ranks¶
Computes the ranks, normal scores, or exponential scores for a vector of observations.
Synopsis¶
ranks (x)
Required Arguments¶
- float
x[](Input) - Array of length
nObservationscontaining the observations to be ranked.
Return Value¶
A vector of length nObservations containing the rank (or optionally, a
transformation of the rank) of each observation.
Optional Arguments¶
averageTie, or
highest, or
lowest, or
randomSplit- Exactly one of these optional arguments can be used to change the method used to assign a score to tied observations.
| Argument | Method |
|---|---|
averageTie |
average of the scores of the tied observations (default) |
highest |
highest score in the group of ties |
lowest |
lowest score in the group of ties |
randomSplit |
tied observations are randomly split using a random number generator |
fuzz, float (Input)Value used to determine when two items are tied. If
abs(x[i] −x[j]) is less than or equal tofuzz, thenx[i]andx[j]are said to be tied.Default:
fuzz= 0.0
ranks, or
blomScores, or
tukeyScores, or
vanDerWaerdenScores, or
expectedNormalScores, or
savageScores- Exactly one of these optional arguments can be used to specify the type of values returned.
| Argument | Result |
|---|---|
ranks |
ranks (default) |
blomScores |
Blom version of normal scores |
tukeyScores |
Tukey version of normal scores |
vanDerWaerdenScores |
Van der Waerden version of normal scores |
expectedNormalScores |
expected value of normal order statistics (for tied observations, the average of the expected normal scores) |
savageScores |
Savage scores (the expected value of exponential order statistics) |
Description¶
Ties¶
In data without ties, the output values are the ordinary ranks (or a
transformation of the ranks) of the data in x. If x[i] has the
smallest value among the values in x and there is no other element in
x with this value, then ranks [i] = 1. If both x[i]
and x[j] have the same smallest value, the output value depends on
the option used to break ties.
| Argument | Result |
|---|---|
averageTie |
ranks[i] = ranks[j] = 1.5 |
highest |
ranks[i] = ranks[j] = 2.0 |
lowest |
ranks[i] = ranks[j] = 1.0 |
randomSplit |
or, randomly,
|
When the ties are resolved randomly, function randomUniform (Random Number Generation) is used to generate random numbers. Different results may occur from different executions of the program unless the “seed” of the random number generator is set explicitly by use of the function randomSeedSet (Random Number Generation).
Scores¶
As an option, normal and other functions of the ranks can be returned. Normal
scores can be defined as the expected values, or approximations to the
expected values, of order statistics from a normal distribution. The simplest
approximations are obtained by evaluating the inverse cumulative normal
distribution function, function normalInverseCdf
(Probability Distribution Functions and Inverses), at the ranks scaled into the open interval
(0, 1). In the Blom version (see Blom 1958), the scaling transformation for
the rank \(r_i\) (\(1\leq r_i\leq n\), where n is the sample size,
nObservations) is \((r_i-3/8)/(n+1/4)\). The Blom normal score
corresponding to the observation with rank \(r_i\) is
where Φ(·) is the normal cumulative distribution function.
Adjustments for ties are made after the normal score transformation. That
is, if x [i] equals x [j] (within fuzz) and their value
is the k-th smallest in the data set, the Blom normal scores are
determined for ranks of k and k + 1. Then, these normal scores are
averaged or selected in the manner specified. (Whether the transformations
are made first or ties are resolved first makes no difference except when
averageTie is specified.)
In the Tukey version (see Tukey 1962), the scaling transformation for the rank \(r_i\) is \((r_i-1/3)/(n+1/3)\). The Tukey normal score corresponding to the observation with rank \(r_i\) is as follows:
Ties are handled in the same way as for the Blom normal scores.
In the Van der Waerden version (see Lehmann 1975, p. 97), the scaling transformation for the rank \(r_i\) is \(r_i/(n+1)\). The Van der Waerden normal score corresponding to the observation with rank \(r_i\) is as follows:
Ties are handled in the same way as for the Blom normal scores.
When option expectedNormalScores is used, the output values are the
expected values of the normal order statistics from a sample of size
nObservations. If the value in x[i] is the k-th smallest, the value
output in ranks [i] is \(E(z_k)\), where \(E(\cdot)\) is the
expectation operator and \(z_k\) is the k-th order statistic in a
sample of size nObservations from a standard normal distribution. Ties
are handled in the same way as for the Blom normal scores.
Savage scores are the expected values of the exponential order statistics
from a sample of size nObservations. These values are called Savage
scores because of their use in a test discussed by Savage 1956 (see also
Lehmann 1975). If the value in x[i] is the k-th smallest, the value
output in ranks [i] is \(E(y_k)\), where \(y_k\) is the
k-th order statistic in a sample of size nObservations from a standard
exponential distribution. The expected value of the k-th order statistic
from an exponential sample of size n (nObservations) is as follows:
Ties are handled in the same way as for the Blom normal scores.
Examples¶
Example 1¶
The data for this example, from Hinkley (1977), contains 30 observations. Note that the fourth and sixth observations are tied and that the third and twentieth observations are tied.
from numpy import *
from pyimsl.stat.ranks import ranks
from pyimsl.stat.writeMatrix import writeMatrix
x = [0.77, 1.74, 0.81, 1.20, 1.95, 1.20, 0.47, 1.43,
3.37, 2.20, 3.00, 3.09, 1.51, 2.10, 0.52, 1.62,
1.31, 0.32, 0.59, 0.81, 2.81, 1.87, 1.18, 1.35,
4.75, 2.48, 0.96, 1.89, 0.90, 2.05]
result = ranks(x)
writeMatrix("Ranks", result, writeFormat="%10.1f")
Output¶
Ranks
1 2 3 4 5 6
5.0 18.0 6.5 11.5 21.0 11.5
7 8 9 10 11 12
2.0 15.0 29.0 24.0 27.0 28.0
13 14 15 16 17 18
16.0 23.0 3.0 17.0 13.0 1.0
19 20 21 22 23 24
4.0 6.5 26.0 19.0 10.0 14.0
25 26 27 28 29 30
30.0 25.0 9.0 20.0 8.0 22.0
Example 2¶
This example uses all the score options with the same data set, which contains some ties. Ties are handled in several different ways in this example.
from numpy import *
from pyimsl.stat.randomSeedSet import randomSeedSet
from pyimsl.stat.ranks import ranks
from pyimsl.stat.writeMatrix import writeMatrix
fuzz_value = 0.0
x = [0.77, 1.74, 0.81, 1.20, 1.95, 1.20, 0.47, 1.43,
3.37, 2.20, 3.00, 3.09, 1.51, 2.10, 0.52, 1.62,
1.31, 0.32, 0.59, 0.81, 2.81, 1.87, 1.18, 1.35,
4.75, 2.48, 0.96, 1.89, 0.90, 2.05]
row_labels = ["Blom", "Tukey", "Van der Waerden",
"Expected Value"]
# Blom scores using largest ranks for ties
score0 = ranks(x, highest=True, blomScores=True)
# Tukey normal scores using smallest ranks for ties
score1 = ranks(x, lowest=True, tukeyScores=True)
# Van der Waerden scores using randomly resolved ties
randomSeedSet(123457)
score2 = ranks(x, randomSplit=True,
vanDerWaerdenScores=True)
# Expected value of normal order statistics using
# averaging to break ties
score3 = ranks(x, expectedNormalScores=True)
rank_array = [score0, score1, score2, score3]
writeMatrix("Normal Order Statistics",
rank_array, rowLabels=row_labels,
writeFormat="%9.3f")
# Savage scores using averaging to break ties
score4 = ranks(x, savageScores=True)
writeMatrix("Expected values of exponential order statistics", score4,
writeFormat="%10.3f")
Output¶
Normal Order Statistics
1 2 3 4 5
Blom -1.024 0.209 -0.776 -0.294 0.473
Tukey -1.020 0.208 -0.890 -0.381 0.471
Van der Waerden -0.989 0.204 -0.753 -0.287 0.460
Expected Value -1.026 0.209 -0.836 -0.338 0.473
6 7 8 9 10
Blom -0.294 -1.610 -0.041 1.610 0.776
Tukey -0.381 -1.599 -0.041 1.599 0.773
Van der Waerden -0.372 -1.518 -0.040 1.518 0.753
Expected Value -0.338 -1.616 -0.041 1.616 0.777
11 12 13 14 15
Blom 1.176 1.361 0.041 0.668 -1.361
Tukey 1.171 1.354 0.041 0.666 -1.354
Van der Waerden 1.131 1.300 0.040 0.649 -1.300
Expected Value 1.179 1.365 0.041 0.669 -1.365
16 17 18 19 20
Blom 0.125 -0.209 -2.040 -1.176 -0.776
Tukey 0.124 -0.208 -2.015 -1.171 -0.890
Van der Waerden 0.122 -0.204 -1.849 -1.131 -0.865
Expected Value 0.125 -0.209 -2.043 -1.179 -0.836
21 22 23 24 25
Blom 1.024 0.294 -0.473 -0.125 2.040
Tukey 1.020 0.293 -0.471 -0.124 2.015
Van der Waerden 0.989 0.287 -0.460 -0.122 1.849
Expected Value 1.026 0.294 -0.473 -0.125 2.043
26 27 28 29 30
Blom 0.893 -0.568 0.382 -0.668 0.568
Tukey 0.890 -0.566 0.381 -0.666 0.566
Van der Waerden 0.865 -0.552 0.372 -0.649 0.552
Expected Value 0.894 -0.568 0.382 -0.669 0.568
Expected values of exponential order statistics
1 2 3 4 5 6
0.179 0.892 0.240 0.474 1.166 0.474
7 8 9 10 11 12
0.068 0.677 2.995 1.545 2.162 2.495
13 14 15 16 17 18
0.743 1.402 0.104 0.815 0.555 0.033
19 20 21 22 23 24
0.141 0.240 1.912 0.975 0.397 0.614
25 26 27 28 29 30
3.995 1.712 0.350 1.066 0.304 1.277