Talk:Mean squared error

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[edit] INCONSISTENT ARTICLE TITLE

There is another article published on Wikipedia and titled "Root mean square error". Notice the use of the adjective "square", rather than "squared".

For the sake of consistency, I suggest to use "square" everywhere, including the title of this article, and indicate in the text that "squared" can be also used:

Symbol Preferred name Other name
SS Sum of squares
MS Mean square ("Mean squared" is not correct)
RMS Root mean square ("Root mean squared" is not correct")
MSE Mean square error Mean squared error
RMSE Root mean square error Root mean squared error
MSD Mean square deviation Mean squared deviation
RMSD Root mean square deviation Root mean squared deviation

A disambiguation is also necessary.

It is also true that a google search yields about twice as many hits for "mean square error" as "mean squared error". Cazort (talk) 17:22, 23 December 2007 (UTC)

[edit] MSE

The article does not give explicit formulae of the MSE for the estimators in the example. Could someone fill this in?

Someone has suggested that the page for Root mean square deviation (RMSD) be merged with mean squared error. I do not think that it makes sense to do this for several reasons: 1. MSE is a measure of error, whereas RMSD method for comparing two biological structures. 2. RMSD is used almost exclusively in the context of protein folding, whereas MSE is used to describe statistics 3. Merging the articles would result in losing the meaning of the RMSD article.

Note that root mean squared deviation is different than root mean squared error.

  1. RMSE = estimator of average error, RMSD = estimator of average distance. They're measuring the same thing: differences or variation.
  2. RMSD is used in disciplines other than bioinformatics/biostatistics—try googling RMSD and "electrical engineering", for example.
  3. Merging the articles should preserve the RMSD tie-in.
  • My opinion: -- PdL -- January 11 2007 (UTC)
  1. D in RMSD typically stands for "deviation", not for "distance". The distance and the difference between two scalar values are not exactly the same thing: the distance is the absolute value of the difference.
  2. Deviation is the difference between the real value of a variable and its estimated or expected or predicted or "desired" value (for instance, the mean).
  3. On the other hand, "error" is the difference between an estimated value of a variable and its real value. There are errors "of estimate" as well as errors "of measurement", and they are all with respect to the (often unknown) real value of the variable.
  4. I conclude that a deviation is the additive opposite of an error. I agree that both words indicate differences, but they have not exactly the same meaning, and it is inappropriate to use theme as synonims.

[edit] Content

Sweeping critique: This article is pretty useless to anyone but a math major.

Specific suggestion: If someone agrees with me on the following statement, then it would be helpful if added into the article--

"MSE is also sometimes called the variance; RMSE is also sometimes called the standard deviation."

I'm pretty sure that's correct, but I won't add it without confirmation.

  • Agreed that the article could be made more friendly to those of use who haven't studied statistical theory. BTW, MSE and RMSE estimate the variance and standard deviation. To equate them would be inaccurate. --DanielPenfield 17:40, 1 November 2006 (UTC)

Some indication of how MSE differs from the variance would be useful.

MSE has a lot in common with variance but they are not the same! As an example, suppose you are trying to estimate the mean of a random variable that has a normal distribution with mean m and variance 1. The mean, m, is a fixed number, but it is unknown. Now, suppose you take a sample from this random variable. If you try to estimate m, your estimator is taking the sample and using it to guess m. A simple case would be to take one sample and have your guess for a be whatever value is picked. But you could also pick many samples and use, say, the median of your sample as the estimator. Or you could just guess the value 0 no matter what (this would be a bad estimator but it would still be an estimator!). The variance of the estimator is going to be the amount by which the estimator varies about ITS mean, not the true mean. The MSE is the amount that the estimator varies about its TRUE mean, which in this example is the number m. For an unbiased estimator, the MSE and the variance are the same. But often, it is not possible to find an unbiased estimator, or in cases a biased estimator might be preferred. I hope this answers the questions given here. Cazort (talk) 17:18, 23 December 2007 (UTC)

"MSE is also sometimes called the variance; RMSE is also sometimes called the standard deviation." Well, the MSE is a random variable itself that needs to be estimated. It's not just a number. If it has been estimated, it gives a measures of the variation of an estimator with repect to a known parameter. But it is not the variance as it also accounts for the bias of the estimator. Squim 10:59, 24 December 2006 (UTC)

In Examples, is it really true that S^2 = \frac{1}{n}\sum_{i=1}^n\left(X_i-\overline{X}\,\right)^2 has a lower MSE than the unbiased estimator S^2 = \frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}\,\right)^2? I agree that it has a lower variance, but this is offset when calculating the MSE by the bias term.

Providing a practical example with real numbers would be desirable. —Preceding unsigned comment added by 131.203.101.15 (talk) 22:14, 16 September 2007 (UTC)

[edit] Squared error loss

Squared error loss redirects to this page, and appropriately so. However, because of this, and because that term is fairly commonly used, and also because this is a question of naming and definitions, I think that the remark about the term squared error loss should remain at the very top of the page, with the definition. Cazort (talk) 23:08, 26 December 2007 (UTC)

[edit] Accessibility of this Article

I first want to say that I am fully committed to making this page accessible, but there are some disagreements about how to do this. In particular, I have been editing the page in order to make it more concise, and removing the explanations/expositions of topics that are duplicated elsewhere. The way I look at things is this:

  • Using technical terms does not necessarily make the article less accessible, nor does replacing them with expanded explanations make it less more so.
  • Defining technical terms within in the article is not the appropriate when it leads to duplication elsewhere on wikipedia. Rather, these terms should be referenced on other pages. This is the whole point of wikipedia! Wikipedia is based around the idea of a web of knowledge, not a more-or-less linear exposition of knowledge like is found in most textbooks.

Cazort (talk) 00:15, 28 December 2007 (UTC)

[edit] Normative statements

I think that a statement "The error is phrased as a mean of squares ... because ..." is problematic because it does not specify what is meant by "because". I think that there are different things going on here, which is that MSE is used in some circumstances solely out of convenience and because the exact choice of loss function has little bearing on the result. However, in other situations, it is used because it approximates some loss function arising in utility theory. In other situations, it might be inappropriate. Still more, there are circumstances where there are compelling theoretical reasons to use it--such as its direct relationship to the expected value, whereas mean absolute error is a more natural way to measure the error of an estimate of a median. On these grounds, I would like to say that I don't think we should avoid normative statements about the loss functions, rather, I think we really ought to include them, and to discuss in more detail exactly why MSE is used in different situations. Cazort (talk) 00:28, 28 December 2007 (UTC)