Wikipedia:Reference desk/Archives/Mathematics/2011 March 19

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 19

Separating variables in a poly nomail in multivariables

what is the criteria that a polynomial with integer coefficients in two variables as f(x,y) can be written as g(x).h(y) — Preceding unsigned comment added by True path finder (talk • contribs) 11:52, 19 March 2011 (UTC)

If the polynomial is $\textstyle \sum _{i,j}a_{ij}x^{i}y^{j}$, then consider the matrix $(a_{ij})$ formed by the coefficients. This matrix must be the product of a single column vector with a single row vector. Or, equivalently, all the rows (or columns) must be integer multiples of a single vector -- which, if it exists, you can find by taking any nonzero row (or column) and dividing through by the greatest common divisor of its elements. –Henning Makholm (talk) 12:45, 19 March 2011 (UTC)

Linear equation system with two solutions

there is a system of linear equation with exactly two solution, state true or false if false give reason —Preceding unsigned comment added by 219.65.191.75 (talk) 14:18, 19 March 2011 (UTC)

Yes – for example the single equation x+y=0 over F₂. –Henning Makholm (talk) 21:50, 19 March 2011 (UTC)

homomorphism from S₄ to S₃

Can someone tell me an explicit homomorphism from S₄ to S₃ which has kernel V₄, the Kleins four group? Thanks---Shahab (talk) 16:25, 19 March 2011 (UTC)

There are three ways of partitioning a set of four objects into subsets of cardinality two. S4 acts by permutations on the three element set of such partitions. Sławomir Biały (talk) 16:46, 19 March 2011 (UTC)

Can you explain by an example how the action occurs? Its still not clear to me. Thanks--Shahab (talk) 18:03, 19 March 2011 (UTC)

The three partitions are represented by the 2-cycles a=(12)(34), b=(14)(23), c=(13)(24). Conjugation by S4 permutes these 2-cycles. For instance, conjugating by (1234) interchanges a with b and fixes c, so it goes over to the permutation (ab)c in S3. Sławomir Biały (talk) 18:28, 19 March 2011 (UTC)

Calculating one variable from six others

Hello. I have data for six different variables. A seventh variable is then calculated or computed based on the data from the first six variables. I do not know exactly how the seventh variable is calculated, however. (That is, I don't know the exact formula.) It may be based on one, or two, or some, or all of the other six variables. But, it is definitely based on them, somehow. Is there some way to figure this out ... some type of correlation or regression (or whatever mathematical term)? I am 100% sure that the seventh variable is a function of (one, or some, or all) of the other six variables. That is, there is a 100% correlation. Is there a way to go about figuring this out? Is there some way to determine the exact formula used to calculate the seventh variable from the other six? Please keep any statistical (or even, mathematical) explanations at a pretty basic and elementary level, nothing too complicated. I can supply the actual data, if necessary. There are only about 10 pieces of data for each of the seven variables. Thank you. (Joseph A. Spadaro (talk) 18:57, 19 March 2011 (UTC))

Ideally you could change one input variable while keeping the other 5 identical, to determine if this has any effect on the calculated value. However, with only ten "runs", you aren't likely to have this case unless each run is specifically designed that way. Having random values in each run makes this a much harder problem which would likely require a computer program. StuRat (talk) 20:28, 19 March 2011 (UTC)

I'm sorry ... I did not understand a word of what you said. Should / can I clarify my original question in some way? Thanks. (Joseph A. Spadaro (talk) 20:39, 19 March 2011 (UTC))

OK, here's an example:

       I N P U T   V A R I B L E S     O U T P U T   
      VAR1 VAR2 VAR3 VAR4 VAR5 VAR6        VAR7
----  -----------------------------    ----------
RUN1   1.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN2   2.0  2.0  3.0  4.0  5.0  6.0        -2.0 
RUN3   3.0  2.0  3.0  4.0  5.0  6.0        -3.0 
RUN4   4.0  2.0  3.0  4.0  5.0  6.0        -4.0 
RUN5   5.0  2.0  3.0  4.0  5.0  6.0        -5.0 
RUN6   6.0  2.0  3.0  4.0  5.0  6.0        -6.0 
RUN7   7.0  2.0  3.0  4.0  5.0  6.0        -7.0 
RUN8   8.0  2.0  3.0  4.0  5.0  6.0        -8.0 
RUN9   9.0  2.0  3.0  4.0  5.0  6.0        -9.0 
RUN10 10.0  2.0  3.0  4.0  5.0  6.0       -10.0 

In this case we held VAR2-VAR6 constant, while varying VAR1. Thus, we can tell that VAR7 does depend on VAR1. Here's another example:

       I N P U T   V A R I B L E S     O U T P U T   
      VAR1 VAR2 VAR3 VAR4 VAR5 VAR6        VAR7
----  -----------------------------    ----------
RUN1   1.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN2   2.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN3   3.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN4   4.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN5   5.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN6   6.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN7   7.0  2.0  3.0  4.0  5.0  6.0        -1.0  
RUN8   8.0  2.0  3.0  4.0  5.0  6.0        -1.0 
RUN9   9.0  2.0  3.0  4.0  5.0  6.0        -1.0  
RUN10 10.0  2.0  3.0  4.0  5.0  6.0        -1.0 

In this case, VAR7 does not appear to depend on VAR1 (technically it still could, but, for your purposes, we could ignore that). However, I doubt if your data will look anything like this. I suggest you show us the data you have, so we can tailor our responses accordingly. StuRat (talk) 21:53, 19 March 2011 (UTC)

I don't understand your question very clearly. It sounds as if you are unable to calculate values of the seventh variable for arbitrary values of the other six, but only have limited precalculated data. Is that right? What do you mean by "ten pieces of data for each of the seven variables"? Do you mean that you have ten precalculated values of the seventh variable, and for each of these you know the values of the other six variables used to derive it? If that's all you know then the task seems pretty hopeless unless you are lucky that there is a very simple pattern (like variable 7 is a linear function of the other six, say). 86.160.211.135 (talk) 21:25, 19 March 2011 (UTC)

It would help if you could copy some or all of your data here. 92.24.178.214 (talk) 21:57, 19 March 2011 (UTC)

Actual data

Thanks. OK, let me re-phrase my question. Here is the data (below). I purchased a body measurement scale, and these are the results of weighing myself over the past 11 weeks. See here ([1]) for the description of the product and its user's manual. Before using the scale, I input my age (46) and my height (5 feet and 7.5 inches). In the chart below, this is what is happening.

• Column 1 (Weight): the scale physically measures this.
• Column 2 (Body mass index): the scale computes this, using my height and my weight from Column 1.
• Column 3 (Body fat percentage): the scale physically measures this.
• Column 4 (Muscle percentage): the scale physically measures this.
• Column 5 (Visceral fat level): the scale physically measures this and gives a relative (not absolute) index "score" (level); this is a whole number between 0 and 30.
• Column 6 (Body age): the scale calculates this based on the other data.
• Column 7 (Basal metabolic rate): the scale calculates this based on the other data.

Weight	Body mass index	Body fat percentage	Muscle percentage	Visceral fat level	Body age	Basal metabolic rate
159.0	24.5	17.2	39.6	8	41	1,634
158.4	24.4	17.1	39.7	8	41	1,631
156.6	24.1	16.1	40.2	7	39	1,620
156.2	24.1	14.7	41.0	7	39	1,621
154.4	23.8	15.3	40.7	7	38	1,607
154.8	23.9	16.0	40.2	7	38	1,608
155.4	24.0	15.5	40.5	7	39	1,614
154.4	23.8	15.8	40.4	7	38	1,606
153.2	23.6	15.4	40.5	7	37	1,599
153.2	23.6	15.7	40.4	7	37	1,599
154.2	23.8	15.4	40.6	7	38	1,606

So, ultimately, my questions boil down to these. What is the formula by which this scale is computing my Column 6 (Body age) results? What is the formula by which this scale is computing my Column 7 (Basal metabolic rate) results? I have tried analyzing the data and the numbers. I come up with some "close" results, but nothing that seems to "hold up" overall. So, is there any way to figure out exactly what formula produces the results in Column 6 and Column 7? I can only assume that the scale uses the other data (the other 5 columns and, perhaps, my age and height) ... as this is the only information at its disposal. Any thoughts? And is there any mathematical way (regression, correlation, whatever) to know with certainty what exact formula produces the Column 6 and Column 7 results? By the way ... the blue wiki links indicate that I read the relevant articles; I also read the product owner's manual; I also called the company that manufactures this product. So, despite these resources and my efforts, I am still left with these questions. Thanks! (Joseph A. Spadaro (talk) 22:14, 19 March 2011 (UTC))

It might be doing something like rounding or truncating variables before it uses them for the final calculation, and this can make it more difficult to determine the formula. It might also be using "look-up tables", making it even worse. If you list the formula you derived, maybe we can comment on that. StuRat (talk) 22:40, 19 March 2011 (UTC)

Here is one example / for instance. I noticed that "Basal metabolic rate" divided by "Muscle percentage" tends to approximate "Body age". Or "Basal metabolic rate" divided by "Body age" tends to approximate "Muscle percentage" (kinda/sorta). That seemed to work a bit, but then the numbers get skewed a bit. So, it doesn't quite really work. That's one example, off the top of my head. Also ... I think that the rounding and truncating issue will be easily recognized; the decimal numbers will simply be "off" by just a little bit. Also, I very much doubt that look-up tables are involved. I think it's just a straight-forward formula involved. (Joseph A. Spadaro (talk) 22:48, 19 March 2011 (UTC))

Also, the "Basal metabolic rate" seems to go up when the "Body weight" goes up ... and vice versa (when they both seem to go down). But, I can't seem to find some direct proportion (or constant) there? (Joseph A. Spadaro (talk) 22:54, 19 March 2011 (UTC))

Also, you should include your age and height as fundamental inputs, and remove "body mass index" and "body age", since those are derived values. So, even if they are used in the final calc, you could just as well skip them and use their constituent parts from the fundamental input variables. StuRat (talk) 22:44, 19 March 2011 (UTC)

Note that you could use my approach, at least on some of the variables. That is, you could enter different heights while keeping everything else constant, and then repeat for different ages. You might also be able to fake different weights by carrying different objects, while everything else remains constant. StuRat (talk) 22:47, 19 March 2011 (UTC)

Stu Rat ... I appreciate the input. But, your approach is way over my head. All that I am really doing here is fiddling around in Excel with the columns ... and seeing if I can come up with some consistent formula (prediction) that works. (Joseph A. Spadaro (talk) 22:59, 19 March 2011 (UTC))

This is hopeless. There are statistical techniques for dealing with problems like this but they are very complicated and they require far more data than you have. You barely have enough data to determine whether a single variable has a meaningful relationship to a single other variable. Looie496 (talk) 23:36, 19 March 2011 (UTC)

To Looie: This is clearly not hopeless. See Stu Rat's conclusions below. Also, it is false to say that "You barely have enough data to determine whether a single variable has a meaningful relationship to a single other variable". Again, see Stu Rat's conclusions below. Thanks for your, umm, input, though. (Joseph A. Spadaro (talk) 01:09, 20 March 2011 (UTC))

OK, using your data, I think I see the following relationship between Body Age and Basal Metabolic Rate:

BMR = 1276 + (8.75)BA

This works out correctly if we assume that both values have been rounded:

Body Age (rounded)↓   Body Age (not rounded)↓   Basal Metabolic Rate↓
-------------------   -----------------------   ---------------------
     41                   40.9                          1,634
     41                   40.6                          1,631
     39                   39.4                          1,621
     39                   39.3                          1,620
     39                   38.6                          1,614
     38                   37.9                          1,608
     38                   37.8                          1,607
     38                   37.7                          1,606
     38                   37.7                          1,606
     37                   36.9                          1,599
     37                   36.9                          1,599

Of course, we still would need to work out the other relationships. StuRat (talk) 23:40, 19 March 2011 (UTC)

Thanks, Stu Rat. Yes, I think you hit the nail on the head! Thanks! May I ask ... what process led you to that algebraic equation? How did you come up with that? And do you have any ideas for where the BA comes from? Also ... yes, I agree. Many of these numbers have been rounded. Thanks! (Joseph A. Spadaro (talk) 01:06, 20 March 2011 (UTC))

I started with what you said, that Muscle Percentage times Body Age = BMR. I listed those three in a table, and sorted by BMR. I found that BA went up and down with BMR, but that MP did not, so I decided to just look for a relationship between BA and BMR, directly. Next I found the ratio between the highest BMR and lowest (1.022) and between the highest BA and lowest (1.108). This told me that it's not simply a multiplication factor which is applied (or those two numbers would be the same). The next simplest formula to apply is a multiplication factor plus a constant. Since the range of values for BMR is (1634-1599) or 35, and the range of BA values is (41-37) or 4, that makes the multiplication factor 35/4 or 8.75. I applied that to the top case, and got 41(8.75) + X = 1634. Solving for X gave me 1275.25. From there I played around with the rounding a bit to come up with the final answer. StuRat (talk) 03:47, 20 March 2011 (UTC)

Also, BMI is a known linear function of weight. See body mass index. Another a priori relationship is that body fat percentage and muscle percentage are also linearly related. Sławomir Biały (talk) 00:14, 20 March 2011 (UTC)

To Sławomir Biały: Thanks. Yes, my original post indicates that BMI is a function of weight. It also indicates that I already read the article at body mass index. When you say that "body fat percentage and muscle percentage are also linearly related", what do you mean exactly? Thanks! (Joseph A. Spadaro (talk) 01:14, 20 March 2011 (UTC))

To go a step further, there seems to be a simple multiplicative factor between Visceral Fat Level and Body Age:

BA = (16/3)VFL

As before, it's necessary to look at the non-rounded figures to get it to work out:

Visceral Fat Level (rounded and not rounded)↓   Body Age (not rounded and rounded)↓
---------------------------------------------   ----------------------------------
                     8     7.66850                   40.9    41
                     8     7.61500                   40.6    41
                     7     7.38750                   39.4    39
                     7     7.36875                   39.3    39
                     7     7.23750                   38.6    39
                     7     7.10625                   37.9    38
                     7     7.08750                   37.8    38
                     7     7.06875                   37.7    38
                     7     7.06875                   37.7    38
                     7     6.91875                   36.9    37
                     7     6.91875                   36.9    37

Note that we can then derive a direct relationship between VFL and BMR, by substitution:

BMR = 1276 + (8.75)BA

BA = (16/3)VFL

BMR = 1276 + (8.75)(16/3)VFL

BMR =  1276 + (35/4)(16/3)VFL

BMR =  1276 + (140/3)VFL

One general caution I should add is that, since your data is all in a fairly narrow range of values, the relationships I've found are really only accurate within that range (interpolation), and would become increasingly less accurate as you apply them farther outside that range (extrapolation). StuRat (talk) 09:34, 20 March 2011 (UTC)

Wow, thanks! You're a genius! I think you hit all the nails on the head, exactly! Thank you so much! Now, what is the mathematical name for all of this? Isn't it called linear regression (or the line of best fit) or some such? Is there not some easy program or website on the internet where someone can simply enter in the data, and the program (or website) crunches all the numbers and determines that line of best fit? Thank you so much for your time and efforts ... and for explaining it all so well to me! (Joseph A. Spadaro (talk) 14:39, 20 March 2011 (UTC))

You're quite welcome. And congrats, BTW, on losing weight (assuming that the order you provided the info has the older weights listed first). StuRat (talk) 19:35, 20 March 2011 (UTC)

I'm surprised that nobody just used multiple regression. Edit: I think its dumb that that redirects to linear regression. There ought to be a seperate multiple regression article, as it is a common technique. 92.28.241.202 (talk) 16:48, 20 March 2011 (UTC)

Since we don't seem to have an article, here's a link: [2]. StuRat (talk) 19:42, 20 March 2011 (UTC)

Our linear regression article really talks about multiple regression. The single variable case is discussed in Simple linear regression. This is so because multivariate linear regression is more important. -- Meni Rosenfeld (talk) 10:16, 21 March 2011 (UTC)

Something else I should mention is that there may be other relationships which can't be established with the limited data provided. For example, the formula:

BMR = 1276 + (8.75)BA

Could really be:

BMR = 1230 + Age + (8.75)BA

Since all of your data points contain the same age (46), there's no way to determine this from the data. StuRat (talk) 01:51, 22 March 2011 (UTC)