[CF-metadata] Usage of histogram_of_X_over_Z

From: martin.juckes at stfc.ac.uk <martin.juckes>
Date: Thu, 27 Oct 2016 15:51:49 +0000

Dear Jonathan,

thanks for that detailed overview. I accept your justification for having the the key physical quantity of interest in the standard name.

In broad usage, I have the impression that a "histogram" can be expressed as either a count or a percentage, so we should be explicit in the convention if we want a narrower definition here. A narrower definition is probably needed, as there would otherwise be no way of distinguishing between the two.

Would you support the addition of a paragraph in the convention to explain the usage you have described below.

There are two further CMIP variables, both or which are bi-variate distributions, with bins of spectral bands and cloud top height ranges, which I'd like to bring into the discussion, but it might be useful to transfer the conclusions of the exchange so far into a ticket first. I think the two additional variables could be covered by a simple extension to "probability_density_function_of_X_and_Y" ... though you might want to insert "joint_" at the beginning of the term.

regards,
Martin



Dear Martin and Alejandro (following off-list discussions)

> The CF definitions say ''"histogram_of_X[_over_Z]" means histogram (i.e. number of counts for each range of X) of variations (over Z) of X.'

Yes, that's in the guidelines for construction of standard names, and there
are only two of them at present, as you say. The simplest case is when you
have some quantity Q depending on only one dimension, Q(Z). Then the histogram
H(Q) is the number of values of Q which fall into each interval of Q,
considering variation over Z. In general there could be more than one
dimension retained, and more than one removed. If the original field was
Q(P,Y,Z,T), we might construct a histogram H(Q,Z,T), for instance, containing
the frequencies of values of Q falling into joint intervals of Q, Z and T, for
variation over P and Y. Following the guideline above, we would call this a
histogram of Q over P and Y, I think.

It is not necessary to indicate in the standard name the dimensions which
the histogram depends on (Z and T in my example) because the coordinate
variables (of Z and T) make that clear. Martin suggests that by this argument
we could also omit Q from the standard name, and just call it a histogram
(or frequency distribution) rather than a histogram of Q, where Q is air
temperature, precipitation amount, backscattering ratio, etc. I think there
are two reasons why we include Q in the standard name,

* I think a histogram of air temperature is not the same geophysical quantity
as a histogram of precipitation amount, for instance, so they should be
distinguished by standard name.

* Although histograms are pure numbers, and so are probabilities, probability
densities are not. Histograms, probability distributions and probability
density functions are all related ways of expressing the same information.
In the guidelines, we foresee that we might need names for all of them (though
so far we have only histograms) and it would make sense to give them consistent
names. The probability density function of air temperature has units of K-1,
and of precipitation amount kg-1 m2, for instance. Because they have different
canonical units, they must have different standard names, so Q needs to be
included in the standard name.

Cell methods describe how the values represent variation within the cells.
The transformation from the values of a quantity to a histogram of the
quantity makes the original quantity into a dimension. This seems more of
a radical transformation than computing a mean or a standard deviation, which
doesn't change the dimensions of the variable, but just reduces their size
(to unity if completely collapsed). A frequency distribution of Q is
regarded as a different geophysical quantity from Q itself, so we have not
used cell methods to describe the relationship. Of course, this is a bit
arbitrary (like everything else in the CF convention!).

I agree with Martin that we could omit the "over" part of the standard name for
histograms, probabilities and probability densities. It is useful to retain the
collapsed dimensions as size-1 dimensions, so that their original range can
be recorded. They could be assigned cell_method of "sum", the default for
extensive quantities, because the histogram applies to their entire range.
The same applies to the variable with has been histogrammed and is now a
dimension; the histogram is a sum for each of its cells.

For example, in the 1D case, suppose the original field is air_temperature
as a function of time only. Then the histogram variable is
  float hair(tair);
    hair:standard_name="histogram_of_air_temperature";
    hair:units="1";
    hair:cell_methods="time: sum tair: sum";
    hair:coordinates="time";
  float time; // scalar coordinate variable with bounds
  float tair(tair);
    tair:units="K";

As a multidimensional example, suppose the original field is
  float tair(time,altitude,latitude,longitude);
    tair:units="K";
    tair:standard_name="air_temperature";
    tair:cell_methods="altitude: mean area: mean time: mean";
from which we might construct
  float pair(tair,time,altitude);
    pair:standard_name="probability_density_function_of_air_temperature";
    pair:units="K-1";
    pair:cell_methods="altitude: mean time: mean area: sum tair: mean";
    pair:coordinates="latitude longitude"; // to record the ranges
Here, I suggest that the cell_method for area is "sum", because the PDF
applies to the whole area, which is an extensive quantity. For air temperature
it seems more sense to interpret a PDF as a mean within cells, since a PDF is
an intensive quantity - you can interpolate it, for example - but not a point
quantity if it's calculated from a histogram with finite bin-widths.

Best wishes

Jonathan
Received on Thu Oct 27 2016 - 09:51:49 BST