Solved – Struggling with data transformations that can produce negative values

data transformation

My question:
Which data transformations of positive values produces negative values on the transformed scale?

The context:
I'm looking at a recently published paper where there is a figure that reports that the x-axis is in meters on a log-10 transformed scale. I'm very interested what that actual range of the raw distance data was. The original values should have varied from 0 to perhaps 5000 m (though they don't indicate the actual distances in the article). However, the values on the scale range from -2.0 to 0.5 (see figure reproduced below), and neither log nor natural log transformation can produce negative values. Log values should vary from 0 to ~3.5 and natural log from 0 to ~8.5. They used an arcsine transformation for percentage data elsewhere in the paper, but that also cannot produce negative values (and cannot deal with values great than 1, I believe). Square root transformations obviously cannot produce negative values either.

enter image description here

Best Answer

I see no need to answer the general question, as I think your post just arises from a misunderstanding of logarithms.

The logarithm of any positive number to any base is negative when that number is below 1. log10(0.01) is -2, for example. By eye the smallest logarithm on the graph is about -1.7, which would be about 0.02 metres.

I am not impressed by the graph. It is easy enough in decent software to use a log scale and also show values in the original units. As your question shows, that will be clearer to many statistics users.

[LATER] Matt Krause seems very likely to be right about miles, not metres. What meaning is there in saying that agriculture is 20 mm away? I suppose you could be standing at the farm gate....

Related Solutions

Solved – Interpretation of log transformed negatively skewed dependent variable

Your original x and your transformed x are inversely related, so, naturally, any relationship that is positive on one will be negative on the other. One way to see this inverse relationship is

x <- rnorm(100)
x2 <- log(1 + max(x) - x)
plot(x, x2)

(I used a normally distributed X here, but it does not matter for these purposes; you could substitute your variable and its transformation).

Further explanation after reading one of @Beka 's comments above.

x2 is a transformed version of x, using the transformation you used. Then I plotted x vs. x2. When x goes up, x2 goes down. So, any relationship between x and some other variable will be reversed between x2 and that variable.

In other words, your findings do not contradict earlier work.

Solved – Which data transformation can improve the performance of MLP neural networks for classification

I don't think left- or reight-skewness is your concern but rather high variance between features. Also, output of transformation doesn't need to be at the same scale. Similar scales also works fine. I suggest you to use zero-mean unit variance on you feature matrix. But why each of your features has 2 different axes? What are corresponding labels of x- and y- axes?

Best Answer

Related Solutions

Solved – Interpretation of log transformed negatively skewed dependent variable

Solved – Which data transformation can improve the performance of MLP neural networks for classification

Related Question