Difference between revisions of "Recipe/Theory/Statistics exploration"

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Finding the [[Recipe/Definition|recipe]] of a product based on the list of ingredients and nutritional values, is a statistical problem of [https://en.wikipedia.org/wiki/Linear_regression Linear regression] for which a lot of theory exists. The recipe fitting problem is very specific and it is a bit difficult to find the right statistical approach in all the dense wikipedia pages.
 
Finding the [[Recipe/Definition|recipe]] of a product based on the list of ingredients and nutritional values, is a statistical problem of [https://en.wikipedia.org/wiki/Linear_regression Linear regression] for which a lot of theory exists. The recipe fitting problem is very specific and it is a bit difficult to find the right statistical approach in all the dense wikipedia pages.
== Terminology mapping ==
+
== Simple regression ==
 
We are not used to the terminology used in linear regression statistics, so some translation is needed:
 
We are not used to the terminology used in linear regression statistics, so some translation is needed:
* dependent variables / observed values ('''y''') - these comparable to a nutrient of the product;
+
* dependent variables / observed values ('''y''') - these comparable to a nutrient of the product. Usually there are many values '''y''' observed (n dimensions);
* regressors / independent variable ('''x''') - this is a nutrient in an ingredient;
+
* regressors / independent variable ('''x''') - this is a nutrient in an ingredient. This independent variable kan have multiple dimensions p;
* regression coefficients ('''β''') - the recipe fraction;
+
* regression coefficients ('''β''') - the recipe fraction. In a general situation '''β''' can have p values for each observation (making a matrix);
* error term (ε) - the deviation from the fit;
+
* error term (ε) - the deviation from the fit for each observation;
 +
 
  
 
This gives the relation:
 
This gives the relation:
  
  '''y''' = '''X''' '''β''' + '''ε'''
+
  (1)c'''y''' = '''X''' '''β''' + '''ε'''
 +
 
 +
The recipe estimation is much simpler:
 +
* there is only one product, so only 1 observed value, so n = 1;
 +
* but also p = 1, as there is only one percentage per ingredient;
 +
 
 +
This simplifies the relation (1) to a simple multiplication for products with one nutrient. And in that case '''ε''' = 0.
  
In this matrix equation the '''X''' is the independent variable '''x''' corresponding to each '''β''. This is where the correspondence with [[Recipe/Estimation|recipe estimation]] starts to break. The recipe estimation is much simpler:
+
== Complicating things ==
* there is only one product, so only 1 observed value;
 
* as there is only 1 observed value, we have only one independent variable
 
* and with
 

Revision as of 14:08, 21 January 2025

Finding the recipe of a product based on the list of ingredients and nutritional values, is a statistical problem of Linear regression for which a lot of theory exists. The recipe fitting problem is very specific and it is a bit difficult to find the right statistical approach in all the dense wikipedia pages.

Simple regression

We are not used to the terminology used in linear regression statistics, so some translation is needed:

  • dependent variables / observed values (y) - these comparable to a nutrient of the product. Usually there are many values y observed (n dimensions);
  • regressors / independent variable (x) - this is a nutrient in an ingredient. This independent variable kan have multiple dimensions p;
  • regression coefficients (β) - the recipe fraction. In a general situation β can have p values for each observation (making a matrix);
  • error term (ε) - the deviation from the fit for each observation;


This gives the relation:

(1)cy = X β + ε

The recipe estimation is much simpler:

  • there is only one product, so only 1 observed value, so n = 1;
  • but also p = 1, as there is only one percentage per ingredient;

This simplifies the relation (1) to a simple multiplication for products with one nutrient. And in that case ε = 0.

Complicating things