Libinclude file `rank.gms`

for ranking one-dimensional numeric data.

Developed by Thomas F. Rutherford, Department of Economics, University of Colorado

Usage:

```
$libInclude rank v s r [p]
```

Where:

Type Argument Description Input:v(s)Array of values to be ranked. sA one-dimensional set, the domain of array v.Output:r(s)Rank order of element v(s), an integer between 1 and card(s), ranking from smallest to largest.Optional: (input and output)p(*)On input this vector specifies percentile levels to be computed. On output, it returns the linearly interpolated percentiles.

Note:

`rank`

only works for numeric data. You cannot sort sets.- The first invocation must be outside of a loop or if block. This routine may be used within a
`loop`

or`if`

block only if it is first initialized with blank invocations (`"$libInclude rank"`

in a context where set and parameter declarations are permitted (See Example 3). - The following names are used within these routines and may not be used in the calling program:
rank_tmp rank_u rank_p

- This routine returns rank values and does not return
`sorted vectors`

, however rank values are easily used to produce a sorted array. This can be done using computed "leads" and "lags" in GAMS' ordered set syntax, as illustrated in examples 1 and 3 below.

# Examples

## Example 1: Rank a vector, and display the data in sorted order

```
Set
i 'Set on which random data are defined' / a, b, d, c, e, f /
k 'Ordered set for displaying sorted data' / 1*6 /;
Parameter
x(i) 'Random data to be sorted'
r(i) 'Rank values'
s(k,i) 'Sorted data';
x(i) = uniform(0,1);
$libInclude rank x i r
display x;
* Generate a sorted list using the ordered set k.
* This assignment statement illustrates how the rank orders
* can be used to sort output for display in proper order. This
* statement uses GAMS support for computed "leads" and "lags"
* on the ordered set k. The loop is used to improve execution
* speed for larger dimensional sets:
loop(k$sameas(k,"1"),
s(k+(r(i)-1),i) = x(i);
);
option s:3:0:1;
display s;
```

This example is also part of the GAMS Data Utilities Library, see model [rank01] for reference. It writes the following lines to ex1.lst:

---- 11 PARAMETER x Random data to be sorted a 0.172, b 0.843, d 0.550, c 0.301, e 0.292, f 0.224 ---- 75 PARAMETER s Sorted data 1.a 0.172 2.f 0.224 3.e 0.292 4.c 0.301 5.d 0.550 6.b 0.843

The `rank`

libinclude can be used to also sort variable levels as the following example shows:

```
Set
i 'Set on which random data are defined' / a, b, d, c, e, f /
k 'Ordered set for displaying sorted data' / 1*6 /;
Variable
x(i) 'Random data to be sorted'
Parameter
r(i) 'Rank values'
s(k,i) 'Sorted data';
x.l(i) = uniform(0,1);
$onDotL
$libInclude rank x i r
display x.l;
loop(k$sameas(k,"1"),
s(k+(r(i)-1),i) = x(i);
);
option s:3:0:1;
display s;
```

Before the `"$libInclude rank"`

the dollar control option $onDotL needs to be activated.

## Example 2: Generate percentiles for a random vector

```
Set
i 'Set on which random data are defined' / a, b, d, c, e /
p 'Percentiles (all of them)' / 0*100 /;
Parameter x(i) 'Random data to be sorted';
* Generate the random data on set i:
x(i) = uniform(0,1);
display x;
Parameter
r(i) 'Rank values'
pct(*) 'Percentiles to be computed' / 20 20.0, median 50.0, 75 75.0 /;
* Generate ranks and compute the specified percentiles (Note that
* the rank array, r, is required, even if the values are not used.)
$libInclude rank x i r pct
* Display three percentiles:
display pct;
```

This example is also part of the GAMS Data Utilities Library, see model [rank02] for reference. The random data are displayed as follows in the listing file:

---- 11 PARAMETER x Random data to be sorted a 0.172, b 0.843, d 0.550, c 0.301, e 0.292

The interpolated percentiles are computed as follows:

---- 103 PARAMETER pct Percentiles 20 0.268, 75 0.550, median 0.301

The following code evaluates a full set of percentiles, from 1 to 100. The GAMS special value of EPS is used to represent zero in the percentile calculation. (Percentiles between zero and one are not permitted to avoid misunderstandings about how percentiles are scaled.) The code makes use of Tom Rutherfords plot libInclude available at http://www.mpsge.org/gnuplot/index.html

```
pct(p) = (ord(p) - 1) + eps;
pct("median") = 0;
display pct;
$libInclude rank x i r pct
display pct;
* Plot the results using GNUPLOT:
Set pl(p) / 20, 40, 60, 80, 100 /;
$setGlobal domain p
$setGlobal labels pl
$libInclude plot pct
```

This is the generated plot:

## Example 3: Use RANK to report multisectoral Monte Carlo results

One of the most perplexing challenges in economic modeling with GAMS is to present multisectoral results in an easily interpreted format. One simple idea is to present sectoral results in a sorted sequence to make it easier to identify the most seriously affected sectors. The presentation of results in a multisectoral model is made even more challenging when model results are generated for a randomized set of scenarios. A summary of Monte Carlo results involves reporting both mean results and their sensitivity. One means of characterizing the sensitivity of model results is to report functions of the sample distribution such as the upper and lower quartiles.

This example illustrates how `rank`

can be used to help report results from the Monte Carlo analysis of a multisectoral model.

The complete model used as example here is part of the GAMS Data Utilities Library, see model [rank03] for reference. The following extract of that model generates the result we want to look at.

```
* Determine ranking of sectors by mean impact:
mean(i) = qvalue(i,"mean");
$libInclude rank mean i meanrank
* The following statement creates a tuple matching the ordered
* set, ki, to the set of sectors, i. In this tuple, the sequence of
* assignments corresponds to increasing mean impacts:
imap(ki+(meanrank(i)-ord(ki)),i) = yes;
* Evaluate quartiles of sectoral impacts for each sector:
loop(i,
x(s) = v(s,i);
* Load quartile with the perctiles to be
* evaluated (25th, 50th and 75th):
quartile(qtl) = qv(qtl);
$ libInclude rank x s r quartile
* Save the quartile values:
qvalue(i,qtl) = quartile(qtl);
);
display qvalue;
Parameter results(ki,i,*) 'Summary of impacts (sorted)';
results(ki,i,"mean")$imap(ki,i) = mean(i);
results(ki,i,qtl)$imap(ki,i) = qvalue(i,qtl);
display results;
```

The program produces the following display output:

q25 q50 q75 mean 1 .FOO -13.767 -12.110 -10.671 -12.259 2 .MWO -13.496 -11.982 -10.512 -12.035 3 .SCS -12.244 -10.418 -8.738 -10.588 4 .CLO -8.763 -7.407 -6.158 -7.480 5 .ADM -7.865 -5.601 -3.470 -5.812 6 .CNM -6.437 -5.320 -4.404 -5.461 7 .OTH -5.732 -4.781 -4.012 -4.892 8 .PIP -3.142 -1.930 -0.836 -2.145 9 .OIN -2.124 -1.339 -0.563 -1.358 10.TPP -2.527 -0.988 0.521 -1.041 11.GEO -1.542 -0.826 -0.196 -0.883 12.AGF -1.178 -0.624 -0.072 -0.625 13.ECM 0.137 0.466 0.796 0.441 14.SSM 0.303 0.711 1.079 0.684 15.CON 0.604 1.071 1.479 1.041 16.PST 0.814 1.924 2.966 1.858 17.RLW 1.885 2.787 3.632 2.738 18.OFU 2.789 3.300 3.829 3.330 19.OLE 3.155 3.553 3.964 3.554 20.ELE 3.480 3.896 4.318 3.892 21.PSM 3.733 4.134 4.591 4.168 22.CAT 3.216 4.417 5.461 4.369 23.TRO 3.953 5.123 6.549 5.282 24.OLP 4.614 5.262 5.909 5.295 25.TRD 5.694 6.454 7.250 6.476 26.AIR 5.227 7.182 9.276 7.320 27.FIN 4.919 7.301 10.175 7.640 28.COA 6.581 7.818 9.125 7.878 29.TMS 5.668 8.530 11.086 8.591 30.CHM 8.071 9.299 10.534 9.319 31.TRK 7.031 9.367 11.851 9.577 32.MAR 9.485 13.072 16.424 13.049 33.GAS 4.437 13.183 22.311 13.462 34.FME 14.794 16.617 18.382 16.642 35.NFM 23.299 26.398 29.281 26.263

The tabular report is helpful, but it does not convey the results as immediately as a picture. GNUPLOT's errorbar plot format is a convenient graphical format for portraying this information. The libinclude interface to GNUPLOT does not support this type of plot, so the continuation of the program produces the GNUPLOT command and data files before invoking the GNUPLOT program:

```
* Write out a GNUPLOT file to generate a chart of the results:
File kplt / ex3.gnu /;
put kplt;
kplt.lw = 0;
put "reset"/;
put 'set title "Sectoral Impacts with Quartiles"'/;
put "set linestyle 1 lt 8 lw 1 pt 8 ps 0.5"/;
put "set grid"/;
put 'set ylabel "% change"'/;
put "set xzeroaxis"/;
put "set bmargin 4"/;
put "set xlabel 'sector'"/;
put 'set xrange [1:',card(i),']'/;
put 'set xtics rotate (';
loop(ki, loop(i$imap(ki,i), put '\'/' "',i.tl,'" ', ord(ki):0,',';));
put @(file.cc-1) ')'/;
put "plot 'ex3.dat' notitle with errorbars ls 1"/;
putClose;
File kpltdata / ex3.dat /;
put kpltdata;
kpltdata.nr = 2;
kpltdata.nw = 14;
kpltdata.nd = 6;
loop(ki, loop(i$imap(ki,i), put ord(ki):0,qvalue(i,"mean"), qvalue(i,"q25"), qvalue(i,"q75")/;));
putClose;
execute 'wgnupl32 ex3.gnu -';
```

This is the generated plot:

## Example 4: Repeated computation of percentiles within a loop

The following code extract shows how to do repeated computation of percentiles within a loop while the data is changing. This is an extract of the model [rank04] from the GAMS Data Utilities Library.

```
* Do several iterations, computing percentiles in each step:
loop(iter,
* Substitute a call to the NLP solver by a call to the random
* number generator. In many applications, this substitution
* produces profoundly more sensible results.
*
* solve catchment using nlp maximizing max;
z(week) = uniform(0,1);
* If you want to retrieve percentile values, you need to reassign
* the percentiles that you wish to retrieve at this point in the
* program. If pct() were not reassigned at this point, the INPUT
* values would correspond to the OUTPUTs from the previous call.
pct(pctl) = pct0(pctl);
$ libInclude rank z week rnk pct
pctval(iter,pctl) = pct(pctl);
);
display pctval;
```

Output:

---- 58 Here are the INPUT values of PCT0 and PCT prior to the call to rank: ---- 58 PARAMETER pct0 Percentiles to be computed 50 50.000, 75 75.000, 80 80.000, 95 95.000 ---- 58 PARAMETER pct Percentiles to be computed (input) and those values (output) 50 50.000, 75 75.000, 80 80.000, 95 95.000 ---- 153 Here are the values of PCT0 and PCT after the call to rank: ---- 153 PARAMETER pct0 Percentiles to be computed 50 50.000, 75 75.000, 80 80.000, 95 95.000 ---- 153 Note that rank has changed the OUTPUT value of pct ---- 153 PARAMETER pct Percentiles to be computed (input) and those values (output) 50 0.424, 75 0.662, 80 0.712, 95 0.864 ---- 273 PARAMETER pctval Percentile values in successive iterations 50 75 80 95 iter1 0.345 0.594 0.638 0.922 iter2 0.385 0.633 0.672 0.941 iter3 0.474 0.705 0.766 0.962 iter4 0.627 0.796 0.823 0.978 iter5 0.428 0.682 0.793 0.904 iter6 0.422 0.690 0.729 0.958 iter7 0.558 0.716 0.756 0.902 iter8 0.451 0.638 0.726 0.942 iter9 0.464 0.704 0.755 0.916 iter10 0.564 0.805 0.831 0.974

## Example 5: Generating percentiles for heterogenous households.

```
$title Percentile ranking of household expenditure data with heterogenous household size
Set h / 0*100 /;
Parameter
y(h) 'Aggregate expenditure associated with household type h'
n(h) 'Number of persons associated with household type h'
ypc(h) 'Per-capita expenditure of household type h'
rank(h) 'Rank of household in per-capita expenditure';
* Assign some random values:
y(h) = uniform(0.2,1.2);
n(h) = uniform(1,6);
ypc(h) = y(h)/n(h);
* Assign ranks to household based on per-capita expenditures:
$libInclude rank ypc h rank
* Now determine percentile ranking of the households taking into account
* differences in numbers of members and household representation:
Set r 'Temporary set used for ranking' / r0*r100 /;
Parameter
pcttmp(r) 'Temporary array for computing percentiles'
pct(h) 'Percentile rankings for households';
Set r0(r) / r0 /;
* First, create an array with households assigned
loop((r0(r),h), pcttmp(r+(rank(h)-1)) = n(h););
loop(r, pcttmp(r) = pcttmp(r) + pcttmp(r-1););
pcttmp(r) = pcttmp(r)/sum(h, n(h));
loop((r0(r),h), pct(h) = pcttmp(r+(rank(h)-1)););
Parameter ranking 'Ranking of households and expenditures';
loop((r0(r),h),
ranking(r+(rank(h)-1),h,"n") = n(h);
ranking(r+(rank(h)-1),h,"ypc") = ypc(h);
ranking(r+(rank(h)-1),h,"pct") = pct(h);
);
display ranking;
```

This example is also part of the GAMS Data Utilities Library, see model [rank05] for reference. It produces the following output:

n ypc pct r0 .43 5.662 0.044 0.018 r1 .8 5.682 0.047 0.036 r2 .58 4.477 0.052 0.050 r3 .14 5.742 0.058 0.068 r4 .55 4.815 0.063 0.083 r5 .65 3.702 0.063 0.094 r6 .61 5.880 0.064 0.113 r7 .54 4.463 0.064 0.127 .... r98 .62 1.134 0.640 0.991 r99 .10 1.673 0.716 0.997 r100.73 1.053 1.076 1.000