NAG Library Function Document
nag_best_subset_given_size_revcomm (h05aac)
1 Purpose
Given a set of features and a scoring mechanism for any subset of those features, nag_best_subset_given_size_revcomm (h05aac) selects the best subsets of size using a reverse communication branch and bound algorithm.
2 Specification
#include <nag.h> |
#include <nagh.h> |
void |
nag_best_subset_given_size_revcomm (Integer *irevcm,
Integer mincr,
Integer m,
Integer ip,
Integer nbest,
Integer *drop,
Integer *lz,
Integer z[],
Integer *la,
Integer a[],
double bscore[],
Integer bz[],
Integer mincnt,
double gamma,
const double acc[],
Integer icomm[],
Integer licomm,
double rcomm[],
Integer lrcomm,
NagError *fail) |
|
3 Description
Given , a set of unique features and a scoring mechanism defined for all then nag_best_subset_given_size_revcomm (h05aac) is designed to find , an optimal subset of size . Here denotes the cardinality of , the number of elements in the set.
The definition of the optimal subset depends on the properties of the scoring mechanism, if
then the optimal subset is defined as one of the solutions to
else if
then the optimal subset is defined as one of the solutions to
If neither of these properties hold then nag_best_subset_given_size_revcomm (h05aac) cannot be used.
As well as returning the optimal subset,
, nag_best_subset_given_size_revcomm (h05aac) can return the best
solutions of size
. If
denotes the
th best subset, for
, then the
th best subset is defined as the solution to either
or
depending on the properties of
.
The solutions are found using a branch and bound method, where each node of the tree is a subset of
. Assuming that
(1) holds then a particular node, defined by subset
, can be trimmed from the tree if
where
is the
th highest score we have observed so far for a subset of size
, i.e., our current best guess of the score for the
th best subset. In addition, because of
(1) we can also drop all nodes defined by any subset
where
, thus avoiding the need to enumerate the whole tree. Similar short cuts can be taken if
(2) holds. A full description of this branch and bound algorithm can be found in
Ridout (1988).
Rather than calculate the score at a given node of the tree nag_best_subset_given_size_revcomm (h05aac) utilizes the fast branch and bound algorithm of
Somol et al. (2004), and attempts to estimate the score where possible. For each feature,
, two values are stored, a count
and
, an estimate of the contribution of that feature. An initial value of zero is used for both
and
. At any stage of the algorithm where both
and
have been calculated (as opposed to estimated), the estimated contribution of the feature
is updated to
and
is incremented by
, therefore at each stage
is the mean contribution of
observed so far and
is the number of observations used to calculate that mean.
As long as , for the user-supplied constant , then rather than calculating this function estimates it using or if has been estimated, where is a user-supplied scaling factor. An estimated score is never used to trim a node or returned as the optimal score.
Setting
in this function will cause the algorithm to always calculate the scores, returning to the branch and bound algorithm of
Ridout (1988). In most cases it is preferable to use the fast branch and bound algorithm, by setting
, unless the score function is iterative in nature, i.e.,
must have been calculated before
can be calculated.
4 References
Narendra P M and Fukunaga K (1977) A branch and bound algorithm for feature subset selection IEEE Transactions on Computers 9 917–922
Ridout M S (1988) Algorithm AS 233: An improved branch and bound algorithm for feature subset selection Journal of the Royal Statistics Society, Series C (Applied Statistics) (Volume 37) 1 139–147
Somol P, Pudil P and Kittler J (2004) Fast branch and bound algorithms for optimal feature selection IEEE Transactions on Pattern Analysis and Machine Intelligence (Volume 26) 7 900–912
5 Arguments
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument
irevcm. Between intermediate exits and re-entries,
all arguments other than bscore must remain unchanged.
- 1:
irevcm – Integer *Input/Output
On initial entry: must be set to .
On intermediate exit:
and before re-entry the scores associated with
la subsets must be calculated and returned in
bscore.
The
la subsets are constructed as follows:
- The th subset is constructed by dropping the features specified in the first lz elements of z and the single feature given in from the full set of features, . The subset will therefore contain features.
- The th subset is constructed by adding the features specified in the first lz elements of z and the single feature specified in to the empty set, . The subset will therefore contain features.
In both cases the individual features are referenced by the integers
to
m with
indicating the first feature,
the second, etc., for some arbitrary ordering of the features. The same ordering must be used in all calls to nag_best_subset_given_size_revcomm (h05aac).
If
, the score for a single subset should be returned. This subset is constructed by adding or removing only those features specified in the first
lz elements of
z.
If , this subset will either be or .
The score associated with the th subset must be returned in .
On intermediate re-entry:
irevcm must remain unchanged.
On final exit: , and the algorithm has terminated.
Constraint:
or .
- 2:
mincr – IntegerInput
On entry: flag indicating whether the scoring function
is increasing or decreasing.
- , i.e., the subsets with the largest score will be selected.
- , i.e., the subsets with the smallest score will be selected.
For all
and
.
Constraint:
or .
- 3:
m – IntegerInput
On entry: , the number of features in the full feature set.
Constraint:
.
- 4:
ip – IntegerInput
On entry: , the number of features in the subset of interest.
Constraint:
.
- 5:
nbest – IntegerInput
On entry:
, the maximum number of best subsets required. The actual number of subsets returned is given by
la on final exit. If on final exit
then
NE_TOO_MANY is returned.
Constraint:
.
- 6:
drop – Integer *Input/Output
On initial entry:
drop need not be set.
On intermediate exit:
flag indicating whether the intermediate subsets should be constructed by dropping features from the full set (
) or adding features to the empty set (
). See
irevcm for details.
On intermediate re-entry:
drop must remain unchanged.
On final exit:
drop is undefined.
- 7:
lz – Integer *Input/Output
On initial entry:
lz need not be set.
On intermediate exit:
the number of features stored in
z.
On intermediate re-entry:
lz must remain unchanged.
On final exit:
lz is undefined.
- 8:
z[] – IntegerInput/Output
On initial entry:
z need not be set.
On intermediate exit:
, for
, contains the list of features which, along with those specified in
a, define the subsets whose score is required. See
irevcm for additional details.
On intermediate re-entry:
z must remain unchanged.
On final exit:
z is undefined.
- 9:
la – Integer *Input/Output
On initial entry:
la need not be set.
On intermediate exit:
if
, the number of subsets for which a score must be returned.
If
, the score for a single subset should be returned. See
irevcm for additional details.
On intermediate re-entry:
la must remain unchanged.
On final exit: the number of best subsets returned.
- 10:
a[] – IntegerInput/Output
On initial entry:
a need not be set.
On intermediate exit:
, for
, contains the list of features which, along with those specified in
z, define the subsets whose score is required. See
irevcm for additional details.
On intermediate re-entry:
a must remain unchanged.
On final exit:
a is undefined.
- 11:
bscore[] – doubleInput/Output
On initial entry:
bscore need not be set.
On intermediate exit:
bscore is undefined.
On intermediate re-entry:
must hold the score for the
th subset as described in
irevcm.
On final exit: holds the score for the
la best subsets returned in
bz.
- 12:
bz[] – IntegerInput/Output
-
Note: where appears in this document, it refers to the array element
.
On initial entry:
bz need not be set.
On intermediate exit:
bz is used for storage between calls to nag_best_subset_given_size_revcomm (h05aac).
On intermediate re-entry:
bz must remain unchanged.
On final exit: the th best subset is constructed by dropping the features specified in
, for and , from the set of all features, . The score for the th best subset is given in .
- 13:
mincnt – IntegerInput
On entry:
, the minimum number of times the effect of each feature,
, must have been observed before
is estimated from
as opposed to being calculated directly.
If then is never estimated. If then is set to .
- 14:
gamma – doubleInput
On entry: , the scaling factor used when estimating scores. If then is used.
- 15:
acc[] – const doubleInput
On entry: a measure of the accuracy of the scoring function,
.
Letting
, then when confirming whether the scoring function is strictly increasing or decreasing (as described in
mincr), or when assessing whether a node defined by subset
can be trimmed, then any values in the range
are treated as being numerically equivalent.
If then , otherwise .
If then , otherwise .
In most situations setting both and to zero should be sufficient. Using a nonzero value, when one is not required, can significantly increase the number of subsets that need to be evaluated.
- 16:
icomm[licomm] – IntegerCommunication Array
On initial entry:
icomm need not be set.
On intermediate exit:
icomm is used for storage between calls to nag_best_subset_given_size_revcomm (h05aac).
On intermediate re-entry:
icomm must remain unchanged.
On final exit:
icomm is not defined. The first two elements,
and
contain the minimum required value for
licomm and
lrcomm respectively.
- 17:
licomm – IntegerInput
On entry: the length of the array
icomm. If
licomm is too small and
then
NE_ARRAY_SIZE is returned and the minimum value for
licomm and
lrcomm are given by
and
respectively.
Constraints:
- if , ;
- otherwise .
- 18:
rcomm[lrcomm] – doubleCommunication Array
On initial entry:
rcomm need not be set.
On intermediate exit:
rcomm is used for storage between calls to nag_best_subset_given_size_revcomm (h05aac).
On intermediate re-entry:
rcomm must remain unchanged.
On final exit:
rcomm is not defined.
- 19:
lrcomm – IntegerInput
On entry: the length of the array
rcomm. If
lrcomm is too small and
then
NE_ARRAY_SIZE is returned and the minimum value for
licomm and
lrcomm are given by
and
respectively.
Constraints:
- if , ;
- otherwise .
- 20:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_ARRAY_SIZE
-
On entry,
,
.
Constraint:
,
.
The minimum required values for
licomm and
lrcomm are returned in
and
respectively.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_ILLEGAL_COMM
-
icomm has been corrupted between calls.
rcomm has been corrupted between calls.
- NE_INT
-
On entry, .
Constraint: or .
On entry, .
Constraint: .
On entry, .
Constraint: or .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INT_CHANGED
-
drop has changed between calls.
On intermediate entry,
.
On initial entry,
.
ip has changed between calls.
On intermediate entry,
.
On initial entry,
.
la has changed between calls.
On entry,
.
On previous exit,
.
lz has changed between calls.
On entry,
.
On previous exit,
.
m has changed between calls.
On intermediate entry,
.
On initial entry,
.
mincnt has changed between calls.
On intermediate entry,
.
On initial entry,
.
mincr has changed between calls.
On intermediate entry,
.
On initial entry,
.
nbest has changed between calls.
On intermediate entry,
.
On initial entry,
.
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
- NE_REAL
-
, which is inconsistent with the score for the parent node. Score for the parent node is .
- NE_REAL_CHANGED
-
has changed between calls.
On intermediate entry, .
On initial entry, .
has changed between calls.
On intermediate entry, .
On initial entry, .
gamma has changed between calls.
On intermediate entry,
.
On initial entry,
.
- NE_TOO_MANY
-
On entry, .
But only best subsets could be calculated.
- NE_TOO_SMALL
-
On entry,
,
.
Constraint:
,
.
icomm is too small to return the required array sizes.
7 Accuracy
The subsets returned by nag_best_subset_given_size_revcomm (h05aac) are guaranteed to be optimal up to the accuracy of your calculated scores.
8 Parallelism and Performance
nag_best_subset_given_size_revcomm (h05aac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
Users' Note for your implementation for any additional implementation-specific information.
The maximum number of unique subsets of size
from a set of
features is
. The efficiency of the branch and bound algorithm implemented in nag_best_subset_given_size_revcomm (h05aac) comes from evaluating subsets at internal nodes of the tree, that is subsets with more than
features, and where possible trimming branches of the tree based on the scores at these internal nodes as described in
Narendra and Fukunaga (1977). Because of this it is possible, in some circumstances, for more than
subsets to be evaluated. This will tend to happen when most of the features have a similar effect on the subset score.
If multiple optimal subsets exist with the same score, and
nbest is too small to return them all, then the choice of which of these optimal subsets is returned is arbitrary.
10 Example
This example finds the three linear regression models, with five variables, that have the smallest residual sums of squares when fitted to a supplied dataset. The data used in this example was simulated.
10.1 Program Text
Program Text (h05aace.c)
10.2 Program Data
Program Data (h05aace.d)
10.3 Program Results
Program Results (h05aace.r)