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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains procedures and generic interfaces for obtaining the Ordinal Ranking of the elements of arrays of various types. More...
Data Types | |
| type | dense_type |
| This is a concrete derived type whose instances are exclusively used to request dense ranking of a given array within an interface of a procedure of the ParaMonte library. More... | |
| type | fractional_type |
| This is a concrete derived type whose instances are exclusively used to request the fractional ranking of a given array within an interface of a procedure of the ParaMonte library. More... | |
| interface | getRankDense |
Generate and return the Dense rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | getRankFractional |
Generate and return the Fractional rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | getRankModified |
Generate and return the Modified rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | getRankOrdinal |
Generate and return the ordinal rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | getRankStandard |
Generate and return the Standard rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| type | modified_type |
| This is a concrete derived type whose instances are exclusively used to request the modified ranking of a given array within an interface of a procedure of the ParaMonte library. More... | |
| type | ordinal_type |
| This is a concrete derived type whose instances are exclusively used to request the ordinal ranking of a given array within an interface of a procedure of the ParaMonte library. More... | |
| type | rank_type |
This is an abstract derived type for constructing concrete derived types to distinguish various procedure signatures that require different forms of array ranking (dense, fractional, modified, ordinal, standard, ...).More... | |
| interface | setRankDense |
Return the Dense rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | setRankFractional |
Return the Fractional rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | setRankModified |
Return the Modified rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | setRankOrdinal |
Return the ordinal rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| interface | setRankStandard |
Return the Standard rank of the input scalar string or contiguous array of rank 1 in ascending order or in the order specified by the input procedure isSorted() using the Quicksort algorithm such that array(rank) will be in ascending order (or in the requested order as specified by isSorted(). More... | |
| type | standard_type |
| This is a concrete derived type whose instances are exclusively used to request the standard ranking of a given array within an interface of a procedure of the ParaMonte library. More... | |
Variables | |
| character(*, SK), parameter | MODULE_NAME = "@pm_arrayRank" |
| type(dense_type), parameter | dense = dense_type() |
This is a scalar parameter object of type dense_type that is exclusively used to request the dense ranking of a given array within an interface of a procedure of the ParaMonte library.More... | |
| type(ordinal_type), parameter | ordinal = ordinal_type() |
This is a scalar parameter object of type ordinal_type that is exclusively used to request the ordinal ranking of a given array within an interface of a procedure of the ParaMonte library.More... | |
| type(modified_type), parameter | modified = modified_type() |
This is a scalar parameter object of type modified_type that is exclusively used to request the modified ranking of a given array within an interface of a procedure of the ParaMonte library.More... | |
| type(standard_type), parameter | standard = standard_type() |
This is a scalar parameter object of type standard_type that is exclusively used to request the standard ranking of a given array within an interface of a procedure of the ParaMonte library.More... | |
| type(fractional_type), parameter | fractional = fractional_type() |
This is a scalar parameter object of type fractional_type that is exclusively used to request the fractional ranking of a given array within an interface of a procedure of the ParaMonte library.More... | |
This module contains procedures and generic interfaces for obtaining the Ordinal Ranking of the elements of arrays of various types.
Depending on the applications, the rank of the elements of an array can be defined in different ways:
Ordinal ranking (1234) ranking: getRankOrdinal or setRankOrdinal
This kind of ranking of values is widely known as ordinal (1234) ranking.
In ordinal ranking, all items receive distinct ordinal numbers, including items that compare equal.
The assignment of distinct ordinal numbers to items that compare equal can be done at random, or arbitrarily, but it is generally preferable to use a system that is arbitrary but consistent, as this gives stable results if the ranking is done multiple times.
In computer data processing, ordinal ranking is also referred to as row numbering. That is, if A < B == C < D, then the sequence ABCD has the ordinal ranking 1234.
Standard competition (1224) ranking: getRankStandard or setRankStandard
This kind of ranking of values is widely known as Standard Competition (1224) ranking.
In Standard Competition ranking, items that compare equal receive the same ranking number, and then a gap is left in the ranking numbers. The number of ranking numbers that are left out in this gap is one less than the number of items that compared equal.
Equivalently, the ranking number of each item is 1 plus the number of items ranked above it.
This ranking strategy is frequently adopted for competitions, as it means that if two (or more) competitors tie for a position in the ranking, and the position of all those ranked below them is unaffected (i.e., a competitor only comes second if exactly one person scores better than them, third if exactly two people score better than them, fourth if exactly three people score better than them, etc.).
Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 (first), B gets ranking number 2 (joint second), C also gets ranking number 2 (joint second) and D gets ranking number 4 (fourth).
That is, if A < B == C < D, then the sequence ABCD has the Standard Competition ranking 1224.
Modified competition (1334) ranking: getRankModified or setRankModified
This kind of ranking of values is widely known as Modified Competition (1334) ranking.
Sometimes, competition ranking is done by leaving the gaps in the ranking numbers before the sets of equal-ranking items (rather than after them as in Standard Competition ranking).
The number of ranking numbers that are left out in this gap remains one less than the number of items that compared equal.
Equivalently, the ranking number of each item is equal to the number of items ranked equal to it or above it.
This ranking ensures that a competitor only comes second if they score higher than all but one of their opponents, third if they score higher than all but two of their opponents, etc.
Thus if A ranks ahead of B and C (which compare equal) which are both ranked head of D, then A gets ranking number 1 (first), B gets ranking number 3 (joint third), C also gets ranking number 3 (joint third) and D gets ranking number 4 (fourth). In this case, nobody would get ranking number 2 (second) (left as a gap).
That is, if A < B == C < D, then the sequence ABCD has the Modified Competition ranking 1334.
Dense (1223) ranking: getRankDense or setRankDense
This kind of ranking of values is widely known as dense (1223) ranking.
In Dense ranking, items that compare equally receive the same ranking number, and the next items receive the immediately following ranking number.
Equivalently, the ranking number of each item is 1 plus the number of items ranked above it that are distinct with respect to the ranking order.
Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 (first), B gets ranking number 2 (joint second), C also gets ranking number 2 (joint second) and D gets ranking number 3 (Third).
That is, if A < B == C < D, then the sequence ABCD has the Dense ranking 1223.
Dense ranking effective factorizes the array into classes of unique values.
Therefore, the Dense rank of each element of the array is simply its class level.
Fractional (1 2.5 2.5 4) ranking: getRankFractional or setRankFractional
This kind of ranking of values is widely known as fractional (1 2.5 2.5 4) ranking.
In Fractional ranking, items that compare equal receive the same ranking number, which is the mean of what they would have under ordinal rankings;
Equivalently, the ranking number of 1 plus the number of items ranked above it plus half the number of items equal to it.
This strategy has the property that the sum of the ranking numbers is the same as under ordinal ranking.
For this reason, it is used in computing Borda counts and ranking statistics (e.g., Spearman Correlation).
Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 (first), B and C each get ranking number 2.5 (average of joint second/third) and D gets ranking number 4 (fourth).
That is, if A < B == C < D, then the sequence ABCD has the Fractional ranking 1223.
Example:
Suppose the data set is 1.0, 1.0, 2.0, 3.0, 3.0, 4.0, 5.0, 5.0, 5.0.
The ordinal ranks are 1, 2, 3, 4, 5, 6, 7, 8, 9.
For v = 1.0, the Fractional rank is the average of the ordinal ranks: (1 + 2) / 2 = 1.5.
In a similar manner, for v = 5.0, the Fractional rank is (7 + 8 + 9) / 3 = 8.0.
Thus the Fractional ranks are: 1.5, 1.5, 3.0, 4.5, 4.5, 6.0, 8.0, 8.0, 8.0
sorted must be added to all interfaces within this module to allow fast computation of the rankings of previously sorted arrays without redundant resorting.getRank() and setRank().real ranks of default kind RK) with the rest (which output integer ranks of default kind IK).
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
| type(dense_type), parameter pm_arrayRank::dense = dense_type() |
This is a scalar parameter object of type dense_type that is exclusively used to request the dense ranking of a given array within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Definition at line 234 of file pm_arrayRank.F90.
| type(fractional_type), parameter pm_arrayRank::fractional = fractional_type() |
This is a scalar parameter object of type fractional_type that is exclusively used to request the fractional ranking of a given array within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Definition at line 482 of file pm_arrayRank.F90.
| type(modified_type), parameter pm_arrayRank::modified = modified_type() |
This is a scalar parameter object of type modified_type that is exclusively used to request the modified ranking of a given array within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Definition at line 358 of file pm_arrayRank.F90.
| character(*,SK), parameter pm_arrayRank::MODULE_NAME = "@pm_arrayRank" |
Definition at line 142 of file pm_arrayRank.F90.
| type(ordinal_type), parameter pm_arrayRank::ordinal = ordinal_type() |
This is a scalar parameter object of type ordinal_type that is exclusively used to request the ordinal ranking of a given array within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Definition at line 296 of file pm_arrayRank.F90.
| type(standard_type), parameter pm_arrayRank::standard = standard_type() |
This is a scalar parameter object of type standard_type that is exclusively used to request the standard ranking of a given array within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Definition at line 420 of file pm_arrayRank.F90.
1.9.3