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Suppose I'm writing a report where I often refer to a particular training dataset and a particular control dataset.

Would it be grammatically correct to refer to them as to the Train dataset and the Control dataset, respectively?

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    "Training" and "Control" yes. This is common. – Andrew Dec 4 '17 at 16:25
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There is a useful convention in legal writing on creating specially defined nouns for the purpose of a single document or a single set of related documents.

"The data set numbered 123 and entitled "Training Dataset for Introductury Calculus, catalog number M101-M102" (hereafter the "Training Dataset") ..." exemplifies a device that lets you exactly define a word or phrase outside its accepted meaning and still use the same word or phrase in its accepted sense by avoiding the initial capital letters.

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