The owner of a small grocery store wishes to understand their customers better. Specifically, they'd like...

 

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The owner of a small grocery store wishes to understand their customers better. Specifically, they'd like to segment their customers into a small number of groups with similar purchasing characteristics so as to better target their advertising. They hire you to do this. So, you do the following: • You identify n₁ main categories of product (Bread, Milk, Eggs, ...). • You randomly select n₂ customers and (legally and ethically) track their purchasing habits for a month. Specifically you record the amount that they spend on each category of product in the month. This data is presented in matrix X € RX². The first column corresponds to customer 1, the second to customer 2 and so on. The rows corresponds to the product categories. You decompose the matrix X≈ WH into a product of two nonnegative low-dimensional matrices W = R¹, H = RX² using NMF with r <min{n₁, n₂}. 20 (a) What does the value r represent? (b) What do the columns of W represent? (c) What do the columns of H represent? The owner of a small grocery store wishes to understand their customers better. Specifically, they'd like to segment their customers into a small number of groups with similar purchasing characteristics so as to better target their advertising. They hire you to do this. So, you do the following: • You identify n₁ main categories of product (Bread, Milk, Eggs, ...). • You randomly select n₂ customers and (legally and ethically) track their purchasing habits for a month. Specifically you record the amount that they spend on each category of product in the month. This data is presented in matrix X € RX². The first column corresponds to customer 1, the second to customer 2 and so on. The rows corresponds to the product categories. You decompose the matrix X≈ WH into a product of two nonnegative low-dimensional matrices W = R¹, H = RX² using NMF with r <min{n₁, n₂}. 20 (a) What does the value r represent? (b) What do the columns of W represent? (c) What do the columns of H represent?

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