stdlib_stats_descriptive:local:
:depth: 2
corr - Pearson correlation of array elementsExperimental
Returns the Pearson correlation of the elements of array along dimension dim if the corresponding element in mask is true.
The Pearson correlation between two rows (or columns), say x and y, of array is defined as:
corr(x, y) = cov(x, y) / sqrt( var(x) * var(y))
result = {f:iface}corr (array, dim [, mask])
Generic subroutine
array: Shall be a rank-1 or a rank-2 array of type integer, real, or complex. It is an intent(in) argument.
dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array. It is an intent(in) argument.
mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array. It is an intent(in) argument.
If array is of rank 1 and of type real or complex, the result is of type real and has the same kind as array.
If array is of rank 2 and of type real or complex, the result is of the same type and kind as array.
If array is of type integer, the result is of type real(dp).
If array is of rank 1 and of size larger than 1, a scalar equal to 1 is returned. Otherwise, IEEE NaN is returned.
If array is of rank 2, a rank-2 array with the corresponding correlations is returned.
If mask is specified, the result is the Pearson correlation of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.
```{literalinclude} ../../example/stats/example_corr.f90 :language: fortran
## `cov` - covariance of array elements
### Status
Experimental
### Description
Returns the covariance of the elements of `array` along dimension `dim` if the corresponding element in `mask` is `true`.
Per default, the covariance is defined as:
cov(array) = 1/(n-1) sum_i (array(i) - mean(array) * (array(i) - mean(array)))
where `n` is the number of elements.
The scaling can be changed with the logical argument `corrected`. If `corrected` is `.false.`, then the sum is scaled with `n`, otherwise with `n-1`.
### Syntax
`result = ` {f:iface}`cov` `(array, dim [, mask [, corrected]])`
### Class
Generic subroutine
### Arguments
`array`: Shall be a rank-1 or a rank-2 array of type `integer`, `real`, or `complex`. It is an `intent(in)` argument.
`dim`: Shall be a scalar of type `integer` with a value in the range from 1 to `n`, where `n` is the rank of `array`. It is an `intent(in)` argument.
`mask` (optional): Shall be of type `logical` and either a scalar or an array of the same shape as `array`. It is an `intent(in)` argument.
`corrected` (optional): Shall be a scalar of type `logical`. If `corrected` is `.true.` (default value), the sum is scaled with `n-1`. If `corrected` is `.false.`, then the sum is scaled with `n`. It is an `intent(in)` argument.
### Return value
If `array` is of rank 1 and of type `real` or `complex`, the result is of type `real` corresponding to the type of `array`.
If `array` is of rank 2 and of type `real` or `complex`, the result is of the same type as `array`.
If `array` is of type `integer`, the result is of type `real(dp)`.
If `array` is of rank 1, a scalar with the covariance (that is the variance) of all elements in `array` is returned.
If `array` is of rank 2, a rank-2 array is returned.
If `mask` is specified, the result is the covariance of all elements of `array` corresponding to `true` elements of `mask`. If every element of `mask` is `false`, the result is IEEE `NaN`.
### Example
```{literalinclude} ../../example/stats/example_cov.f90
:language: fortran
mean - mean of array elementsExperimental
Returns the mean of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.
result = {f:iface}mean (array [, mask])
result = {f:iface}mean (array, dim [, mask])
Generic subroutine
array: Shall be an array of type integer, real, or complex. It is an intent(in) argument.
dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array. It is an intent(in) argument.
mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array. It is an intent(in) argument.
If array is of type real or complex, the result is of the same type as array.
If array is of type integer, the result is of type real(dp).
If dim is absent, a scalar with the mean of all elements in array is returned. Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension dim dropped is returned.
If mask is specified, the result is the mean of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.
```{literalinclude} ../../example/stats/example_mean.f90 :language: fortran
## `median` - median of array elements
### Status
Experimental
### Description
Returns the median of all the elements of `array`, or of the elements of `array`
along dimension `dim` if provided, and if the corresponding element in `mask` is
`true`.
The median of the elements of `array` is defined as the "middle"
element, after that the elements are sorted in an increasing order, e.g. `array_sorted =
sort(array)`. If `n = size(array)` is an even number, the median is:
median(array) = array_sorted( floor( (n + 1) / 2.))
and if `n` is an odd number, the median is:
median(array) = mean( array_sorted( floor( (n + 1) / 2.):floor( (n + 1) / 2.) + 1 ) )
The current implementation relies on a selection algorithm applied on a copy of
the whole array, using the subroutine `select`
provided by the `stdlib_selection` module.
### Syntax
`result = ` {f:iface}`median` `(array [, mask])`
`result = ` {f:iface}`median` `(array, dim [, mask])`
### Class
Generic subroutine
### Arguments
`array`: Shall be an array of type `integer` or `real`. It is an `intent(in)` argument.
`dim`: Shall be a scalar of type `integer` with a value in the range from 1 to `n`, where `n` is the rank of `array`. It is an `intent(in)` argument.
`mask` (optional): Shall be of type `logical` and either a scalar or an array of the same shape as `array`. It is an `intent(in)` argument.
### Return value
If `array` is of type `real`, the result is of type `real` with the same kind as `array`.
If `array` is of type `real` and contains IEEE `NaN`, the result is IEEE `NaN`.
If `array` is of type `integer`, the result is of type `real(dp)`.
If `dim` is absent, a scalar with the median of all elements in `array` is returned. Otherwise, an array of rank `n-1`, where `n` equals the rank of `array`, and a shape similar to that of `array` with dimension `dim` dropped is returned.
If `mask` is specified, the result is the median of all elements of `array` corresponding to `true` elements of `mask`. If every element of `mask` is `false`, the result is IEEE `NaN`.
### Example
```{literalinclude} ../../example/stats/example_median.f90
:language: fortran
moment - central moments of array elementsExperimental
Returns the k-th order central moment of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.
If a scalar or an array center is provided, the function returns the k-th order moment about ‘center’, of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.
The k-th order central moment is defined as :
moment(array) = 1/n sum_i (array(i) - mean(array))^k
where n is the number of elements.
The k-th order moment about center is defined as :
moment(array) = 1/n sum_i (array(i) - center)^k
result = {f:iface}moment (array, order [, center [, mask]])
result = {f:iface}moment (array, order, dim [, center [, mask]])
Generic subroutine
array: Shall be an array of type integer, real, or complex.
order: Shall be an scalar of type integer.
dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array.
center (optional): Shall be a scalar of the same type of result if dim is not provided. If dim is provided, center shall be a scalar or an array (with a shape similar to that of array with dimension dim dropped) of the same type of result.
mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array.
If array is of type real or complex, the result is of the same type as array.
If array is of type integer, the result is of type real(dp).
If dim is absent, a scalar with the k-th (central) moment of all elements in array is returned. Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension dim dropped is returned.
If mask is specified, the result is the k-th (central) moment of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.
```{literalinclude} ../../example/stats/example_moment.f90 :language: fortran
## `pca` - Principal Component Analysis
### Status
Experimental
### Description
Performs Principal Component Analysis (PCA) on a 2D array of observations and features.
The input matrix `x` has shape `(m, n)`, where `m` is the number of observations and `n` is the number of features.
The subroutine computes the principal components (loadings), the singular values, and optionally the feature means.
Two methods are supported:
- `"svd"`: (Default) Computes PCA via Singular Value Decomposition of the centered data. This is generally more numerically stable.
- `"eig"` or `"cov"`: Computes PCA via Eigendecomposition of the covariance matrix.
### Syntax
`call ` {f:subr}`pca` `(x, components, singular_values [, x_mean, method, overwrite_x, err])`
### Class
Generic subroutine
### Arguments
`x`: Shall be a rank-2 real array with shape `(m, n)`. It is an `intent(inout)` argument. If `overwrite_x` is `.true.`, `x` may be modified during computation.
`components`: Shall be a rank-2 real array with shape `(n_components, n)`. It stores the principal components as rows. It is an `intent(out)` argument.
`singular_values`: Shall be a rank-1 real array with shape `(n_components)`. It stores the singular values in descending order. It is an `intent(out)` argument.
`x_mean` (optional): Shall be a rank-1 real array with shape `(n)`. It stores the mean of each feature (column). It is an `intent(out)` argument.
`method` (optional): Shall be a character string. Either `"svd"` or `"eig"`/`"cov"`. It is an `intent(in)` argument.
`overwrite_x` (optional): Shall be a scalar of type `logical`. If `.true.`, the input matrix `x` can be used as a workspace and modified. It is an `intent(in)` argument.
`err` (optional): Shall be of type `linalg_state_type`. It is an `intent(out)` argument.
### Example
```{literalinclude} ../../example/stats/example_pca.f90
:language: fortran
pca_transform - Projects data into principal component spaceExperimental
Projects the input data x into the reduced dimensional space defined by the provided principal components.
The transformation is defined as x_transformed = (x - x_mean) * components^T.
call {f:subr}pca_transform (x, components, x_transformed [, x_mean])
Generic subroutine
x: Shall be a rank-2 real array with shape (m, n). It is an intent(in) argument.
components: Shall be a rank-2 real array with shape (nc, n). It stores the principal components as rows. It is an intent(in) argument.
x_transformed: Shall be a rank-2 real array with shape (m, nc). It stores the projected data. It is an intent(out) argument.
x_mean (optional): Shall be a rank-1 real array with shape (n). It stores the feature means to subtract. It is an intent(in) argument.
pca_inverse_transform - Reconstructs original data from principal component spaceExperimental
Reconstructs the original data representation from the principal component space.
The reconstruction is defined as x_reconstructed = x_reduced * components + x_mean.
call {f:subr}pca_inverse_transform (x_reduced, components, x_reconstructed [, x_mean])
Generic subroutine
x_reduced: Shall be a rank-2 real array with shape (m, nc). It is an intent(in) argument.
components: Shall be a rank-2 real array with shape (nc, n). It stores the principal components as rows. It is an intent(in) argument.
x_reconstructed: Shall be a rank-2 real array with shape (m, n). It stores the reconstructed data. It is an intent(out) argument.
x_mean (optional): Shall be a rank-1 real array with shape (n). It stores the feature means to add back. It is an intent(in) argument.
var - variance of array elementsExperimental
Returns the variance of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.
Per default, the variance is defined as the best unbiased estimator and is computed as:
var(array) = 1/(n-1) sum_i (array(i) - mean(array))^2
where n is the number of elements.
The use of the term n-1 for scaling is called Bessel ‘s correction. The scaling can be changed with the logical argument corrected. If corrected is .false., then the sum is scaled with n, otherwise with n-1.
result = {f:iface}var (array [, mask [, corrected]])
result = {f:iface}var (array, dim [, mask [, corrected]])
Generic subroutine
array: Shall be an array of type integer, real, or complex. It is an intent(in) argument.
dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array. It is an intent(in) argument.
mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array. It is an intent(in) argument.
corrected (optional): Shall be a scalar of type logical. If corrected is .true. (default value), the sum is scaled with n-1. If corrected is .false., then the sum is scaled with n. It is an intent(in) argument.
If array is of type real or complex, the result is of type real corresponding to the type of array.
If array is of type integer, the result is of type real(dp).
If dim is absent, a scalar with the variance of all elements in array is returned. Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension dim dropped is returned.
If mask is specified, the result is the variance of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.
If the variance is computed with only one single element, then the result is IEEE NaN if corrected is .true. and is 0. if corrected is .false..
{literalinclude} ../../example/stats/example_var.f90
:language: fortran