Currently, there are 8 functions associated with the
`calculate`

verb in the `sgsR`

package:

Algorithm | Description |
---|---|

`calculate_representation()` |
Determine representation of strata in existing sample unit |

`calculate_distance()` |
Per pixel distance to closest `access`
vector |

`calculate_pcomp()` |
Principal components of input
`mraster` |

`calculate_sampsize()` |
Estimated sample sizes based on relative standard error |

`calculate_allocation()` |
Proportional / optimal / equal / manual allocation |

`calculate_coobs()` |
Detail how `existing` sample units are
distributed among `mraster` |

`calculate_pop()` |
Population level information (PCA / quantile matrix /
covariance matrix) of `mraster` |

`calculate_lhsOpt()` |
Determine optimal Latin hypercube sampling parameters including sample size |

`calculate_representation()`

`calculate_representation()`

function allows the users to
verify how a stratification is spatially represented in an
`existing`

sample. Result in tabular and graphical (if
`plot = TRUE`

) outputs that compare strata coverage frequency
and sampling frequency.

```
#--- quantile sraster ---#
quantiles <- strat_quantiles(
mraster = mraster$zq90,
nStrata = 8
)
#--- random samples ---#
srs <- sample_srs(
raster = sraster,
nSamp = 50
)
#--- calculate representation ---#
calculate_representation(
sraster = quantiles,
existing = srs,
plot = TRUE
)
```

```
#> # A tibble: 8 × 6
#> strata srasterFreq sampleFreq diffFreq nSamp need
#> <dbl> <dbl> <dbl> <dbl> <int> <dbl>
#> 1 1 0.13 0.18 0.05 9 -2
#> 2 2 0.13 0.14 0.0100 7 0
#> 3 3 0.12 0.12 0 6 0
#> 4 4 0.13 0.04 -0.09 2 5
#> 5 5 0.13 0.14 0.0100 7 0
#> 6 6 0.12 0.16 0.04 8 -2
#> 7 7 0.13 0.12 -0.0100 6 1
#> 8 8 0.12 0.1 -0.0200 5 1
```

The tabular output presents the frequency of coverage for each strata
(`srasterFreq`

) (what % of the landscape does the strata
cover) and the sampling frequency within each strata
(`sampleFreq`

) (what % of total `existing`

samples
are in the strata). The difference (`diffFreq`

) between
coverage frequency and sampling frequency determines whether the values
are over-represented (positive numbers) or under-represented (negative
numbers). This value translates to a discrete `need`

attribute that defines whether there is a need to add or remove samples
to meet the number of samples necessary to be considered representative
of the spatial coverage of strata inputted in `sraster`

.

Performing the algorithm on a sample set derived using
`sample_strat()`

exhibits proportional sampling to strata
coverage given that samples were allocated proportionally to spatial
coverage.

```
#> # A tibble: 4 × 6
#> strata srasterFreq sampleFreq diffFreq nSamp need
#> <dbl> <dbl> <dbl> <dbl> <int> <dbl>
#> 1 1 0.25 0.26 0.0100 51 -1
#> 2 2 0.25 0.25 0 50 0
#> 3 3 0.25 0.24 -0.0100 49 1
#> 4 4 0.25 0.25 0 50 0
```

**TIP!**

Presence of very small (negligible) differences between
`srasterFreq`

and `sampleFreq`

is common.
**In these situations, it is important for the user to determine
whether to add or remove the samples.**

`calculate_distance`

`calculate_distance()`

uses input `raster`

and
`access`

data and outputs the per pixel distance to the
nearest access point. This algorithm has a specific value for
constraining sampling protocols, such as with
`sample_clhs()`

, where the output raster layer can be used as
the `cost`

for the constraint. The output raster consists of
the input appended with the calculated distance layer
(`dist2access`

). The `slope`

parameters also
exists to calculate slope distance instead of geographic distance, which
becomes very handy in the case of steep mountainous areas and is faster
from a computational point of view. If `slope == TRUE`

, the
`mraster`

provided should be a digital terrain model.

```
calculate_distance(
raster = sraster, # input
access = access, # define access road network
plot = TRUE
) # plot
#>
|---------|---------|---------|---------|
=========================================
```

```
#> class : SpatRaster
#> dimensions : 277, 373, 2 (nrow, ncol, nlyr)
#> resolution : 20, 20 (x, y)
#> extent : 431100, 438560, 5337700, 5343240 (xmin, xmax, ymin, ymax)
#> coord. ref. : UTM Zone 17, Northern Hemisphere
#> source(s) : memory
#> varnames : mraster
#> mraster
#> names : strata, dist2access
#> min values : 1, 0.3438
#> max values : 4, 1061.6599
```

*This function performs considerably slower when
access has many features. Consider mergeing features for
improved efficiency.*

`calculate_pcomp`

`calculate_pcomp()`

uses `mraster`

as the input
and performs principal component analysis. The number of components
defined by the `nComp`

parameter specifies the number of
components that will be rasterized.

```
calculate_pcomp(
mraster = mraster, # input
nComp = 3, # number of components to output
plot = TRUE, # plot
details = TRUE
) # details about the principal component analysis appended
```

```
#> $pca
#> Standard deviations (1, .., p=3):
#> [1] 1.5479878 0.7359109 0.2493371
#>
#> Rotation (n x k) = (3 x 3):
#> PC1 PC2 PC3
#> zq90 0.6286296 -0.1795433 -0.7566961
#> pzabove2 0.5104140 0.8293596 0.2272450
#> zsd 0.5867729 -0.5290812 0.6130014
#>
#> $raster
#> class : SpatRaster
#> dimensions : 277, 373, 3 (nrow, ncol, nlyr)
#> resolution : 20, 20 (x, y)
#> extent : 431100, 438560, 5337700, 5343240 (xmin, xmax, ymin, ymax)
#> coord. ref. : UTM Zone 17, Northern Hemisphere
#> source(s) : memory
#> varnames : mraster
#> mraster
#> mraster
#> names : PC1, PC2, PC3
#> min values : -4.402269, -5.357801, -1.446156
#> max values : 5.282663, 2.155242, 1.510955
```

`calculate_sampsize`

`calculate_sampsize()`

function allows the user to
estimate an appropriate sample size using the relative standard error
(`rse`

) of input metrics. If the input `mraster`

contains multiple layers, the sample sizes will be determined for all
layers. If `plot = TRUE`

and `rse`

is defined, a
sequence of `rse`

values will be visualized with the
indicators and the values for the matching sample size.

```
#--- determine sample size based on relative standard error (rse) of 1% ---#
calculate_sampsize(
mraster = mraster,
rse = 0.01
)
#> nSamp rse var
#> 1 1394 0.01 zq90
#> 2 1341 0.01 pzabove2
#> 3 1859 0.01 zsd
```

```
#--- change default threshold sequence values ---#
#--- if increment and rse are not divisible the closest value will be taken ---#
p <- calculate_sampsize(
mraster = mraster,
rse = 0.025,
start = 0.01,
end = 0.08,
increment = 0.01,
plot = TRUE
)
#> 'rse' not perfectly divisible by 'increment'. Selecting closest sample size (rse = 0.03) based on values.
p
#> $nSamp
#> # A tibble: 3 × 3
#> # Groups: var [3]
#> nSamp rse var
#> <dbl> <dbl> <chr>
#> 1 157 0.03 zq90
#> 2 151 0.03 pzabove2
#> 3 211 0.03 zsd
#>
#> $plot
```

`calculate_allocation`

`calculate_allocation()`

determines how sample units are
allocated based on the desired sample size (`nSamp`

) and the
input `sraster`

. It is used internally in a number of
algorithms, including `sample_strat`

.
Currently, there are four allocation methods: proportional
(`prop`

; default), optimal (`optim`

), equal
(`equal`

), and manual (`manual`

).

Samples are allocated based on the spatial coverage area of the strata. This is the default allocation method.

```
#--- perform grid sampling ---#
calculate_allocation(
sraster = sraster,
nSamp = 200
)
#> strata total
#> 1 1 51
#> 2 2 50
#> 3 3 49
#> 4 4 50
```

In this case the sraster has equally sized strata, so the number of allocated sample units is always the same.

```
#--- calculate existing samples to include ---#
e.sr <- extract_strata(
sraster = sraster,
existing = existing
)
calculate_allocation(
sraster = sraster,
nSamp = 200,
existing = e.sr
)
#> strata total need
#> 1 1 0 51
#> 2 2 0 50
#> 3 3 0 49
#> 4 4 0 50
```

At times, values under `total`

can be negative. Negative
values indicate that `existing`

sample units over represent
those strata and that some sample units could removed to prevent
over-representation. `$total`

indicates the number of sample
units that could be added or removed.

Sample units are allocated based on within strata variance. The
optimal allocation method uses the variation within the strata metric to
allocate sample units. This means that in addition to providing an
`sraster`

, that a specific metric (`mraster`

) must
be provided to calculate variation to optimally allocate sample
units.

```
calculate_allocation(
sraster = sraster, # stratified raster
nSamp = 200, # desired sample number
existing = e.sr, # existing samples
allocation = "optim", # optimal allocation
mraster = mraster$zq90, # metric raster
force = TRUE
) # force nSamp number
#> # A tibble: 4 × 3
#> # Rowwise:
#> strata total need
#> <dbl> <dbl> <dbl>
#> 1 1 27 78
#> 2 2 -14 36
#> 3 3 -25 24
#> 4 4 12 62
```

There may be situations where the user wants an equal number of
sample units allocated to each strata. In these situations use
`allocation = equal`

. In this case, `nSamp`

refers
to the total number of sample units per strata, instead of the overall
total number.

```
calculate_allocation(
sraster = sraster, # stratified raster
nSamp = 20, # desired sample number
allocation = "equal"
) # optimal allocation
#> Implementing equal allocation of samples.
#> # A tibble: 4 × 2
#> strata total
#> <dbl> <dbl>
#> 1 1 20
#> 2 2 20
#> 3 3 20
#> 4 4 20
```

The code in the demonstration above yields a total of 80 samples (20
`nSamp`

for each of the 4 strata in
`sraster`

).

The user may wish to manually assign weights to strata. In this case,
`allocation = manual`

can be used and `weights`

must be provided as a numeric vector
(e.g. `weights = c(0.2, 0.2, 0.2, 0.4)`

where
`sum(weights) == 1`

). In this case, `nSamp`

will
be allocated based on `weights`

.

```
weights <- c(0.2, 0.2, 0.2, 0.4)
calculate_allocation(
sraster = sraster, # stratified raster
nSamp = 20, # desired sample number
allocation = "manual", # manual allocation
weights = weights
) # weights adding to 1
#> Implementing allocation of samples based on user-defined weights.
#> strata total
#> 1 1 4
#> 2 2 4
#> 3 3 4
#> 4 4 8
```

The code above yields a total of 20 sample units with plots being
allocated based on the `weights`

provided in ascending strata
order.

The following algorithms were initially developed by Dr. Brendan Malone from the University of Sydney. Dr. Malone and his colleagues supplied an in depth description of the functionality of these algorithms, which were originally developed to improve soil sampling strategies. These functions were modified and implemented to be used for structurally guided sampling approaches. Many thanks to Dr. Malone for being a proponent of open source algorithms.

Please consult the original reference for these scripts and ideas as their publication holds helpful and valuable information to understand their rationale for sampling and algorithm development.

*Malone BP, Minansy B, Brungard C. 2019. Some methods to improve
the utility of conditioned Latin hypercube sampling. PeerJ 7:e6451 DOI
10.7717/peerj.6451*

`calculate_coobs`

`calculate_coobs()`

function performs the
**CO**unt of **OBS**ervations (coobs)
algorithm using an `existing`

sample and
`mraster`

. This algorithm helps the user understand how an
`existing`

sample is distributed among the landscape in
relation to `mraster`

data.

**TIP!**

The output coobs raster can be used to constrain clhs sampling using
the `sample_clhs()`

function to the areas that are
under-represented.

The coobs raster determines how many observations are similar in terms of the covariate space at every pixel, and takes advantage of parallel processing routines.

The following 2 algorithms present the means to maximize the effectiveness of the latin hypercube sampling protocols.

`calculate_pop`

`calculate_pop()`

calculates population level statistics
of an `mraster`

, including calculating the principal
components, quantile & covariate distributions, and Kullback-Leibler
divergence testing.

The outputs produced from this functions are required to use the
`calculate_lhsOpt()`

function described in the following
section.

**TIP!**

This algorithm can be pre-emptively used to calculate
`matQ`

and `MatCov`

, two values that are used for
the `sample_ahels()`

function. This will save processing time
during sampling.

The output list contains the following:

`$values`

- Pixel values from`mraster`

`$pcaLoad`

- PCA loadings`$matQ`

- Quantile matrix`$matCov`

- Covariate matrix

```
#--- statistical analyses can be chosen by setting their parameter to `FALSE` ---#
mat <- calculate_pop(
mraster = mraster, # input
nQuant = 10
) # desired number of quantiles
#--- use matrix output within sample ahels algorithm ---#
sample_ahels(
mraster = mraster, # input
existing = existing, # existing sample
nQuant = 10, # number of desired quantiles
nSamp = 50, # desired sample size
matCov = mat
) # covariance matrix
```

`calculate_lhsOpt`

`calculate_lhsOpt()`

function performs a bootstrapped
latin hypercube sampling approach where population level analysis of
`mraster`

data is performed to determine the optimal latin
hypercube sample size.

`calculate_pop()`

and varying sample sizes defined by
`minSamp`

, `maxSamp`

, `step`

and
`rep`

. Sampling protocols are conducted and statistical
effectiveness of those sampling outcomes are evaluated to determine
where sample size is minimized and statistical representation is
maximized.