QUANTA <data>,<variable>,k,L
estimates k quanta from the given data set of one variable.
Consider a data set x_1, x_2,..., x_n where each observation is
an approximate integral multiple of one of positive numbers
q_1, q_2,..., q_k where typically k=1 or another small integer.
Our task is to estimate the values of quanta q_1, q_2,..., q_k on the
condition that each of them exceeds a certain minimum value q_min.
D.G.Kendall has in his paper "Hunting quanta" (Royal Society of London.
Mathematical and Physical Sciences A 276, 231-266) proposed using a
"cosine quantogram" of the form
n
phi(q) = sqrt(2/n)* SUM cos(2*pi*eps(i)/q) (Kendall)
i=1
where 0<=eps(i)<q is the remainder when x_i is divided by q.
The q-values of highest upward peaks of this function will be considered
as candidates for quanta.
Our idea is that the quanta are estimated by a selective, conditional
least squares method where the sum
n
ss(q_1,...,q_k) = SUM min[g(x_i,q_1)^2,...,g(x_i,q_k)^2] (SLS)
i=1
where g(x,q) in the least absolute remainder when x is divided by q,
is to be minimized with respect of q_1,...,q_k on the condition that
each q_i is at least q_min.
The default method is SLS.
Then the permitted range of quanta is given by
RANGE=<lower_limit>(<step>)<upper_limit>
and least possible value of a quant is given by
Q_MIN=<q_min>.
All combinations of values given by RANGE are used as starting values
for q_1,...,q_k for a minimization process of ss(q_1,...,q_k) and it
is performed by Powell's conjugate gradient method.
By RES=<quant_number>,<residual>,<coeff> three extra variables are given
for saving corresponding information about the results for each case.
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Example:
DATA X: 3 6 7 9 12 14 15 18 21 24 27 28 30 33 35 END
RANGE=2(0.2)8 Q_MIN=2
QUANTA X,X,2,CUR+1
Data: X Variable: X N=15
ss=0
quant # matches
1 3.000000 11
2 3.500000 4
By specification METHOD=Kendall the cosine quantogram and its highest
peaks are computed.
Then
RANGE=<lower_limit>(<step>)<upper_limit>
gives values for which phi(q) is computed. The q and phi(q) values
are save as a Survo data file COSQUANT and QUANTA creates a simple GPLOT
scheme for plotting the quantogram. Also the highest peaks exceeding
a value given by SCORE_MIN will be listed.
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Example:
DATA X: 3 6 7 9 12 14 15 18 21 24 27 28 30 33 35 END
METHOD=Kendall RANGE=2(0.0001)8 SCORE_MIN=1.5
QUANTA X,X,0,CUR+1 / The third parameter has no meaning in this case.
Data: X Variable: X N=15
GPLOT COSQUANT,quant,score / LINE=1 MODE=SVGA Plot the quantogram!
Peaks of Kendall's Cosine Quantogram:
quant score
2.3350 1.5076
2.9950 3.2941
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