# Interpolation with mixed types on range

I have the following process: an item of certain type (there are about 20) has real value. After going though processing (which I don't have any information about) changes its value. It may be smaller or larger, but with two properties: 1. If initially zero then stays zero. 2. Always positive

I need to estimate the change in value based on examples. My example data however is not for item by item. I have sets of items each has various types for which I know the initial value of each item and the total value (summed) of the set that contain the items after processing.

• I know how much of each type I have in each set.
• I don't know how much each type contributes the the total sum of each set.
• I can't test all my data, I have billions of items to work with
• The resulting estimation should be very efficient.

My questions are:

1. What method should I use?
2. Is this a textbook problem with textbook methods of analysis and solutions?
3. How do I determine the sample size?

Thank You

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 I find the question rather hard to follow. Is it correctly rephrased as follows? You have sets of items. The items are of various types, and for each set you know how many items of each type it contains. The real value of an item depends only on its type. All you know about the real value of a type is whether it is zero or not, and that it is positive. For each set, you know the total real value of all items in the set. You want to find the real value of each type. You have about 20 types and billions of items. Is that correct? And about how many sets do you have? – joriki Apr 6 '12 at 7:26 I can explain with the following metaphor: I have billions of objects of about 20 types. Each object has a weight before the process. After going through the process the weight changes in a manner that depends only on its type. Then I throw the item into a bucket. I know how many items I have thrown in the bucket, their initial size and their types. I can weigh the bucket but not each item individually. I have a few hundred buckets with varying size containing a total of billions of items. I want to estimate the resulting weight of each item at the beginning of the process. – Sonia Apr 6 '12 at 7:45 That means that if I have $N$ types I would have $N$ functions $\phi_i : [0, \infty)->[0, \infty)$ with $\phi_i (0) = 0$ – Sonia Apr 6 '12 at 7:48 The idea of rephrasing the question was that it might be easier to clarify things if you point out what I got wrong. Since you ignored my rephrasing and instead responded with another description that I find hard to follow and that appears to contradict your original description, we haven't really made any progress in clarifying the question. In the question as posted, you wanted to estimate the change in value, in the new formulation you want to estimate the weight at the beginning of the process. Please either post a clearer description, or preferably respond to my attempt at rephrasing. – joriki Apr 6 '12 at 9:21 joriki, I was trying to be as clear as I can. I appreciate your help a lot. Now, as to your first comment. That is true. I am trying to estimate the portion of each type in the mixed set. But I do know the true initial size (before I add it to the set after processing) of each and every type. I also know that before and after processing the value stays positive, but If initially zero, stays zero, as you wrote. – Sonia Apr 6 '12 at 12:44