# Measuring growth rate when values are negative

I am studying nightlights data and trying to model cross-sectional convergence in nightlights in India. However data has some stray light corrections, which has made some of the measurements below zero. Another point being, that the value of 0 has a representative meaning.

Hence any conventional way of measuring growth rates in literature is not applicable here.

In simple words: Nightlights are measured on a relative scale from 0 to 63. The corrections for cloud cover and ephemeral lights makes some of the data points negative. So:

It is analogous to ask what is the percentage growth in nightlights if it increases from - 1 DN to 0 DN

Even more problematic question is what is the percentage growth in nightlights if it increases from 0 DN to 2 DN.

I have already thought of adding an arbitrary but reasonable positive number to data points, to bring everything into a positive domain. But this is a very crude and not an elegant way to do this.

For negative values alone you can define relative change as:

$$\frac{X_t-X_{t-1}}{|X_{t-1}|}$$

This is quite common way to deal with rates of change when you have negative numbers. However, when the denominator is zero then the growth rate is not defined. This can solve any issue when zero is not a base.

When zero is the base unfortunately there is no way how you can calculate growth rate while keeping the zero there. This is because if you start from zero any change represents $$\infty$$ increase. Going from 0 to 1 is $$\infty$$ growth rate as well as going from 0 to 10 or 0 to 1000. You have to either get rid of a zero in some way or another or simply do not calculate growth rate but some other measure of change (e.g. simple change $$X_t -X_{t-1}$$).

I have already thought of adding an arbitrary but reasonable positive number to data points, to bring everything into a positive domain. But this is a very crude and not an elegant way to do this.

Yes, this is very crude way of doing it but perhaps there are more elegant alternatives. Using the temperature as an example a solution would be to convert degrees Celsius to Kelvin (with $$T_{(K)} = T_{(°C)} + 273.15$$) which would solve the problem.

Perhaps there is an another way to measure the quantity you want. Otherwise, adding an arbitrary constant would be a solution. It is hard to recommend any specific way without knowing all the details of the research.

• Sir, Thank you for your answer. The definition you gave for the relative change still has a problem of dealing with zeros. Value of zero has a substantive meaning and I cannot remove the data points with a value of zero from my sample. Jan 9, 2021 at 11:44
• The temperature was just an example to explain the problem clearly so that my issue is understandable to a large audience. Nightlights are measured on a relative scale by an unit called Digital Number (DN). Jan 9, 2021 at 11:46
• @DrStrangeLove oh ok, I though you actually had a problem with temperature since you been vague about it. Well, if you have zero as a base growth rates are simply not defined, there is no direct way how to apply it. As a matter of fact if your base is zero any increase from zero represents $\infty$ growth. You either need to find some alternative transformation like in the example with temperature (or worst case scenario just adding any constant), or what is the reason why you want to calculate changes of that variable? There might be some alternative solution for that reason
– 1muflon1
Jan 9, 2021 at 11:49
• @DrStrangeLove Maybe you should improve your question then. In your question 80% of what you write are about temperature. Sure, it is only an example but when you offer no real description of what kind of data on nightlights you are considering, it is impossible to help you in any other way than dealing with the example. If the example does not match the data you have in some way, maybe you have chosen a bad example. Jan 9, 2021 at 11:51
• @Jesper Sure I will do that. But the problem I am raising is more general and not specific to me. How do you calculate growth rates when you have negative values and zeroes in your dataset? Jan 9, 2021 at 12:01