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Re: st: egen and Mata

From (William Gould, StataCorp LP)
Subject   Re: st: egen and Mata
Date   Wed, 16 Jul 2008 09:25:00 -0500

Paulo Guimaraes <> reports, 

> [...] I tried to code in Mata the equivalent to the following Stata
> command:
>           . egen newvar=mean(var), by(id)

His goal was to see if he could write code in Mata that would be faster 
then -egen- (an ado-file), and he failed.  He made two attempts.  He reported
these results, and I have duplicated them:

                        ---- Timings by ----         Ratio 
                        Guimaraes      Gould    Gould/Guimaraes
         egen              0.59         0.24          .41
         Attempt 1        65.81        93.94         1.43        (1) 
         Attempt 2         0.86         0.40          .47
           1.  Gould's attempt to reproduce Attempt 1 also 
               resulted in incorrect results.

           2. Times are in seconds.

           3.  Evidently I (Gould) have a faster computer than Paulo, 
               although mine is a middle of the road computer by modern
               standards.  Mine is an Intel 2.13 GHz Core 2 Duo running
               64-bit Stata under Linux.

One of Paulo's questions was why Attempt 1 is running so slowly.  I haven't 
looked into that because, in addition to running slowly, Attempt 1 is 
producing incorrect results on my computer, and I assume it is 
doing the same on Paulo's.  The program has a bug.

Attempt 2, on the other hand, works perfectly and, in fact, is running 
pretty quickly.  Paulo finds Attempt 2 takes 1.46 times as long as egen, 
and on my comp[uter, the multiplier is 1.15.

Paulo wants to know how speed up Attempt 2, too.  Answer:  I don't know 
how.  I made two attempts and produced code that I considered prettier 
but in fact ran more slowly.  Actually, I wasn't surprised by that. 

My expectation was based on a comparison of what is true and what is
theoretically possible in comparisons of C and Mata.  Mata produces 
P-code, which must be interpreted.  C produces machine language.  
Mata's low-level functions (i=0, i++, etc.) run 2 to 4 times slower than 
compiled C code.  Said differently, they run at an overhead of 
of 1:1 to 3:1.  One-to-one (1:1) means that, to execute one machine 
instruction on behalf of the user, Mata executes one additional instruction
just to figure out what needs to be done.  Three-to-one means three additional
instructions are executed to execute the one instruction on behalf of the

1:1 may seem like a theoretical lower bound, but it is not.  Consider 
a high-level function such as matrix inversion.  It might take the Mata 
P-code interpreter 5 or even 10 machine instructions to figure out what 
needs to be done, and then it would execute a million machine instructions 
that were needed for the problem, the same million instructions C would 
have to execute, too.  Thus, the overhead would be .000005:1.  

But for very low level instructions for which there is a one-to-one matching 
with the instruction set on the chip, such as i++, 1:1 is a theoretical 
lower bound.  So why not produce machine language rather than P-code?
Because in that case, the compiled code would not be portable. 

All the above has to do with a comparison of Mata and C.  Paulo is comparing
Mata to ado-code, however.  The ado-code to which Paulo is comparing, 
however, is not typical or rather, it is what should be typical when 
ado-code is used for the appropriate problem.  In this case, most of the 
ado-code time is being spent in just a few highly efficient Stata 
commands which themselves are implemented in C.  

The problem Paulo set up,

            . egen newvar=mean(var), by(id)                      (1)

could have been solved via 

            . sort id 
            . by id: gen newvar = sum(var)/_n                     (2)
            . by id: replace newvar = newvar[_N] 

On my computer, the time required for solution (1) was 0.24 seconds. 
Solution (2) would have taken 0.11 seconds.  In fact, solution (2) 
is exactly what -egen- implements.  So -egen- is running at 
45.8% efficiency.  What could have been done by Stata's C code in 
.11 seconds, took -egen- .24 seconds.

As an aside, let me add that -sort-, -by id:-, -gen-, and -replace-
are four features we have worked very hard on to make as efficient 
as possible.  There are 4,901 lines of C code involved in implementing 
these functions for the problem given.

In Paulo's Alternative 2, he wrote 27 lines of code, and on my computer, it
ran in .47 seconds.  Thus, the ratio is .47/.11 = 4.27.  This is about the
ratio we expect to see on low-level code.

Paulo did as well as could be expected.

Note well, when Mata is used on problems requiring matrix inversion or other
heavy lifting, which includes implementation of statistical commands for which
Mata was designed, this multiplier of 4.27 falls, and can even fall to 2 or
even just a little more than 1.  That is why we use Mata for our statistical
development.  The fact remains, however, that for problems where the solution
involves only Mata's low-level features, we expect a multiplier of 4.

So when I saw Paulo's question, I knew I was unlikely to have a better 
answer for him.  Nonetheless, I tried.

Here, for Paulo's information, was my final result:

        void function mean_by2(string scalar var, string scalar groupid)
                real scalar     i, j, j0, j1, sum, mean
                real colvector  id, y, p, y2
                transmoprhic        info

                st_view(id, ., groupid)
                st_view(y, ., var)
                p = order(id, 1)

                info=panelsetup(my_y, 1)
                st_view(y2, ., st_addvar("double", "_y2"))

                for (i=1; i<=rows(info); i++) {
                        j0 = info[i, 1]
                        j1 = info[i, 2]
                        sum = 0
                        mean = mean(my_y[|j0\j1|])
                        for (j=j0; j<=j1; j++) y2[p[j]] = mean

My embarrassment is that this took 0.77 seconds to run on my computer, 
versus .47 seconds for Paulo's Attempt 2.  I am nonetheless convinced 
that a combination of Paulo's code and mine could run faster.  Who knows, 
we might even get down to .40 seconds from Paulo's .47.  

Paulo, my one comment about your Attempt 2 is putting all the data together 
in one matrix and then sorting it.  In my attempt, I keep the data in 
separate vectors, and rather than using sort(), I use order() to obtain 
the permutation vector and then use it where I need it.
My goal was to prevent coding things like data[,2], which requires 
exectracting the second column from a matrix, and will require 
extra time to execute.

-- Bill
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