Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
This item sequence assumes an enemy team with a roughly even magic/physical damage composition, slightly favoring magic damage -- hence my getting the Negatron Cloak, Quicksilver Sash, and Force of Nature before the Chain Vest, Glacial Shroud, and Frozen Heart.
If the other team has more four or even five champions that deal mostly magic damage, feel free to substitute a Banshee's Veil for the Frozen Heart. Likewise, if they have four or five physical damage champions, drop the Quicksilver Sash and/or Spirit Visage for a Thornmail/Sunfire Cape. Use your best judgement.
Now, for some justifications:
Wriggle's Lantern over Madred's Bloodrazor: There are two factors for this choice. First, I always find myself appreciating the life steal provided by the Lantern. Second, it costs a LOT less than Madred's. However, when the other team is heavy in tanks or high-health champions, I will save up for a Bloodrazor to up my DPS.
Force of Nature over Banshee's Veil: This is just a matter of preference. Banshee's Veil's spell block and bonus health/mana make up for the lesser magic resistance, but I love the health regeneration from FoN. With it, you will have to go back to base less for healing after a fight, and thus you can farm more and generally have a better map presence.
Frozen Heart over Randuin's Omen: Since the last patch, Frozen Heart overs a full 19 more armor than Randuin's. Plus, it's cheaper, has a slowing effect that isn't too much worse, and the CDR is very, very handy on Udyr. With this and Spirit Visage, your CDR will be almost maxed, letting you stance dance all the more.