# Introduction

This vignette of package `groupedHyperframe` (`CRAN`, Github, RPubs) documents the creation of `groupedHyperframe` object, the batch processes for a `groupedHyperframe`, and aggregations of various statistics over multi-level grouping structure.

## Prerequisite

Experimental (and maybe unstable) features are implemented extremely frequently on Github. Active developers should use the Github version; suggestions and bug reports are welcome!

    remotes::install_github('tingtingzhan/groupedHyperframe')

Stable releases to `CRAN` are typically updated every 2 to 3 months, or when the authors have an upcoming manuscript in the peer-reviewing process. Developers should not use the `CRAN` version!

    utils::install.packages('groupedHyperframe') # Developers, do NOT use!!

Package `groupedHyperframe` may require the development versions of the `spatstat` family.

    remotes::install_github('spatstat/spatstat', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.data', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.explore', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.geom', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.linnet', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.model', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.random', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.sparse', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.univar', upgrade = 'always')
    remotes::install_github('spatstat/spatstat.utils', upgrade = 'always')

### Dependencies

Package `groupedHyperframe` `Imports` packages

- `cli` [Csárdi (2025), version 3.6.5], for attractive command line interfaces (CLIs)
- `matrixStats` (Bengtsson 2025, version 1.5.0, key dependency), for matrix arithmetic
- `parallel` (shipped with vanilla `R`, version 4.5.1), for parallel computing
- `pracma` (Borchers 2023, version 2.4.4, key dependency), for (cumulative) trapezoidal integration
- `spatstat.explore` (version 3.4.3.7) and `spatstat.geom` (version 3.4.1.14), Baddeley, Rubak, and Turner (2015), key dependency, for spatial statistics
- `SpatialPack` (Vallejos, Osorio, and Bevilacqua 2020, version 0.4.1, key dependency), for Tjøstheim’s coefficient of spatial association

## Getting Started

Examples in this vignette require that the `search` path has

    library(groupedHyperframe)
    library(survival) # to help hyperframe understand Surv object

## Terms and Abbreviations

Term / Abbreviation | Description
--- | ---
`|>` | Forward pipe operator introduced in `R` 4.1.0
`.Machine` | Numerical characteristics of the machine `R` is running on, e.g., 32-bit integers and IEC 60559 floating-point (double precision) arithmetic
`attr`, `attributes` | Attributes
`CRAN`, `R` | The Comprehensive R Archive Network
`cor` | Correlation matrix
`cor.spatial` | Tjøstheim’s nonparametric correlation coefficient, from package `SpatialPack` (Vallejos, Osorio, and Bevilacqua 2020)
`cov`, `cov2cor` | Variance-covariance matrix, and conversion to correlation matrix
`data.frame` | Data frame
`diag` | Matrix diagonals
`dist` | Distance matrix; to take advantage of `stats:::as.matrix.dist`
`file.size` | File size in bytes
`formula` | Formula
`fv`, `fv.object`, `plot.fv` | (Plot of) function value table
`groupedData`, `~ g1/.../gm` | Grouped data frame; nested grouping structure, from package `nlme` (Pinheiro, Bates, and R Core Team 2025)
`groupedHyperframe` | Grouped hyper data frame
`hypercolumns`, `hyperframe` | (Hyper columns of) hyper data frame, from package `spatstat.geom` (Baddeley and Turner 2005)
`inherits` | Class inheritance
`kerndens` | Kernel density, `stats::density.default()$y`
`Inf` | Positive infinity \infty
`kmeans` | k-means clustering (Hartigan and Wong 1979)
`list`, `listof` | Lists of objects
`markformat` | Storage mode of `marks`
`marks`, `marked` | Marks of a point pattern
`merge` | Merge two `data.frame`s
`mc.cores` | Number of CPU cores to use for parallel computing
`message` | Diagnostic message printed in `R` console
`multitype` | Multitype spatial object
`NaN` | Not-a-Number
`object.size` | Memory allocation
`pmean`, `pmedian` | Parallel, or point-wise, mean and median, `groupedHyperframe::pmean`; `groupedHyperframe::pmedian`
`pmax`, `pmin` | Parallel, or point-wise, maxima and minima
`ppp`, `ppp.object` | (Marked) point pattern
`quantile` | Quantile
`save`, `saveRDS`, `xz` | Save with `xz` compression
`S3`, `generic`, `methods` | `S3` object oriented system, `UseMethod`; `getS3method`; https://adv-r.hadley.nz/s3.html
`S4`, `generic`, `methods` | `S4` object oriented system, `isS4`; `setClass`; `setMethod`; `getMethod`; https://adv-r.hadley.nz/s4.html
`sd` | Standard deviation
`search` | Search path
`Surv` | Survival, i.e., time-to-event, object
`trapz`, `cumtrapz` | (Cumulative) trapezoidal integration, from package `pracma` (Borchers 2023)
`vector` | Vector

## Acknowledgement

This work is supported by National Institutes of Health, U.S. Department of Health and Human Services grants

- R01CA222847 (I. Chervoneva, T. Zhan, and H. Rui)
- R01CA253977 (H. Rui and I. Chervoneva).

# Grouped Hyper Data Frame

We introduce a new `S3` class `groupedHyperframe` for grouped hyper data frame, which `inherits` from the hyper data frame `hyperframe` class from package `spatstat.geom` (Baddeley, Rubak, and Turner 2015; Baddeley and Turner 2005). A `hyperframe` contains columns either as `vector`s like in a `data.frame`, or as `list`s of objects of the same class, a.k.a, the `hypercolumns`. This data structure is particularly useful in spatial analysis, e.g., with medical images, where the spatial information in each image would be represented by one element in a `hypercolumn`. The derived class `groupedHyperframe` has additional `attributes`

- `attr(., 'group')`, a `formula` of the (nested) grouping structure, e.g., `~patient/image` when each patient has one or more images

The grammar of the nested grouping structure g_1/.../g_m (`~g1/.../gm`) follows that of the parameter `random` of functions `nlme::lme()` and `nlme::nlme()`. In fact, the `'grouped'` extension of a `hyperframe` is inspired by the `nlme::groupedData` class which inherits from `data.frame` (Pinheiro, Bates, and R Core Team 2025).

In this section, we introduce several `S3` method dispatches of the `S3` generic `as.groupedHyperframe()` to convert various classes into a `groupedHyperframe`. We also introduce aggregation functions `aggregate_*()` of the hypercolumns in a `groupedHyperframe`, at either one of the nested grouping levels g_1,\cdots,g_{m-1}. Aggregation at the lowest grouping level g_m is ignored, i.e., no aggregation to be performed. Available aggregation methods are the parallel minima `base::pmin()`, parallel maxima `base::pmax()`, parallel means `pmean()` (default) and parallel medians `pmedian()`.

## From `data.frame`

User may convert a `data.frame` with substantial amount of duplicated information into a `groupedHyperframe` using the `S3` method dispatch `as.groupedHyperframe.data.frame()`. This function

1. inspects the input `data.frame` by the user-specified (nested) `group`ing structure;
2. identifies the column(s) with non-identical elements within the lowest group, and converts them into hypercolumn(s);
3. returns a `groupedHyperframe` with the user-specified (nested) grouping structure.

In the following example, consider a toy data set `wrobel_lung0` with non-identical column `hladr` in the lowest group `image_id` of the nested grouping structure `~patient_id/image_id`.

    wrobel_lung0 = wrobel_lung |>
      within.data.frame(expr = {
        x = y = NULL
        dapi = phenotype = tissue = NULL
      })

    wrobel_lung0 |> head()
    #>            image_id    patient_id gender stage_numeric pack_years
    #> 1 [40864,18015].im3 #01 0-889-121      F             1         60
    #> 2 [40864,18015].im3 #01 0-889-121      F             1         60
    #> 3 [40864,18015].im3 #01 0-889-121      F             1         60
    #> 4 [40864,18015].im3 #01 0-889-121      F             1         60
    #> 5 [40864,18015].im3 #01 0-889-121      F             1         60
    #> 6 [40864,18015].im3 #01 0-889-121      F             1         60
    #>   adjuvant_therapy hladr    OS mhcII stage age
    #> 1               No 0.115 3488+   low    IA  85
    #> 2               No 0.239 3488+   low    IA  85
    #> 3               No 0.268 3488+   low    IA  85
    #> 4               No 0.245 3488+   low    IA  85
    #> 5               No 0.127 3488+   low    IA  85
    #> 6               No 0.136 3488+   low    IA  85

By converting `wrobel_lung0` into a `groupedHyperframe`, the numeric `hladr` from each `~patient_id/image_id` are converted into elements of the numeric-hypercolumn `hladr` in the returned `wrobel_lung0g`. Each row of a `groupedHyperframe` represents the lowest group of the nested grouping structure. The `R` console output (`S3` method dispatch `print.groupedHyperframe()`) highlights the nested grouping structure, number of clusters at each grouping level, as well as the first 10 (or less) rows of the `groupedHyperframe`.

    (wrobel_lung0g = wrobel_lung0 |> as.groupedHyperframe(group = ~ patient_id/image_id))
    #> Grouped Hyperframe: ~patient_id/image_id
    #>
    #> 15 image_id nested in
    #> 3 patient_id
    #>
    #> Preview of first 10 (or less) rows:
    #>
    #>        hladr          image_id    patient_id gender stage_numeric pack_years
    #> 1  (numeric) [40864,18015].im3 #01 0-889-121      F             1         60
    #> 2  (numeric) [42689,19214].im3 #01 0-889-121      F             1         60
    #> 3  (numeric) [42806,16718].im3 #01 0-889-121      F             1         60
    #> 4  (numeric) [44311,17766].im3 #01 0-889-121      F             1         60
    #> 5  (numeric) [45366,16647].im3 #01 0-889-121      F             1         60
    #> 6  (numeric) [56576,16907].im3 #02 1-037-393      M             1         30
    #> 7  (numeric) [56583,15235].im3 #02 1-037-393      M             1         30
    #> 8  (numeric) [57130,16082].im3 #02 1-037-393      M             1         30
    #> 9  (numeric) [57396,17896].im3 #02 1-037-393      M             1         30
    #> 10 (numeric) [57403,16934].im3 #02 1-037-393      M             1         30
    #>    adjuvant_therapy    OS mhcII stage age
    #> 1                No 3488+   low    IA  85
    #> 2                No 3488+   low    IA  85
    #> 3                No 3488+   low    IA  85
    #> 4                No 3488+   low    IA  85
    #> 5                No 3488+   low    IA  85
    #> 6                No  1605   low    IA  66
    #> 7                No  1605   low    IA  66
    #> 8                No  1605   low    IA  66
    #> 9                No  1605   low    IA  66
    #> 10               No  1605   low    IA  66

### Reducing memory allocation

Converting a `data.frame` with substantial amount of duplicated information like `wrobel_lung0` into a `groupedHyperframe` greatly reduces the memory allocation.

    unclass(object.size(wrobel_lung0g)) / unclass(object.size(wrobel_lung0))
    #> [1] 0.113451

A `groupedHyperframe`, however, would not reduce much the `save`d `file.size` compared to a `data.frame`, if `xz` compression is used for both.

    f_g = tempfile(fileext = '.rds')
    wrobel_lung0g |> saveRDS(file = f_g, compress = 'xz')
    f = tempfile(fileext = '.rds')
    wrobel_lung0 |> saveRDS(file = f, compress = 'xz')
    file.size(f_g) / file.size(f) # not much reduction
    #> [1] 0.9386422

### Aggregation of numeric-hypercolumn

We use function `aggregate_quantile()` to aggregate the `quantile`s of each element in the numeric-hypercolumn `hladr` in `wrobel_lung0g` by point-wise means (default of parameter `f_aggr_`) at the second-lowest group `~patient_id`. The returned object is a `hyperframe` instead of a `groupedHyperframe`, as we have one aggregated `hladr.quantile` per `~patient_id`, thus eliminates the need for a grouping structure.

    wrobel_lung0g |>
      aggregate_quantile(by = ~ patient_id, probs = seq.int(from = .01, to = .99, by = .01))
    #> Hyperframe:
    #>      patient_id gender stage_numeric pack_years adjuvant_therapy    OS mhcII
    #> 1 #01 0-889-121      F             1         60               No 3488+   low
    #> 2 #02 1-037-393      M             1         30               No  1605   low
    #> 3 #03 2-080-378      M             3         50               No   176  high
    #>   stage age hladr.quantile
    #> 1    IA  85      (numeric)
    #> 2    IA  66      (numeric)
    #> 3  IIIA  84      (numeric)

In this package, we have include a `groupedHyperframe` example `Ki67` with a numeric-hypercolumn `logKi67` and a nested grouping structure `~patientID/tissueID`.

    data(Ki67, package = 'groupedHyperframe')
    Ki67
    #> Grouped Hyperframe: ~patientID/tissueID
    #>
    #> 645 tissueID nested in
    #> 622 patientID
    #>
    #> Preview of first 10 (or less) rows:
    #>
    #>      logKi67 tissueID Tstage  PFS recfreesurv_mon recurrence adj_rad adj_chemo
    #> 1  (numeric) TJUe_I17      2 100+             100          0   FALSE     FALSE
    #> 2  (numeric) TJUe_G17      1   22              22          1   FALSE     FALSE
    #> 3  (numeric) TJUe_F17      1  99+              99          0   FALSE        NA
    #> 4  (numeric) TJUe_D17      1  99+              99          0   FALSE      TRUE
    #> 5  (numeric) TJUe_J18      1  112             112          1    TRUE      TRUE
    #> 6  (numeric) TJUe_N17      4   12              12          1    TRUE     FALSE
    #> 7  (numeric) TJUe_J17      2  64+              64          0   FALSE     FALSE
    #> 8  (numeric) TJUe_F19      2  56+              56          0   FALSE     FALSE
    #> 9  (numeric) TJUe_P19      2  79+              79          0   FALSE     FALSE
    #> 10 (numeric) TJUe_O19      2   26              26          1   FALSE      TRUE
    #>    histology  Her2   HR  node  race age patientID
    #> 1          3  TRUE TRUE  TRUE White  66   PT00037
    #> 2          3 FALSE TRUE FALSE Black  42   PT00039
    #> 3          3 FALSE TRUE FALSE White  60   PT00040
    #> 4          3 FALSE TRUE  TRUE White  53   PT00042
    #> 5          3 FALSE TRUE  TRUE White  52   PT00054
    #> 6          2  TRUE TRUE  TRUE Black  51   PT00059
    #> 7          3 FALSE TRUE  TRUE Asian  50   PT00062
    #> 8          2  TRUE TRUE  TRUE White  37   PT00068
    #> 9          3  TRUE TRUE FALSE White  68   PT00082
    #> 10         2  TRUE TRUE FALSE Black  55   PT00084

Similarly, we use function `aggregate_quantile()` to aggregate the `quantile`s of each element in the numeric-hypercolumn `logKi67` at the second-lowest group `~patientID`.

    s = Ki67 |>
      aggregate_quantile(by = ~ patientID, probs = seq.int(from = .01, to = .99, by = .01))
    s |> head()
    #> Hyperframe:
    #>   Tstage  PFS recfreesurv_mon recurrence adj_rad adj_chemo histology  Her2   HR
    #> 1      2 100+             100          0   FALSE     FALSE         3  TRUE TRUE
    #> 2      1   22              22          1   FALSE     FALSE         3 FALSE TRUE
    #> 3      1  99+              99          0   FALSE        NA         3 FALSE TRUE
    #> 4      1  99+              99          0   FALSE      TRUE         3 FALSE TRUE
    #> 5      1  112             112          1    TRUE      TRUE         3 FALSE TRUE
    #> 6      4   12              12          1    TRUE     FALSE         2  TRUE TRUE
    #>    node  race age patientID logKi67.quantile
    #> 1  TRUE White  66   PT00037        (numeric)
    #> 2 FALSE Black  42   PT00039        (numeric)
    #> 3 FALSE White  60   PT00040        (numeric)
    #> 4  TRUE White  53   PT00042        (numeric)
    #> 5  TRUE White  52   PT00054        (numeric)
    #> 6  TRUE Black  51   PT00059        (numeric)

Users are encouraged to learn more about the applications of the aggregated quantiles of `Ki67` data from package `hyper.gam` vignettes (RPubs, `CRAN`), section Quantile Index, as well as from our peer-reviewed publications Yi et al. (2025); Yi et al. (2023b); Yi et al. (2023a).

## From `hyperframe`

Users may convert a `hyperframe` provided in the package `spatstat.data` into a `groupedHyperframe` using the `S3` method dispatch `as.groupedHyperframe.hyperframe()`. This function simply inspects and adds a (nested) grouping structure to the input `hyperframe`.

In the following example, we inspect the data set `spatstat.data::osteo`, which has the serial number of sampling volume `brick` nested in the bone sample `id`, and add the nested grouping structure `~id/brick` to it.

    spatstat.data::osteo |>
      as.groupedHyperframe(group = ~ id/brick)
    #> Grouped Hyperframe: ~id/brick
    #>
    #> 40 brick nested in
    #> 4 id
    #>
    #> Preview of first 10 (or less) rows:
    #>
    #>        id shortid brick   pts depth
    #> 1  c77za4       4     1 (pp3)    45
    #> 2  c77za4       4     2 (pp3)    60
    #> 3  c77za4       4     3 (pp3)    55
    #> 4  c77za4       4     4 (pp3)    60
    #> 5  c77za4       4     5 (pp3)    85
    #> 6  c77za4       4     6 (pp3)    90
    #> 7  c77za4       4     7 (pp3)    95
    #> 8  c77za4       4     8 (pp3)    65
    #> 9  c77za4       4     9 (pp3)   100
    #> 10 c77za4       4    10 (pp3)   100

In this vignette, we do not place much emphasize on the objects provided in package `spatstat.data`, for now.

# Grouping Structure on `ppp`-Hypercolumn

In this section, we introduce the creation of `groupedHyperframe` with one-and-only-one point pattern (`ppp`) hypercolumn, as well as the batch processes of spatial point pattern analyses on the one-and-only-one `ppp`-hypercolumn of a `hyperframe` (and/or `groupedHyperframe`).

These batch processes are not intended for a `hyperframe` (and/or `groupedHyperframe`) with multiple `ppp`-hypercolumn in the foreseeable future, as that would require checking for name clashes in the `marks` from multiple `ppp`-hypercolumn.

## Grouped hyper data frame with `ppp`-hypercolumn

Function `grouped_ppp()` creates a `groupedHyperframe` with one-and-only-one `ppp`-hypercolumn.

In the following example, the argument `formula` specifies

- the point pattern `marks`, e.g., numeric mark `hladr` and `multitype` mark `phenotype`, on the left-hand-side
- the additional predictors and/or endpoints for downstream analysis, e.g., `OS`, `gender` and `age`, before the `|` separator on the right-hand-side
- the (nested) grouping structure, e.g., `image_id` nested in `patient_id`, after the `|` separator on the right-hand-side.

    (s = wrobel_lung |>
       grouped_ppp(formula = hladr + phenotype ~ OS + gender + age | patient_id/image_id))
    #> Grouped Hyperframe: ~patient_id/image_id
    #>
    #> 15 image_id nested in
    #> 3 patient_id
    #>
    #> Preview of first 10 (or less) rows:
    #>
    #>       OS gender age    patient_id          image_id  ppp.
    #> 1  3488+      F  85 #01 0-889-121 [40864,18015].im3 (ppp)
    #> 2  3488+      F  85 #01 0-889-121 [42689,19214].im3 (ppp)
    #> 3  3488+      F  85 #01 0-889-121 [42806,16718].im3 (ppp)
    #> 4  3488+      F  85 #01 0-889-121 [44311,17766].im3 (ppp)
    #> 5  3488+      F  85 #01 0-889-121 [45366,16647].im3 (ppp)
    #> 6   1605      M  66 #02 1-037-393 [56576,16907].im3 (ppp)
    #> 7   1605      M  66 #02 1-037-393 [56583,15235].im3 (ppp)
    #> 8   1605      M  66 #02 1-037-393 [57130,16082].im3 (ppp)
    #> 9   1605      M  66 #02 1-037-393 [57396,17896].im3 (ppp)
    #> 10  1605      M  66 #02 1-037-393 [57403,16934].im3 (ppp)

## Batch processes

### Batch processes to return `fv`-hypercolumn

In this section, we discuss the batch processes that return a function value table (`fv`) hypercolumn, i.e., a hypercolumn which consists of a `list` of `fv.object`s.

#### Batch processes applicable to numeric `marks`

Batch Process | Workhorse in package `spatstat.explore` | `fv`-hypercolumns Suffix
--- | --- | ---
`Emark_()` and `Vmark_()` | `Emark` and `Vmark`, conditional mean E(r) and conditional variance V(r), diagnostics for dependence between the points and the marks (Schlather, Ribeiro, and Diggle 2003) | `.E` and `.V`
`markcorr_()` | `markcorr`, marked correlation k_{mm}(r) or generalized mark correlation k_f(r) (Stoyan and Stoyan 1994) | `.k`
`markvario_()` | `markvario`, mark variogram \gamma(r) (Wälder and Stoyan 1996) | `.gamma`
`Kmark_()` | `Kmark`, mark-weighted K_f(r) function (Penttinen, Stoyan, and Henttonen 1992) | `.K`

##### Exception handling

In package `spatstat.explore` (up to version `3.4.3`, 2025-05-21), function `markcorr()` is the workhorse inside functions `Emark()`, `Vmark()` and `markvario()`. Function `markcorr()` relies on the un-exported workhorse function `spatstat.explore:::sewsmod()`, whose default `method = "density"` contains the calculation of the ratio of two kernel densities. Due to the floating-point precision of `R`, such density ratios may have exceptional returns of

- `0`, from 0/\delta, where \delta\geq (approximately) `2.6e-324`
- `NaN`, from 0/\varepsilon, where \varepsilon\leq (approximately) `2.5e-324`
- `Inf`, from \delta/0, where \delta\geq (approximately) `2.6e-324`

See for yourself

    0 / c(2.6e-324, 2.5e-324)
    #> [1]   0 NaN
    c(2.5e-324, 2.6e-324) / 0
    #> [1] NaN Inf

Function `markcorr()` provides a default argument of parameter r, at which the mark correlation function k_f(r) are evaluated, using function `spatstat.geom::handle.r.b.args()`. The S3 method dispatch `spatstat.explore::print.fv()` prints the recommended range and available range of the argument r.

    spatstat.data::spruces |>
      spatstat.explore::markcorr()
    #> Function value object (class 'fv')
    #> for the function r -> k[mm](r)
    #> ................................................................................
    #>       Math.label
    #> r     r
    #> theo  {k[mm]^{iid}}(r)
    #> trans {hat(k)[mm]^{trans}}(r)
    #> iso   {hat(k)[mm]^{iso}}(r)
    #>       Description
    #> r     distance argument r
    #> theo  theoretical value (independent marks) for k[mm](r)
    #> trans translation-corrected estimate of k[mm](r)
    #> iso   Ripley isotropic correction estimate of k[mm](r)
    #> ................................................................................
    #> Default plot formula:  .~r
    #> where "." stands for 'iso', 'trans', 'theo'
    #> Recommended range of argument r: [0, 9.5]
    #> Available range of argument r: [0, 9.5]
    #> Unit of length: 1 metre

We may observe exceptional returns if we go beyond the recommended range and/or available range. In the following example, we see that the mark correlation k_f(r) (column `iso`) having value `NaN` at r=81,88,89,90, value `0` at r=82 and value `Inf` at r=83,84,87.

    spatstat.data::spruces |>
      spatstat.explore::markcorr(r = 0:90) |>
      spatstat.explore::as.data.frame.fv() |>
      utils::tail(n = 10L)
    #>     r theo      trans        iso
    #> 82 81    1 0.83312629 0.05832148
    #> 83 82    1 7.85844905 0.00000000
    #> 84 83    1 0.91665313 0.51369152
    #> 85 84    1 1.34455861 0.23996942
    #> 86 85    1 2.45829352 0.00000000
    #> 87 86    1 1.62990007 0.24344830
    #> 88 87    1 0.00000000 0.32811666
    #> 89 88    1        Inf 0.00000000
    #> 90 89    1 0.03699099 0.01725363
    #> 91 90    1        NaN 0.00000000

The batch process `markcorr_()`, as well as `Emark_()`, `Vmark_()` and `markvario_()` which rely on the workhorse `markcorr()`, prints the Recommended r_\text{max} from each of the `fv`-returns, no matter a user-specified argument for parameter `r` is provided or not.

    s |>
      Emark_(correction = 'none')
    #>
    #> Recommended rmax for hladr.E are 15⨯ 125.625

When a user-specified `r` is provided for a batch process on all `ppp.object`s in the `ppp`-hypercolumn, inevitably some of the `fv`-returns may have exceptional values. We discuss this exception handling in the next section Aggregation over nested grouping structure.

#### Batch processes applicable to `multitype` `marks`

Batch Process | Workhorse in package `spatstat.explore` | `fv`-hypercolumns Suffix
--- | --- | ---
`Gcross_()` | `Gcross`, multitype nearest-neighbour distance G_{ij}(r) | `.G`
`Kcross_()` | `Kcross`, multitype K_{ij}(r) | `.K`
`Jcross_()` | `Jcross`, multitype J_{ij}(r) (Van Lieshout and Baddeley 1999) | `.J`
`Lcross_()` | `Lcross`, multitype L_{ij}(r)=\sqrt{\frac{K_{ij}(r)}{\pi}} | `.L`

### Batch processes to return numeric-hypercolumn

Batch Process | Workhorse in package `spatstat.geom` | Applicable to | numeric-hypercolumns Suffix
--- | --- | --- | ---
`nncross_()` | `nncross.ppp``(., what = 'dist')`, nearest neighbour distance | `multitype` marks | `.nncross`

### Batch processes in a pipeline

Multiple batch processes may be applied to a `hyperframe` (and/or `groupedHyperframe`) in a pipeline using the native pipe operator `|>` introduced since `R` 4.1.0.

    r = seq.int(from = 0, to = 250, by = 10)
    out = s |>
      Emark_(r = r, correction = 'none') |> # slow
      # Vmark_(r = r, correction = 'none') |> # slow
      # markcorr_(r = r, correction = 'none') |> # slow
      # markvario_(r = r, correction = 'none') |> # slow
      # Kmark_(r = r, correction = 'none') |> # fast
      Gcross_(i = 'CK+.CD8-', j = 'CK-.CD8+', r = r, correction = 'none') |> # fast
      # Kcross_(i = 'CK+.CD8-', j = 'CK-.CD8+', r = r, correction = 'none') |> # fast
      nncross_(i = 'CK+.CD8-', j = 'CK-.CD8+', correction = 'none') # fast
    #>
    #> Recommended rmax for hladr.E are 15⨯ 125.625
    #>

The returned `hyperframe` (or `groupedHyperframe`) has

- `fv`-hypercolumn `hladr.E`, created by function `Emark_()` on numeric mark `hladr`
- `fv`-hypercolumn `phenotype.G`, created by function `Gcross_()` on `multitype` mark `phenotype`
- numeric-hypercolumn `phenotype.nncross`, created by function `nncross_()` on `multitype` mark `phenotype`

    out
    #> Grouped Hyperframe: ~patient_id/image_id
    #>
    #> 15 image_id nested in
    #> 3 patient_id
    #>
    #> Preview of first 10 (or less) rows:
    #>
    #>       OS gender age    patient_id          image_id  ppp. hladr.E phenotype.G
    #> 1  3488+      F  85 #01 0-889-121 [40864,18015].im3 (ppp)    (fv)        (fv)
    #> 2  3488+      F  85 #01 0-889-121 [42689,19214].im3 (ppp)    (fv)        (fv)
    #> 3  3488+      F  85 #01 0-889-121 [42806,16718].im3 (ppp)    (fv)        (fv)
    #> 4  3488+      F  85 #01 0-889-121 [44311,17766].im3 (ppp)    (fv)        (fv)
    #> 5  3488+      F  85 #01 0-889-121 [45366,16647].im3 (ppp)    (fv)        (fv)
    #> 6   1605      M  66 #02 1-037-393 [56576,16907].im3 (ppp)    (fv)        (fv)
    #> 7   1605      M  66 #02 1-037-393 [56583,15235].im3 (ppp)    (fv)        (fv)
    #> 8   1605      M  66 #02 1-037-393 [57130,16082].im3 (ppp)    (fv)        (fv)
    #> 9   1605      M  66 #02 1-037-393 [57396,17896].im3 (ppp)    (fv)        (fv)
    #> 10  1605      M  66 #02 1-037-393 [57403,16934].im3 (ppp)    (fv)        (fv)
    #>    phenotype.nncross
    #> 1          (numeric)
    #> 2          (numeric)
    #> 3          (numeric)
    #> 4          (numeric)
    #> 5          (numeric)
    #> 6          (numeric)
    #> 7          (numeric)
    #> 8          (numeric)
    #> 9          (numeric)
    #> 10         (numeric)

## Aggregation over nested grouping structure

### Of `fv`-hypercolumn(s)

Function `aggregate_fv()` aggregates

- the function values, i.e., the black-solid-curve of `plot.fv`.
- the cumulative trapezoidal integration under the black-solid-curve.

In the following example, we have

- numeric-hypercolumns `hladr.E.value` and `phenotype.G.value`, the aggregated function values from `fv`-hypercolumns `hladr.E` and `phenotype.G`, respectively.
- numeric-hypercolumns `hladr.E.cumtrapz` and `phenotype.G.cumtrapz`, the aggregated cumulative trapezoidal integrations from `fv`-hypercolumns `hladr.E` and `phenotype.G`, respectively.

    (afv = out |>
      aggregate_fv(by = ~ patient_id, f_aggr_ = pmean))
    #> Hyperframe:
    #>      OS gender age    patient_id hladr.E.value hladr.E.cumtrapz
    #> 1 3488+      F  85 #01 0-889-121     (numeric)        (numeric)
    #> 2  1605      M  66 #02 1-037-393     (numeric)        (numeric)
    #> 3   176      M  84 #03 2-080-378     (numeric)        (numeric)
    #>   phenotype.G.value phenotype.G.cumtrapz
    #> 1         (numeric)            (numeric)
    #> 2         (numeric)            (numeric)
    #> 3         (numeric)            (numeric)

Each of the numeric-hypercolumns contains tabulated values on the common grid of `r`. One “slice” of this grid may be extracted by

    afv$hladr.E.cumtrapz |> .slice(j = '50')
    #>        1        2        3
    #> 10.60151 10.49337 31.05115

#### Exception handling

As we have mentioned in the previous section Batch processes, a same user-specified argument of `r` will be used for all `ppp.object`s in the `ppp`-hypercolumn. Suppose a naive user uses an r-vector well beyond the recommended range and/or available range. In this case, function `aggregate_fv()` prints a `message` of Legal r_\text{max}, which is determined by the last value of r, that no value of `NaN` and/or `Inf` appears in any of the `fv`-returns, e.g., in the hypercolumn `hladr.E`. Note that the `0`-values in an `fv`-return are typically a sign of degeneration as well, but function `aggregate_fv()` does not eliminate `0`-values from the determination of legal r_\text{max}.

    r = seq.int(from = 0, to = 1000, by = 50)
    s |>
      Emark_(r = r, correction = 'none') |>
      aggregate_fv(by = ~ patient_id, f_aggr_ = pmean)
    #>
    #> Recommended rmax for hladr.E are 15⨯ 125.625
    #> Legal rmax for hladr.E is 800

### Of numeric-hypercolumn and numeric marks in `ppp`-hypercolumn

#### On `quantile`s

Function `aggregate_quantile()` aggregates the `quantile`s of the numeric-hypercolumns and the numeric marks in the `ppp`-hypercolumn.

In the following example, we have

- numeric-hypercolumn `phenotype.nncross.quantile`, the aggregated `quantile`s of numeric-hypercolumn `phenotype.nncross`.
- numeric-hypercolumn `hladr.quantile`, the aggregated `quantile`s of numeric mark `hladr` in `ppp`-hypercolumn.

    out |>
      aggregate_quantile(by = ~ patient_id, probs = seq.int(from = 0, to = 1, by = .1))
    #> Hyperframe:
    #>      OS gender age    patient_id phenotype.nncross.quantile hladr.quantile
    #> 1 3488+      F  85 #01 0-889-121                  (numeric)      (numeric)
    #> 2  1605      M  66 #02 1-037-393                  (numeric)      (numeric)
    #> 3   176      M  84 #03 2-080-378                  (numeric)      (numeric)

#### On kernel densities

Function `aggregate_kerndens()` aggregates the kernel density of the numeric-hypercolumns and the numeric marks in the `ppp`-hypercolumn.

In the following example, we have

- numeric-hypercolumn `phenotype.nncross.kerndens`, the aggregated kernel densities of numeric-hypercolumn `phenotype.nncross`.
- numeric-hypercolumn `hladr.kerndens`, the aggregated kernel densities of numeric mark `hladr` in `ppp`-hypercolumn.

    (mdist = out$phenotype.nncross |> unlist() |> max())
    #> [1] 333.2417
    out |>
      aggregate_kerndens(by = ~ patient_id, from = 0, to = mdist)
    #> Hyperframe:
    #>      OS gender age    patient_id phenotype.nncross.kerndens hladr.kerndens
    #> 1 3488+      F  85 #01 0-889-121                  (numeric)      (numeric)
    #> 2  1605      M  66 #02 1-037-393                  (numeric)      (numeric)
    #> 3   176      M  84 #03 2-080-378                  (numeric)      (numeric)

# Appendix A: Minor Functionalities

## k-Means Clustering

The `S3` generic `.kmeans()` performs k-means clustering (workhorse function `stats::kmeans`).

### On `ppp.object`

    data(shapley, package = 'spatstat.data')
    shapley
    #> Marked planar point pattern: 4215 points
    #> Mark variables: Mag, V, SigV
    #> window: polygonal boundary
    #> enclosing rectangle: [192.80185, 216.26227] x [-37.75511, -27.40664] degrees

The `S3` method dispatch `.kmeans.ppp()` performs k-means clustering, with paramters

- `formula`, user-specified coordinate(s) and/or numeric `marks`;
- `centers`, number of clusters, see `?stats::kmeans`
- `clusterSize`, “expected” cluster size.

#### By coordinate(s) and/or `marks`

Example below shows a clustering based on the x- and y-coordinates, as well as the numeric mark `Mag`.

    km = shapley |> .kmeans(formula = ~ x + y + Mag, centers = 3L)
    km |> class()
    #> [1] "kmeans"

Example below shows a clustering based on x-coordinate and `Mag`.

    km1 = shapley |> .kmeans(formula = ~ x + Mag, centers = 3L)
    km1 |> class()
    #> [1] "kmeans"

Example below shows a clustering based on x- and y-coordinates only.

    km2 = shapley |> .kmeans(formula = ~ x + y, centers = 3L)
    km2 |> class()
    #> [1] "kmeans"

#### By `clusterSize`

Example below shows a clustering specified by `clusterSize`.

    km3 = shapley |> .kmeans(formula = ~ x + y, clusterSize = 1e3L)
    km3 |> class()
    #> [1] "kmeans"
    km3$centers # 5 clusters needed
    #>          x         y
    #> 1 210.2438 -31.12496
    #> 2 194.6948 -29.94795
    #> 3 202.1755 -31.00384
    #> 4 205.7199 -33.97703
    #> 5 198.9476 -34.85859
    km3$cluster |> table()
    #>
    #>    1    2    3    4    5
    #>  640 1099 1475  565  436

## Split by k-Means Clustering

The `S3` generic `split_kmeans()` `split`s `ppp.object`, `listof` `ppp.object`s, and `hyperframe` by k-means clustering.

Note that many functions in package `groupedHyperframe` require a `'dataframe'` `markformat` for `ppp.object`s.

    data(flu, package = 'spatstat.data')
    flu$pattern[[1L]] |>
      spatstat.geom::markformat()
    #> [1] "vector"

User may convert a `'vector'` `markformat` to `'dataframe'` using the syntactic sugar ``mark_name<-``,

    flu$pattern[] = flu$pattern |>
      lapply(FUN = `mark_name<-`, value = 'stain') # read ?flu carefully
    flu$pattern[[1L]] |>
      spatstat.geom::markformat()
    #> [1] "dataframe"

### Split a `ppp.object`

The `S3` method dispatch `split_kmeans.default()` splits a `ppp.object` by k-means clustering.

    flu$pattern[[1L]] |> split_kmeans(formula = ~ x + y, centers = 3L)
    #> Point pattern split by factor
    #>
    #> 1:
    #> Marked planar point pattern: 145 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> 2:
    #> Marked planar point pattern: 169 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> 3:
    #> Marked planar point pattern: 157 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm

### Split a `listof` `ppp.object`s

The `S3` method dispatch `split_kmeans.listof()` splits a `listof` `ppp.object`s by k-means clustering.

The returned object has `attributes`

- `attr(.,'.id')`, indices of the `ppp.object`s before splitting.
- `attr(.,'.cluster')`, indices of k-means clusters, nested in `.id`.

    flu$pattern[1:2] |> split_kmeans(formula = ~ x + y, centers = 3L)
    #> $`wt M2-M1 13.1`
    #> Marked planar point pattern: 157 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> $`wt M2-M1 13.2`
    #> Marked planar point pattern: 169 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> $`wt M2-M1 13.3`
    #> Marked planar point pattern: 145 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> $`wt M2-M1 22.1`
    #> Marked planar point pattern: 77 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> $`wt M2-M1 22.2`
    #> Marked planar point pattern: 57 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> $`wt M2-M1 22.3`
    #> Marked planar point pattern: 83 points
    #> Multitype, with levels = M2, M1
    #> window: rectangle = [0, 3331] x [0, 3331] nm
    #>
    #> attr(,"id")
    #> [1] 1 1 1 2 2 2
    #> attr(,"cluster")
    #> [1] 1 2 3 1 2 3

### Split a `hyperframe` and/or `groupedHyperframe`

The `S3` method dispatch `split_kmeans.hyperframe()` splits a `hyperframe` and/or `groupedHyperframe` by k-means clustering of the one-and-only-one `ppp`-hypercolumn.

The returned object is a `groupedHyperframe` with grouping structure

- `~.id/.cluster`, if the input is a `hyperframe`
- `~ existing/grouping/structure/.cluster`, if the input is a `groupedHyperframe`. Note that the grouping level `.id` is believed to be equivalent to the lowest level of existing grouping structure.

    flu[1:2,] |> split_kmeans(formula = ~ x + y, centers = 3L)
    #> Grouped Hyperframe: ~.id/.cluster
    #>
    #> 6 .cluster nested in
    #> 2 .id
    #>
    #> Preview of first 10 (or less) rows:
    #>
    #>   pattern .id .cluster virustype stain frameid
    #> 1   (ppp)   1        1        wt M2-M1      13
    #> 2   (ppp)   1        2        wt M2-M1      13
    #> 3   (ppp)   1        3        wt M2-M1      13
    #> 4   (ppp)   2        1        wt M2-M1      22
    #> 5   (ppp)   2        2        wt M2-M1      22
    #> 6   (ppp)   2        3        wt M2-M1      22

## Pairwise Tjøstheim’s Coefficient

The `S3` generic `pairwise_cor_spatial()` calculates the nonparametric, rank-based, Tjøstheim’s correlation coefficients (Tjøstheim 1978; Hubert and Golledge 1982) in a pairwise-combination fashion, using the workhorse function `SpatialPack::cor.spatial()`. All `S3` method dispatches return a object of class `'pairwise_cor_spatial'`, which `inherits` from class `'dist'`.

### Of `ppp.object`

The `S3` method dispatch `pairwise_cor_spatial.ppp()` finds the nonparametric Tjøstheim’s correlation coefficients from the pairwise-combinations of all numeric marks of a `ppp.object`.

    data(finpines, package = 'spatstat.data')
    (r = finpines |> pairwise_cor_spatial())
    #>         diameter
    #> height 0.7287879

The printing of `'pairwise_cor_spatial'` is taken care of by function `stats:::print.dist`.

### Matrix of pairwise Tjøstheim’s coefficient

The `S3` method dispatch `as.matrix.pairwise_cor_spatial()` returns a `matrix` with `diag`onal values of 1.

    r |> as.matrix()
    #>           diameter    height
    #> diameter 1.0000000 0.7287879
    #> height   0.7287879 1.0000000

Note that this matrix is not a `cor`relation matrix, because Tjøstheim’s correlation coefficient

- is nonparametric, i.e., there is no definition of the corresponding `cov`ariance, standard deviation `sd`, nor the conversion `cov2cor` method;
- does not provide a mathematical mechanism to ensure this matrix is positive definite.

# Appendix B: What We Don’t Do

## S4 method dispatch `merge`

The authors plan not to implement an S4 method dispatch `merge` (e.g., S3 method dispatch `base::merge.data.frame()`) for `hyperframe` and/or `groupedHyperframe` classes, for several reasons.

- There is not an S3 method dispatch `merge.hyperframe` in package `spatstat.geom` (as of version 3.4-1);
- Should the authors decide to implement the `merge` functionality for `hyperframe` class, at least two S4 method dispatches need to be written for `setMethod(f = merge, ...)`
  - `signature = c(x = 'hyperframe', y = 'data.frame')`
  - `signature = c(x = 'hyperframe', y = 'hyperframe')`
- Should the authors decide to implement the `merge` functionality for `groupedHyperframe` class, at least three S4 method dispatches need to be written for `setMethod(f = merge, ...)`
  - `signature = c(x = 'groupedHyperframe', y = 'data.frame')`
  - `signature = c(x = 'groupedHyperframe', y = 'hyperframe')`
  - `signature = c(x = 'groupedHyperframe', y = 'groupedHyperframe')`, for which the authors need to consider the (potentially different) grouping structures of `x` and `y` inputs.

The authors suggest users do `base::merge.data.frame()` first, then do `as.groupedHyperframe()` as a workaround.

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