SOS Battles

When you use an Adrenaline Orb to trigger an SOS battle, the opposing pokemon can call for help if they are at low health. The called pokemon tends be that same pokemon as the caller. Sometimes however the pokemon called is different, whether it be an evolution, a predator/prey relationship, or simply unrelated. This happens because they are in the same spot in the equivalent slots. All tables have 8 slots, the encounter slot, and 7 SOS slots, and each slot has 10 spots. Each table has a pattern, and all slots in that table follow that pattern. The patterns are as follows.

20%, 20%, 10%, 10%, 10%, 10%, 10%, 5%, 4%, and 1%.
78%, 20%, 1%, 1%, 0%, 0%, 0%, 0%, 0%, and 0%.
70%, 30%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
100%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
79%, 20%, 1%, 0% 0%, 0%, 0%, 0%, 0%, and 0%.
50%, 30%, 20%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
80%, 20%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
99%, 1%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
50%, 50%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
60%, 20%, 20%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
15%, 10%, 70%, 5%, 0%, 0%, 0%, 0%, 0%, and 0%.
70%, 29%, 1%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
50%, 45%, 5%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
70%, 20%, 10%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
60%, 40%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
90%, 10%, 0%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
50%, 40%, 10%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.
10%, 10%, 20%, 20%, 10%, 10%, 10%, 5%, 4%, and 1%.
59%, 40%, 1%, 0%, 0%, 0%, 0%, 0%, 0%, and 0%.

In the generally seen encounter table compilation though, you don't see these 10 spots. As seen in these pastebins though, you can see the true encounter tables as they work in data. Credits to SciresM.
http://pastebin.com/MwTpYFxn
http://pastebin.com/FpfccU89
http://pastebin.com/z4JZJMWn
http://pastebin.com/ih5SMYe0

Now in Table 1 (Day) for Route 1, you can see this:
Encounters (Levels 2-3): Yungoos (20%), Pikipek (20%), Yungoos (10%), Pikipek (10%), Caterpie (10%), Caterpie (10%), Ledyba (10%), Ledyba (5%), Ledyba (4%), Ledyba (1%)

Simplified, Yungoos appears 30% of the time, Pikipek appears 30% of the time, Caterpie appears 20% of the time, and Ledyba appears 20% of the time. All slots for Table 1 (Day) for Route 1 match this slot makeup, and so if you ever enter an SOS battle with Yungoos/Pikipek/Caterpie/Ledyba in this table, they will only call on themselves. Pretty simple.

If we go to Table 2 (Night) for Route 1 however, you see this:
Encounters (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Metapod (10%), Spinarak (10%), Spinarak (5%), Pichu (4%), Pichu (1%)

Simplified, Alolan Rattata appears 30% of the time, Pikipek appears 20% of the time, Grubbin appears 10% of the time, Caterpie appears 10% of the time, Metapod appears 10% of the time, Spinarak appears 15% of the time, and Pichu appears 5% of the time. Seems simple enough, right?

When you look at the other slots though, you see this:
SOS Slot 1 (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Caterpie (10%), Spinarak (10%), Spinarak (5%), Pikachu (4%), Pikachu (1%)
SOS Slot 2 (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Caterpie (10%), Spinarak (10%), Spinarak (5%), Pikachu (4%), Pikachu (1%)
SOS Slot 3 (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Caterpie (10%), Spinarak (10%), Spinarak (5%), Pikachu (4%), Pikachu (1%)
SOS Slot 4 (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Caterpie (10%), Spinarak (10%), Spinarak (5%), Pikachu (4%), Pikachu (1%)
SOS Slot 5 (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Caterpie (10%), Spinarak (10%), Spinarak (5%), Happiny (4%), Happiny (1%)
SOS Slot 6 (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Caterpie (10%), Spinarak (10%), Spinarak (5%), Pichu (4%), Pichu (1%)
SOS Slot 7 (Levels 3-5): Rattata (Forme 1) (20%), Pikipek (20%), Rattata (Forme 1) (10%), Grubbin (10%), Caterpie (10%), Caterpie (10%), Spinarak (10%), Spinarak (5%), Pichu (4%), Pichu (1%)

As you can see, it isn't the exact same throughout. In SOS slots 1-7, Metapod (10%) is replaced with another Caterpie (10%). And in slots 2-5 Pichu (4%) and Pichu (1%) is replaced with Pikachu (4%) and Pikachu (1%), in slot 6 the Pichu are replaced with Happiny, and in slots 7 & 8, Pichu returns.

What does this mean? If you encounter an Alolan Rattata, a Pikipek, a Grubbin, a Caterpie, or a Spinarak, it follows the first example. If you encounter a Metapod, and a SOS battle triggers, that Metapod will call a Caterpie. SOS battles call on a pokemon from an equivalent spot on another slot. If you encounter a Pichu, and a SOS battle triggers, that Pichu has some options: it can either call on another Pichu from SOS slots 6 and 7, it can call a Pikachu from SOS slots 1-4, or it can even call a Happiny from SOS slot 5. These all appear in the equivalent spots on the slots for Pichu. It's perfectly possible for Pichu to call some Pichu, some Pikachu, some more Pichu, a Happiny, and then more Pichu, as they share those equivalent spots. So if you want Happiny from this location, you have to trigger a SOS battle with a Pichu, and chain encounters by repeatedly knocking out the called pokemon until a Happiny appears.

Some tables have additional SOS encounters, triggered by weather. These can be found easily by searching for Castform.

If you just want to know what pokemon appear at what frequency, go ahead and use the other lists. If you want a better idea which pokemon call what, use the upper pastebins. They accurately list the slots and spots pokemon appear in, so you can have a better idea what to SOS battle against for that certain rare pokemon. If you want a simplified list of which pokemon call what, check my other pastebin here: http://pastebin.com/sV6rFcfA
As a helpful note to know what tables are being used so you know which patches of grass to look in, check the levels listed next to the slots. So if you're looking for Grubbin in the grass with level 2-3 pokemon, you won't find it.

Unfortunately I don't have good data on the frequency of how often a pokemon picks a certain slot, except that it seems like they tend to prefer their own kind. Any help with this is greatly appreciated.