Anatolian Neolithic farmer I0708 from the
Mathieson et al. 2015 dataset belongs to Y-haplogroup J2a and is the most Caucasus-shifted of the early Anatolian farmers in my Principal Component Analysis (PCA) of West Eurasia (see below). This is unlikely to be a coincidence and provides strong evidence that at least some Neolithic farmers in western Anatolia harbored Caucasus Hunter-Gatherer (CHG) ancestry.
Note that the two CHG genomes sequenced to date courtesy of
Jones et al. 2015, Kotias and Satsurblia, belonged to Y-haplogroups J and J2a. Moreover, J2 today shows peaks in frequency and diversity in and around the Caucasus. In other words, Y-haplogroup J, and in particular J2, appear to represent paternal signals of CHG admixture.
Unfortunately, it's not yet possible to demonstrate with formal tests beyond any doubt that I0708 has CHG admixture.
For instance, the D-stats below, in which a couple of the least Caucasus-shifted Anatolian farmers are Anatolia Neolithic1, while I0708 is Anatolia Neolithic2, fail to reach significance (Z=3). Please note, I ran the stats with the Amerindian and Siberian samples to test for Ancient North Eurasian (ANE) admixture, which appears to be a feature of CHG.
However, the results are all clearly positive, and might reach significance with higher quality data and/or a better reference than Anatolia Neolithic1.


Indeed, the subtle difference in ANE affinity between Anatolia Neolithic1 and Anatolia Neolithic2 is underlined by the D-stats below. Note that here Kotias shows significant signals of admixture when paired with Anatolia Neolithic1, but not when paired with Anatolia Neolithic2. This is despite the fact that Anatolia Neolithic2 is a higher coverage sequence (6.95x vs 2.66x) and offers more markers.

I0708 is unlikely to be the only early western Anatolian farmer with CHG/ANE admixture. The PCA above show that a couple of others are also pulling strongly towards the Caucasus. Indeed, all of the Anatolian and European Neolithic samples might harbor low levels of CHG ancestry. The problem with testing this idea at present is a lack of more basal Near Eastern ancient genomes from core areas of the Near East, like, say, the Levant.
Hopefully they're on their way, but in any case, it's almost certain now that CHG was already expanding west, and in all likelihood east, during the early Neolithic. This probably has some important implications for the peopling of West Eurasia and their linguistic affinities. Feel free to post what these might be in the comments.
Update 11/02/2016: I came up with new Anatolia Neolithic1 and Anatolia Neolithic2 sets using D-stats (by comparing each of the Anatolians to Kotias versus sample I0708). For a breakdown see
here. Anatolia Neolithic2 now shows significant signals of admixture from Kotias, Dai, Surui and Han. This implies that it not only harbors CHG ancestry, but also ANE and East Asian-related admixtures.