# How to measure the magnitude of a change in distribution across time

I have a dataset where rows are individuals and I want to measure the magnitude of the change in the distribution of their spending from year to year.

So for Person A:

• T1 his values are: 30% A 30% B and 40% C

• T2 his values are: 90% A 5% B and 5% C

I want to create a measure that will indicate the magnitude of the change that takes into account the change in proportion for each category.

• I have data for around 80K observations and 15 time points for each observation. Essentially I want to identify meaningful deviations in their spending behavior. So, to start, I figured I would need calculate the change from t1 to t2. Later, I except I will have to look at longer periods, but for right now I wanted to keep it as simple as possible

• I'm using this measure as a way to gauge an individuals priorities in terms of how they allocate their money. So if someone who has allocated just 10% of their spending on Category A in T1 but then it becomes 90% in T2. This is more meaningful to me then a person to goes from 70% to 90% during the same time period. I know I could look at simple percent change for each category, but I wasn't sure if there was a more robust way to identify how large the change in behavior is across all of the categories

• Without knowing what would count as a good measure for your scientific question this is very broad. Can you expand on what properties you are aiming for? Nov 1, 2016 at 16:54
• How many years of data do you have? How many individuals are there? By analogy to time series of normal data, there can be a great deal of volatility around a relatively stable trend & little volatility around a rapidly changing trend. Which are you looking for here? W/ 3 categories, the data are multidimensional which means there are different kinds of changes that can occur. Can you specify what type might interest you? (Eg, if you were interested in A vs ~A, then [30, 20, 50] = [30, 60, 10].) Nov 1, 2016 at 18:13
• @mdewey I have data for around 80K observations and 15 time points for each observation. Essentially I want to identify meaningful deviations in their spending behavior. So, to start, I figured I would need calculate the change from t1 to t2. Later, I except I will have to look at longer periods, but for right now I wanted to keep it as simple as possible. Nov 2, 2016 at 17:41
• @gung I'm using this measure as a way to gauge an individuals priorities in terms of how they allocate their money. So if someone who has allocated just 10% of their spending on Category A in T1 but then it becomes 90% in T2. This is more meaningful to me then a person to goes from 70% to 90% during the same time period. I know I could look at simple percent change for each category, but I wasn't sure if there was a more robust way to identify how large the change in behavior is across all of the categories. Nov 2, 2016 at 17:44