Well yes that is a Ti plant.

The problem is probably a combination of things.

Ti Plants do not like it inside the house. They like it outside. Its generally a little too dry for them inside. They will respond to a significant change in their humidity levels by doing just what they are doing...dropping leaves and getting leaves with brown edges.

Depending on the soil in the container and its composition, watering once a week could be too often. You need to check the soil at the bottom of the container where the holes are and see if its waterlogged.

You can try misting the plant several times a day, but the best thing is to put it back outside. They like really good air circulation, and are spider mite magnets in the house.

Ti respond to different lighting conditions depending on the cultivar. Some can sit out in full sun, some like half day sun, and some like bright diffused light. There are several that will lose a lot of their color if they don;t get enough light.

Ti's are hardy to about 30F. We plant them out in the ground here, and they may defoliate in the average frost, or even freeze to the ground in a hard freeze, but they almost never fail to re-grow.

If I were you I would keep it outside as much as I could, like as a porch or entryway plant, and only bring it in when its in danger of freezing

They'll also get tip burn from fluoride buildup, though I agree with bihai that humidity is the more likely issue, if it's started up that quickly.

I don't have that tough of a time with ti plants inside, personally - that could be the particular varieties I've tried, though. Mine have had spider mites a few times, though I wouldn't call them spider mite magnets to the same degree as a croton or English ivy.

All of mine have been indoors for two years; for quite a while now I've had them in a south window, in full sun.

If things get too bad, you can always cut the tops off and root them in water: the base will resprout, and sometimes plants will be better behaved indoors if they start growth already acclimated to it. I don't know that Cordylines are necessarily one of those plants, but it's easy enough to do that you're not risking much, and the stump will usually resprout in more than one spot, making for a fuller plant eventually.

I usually wait til mine get to be about 6-7 ft tall, then I top them to make them branch. They root so easily, you can just stick the cut top into the ground here and it roots into another plant.

I have a few Ti's myself....and they are somewhat difficult to grow indoors.
I've had the best luck with placing the pots of mine (4-6") on top of the soil of larger plants, and near my south window where they get very bright light, with even a little direct light in the morning.

I think the humidity that 'radiates' (for lack of a better word) up from the soil of the larger plant creates a lot of the humidity these need.

I don't mist mine, as I've read that many of the cultivars don't respond well to it.

Call me crazy...but through trial and error, I've also found that they don't like water with a high lime content(which our water has)....so I water mine with distilled water as often as possible.

I've lost some larger Ti's in the past with the same symptoms that yours are showing, so I've tried to steer away from the larger ones and have concentrated on purchasing small ones to give them a better chance to acclimate.

As Bihai stated above, it's likely a combination of factors....and I think your lighting and humidity are the two main ones at work, here.

I know at my house, the *brightest* spaces are always the *driest* spaces!

You could try a 'pebble tray' under this one.