the munnell data set). The idx column can be retrieved using the idx function:
munnell %>% dfidx %>% idx# A tibble: 816 × 2
state year
<chr> <fct>
1 Alabama 1970
2 Alabama 1971
3 Alabama 1972
# ℹ 813 more rows
If the first two columns don’t contain the indexes, the idx argument should be set. If the observations are ordered first by the first index and then by the second one and if the data set is balanced, idx can be an integer, the number of distinct values of the first index:
munnell %>% dfidx(48L)# A tibble: 816 × 13
# Index: 48 (id1) x 17 (id2)
# Balanced: yes
state year region president publiccap highway water utilities
<chr> <int> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
1 Alabama 1970 East-Sout… Nixon 15033. 7326. 1656. 6051.
2 Alabama 1971 East-Sout… Nixon 15502. 7526. 1721. 6255.
3 Alabama 1972 East-Sout… Nixon 15972. 7765. 1765. 6442.
# ℹ 813 more rows
# ℹ 5 more variables: privatecap <dbl>, gsp <int>, labor <dbl>,
# unemp <dbl>, idx <idx>
Then the two indexes are created with the default names id1 and id2. More relevant names can be indicated using the idnames argument and the values of the second index can be indicated, using the levels argument.
munnell %>% dfidx(48L, idnames = c("state", "year"), levels = 1970:1986)# A tibble: 816 × 11
# Index: 48 (state) x 17 (year)
# Balanced: yes
region president publiccap highway water utilities privatecap gsp
<chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 East-S… Nixon 15033. 7326. 1656. 6051. 35794. 28418
2 East-S… Nixon 15502. 7526. 1721. 6255. 37300. 29375
3 East-S… Nixon 15972. 7765. 1765. 6442. 38670. 31303
# ℹ 813 more rows
# ℹ 3 more variables: labor <dbl>, unemp <dbl>, idx <idx>
The idx argument can also be a character of length one or two. In the first case, only the first index is indicated:
munnell %>% dfidx("state", idnames = c(NA, "date"), levels = 1970:1986)# A tibble: 816 × 12
# Index: 48 (state) x 17 (date)
# Balanced: yes
year region president publiccap highway water utilities privatecap
<int> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
1 1970 East-S… Nixon 15033. 7326. 1656. 6051. 35794.
2 1971 East-S… Nixon 15502. 7526. 1721. 6255. 37300.
3 1972 East-S… Nixon 15972. 7765. 1765. 6442. 38670.
# ℹ 813 more rows
# ℹ 4 more variables: gsp <int>, labor <dbl>, unemp <dbl>, idx <idx>
Note that we’ve only provided a name for the second index, the NA in the first position of the idnames argument meaning that we want to keep the original name for the first index. Finally, if the idx argument is a character of length 2, it should contain the name of the two indexes.
munnell %>% dfidx(c("state", "year"))# A tibble: 816 × 11
# Index: 48 (state) x 17 (year)
# Balanced: yes
region president publiccap highway water utilities privatecap gsp
<chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 East-S… Nixon 15033. 7326. 1656. 6051. 35794. 28418
2 East-S… Nixon 15502. 7526. 1721. 6255. 37300. 29375
3 East-S… Nixon 15972. 7765. 1765. 6442. 38670. 31303
# ℹ 813 more rows
# ℹ 3 more variables: labor <dbl>, unemp <dbl>, idx <idx>
dfidxOne or both of the indexes may be nested in another series. In this case, the idx argument is still a character of length two, but the nesting series is indicated as the name of the corresponding index:
mn <- munnell %>% dfidx(c(region = "state", "year"))
mn <- munnell %>% dfidx(c(region = "state", president = "year"))
mn# A tibble: 816 × 9
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region), year (president)
publiccap highway water utilities privatecap gsp labor unemp
<dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl>
1 62201. 26836. 7670. 27696. 146286. 168627 4656. 6.2
2 63096. 27300. 8005. 27791. 150855. 173767 4789. 6
3 63643. 27247. 8491. 27904. 156752. 173817 4880 5.5
# ℹ 813 more rows
# ℹ 1 more variable: idx <idx>
The idx column is now a tibble containing the two indexes and the nesting variables.
mn %>% idx# A tibble: 816 × 4
state region year president
<chr> <chr> <fct> <fct>
1 Illinois East-North Central 1977 Carter
2 Illinois East-North Central 1978 Carter
3 Illinois East-North Central 1979 Carter
# ℹ 813 more rows
idx columnBy default, the column that contains the indexes is called idx and is the first column of the returned data frame. The position and the name of this column can be set using the position and name arguments:
dfidx(munnell, idx = c(region = "state", president = "year"),
name = "index", position = 4)# A tibble: 816 × 9
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region), year (president)
publiccap highway water index utilities privatecap gsp labor
<dbl> <dbl> <dbl> <idx> <dbl> <dbl> <int> <dbl>
1 62201. 26836. 7670. Illi…:1977 27696. 146286. 168627 4656.
2 63096. 27300. 8005. Illi…:1978 27791. 150855. 173767 4789.
3 63643. 27247. 8491. Illi…:1979 27904. 156752. 173817 4880
# ℹ 813 more rows
# ℹ 1 more variable: unemp <dbl>
dfidx can deal with data frames in wide format, i.e., for which each series for a given value of the second index is a column of the data frame. This is the case of the munnell_wide tibble that contains two series of the original data set (gsp and unemp).
data("munnell_wide", package = "dfidx")
munnell_wide <- munnell_wide %>% as_tibbleEach line is now an American state and, apart the indexes, there are now 34 series with names obtained by the concatenation of the name of the series and the year (for example gsp_1988). In this case a supplementary argument called varying should be provided. It is a vector of integers indicating the position of the columns that should be merged in the resulting long formatted data frame. The stats::reshape function is then used and the sep argument can be also provided to indicate the separating character in the names of the series (the default value being ".").
munnell_wide %>% dfidx(varying = 3:36, sep = "_")# A tibble: 816 × 5
# Index: 48 (id1) x 17 (id2)
# Balanced: yes
state region gsp unemp idx
<chr> <chr> <int> <dbl> <idx>
1 Alabama East-South Central 28418 4.7 1:1970
2 Alabama East-South Central 29375 5.2 1:1971
3 Alabama East-South Central 31303 4.7 1:1972
# ℹ 813 more rows
Better results can be obtained using the idx and idnames previously described:
munnell_wide %>% dfidx(idx = c(region = "state"), varying = 3:36,
sep = "_", idnames = c(NA, "year"))# A tibble: 816 × 3
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region)
gsp unemp idx
<int> <dbl> <idx>
1 145792 4.1 Illinois:1970
2 148503 5.1 Illinois:1971
3 154413 5.1 Illinois:1972
# ℹ 813 more rows
The name (and the position) of the idx column can be obtained as a named integer (the integer being the position of the column and the name its name) using the idx_name function:
idx_name(mn)
## idx
## 9To get the name of one of the indexes, the second argument, n, is set either to 1 or 2 to get the first or the second index, ignoring the nesting variables:
idx_name(mn, 2)
## [1] "year"
idx_name(idx(mn), 2)
## [1] "year"Not that idx_name can be in this case applied to a dfidx or to a idx object. To get a nesting variable, the third argument, called m, is set to 2:
idx_name(mn, 1, 1)
## [1] "state"
idx_name(mn, 1, 2)
## [1] "region"To extract one or all the indexes, the idx function is used. This function has already been encountered when one wants to extract the idx column of a dfidx object. The same n and m arguments as for the idx_name function can be used in order to extract a specific series. For example, to extract the region index, which nests the state index:
id_index1 <- idx(mn, n = 1, m = 2)
id_index2 <- idx(idx(mn), n = 1, m = 2)
head(id_index1)
## [1] "East-North Central" "East-North Central" "East-North Central"
## [4] "East-North Central" "East-North Central" "East-North Central"
identical(id_index1, id_index2)
## [1] TRUESubsets of data frames are obtained using the [ and the [[ operators. The former returns most of the time a data frame as the second one always returns a series.
Consider first the use of [. If one argument is provided, it indicates the columns that should be selected. The result is always a data frame, even if a single column is selected. If two arguments are provided, the first one indicates the subset of lines and the second one the subset of columns that should be returned. If only one column is selected, the result depends on the value of the drop argument. If TRUE, a series is returned and if FALSE, a one series data frame is returned. An important difference between tibbles and ordinary data frames is that the default value of drop is FALSE for the former and TRUE for the later. Therefore, with tibbles, the use of [ will always by default return a data frame.
A specific dfidx method is provided for one reason: the column that contains the indexes should be “sticky” (we borrow this idea from the sf package1), which means that it should be always returned while using the extractor operator, even if it is not explicitly selected.
mn[mn$unemp > 10, ]# A tibble: 46 × 9
# Index: 19 (state) x 8 (year)
# Balanced: no
# Nesting: state (region), year (president)
publiccap highway water utilities privatecap gsp labor unemp
<dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl>
1 65064. 27568. 10218 27278. 154806. 159778 4593. 11
2 64752. 27483 10436. 26833. 157096. 160856 4531. 11
3 25109. 10619. 3297. 11193. 82361. 64042 2028 12
# ℹ 43 more rows
# ℹ 1 more variable: idx <idx>
mn[mn$unemp > 10, c("highway", "utilities")]# A tibble: 46 × 3
# Index: 19 (state) x 8 (year)
# Balanced: no
# Nesting: state (region), year (president)
highway utilities idx
<dbl> <dbl> <idx>
1 27568. 27278. Illinois:1982
2 27483 26833. Illinois:1983
3 10619. 11193. Indiana:1982
# ℹ 43 more rows
mn[mn$unemp > 10, "highway"]# A tibble: 46 × 2
# Index: 19 (state) x 8 (year)
# Balanced: no
# Nesting: state (region), year (president)
highway idx
<dbl> <idx>
1 27568. Illinois:1982
2 27483 Illinois:1983
3 10619. Indiana:1982
# ℹ 43 more rows
All the previous commands extract the observations where the unemployment rate is greater than 10% and, in the first case all the series, in the second case two of them and in the third case only one series.
A series can be extracted using any of the following commands:
mn1 <- mn[, "highway", drop = TRUE]
mn2 <- mn[["highway"]]
mn3 <- mn$highway
c(identical(mn1, mn2), identical(mn1, mn3))
## [1] TRUE TRUEThe result is a xseries which inherits the idx column from the data frame it has been extracted from as an attribute :
mn1 %>% print(n = 3)# Index:
48 (state) x 17 (year)
[1] 26835.52 27300.22 27247.22
class(mn1)[1] "xseries" "numeric"
idx(mn1) %>% print(n = 3)# A tibble: 816 × 4
state region year president
<chr> <chr> <fct> <fct>
1 Illinois East-North Central 1977 Carter
2 Illinois East-North Central 1978 Carter
3 Illinois East-North Central 1979 Carter
# ℹ 813 more rows
Note that, except when dfidx hasn’t been used with drop.index = FALSE, a series which defines the indexes is dropped from the data frame (but is one of the column of the idx column of the data frame). It can be therefore retrieved using:
mn$idx$president %>% head[1] Carter Carter Carter Carter Ford Ford
Levels: Carter Ford Nixon Reagan
or
idx(mn)$president %>% head[1] Carter Carter Carter Carter Ford Ford
Levels: Carter Ford Nixon Reagan
or more simply by applying the $ operator as if the series were a stand-alone series in the data frame :
mn$president %>% print(n = 3)# Index:
48 (state) x 17 (year)
[1] Carter Carter Carter
Levels: Carter Ford Nixon Reagan
In this last case, the resulting series is a xseries, ie it inherits the index data frame as an attribute.
While creating the dfidx, a pkg argument can be indicated, so that the resulting dfidx object and its series are respectively of class c("dfidx_pkg", "dfidx") and c("xseries_pkg", "xseries") which enables the definition of special methods for dfidx and xseries objects. For example, consider the hypothetical pnl package for panel data:
mn <- dfidx(munnell, idx = c(region = "state", president = "year"),
pkg = "pnl")
mn1 <- mn$gsp
class(mn)
## [1] "dfidx_pnl" "tbl_dfidx" "dfidx" "tbl_df" "tbl"
## [6] "data.frame"
class(mn1)
## [1] "xseries_pnl" "xseries" "integer"For example, we want to define a lag method for xseries_pnl objects. While lagging there should be a NA not only on the first position of the resulting vector like for time-series, but each time we encounter a new individual. A minimal lag method could therefore be written as:
lag.xseries_pnl <- function(x, ...){
.idx <- idx(x)
class <- class(x)
x <- unclass(x)
id <- .idx[[1]]
lgt <- length(id)
lagid <- c("", id[- lgt])
sameid <- lagid == id
x <- c(NA, x[- lgt])
x[! sameid] <- NA
structure(x, class = class, idx = .idx)
}
lmn1 <- stats::lag(mn1)
lmn1 %>% print(n = 3)# Index:
48 (state) x 17 (year)
[1] NA 168627 173767
class(lmn1)[1] "xseries_pnl" "xseries" "integer"
rbind(mn1, lmn1)[, 1:20] [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
mn1 168627 173767 173817 165722 157366 163112 145792 148503 154413
lmn1 NA 168627 173767 173817 165722 157366 163112 145792 148503
[,10] [,11] [,12] [,13] [,14] [,15] [,16] [,17] [,18]
mn1 163125 161725 166029 159778 160856 173602 178493 183849 68832
lmn1 154413 163125 161725 166029 159778 160856 173602 178493 NA
[,19] [,20]
mn1 71717 72047
lmn1 68832 71717
Note the use of stats::lag instead of lag which ensures that the stats::lag function is used, even if the dplyr (or tidyverse) package is attached.
dfidx supports some of the verbs of dplyr, namely, for the current version:
select to select columns,filter to select some rows using logical conditions,arrange to sort the lines according to one or several variables,mutate and transmute for creating new series,slice to select some rows using their position.dplyr’s verbs don’t work with dfidx objects for two main reasons:
select is an exception), the returned object is a data.frame (or a tibble) and not a dfidx,transmute.Therefore, specific methods are provided for dplyr’s verb. The general strategy consists on:
dfidx object),as.data.frame method,dplyr’s verb,transmute or while selecting a subset of columns which doesn’t contain the index column),The following code illustrates the use of dplyr’s verbs applied to dfidx objects.
select(mn, highway, utilities)# A tibble: 816 × 3
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region), year (president)
highway utilities idx
<dbl> <dbl> <idx>
1 26836. 27696. Illinois:1977
2 27300. 27791. Illinois:1978
3 27247. 27904. Illinois:1979
# ℹ 813 more rows
arrange(mn, desc(unemp))# A tibble: 816 × 9
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region), year (president)
publiccap highway water utilities privatecap gsp labor unemp
<dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl>
1 11079. 7551. 756. 2772. 35933. 20822 582. 18
2 51956. 19881. 10759. 21316. 115911. 108627 3193. 16
3 11073. 7562. 809. 2702. 36068. 21615 597. 15
# ℹ 813 more rows
# ℹ 1 more variable: idx <idx>
mutate(mn, lgsp = log(gsp), lgsp2 = lgsp ^ 2)# A tibble: 816 × 11
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region), year (president)
publiccap highway water utilities privatecap gsp labor unemp
<dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl>
1 62201. 26836. 7670. 27696. 146286. 168627 4656. 6.2
2 63096. 27300. 8005. 27791. 150855. 173767 4789. 6
3 63643. 27247. 8491. 27904. 156752. 173817 4880 5.5
# ℹ 813 more rows
# ℹ 3 more variables: idx <idx>, lgsp <dbl>, lgsp2 <dbl>
transmute(mn, lgsp = log(gsp), lgsp2 = lgsp ^ 2)# A tibble: 816 × 3
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region), year (president)
lgsp lgsp2 idx
<dbl> <dbl> <idx>
1 12.0 145. Illinois:1977
2 12.1 146. Illinois:1978
3 12.1 146. Illinois:1979
# ℹ 813 more rows
filter(mn, unemp > 10, gsp > 150000)# A tibble: 2 × 9
# Index: 1 (state) x 2 (year)
# Balanced: yes
# Nesting: state (region), year (president)
publiccap highway water utilities privatecap gsp labor unemp
<dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl>
1 65064. 27568. 10218 27278. 154806. 159778 4593. 11
2 64752. 27483 10436. 26833. 157096. 160856 4531. 11
# ℹ 1 more variable: idx <idx>
slice(mn, 1:3)# A tibble: 3 × 9
# Index: 1 (state) x 3 (year)
# Balanced: yes
# Nesting: state (region), year (president)
publiccap highway water utilities privatecap gsp labor unemp
<dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl>
1 62201. 26836. 7670. 27696. 146286. 168627 4656. 6.2
2 63096. 27300. 8005. 27791. 150855. 173767 4789. 6
3 63643. 27247. 8491. 27904. 156752. 173817 4880 5.5
# ℹ 1 more variable: idx <idx>
mutate(mn, gsp = ifelse(gsp < 170000, 0, gsp))# A tibble: 816 × 9
# Index: 48 (state) x 17 (year)
# Balanced: yes
# Nesting: state (region), year (president)
publiccap highway water utilities privatecap gsp labor unemp
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 62201. 26836. 7670. 27696. 146286. 0 4656. 6.2
2 63096. 27300. 8005. 27791. 150855. 173767 4789. 6
3 63643. 27247. 8491. 27904. 156752. 173817 4880 5.5
# ℹ 813 more rows
# ℹ 1 more variable: idx <idx>
To extract a series, the pull function can be used:
mn %>% pull(utilities)# Index:
48 (state) x 17 (year)
[1] 27695.71 27791.28 27904.24
The two main steps in R in order to estimate a model are to use the model.frame function to construct a data frame, using a formula and a data frame and then to extract from it the matrix of covariates using the model.matrix function.
The default method of model.frame has as first two arguments formula and data. It returns a data frame with a terms attribute. Some other methods exist in the stats package, for example for lm and glm object with a first and main argument called formula. This is quite unusual and misleading as for most of the generic functions in R, the first argument is called either x or object.
Another noticeable method for model.frame is provided by the Formula package and, in this case, the first argument is a Formula object, which is an extended formula which can contain several parts on the left and/or on the right hand side of the formula.
We provide a model.frame method for dfidx objects, mainly because the idx column should be returned in the resulting data frame. This leads to an unusual order of the arguments, the data frame first and then the formula. The method then first extract (and subset if necessary the idx column), call the formula/Formula method and then add to the resulting data frame the idx column. The resulting data frame is a dfidx object.
mf_mn <- mn %>% model.frame(gsp ~ utilities + highway | unemp | labor,
subset = unemp > 10)
mf_mn# A tibble: 46 × 6
# Index: 19 (state) x 8 (year)
# Balanced: no
# Nesting: state (region), year (president)
gsp utilities highway unemp labor idx
<int> <dbl> <dbl> <dbl> <dbl> <idx>
1 159778 27278. 27568. 11 4593. Illinois:1982
2 160856 26833. 27483 11 4531. Illinois:1983
3 64042 11193. 10619. 12 2028 Indiana:1982
# ℹ 43 more rows
formula(mf_mn)gsp ~ utilities + highway + unemp + labor + (state + region +
year + president)
<environment: 0x64ec783829f8>
Note that the column that contains the indexes is at the end and not at the begining of the returned data frame. This is because the stats::model.response function, which is used to extract the response of a model and is not generic consider that the first column of the model frame is the response.
model.matrix is a generic function and for the default method, the first two arguments are a terms object and a data frame. In lm, the terms attribute is extracted from the model.frame internally constructed using the model.frame function. This means that, at least in this context, model.matrix doesn’t need a formula/term argument and a data.frame, but only a data frame returned by the model frame method, i.e., a data frame with a terms attribute.
We use this idea for the model.matrix method for dfidx object; the only required argument is a dfidx returned by the model.frame function. The formula is then extracted from the dfidx and the Formula or default method is then called. The result is a matrix of class dfidx_matrix, with a printing method that allows the use of the n argument:
mf_mn %>% model.matrix(rhs = 1) %>% print(n = 5)# [46 x 3]
(Intercept) utilities highway
1 1 27277.69 27568.50
2 1 26832.94 27483.00
3 1 11192.68 10618.71
4 1 11141.74 10558.11
5 1 21281.74 19996.38
mf_mn %>% model.matrix(rhs = 2:3) %>% print(n = 5)# [46 x 3]
(Intercept) unemp labor
1 1 11 4593.3
2 1 11 4530.6
3 1 12 2028.0
4 1 11 2029.5
5 1 12 3442.8
Pebesma and Bivand (2023) and Pebesma (2018).↩︎