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Suppose I have two variables x and y. x and y have a non-linear relationship. However, they always move in the same direction. In other words, when x goes up, y goes up. When x goes down, y goes down.

Is it correct to say that the correlation between x and y will surely be positive, although the correlation value is unlikely to be near 1 because they have a non-linear relationship?

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they always move in the same direction. In other words, when x goes up, y goes up. When x goes down, y goes down.

i.e. their relationship is monotonic (specifically, monotonic increasing).

Is it correct to say that the correlation between x and y will surely be positive

For a perfectly monotonic relationship, the linear correlation will be positive, though it may be arbitrarily close to 0. (I presume you specifically intend the Pearson correlation coefficient.)

Some other measures of correlation -- specifically ones which pick up monotonic relationships, like the Kendall and Spearman correlation coefficients -- will be 1.

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