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I am reading Marcos de Prado's Advances in Financial Machine Learning. In a section titled "the ETF Trick", he explains how to calculate periodic price and volume samples for a basket of securities. I have a number of questions about his methodology.

First, he defines the number of units held for instrument i at time t as:

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where Kt is the AUM of the basket, w_it is the weight of instrument i, o_t is the open price, and phi_t is the close price of i in USD. B is referred to as "bars", which as I understand is a time period sampled according to a fixed frequency defined in the units of t. This brings me to my first question: why is h_it defined as a function of the open price, if t is a subset of B?

Second question is likely related to the first. He further defines the change in market value of instrument i, as:

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where p_it is the close price. Why, again, the change in market value is a function of the open price, when t-1 is a subset of B?

Then, he defines the nominal price of the basket of securities, as:

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where d(t) is the dividend at a time t. This equation generally makes sense to me. I am just quoting it here for context.

Last, he defines volume traded for the basket of securities as determined by the "least active member" in the basket:

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Here, I have two questions. (1) Why the least active member? (2) Why normalize by the number of units held for that security, i.e. what is the purpose of the denominator |h_i,t-1|? Thank you!

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