Permission on reproduction of data extracted from a figure I have been studying a recent paper, and wanted to extract the some data from it, in an effort to reproduce it. I wrote to the authors, and got no response, so I got the raw data by digitalizing a figure in the paper where the data is plotted. 

Then I made another plot, where I plot  the raw data together with mine on top of each other, to compare them. If I publish this plot and properly cite their paper, do I need to ask them for permission? 

It looks like I don't, because this is not like copying and pasting a figure of theirs, but I am not 100% sure. 

Please find enclosed the webpage of the publishing company where I plan to publish, which contains information on their policy on the matter. The  chapter relevant to my case seems to be this one.
I plot the raw data together with mine


The plot is yours, the data is theirs. 


  If I publish this plot and properly cite their paper, do I need to ask them for permission? 


No, the citation (and suitable explanation) suffices to attribute the data source. Indeed, as noted by the referenced source (under the heading Reuse of graphs/tables...):

...reuse of raw data does not generally need permission, provided you have plotted 
it into a different graph/table. You should however cite the source of the data.