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It annoys me when people do it, a negative correlation (or relationship) means $y=-x$, and inverse means $y=1/x$.
However, I just googled it to gain some evidence to prove my point, and the first link says "A negative correlation means that there is an inverse relationship between two variables - when one variable decreases, the other increases." (Emphasis mine.)
So, do I have to eat my words? Or can I continue to tell people not to call negative relationships inverse?
In physics, more so than other things, one has occasion to say directly related and inversely related when speaking of proportional relationships. That is inexact language use, of the type often called hand waving, with the advantage of helping students uncomfortable with the concept of proportionality to grasp the essentials of proportionality without using the word. More exact phraseology would be directly proportional and inversely proportional. Similarly, it would be rare to use the phrase directly proportional to the negative of something, as it is easier to grasp a negative slope, and rework a phrase to accommodate that.
The concept of inverse proportionality is often approached at the beginner level by hand-waving in this fashion; in this equation as $x$ increases, $y$ decreases. Although true for definite positive inverse proportionality, this has the disadvantage of not being unique, as a negative slope direct proportionality has that same property. In general, to keep the word inverse from causing confusion, all that need be done is to say inverse ______ <-- what and fill in the blank. What operation is an inverse of what other operation depends on which algebraic procedure is being used, for example, subtraction is inverse addition, division is inverse multiplication, deconvolution is inverse convolution, a matrix inverse is the inverse of an invertible matrix, an inverse Laplace transform is the inverse of a Laplace transform, and so on.
The OP's question is: Is it correct to refer to a negative correlation as an 'inverse correlation'? The answer is no, correlation is intransitive, that is, given a correlation one cannot invert the procedure. For it to be an inverse, it would have to be an inverse of some algebraic operation, and the reason for using it in this context is otherwise, namely that it is inverse hand-waving of "as $x$ increases, $y$ increases type," and hand-waving is also not a defined algebraic procedure. The unambiguous terminology for "as $x$ increases, $y$ increases" is monotonically increasing, and, to affirm that this is not local, the term strictly monotonically increasing is used, the inverse of which is then monotonically non-increasing and not strictly monotonically decreasing. Now note, the inverse of monotonically increasing is not monotonically decreasing, which demonstrates what the semantic problem is.
In that light, then, we can state that the phrase "A negative correlation means that there is an inverse relationship between two variables - when one variable decreases, the other increases." is 1) gibberish of the hand-waving type, that 2) uses the word "inverse" improperly, and which when cleaned up by replacing the all of the improper language could read "A negative correlation means that the normalized covariance is negative," furthermore 3) A negative correlation does not imply monotonicity between discrete random variables, the admixture of continuous and discrete parameter types notwithstanding, and if not totally incorrect it is a stretch to define a correlation via an ordinary least squares in $y$ linear regression model that would have monotonicity.
Finally, the OP questions
So, do I have to eat my words? Or can I continue to tell people not to call negative relationships inverse? have the following answers 1) Yes, you are correct, and should not (never mind cannot) eat your words. Negative relationships imply additive inverses not proportional (i.e., multiplicative) inverses, and use of the word inverse in two separate contexts simultaneously is ill advised.
Short answer: You don't have to eat your words. In the context of ratio-scale measurement, negative correlation does not properly measure the inverse functional relationship.
Correlation only measures the linear component of a statistical relationship, so if there is a nonlinear statistical relationship, it may still show up as correlation, but the correlation will not fully describe the relationship. In the case of an inverse relationship $y=1/x$ (which is nonlinear), random sampling of these values certainly be negatively correlated (assuming they are both positive$^\dagger$) so it is correct to say that an inverse relationship does lead to negative correlation. However, there are many other relationships that lead to negative correlation, so it is not correct to say that negative correlation always indicates an inverse relationship.
As some commentators have pointed out, the word "inverse" can be used in its strict mathematical sense, or in a broader sense that refers generally to relationships that are "opposing" in some broad way. Notwithstanding this diversity of usage, in a mathematical context the term "inverse" does have a strict meaning, and so use of this term invites consideration of that find of function.
Whilst it is true that an inverse relationship will lead to negative correlation (again, assuming positive values), describing negative correlation as "inverse correlation" is not good practice, and you are right to be a bit uncomfortable with it. The reason this is not good practice is that correlation measures the linear part of the relationship, not functions of inverse form; calling it "inverse correlation" suggest that it is measuring the inverse component of the relationship, which is not accurate.
$^\dagger$ In fact, it is possible to obtain any correlation value (including positive values or zero) with data from the inverse function $(x_i, y_i,)$ with $y_i = 1/x_i$. To obtain zero correlation or positive correlation you can take some data from the negative-negative quadrant and some data from the positive-positive quadrant. In the body of the answer we rule this out by assuming both values are positive.
I guess the problem comes from a misunderstanding of the correlation term. It is highly accepted that the term correlation refers to linear correlation. Then, in a mathematical terminology, $y=-x$ the best describes negative correlation. Your term, $y=1/x$ is true if we define correlation in a non-linear space.
To me, inverse relationship (correlation) or negative relationship means the same. When one thing goes up the other goes down. The two ends of a teeter-totter have an inverse relationship (negative relationship).
Yes, it is correct, and yes, you would have "to eat your words".
Arguably, this question is off-topic, as it refers to semantics and consensus, which is not a technical statistical issue, but a cultural one.
The top voted answers are biased because they appeal mostly to the same specific sub-culture of yours and people that visit Cross Validated. They define a narrow context in which words are defined and then make ethnocentric statements involving truth or correctness (whatever that is in a socially mediated process).
The truth is that people other than statisticians or mathematically-oriented people use some concepts. And concepts are used to communicate in many cases between these same people. Correlations are one of those concepts that are extensively used by non-mathematically-trained professionals. Given that the word 'inverse' predates any mathematical-specific definition given to it, it is no surprise that it is frequently (and validly) understood as:
turned in the opposite direction, having an opposite course or tendency
A different question is whether it is an ambiguous concept or not. Or, the question others have replied to here, whether the mathematical definition of it applies.
