---
title: "Grouped Hyper Data Frame"
author: Tingting Zhan
date: "`r format(Sys.time(), '%d %B, %Y')`"
format: 
  html:
    page-layout: full
    html-math-method: katex
toc: true
toc-location: left
toc-depth: 4
toc-title: ''
editor: visual
bibliography: groupedHyperframe.bib
knitr:
  opts_chunk: 
    collapse: true
    comment: "#>" 
vignette: >
  %\VignetteIndexEntry{intro}
  %\VignetteEngine{quarto::html}
  %\VignetteEncoding{UTF-8}
---

# Introduction

This vignette of package **`groupedHyperframe`** ([`CRAN`](https://cran.r-project.org/package=groupedHyperframe), [Github](https://github.com/tingtingzhan/groupedHyperframe), [RPubs](https://rpubs.com/tingtingzhan/groupedHyperframe)) 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](https://github.com/tingtingzhan/groupedHyperframe). [Active developers should use the Github version; suggestions and bug reports are welcome!]{style="background-color: #FFFF00"}

```{r}
#| warning: false
#| eval: false
remotes::install_github('tingtingzhan/groupedHyperframe')
```

Stable releases to [`CRAN`](https://CRAN.R-project.org/package=groupedHyperframe) 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!]{style="background-color: #FFFF00"}

```{r}
#| warning: false
#| eval: false
utils::install.packages('groupedHyperframe') # Developers, do NOT use!!
```

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

```{r}
#| label: prerequisite
#| warning: false
#| eval: false
remotes::install_github('spatstat/spatstat')
remotes::install_github('spatstat/spatstat.data')
remotes::install_github('spatstat/spatstat.explore')
remotes::install_github('spatstat/spatstat.geom')
remotes::install_github('spatstat/spatstat.linnet')
remotes::install_github('spatstat/spatstat.model')
remotes::install_github('spatstat/spatstat.random')
remotes::install_github('spatstat/spatstat.sparse')
remotes::install_github('spatstat/spatstat.univar')
remotes::install_github('spatstat/spatstat.utils')
```

## Getting Started

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

```{r}
#| message: false
library(groupedHyperframe)
library(survival) # to help hyperframe understand Surv object
```

```{r}
#| echo: false
op = par(no.readonly = TRUE)
options(mc.cores = 1L) # for CRAN submission
```

## Terms and Abbreviations

| Term / Abbreviation | Description |
|------------------------------------|------------------------------------|
| [`|>`](https://search.r-project.org/R/refmans/base/html/pipeOp.html) | Forward pipe operator introduced in `R` 4.1.0 |
| [`.Machine`](https://search.r-project.org/R/refmans/base/html/zMachine.html) | Numerical characteristics of the machine `R` is running on, e.g., 32-bit integers and IEC 60559 floating-point (double precision) arithmetic |
| [`attr`](https://search.r-project.org/R/refmans/base/html/attr.html), [`attributes`](https://search.r-project.org/R/refmans/base/html/attributes.html) | Attributes |
| `CRAN`, `R` | [The Comprehensive R Archive Network](https://cran.r-project.org) |
| [`cor`](https://search.r-project.org/R/refmans/stats/html/cor.html) | Correlation matrix |
| [`cor.spatial`](https://search.r-project.org/CRAN/refmans/SpatialPack/html/cor.spatial.html) | Tjøstheim's nonparametric correlation coefficient, from package **`SpatialPack`** [@SpatialPack] |
| [`cov`, `cov2cor`](https://search.r-project.org/R/refmans/stats/html/cor.html) | Variance-covariance matrix, and conversion to correlation matrix |
| [`data.frame`](https://search.r-project.org/R/refmans/base/html/data.frame.html) | Data frame |
| [`diag`](https://search.r-project.org/R/refmans/base/html/diag.html) | Matrix diagonals |
| [`dist`](https://search.r-project.org/R/refmans/stats/html/dist.html) | Distance matrix; to take advantage of `stats:::as.matrix.dist` |
| [`file.size`](https://search.r-project.org/R/refmans/base/html/file.info.html) | File size in bytes |
| [`formula`](https://search.r-project.org/R/refmans/stats/html/formula.html) | Formula |
| [`fv`, `fv.object`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/fv.object.html), [`plot.fv`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/plot.fv.html) | (Plot of) function value table |
| [`groupedData`](https://search.r-project.org/CRAN/refmans/nlme/html/groupedData.html), `~ g1/.../gm` | Grouped data frame; nested grouping structure, from package **`nlme`** [@nlme] |
| [`groupedHyperframe`](https://CRAN.R-project.org/package=groupedHyperframe) | Grouped hyper data frame |
| `hypercolumns`, [`hyperframe`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/hyperframe.html) | (Hyper columns of) hyper data frame, from package **`spatstat.geom`** [@spatstat05] |
| [`inherits`](https://search.r-project.org/R/refmans/base/html/class.html) | Class inheritance |
| `kerndens` | Kernel density, `stats::density.default()$y` |
| [`Inf`](https://search.r-project.org/R/refmans/base/html/is.finite.html) | Positive infinity $\infty$ |
| [`kmeans`](https://search.r-project.org/R/refmans/stats/html/kmeans.html) | $k$-means clustering |
| [`list`](https://search.r-project.org/R/refmans/base/html/list.html), [`listof`](https://search.r-project.org/R/refmans/stats/html/listof.html) | Lists of objects |
| [`markformat`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/spatstat.geom-internal.html) | Storage mode of `marks` |
| [`marks`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/spatstat.geom-internal.html), [`marked`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/is.marked.html) | Marks of a point pattern |
| [`mc.cores`](https://search.r-project.org/R/refmans/parallel/html/mclapply.html) | Number of CPU cores to use for parallel computing |
| [`message`](https://search.r-project.org/R/refmans/base/html/message.html) | Diagnostic message printed in `R` console |
| [`multitype`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/spatstat.geom-internal.html) | Multitype spatial object |
| [`NaN`](https://search.r-project.org/R/refmans/base/html/is.finite.html) | Not-a-Number |
| [`object.size`](https://search.r-project.org/R/refmans/utils/html/object.size.html) | Memory allocation |
| `pmean`, `pmedian` | Parallel, or point-wise, mean and median, `groupedHyperframe::pmean`; `groupedHyperframe::pmedian` |
| [`pmax`, `pmin`](https://search.r-project.org/R/refmans/base/html/Extremes.html) | Parallel, or point-wise, maxima and minima |
| [`ppp`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/ppp.html), [`ppp.object`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/ppp.object.html) | (Marked) point pattern |
| [`quantile`](https://search.r-project.org/R/refmans/stats/html/quantile.html) | Quantile |
| [`save`](https://search.r-project.org/R/refmans/base/html/save.html), [`saveRDS`](https://search.r-project.org/R/refmans/base/html/readRDS.html), `xz` | Save with [`xz`](https://en.wikipedia.org/wiki/XZ_Utils) compression |
| `S3`, `generic`, [`methods`](https://search.r-project.org/R/refmans/utils/html/methods.html) | `S3` object oriented system, [`UseMethod`](https://search.r-project.org/R/refmans/base/html/UseMethod.html); [`getS3method`](https://search.r-project.org/R/refmans/utils/html/getS3method.html); <https://adv-r.hadley.nz/s3.html> |
| [`sd`](https://search.r-project.org/R/refmans/stats/html/sd.html) | Standard deviation |
| [`search`](https://search.r-project.org/R/refmans/base/html/search.html) | Search path |
| [`Surv`](https://search.r-project.org/CRAN/refmans/survival/html/Surv.html) | Survival, i.e., time-to-event, object |
| [`trapz`, `cumtrapz`](https://search.r-project.org/CRAN/refmans/pracma/html/trapz.html) | (Cumulative) [trapezoidal integration](https://en.wikipedia.org/wiki/Trapezoidal_rule), from package **`pracma`** [@pracma] |
| [`vector`](https://search.r-project.org/R/refmans/base/html/vector.html) | Vector |

## Acknowledgement

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

-   R01CA222847 ([I. Chervoneva](https://orcid.org/0000-0002-9104-4505), [T. Zhan](https://orcid.org/0000-0001-9971-4844), and [H. Rui](https://orcid.org/0000-0002-8778-261X))
-   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`** [@spatstat15; @spatstat05]. 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` [@nlme].

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`*.

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

```{r}
wrobel_lung0 |> head()
```

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`.

```{r}
(wrobel_lung0g = wrobel_lung0 |> as.groupedHyperframe(group = ~ patient_id/image_id))
```

### Reducing memory allocation

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

```{r}
unclass(object.size(wrobel_lung0g)) / unclass(object.size(wrobel_lung0))
```

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

```{r}
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
```

### 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.

```{r}
#| message: false
wrobel_lung0g |>
  aggregate_quantile(by = ~ patient_id, probs = seq.int(from = .01, to = .99, by = .01))
```

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

```{r}
data(Ki67, package = 'groupedHyperframe')
Ki67
```

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`.

```{r}
#| message: false
s = Ki67 |>
  aggregate_quantile(by = ~ patientID, probs = seq.int(from = .01, to = .99, by = .01))
s |> head()
```

Users are encouraged to learn more about the applications of the aggregated quantiles of **`Ki67`** data from package **`hyper.gam`** vignettes ([RPubs](https://rpubs.com/tingtingzhan/hyper_gam), [`CRAN`](https://CRAN.R-project.org/package=hyper.gam/vignettes/applications.html)), section *Quantile Index*, as well as from our peer-reviewed publications @Yi25; @Yi23a; @Yi23b.

## 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.

```{r}
spatstat.data::osteo |> 
  as.groupedHyperframe(group = ~ id/brick)
```

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.

```{r}
(s = wrobel_lung |>
   grouped_ppp(formula = hladr + phenotype ~ OS + gender + age | patient_id/image_id))
```

## 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`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/Emark.html), conditional mean $E(r)$ and conditional variance $V(r)$, diagnostics for dependence between the points and the marks [@Emark] | `.E` and `.V` |
| `markcorr_()` | [`markcorr`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/markcorr.html), marked correlation $k_{mm}(r)$ or generalized mark correlation $k_f(r)$ [@markcorr] | `.k` |
| `markvario_()` | [`markvario`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/markvario.html), mark variogram $\gamma(r)$ [@markvario] | `.gamma` |
| `Kmark_()` | [`Kmark`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/Kmark.html), mark-weighted $K_f(r)$ function [@Kmark] | `.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`

```{r}
#| code-fold: true
#| code-summary: "See for yourself"
0 / c(2.6e-324, 2.5e-324)
c(2.5e-324, 2.6e-324) / 0
```

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 $r$.

```{r}
spatstat.data::spruces |> 
  spatstat.explore::markcorr()
```

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 $r=83,84,87$.

```{r}
spatstat.data::spruces |> 
  spatstat.explore::markcorr(r = 0:90) |>
  spatstat.explore::as.data.frame.fv() |>
  utils::tail(n = 10L)
```

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.

```{r}
#| results: hide
s |>
  Emark_(correction = 'none')
```

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`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/Gcross.html), multitype nearest-neighbour distance $G_{ij}(r)$ | `.G` |
| `Kcross_()` | [`Kcross`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/Kcross.html), multitype $K_{ij}(r)$ | `.K` |
| `Jcross_()` | [`Jcross`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/Jcross.html), multitype $J_{ij}(r)$ [@Jcross] | `.J` |
| `Lcross_()` | [`Lcross`](https://search.r-project.org/CRAN/refmans/spatstat.explore/html/Lcross.html), 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`](https://search.r-project.org/CRAN/refmans/spatstat.geom/html/nncross.html)`(., 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}
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
```

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`*

```{r}
out
```

## 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.

```{r}
#| message: false
(afv = out |>
  aggregate_fv(by = ~ patient_id, f_aggr_ = pmean))
```

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

```{r}
afv$hladr.E.cumtrapz |> .slice(j = '50')
```

#### 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}
#| results: hide
r = seq.int(from = 0, to = 1000, by = 50)
s |>
  Emark_(r = r, correction = 'none') |>
  aggregate_fv(by = ~ patient_id, f_aggr_ = pmean)
```

### 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.

```{r}
#| message: false
out |>
  aggregate_quantile(by = ~ patient_id, probs = seq.int(from = 0, to = 1, by = .1))
```

#### 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.

```{r}
#| message: false
(mdist = out$phenotype.nncross |> unlist() |> max())
out |> 
  aggregate_kerndens(by = ~ patient_id, from = 0, to = mdist)
```

# Appendix

## $k$-Means Clustering

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

### On `ppp.object`

```{r}
data(shapley, package = 'spatstat.data')
shapley
```

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`*.

```{r}
km = shapley |> .kmeans(formula = ~ x + y + Mag, centers = 3L)
km |> class()
```

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

```{r}
km1 = shapley |> .kmeans(formula = ~ x + Mag, centers = 3L)
km1 |> class()
```

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

```{r}
km2 = shapley |> .kmeans(formula = ~ x + y, centers = 3L)
km2 |> class()
```

#### By `clusterSize`

Example below shows a clustering specified by `clusterSize`.

```{r}
km3 = shapley |> .kmeans(formula = ~ x + y, clusterSize = 1e3L)
km3 |> class()
km3$centers # 5 clusters needed
km3$cluster |> table()
```

## 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.

```{r}
data(flu, package = 'spatstat.data')
flu$pattern[[1L]] |> 
  spatstat.geom::markformat()
```

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

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

### Split a `ppp.object`

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

```{r}
flu$pattern[[1L]] |> split_kmeans(formula = ~ x + y, centers = 3L)
```

### 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`.

```{r}
flu$pattern[1:2] |> split_kmeans(formula = ~ x + y, centers = 3L) 
```

### 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*.

```{r}
flu[1:2,] |> split_kmeans(formula = ~ x + y, centers = 3L)
```

## Pairwise Tjøstheim's Coefficient

The `S3` generic `pairwise_cor_spatial()` calculates the nonparametric, rank-based, Tjøstheim's correlation coefficients [@Tjostheim78; @Hubert82] 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`.

```{r}
data(finpines, package = 'spatstat.data')
(r = finpines |> pairwise_cor_spatial())
```

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}
r |> as.matrix()
```

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](https://en.wikipedia.org/wiki/Definite_matrix).

# References

::: {#refs}
:::
