public class Ranks extends Object
The class Ranks can be used to compute the ranks, normal scores, or
exponential scores of the data in X. Ties in the data
can be resolved in four different ways, as specified by member function
setTieBreaker. The type of values returned can vary depending
on the member function called:
GetRanks: Ordinary Ranks
For this member function, the values output are the ordinary ranks of the
data in X. If X[i] has the
smallest value among those in X and there is no other
element in X with this value, then getRanks(i) = 1.
If both X[i] and X[j] have the same smallest value, then
if TieBreaker = 0, Ranks[i] = getRanks([j] = 1.5
if TieBreaker = 1, Ranks[i] = Ranks[j] = 2.0
if TieBreaker = 2, Ranks[i] = Ranks[j] = 1.0
if TieBreaker = 3, Ranks[i] = 1.0 and Ranks[j] = 2.0
or Ranks[i] = 2.0 and Ranks[j] = 1.0.
When the ties are resolved by use of function setRandom,
different results may occur when running the same program at different
times unless the "seed" of the random number generator is set explicitly
by use of Random method setSeed. Ordinarily, there
is no need to call the routine to set the seed, even if there are ties in
the data.
getBlomScores: Normal Scores, Blom Version
Normal scores are 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, inverseNormal, at the ranks
scaled into the open interval (0, 1). In the Blom
version (see Blom 1958), the scaling transformation for the rank
\(r_i(1 \le r_i \le n\), where n is
the sample size is \((r_i - 3/8)/(n + 1/4)\).
The Blom normal score corresponding to the observation with rank
\(r_i\) is
$$\Phi ^{ - 1} \left( {\frac{{r_i - 3/8}}{{n + 1/4}}} \right) $$
where \(\Phi(\cdot)\) 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, and then these normal scores are averaged or
selected in the manner specified by TieBreaker, which is set by the
method setTieBreaker. (Whether the transformations are
made first or ties are resolved first makes no difference except when
averaging is done.)
getTukeyScores: Normal Scores, Tukey Version
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
$$\Phi ^{ - 1} \left( {\frac{{r_i - 1/3}}{{n + 1/3}}} \right) $$
Ties are handled in the same way as discussed above for the Blom normal scores.
getVanDerWaerdenScores: Normal Scores, Van der Waerden Version
In the Van der Waerden version (see Lehmann 1975, page 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
$$\Phi ^{ - 1} \left( {\frac{{r_i }}{{n + 1}}} \right) $$
Ties are handled in the same way as discussed above for the Blom normal scores.
getNormalScores: Expected Value of Normal Order Statistics
The method getNormalScores returns the
expected values of the normal order statistics.
If the value in X[i] is the k-th
smallest, then the value getNormalScores[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 NOBS from a standard normal
distribution. Ties are handled in the same way as discussed above for the
Blom normal scores.
getSavageScores: Savage Scores
The method getSavageScores returns
the expected values of the exponential order statistics.
These values are called Savage scores because of
their use in a test discussed by Savage (1956) (see Lehman 1975). If the
value in X[i] is the k-th smallest, then the
i-th output value output is \(E(Y_k)\), where
\(Y_k\) is the k-th order statistic in a sample of
size n from a standard exponential distribution. The
expected value of the k-th order statistic from an exponential sample
of size n is
$$\frac{1}{n} + \frac{1}{{n - 1}} + \ldots + \frac{1}{{n - k + 1}} $$
Ties are handled in the same way as discussed above for the Blom normal scores.
| Modifier and Type | Field and Description |
|---|---|
static int |
TIE_AVERAGE
In case of ties, use the average of the scores of the tied observations.
|
static int |
TIE_HIGHEST
In case of ties, use the highest score in the group of ties.
|
static int |
TIE_LOWEST
In case of ties, use the lowest score in the group of ties.
|
static int |
TIE_RANDOM
In case of ties, use one of the group of ties chosen at random.
|
| Constructor and Description |
|---|
Ranks()
Constructor for the Ranks class.
|
| Modifier and Type | Method and Description |
|---|---|
static double |
expectedNormalOrderStatistic(int i,
int n)
Returns the expected value of a normal order statistic.
|
double[] |
getBlomScores(double[] x)
Gets the Blom version of normal scores for each observation.
|
double[] |
getNormalScores(double[] x)
Gets the expected value of normal order statistics
(for tied observations, the average of the expected normal scores).
|
double[] |
getRanks(double[] x)
Gets the rank for each observation.
|
double[] |
getSavageScores(double[] x)
Gets the Savage scores (the expected value of exponential order statistics).
|
double[] |
getTukeyScores(double[] x)
Gets the Tukey version of normal scores for each observation.
|
double[] |
getVanDerWaerdenScores(double[] x)
Gets the Van der Waerden version of normal scores for each observation.
|
void |
setFuzz(double fuzz)
Sets the fuzz factor used in determining ties.
|
void |
setRandom(Random random)
Sets the
Random object. |
void |
setTieBreaker(int iTie)
Sets the tie breaker for Ranks.
|
public static final int TIE_AVERAGE
public static final int TIE_HIGHEST
public static final int TIE_LOWEST
public static final int TIE_RANDOM
public void setTieBreaker(int iTie)
iTie - an int which represents the tie breakerpublic void setFuzz(double fuzz)
fuzz - a double which represents the fuzz factorpublic void setRandom(Random random)
Random object.random - a Random object used in breaking tiespublic double[] getRanks(double[] x)
x - a double array which contains the observations
to be rankeddouble array which contains the rank for each
observation in xpublic double[] getBlomScores(double[] x)
x - a double array which contains the observations
to be rankeddouble array which contains the Blom version
of normal scores for each observation in xpublic double[] getTukeyScores(double[] x)
x - a double array which contains the observations
to be rankeddouble array which contains the Tukey version
of normal scores for each observation in xpublic double[] getVanDerWaerdenScores(double[] x)
x - a double array which contains the observations
to be rankeddouble array which contains the Van der Waerden version
of normal scores for each observation in xpublic double[] getNormalScores(double[] x)
x - a double array which contains the observationsdouble array which contains the expected value
of normal order statistics for the observations in x
(for tied observations, the average of the expected normal scores)public double[] getSavageScores(double[] x)
x - a double array which contains the observationsdouble array which contains the Savage scores for
the observations in x. (the expected value of exponential order
statistics)public static double expectedNormalOrderStatistic(int i,
int n)
i - an int, the rank of the order statisticn - an int, the sample sizedouble, the expected value of the i-th
order statistic in a sample of size n from the standard
normal distributionCopyright © 2020 Rogue Wave Software. All rights reserved.