non-diatonic chord progression ?

A few general observations before my analysis:

First, If a chord progression is truly non-diatonic, Roman numerals are of no use. That's not important here, because it IS a diatonic progression, as you'll see below.

Second, any Roman numeral analysis of a chord progression that doesn't include "I" is almost always wrong. If you've got a progression, you' should have a tonal center. Here it's pretty clearly Fm - it is the first chord, the last chord, the most frequently used chord, and the closing chord of a V-i cadence at the end.

Third, minor key chord progressions will often have chords that look like they're outside the key. But that's often because you're thinking in terms of major chord progressions... there's only one major scale available in any key, but there are lots of minor scales, which lead to lots more possibilities for a harmony.

Now my analysis:

It's usually helpful to start by writing out the notes of the chords. You've got Fm (F-Ab-C), C (C-E-G), Db (Db-F-Ab), Bb (Bb-D-F), and Eb (Eb-G-Bb).

It's reasonable to assume Fm is your key, for the reasons I stated above. So let's line up all the notes used in the chords starting with F:

F-G-Ab-Bb-C-Db-D-Eb-E

Now let's compare that to the F melodic minor scale:

F-G-Ab-Bb-C-D-E-F (ascending) F-Eb-Db-C-Bb-Ab-G-A (descending)

Every chord tone is represented in the scale. So my first conclusion is that this IS a diatonic progression, and you're clearly in F minor. Now we can apply the Roman numerals:

i - V - bvi - V
i- IV - bVII - i
i - V - bvi - V - i

This makes sense. The first line ends in V - that's called a half cadence. A bvi-V in a minor key is called a Phrygian half cadence. It would be more typical to see v (Cm) or bVII (Eb) as the second chord - that would make things a bit more interesting - but

The second line has IV moving to bVII. A more typical use would be Bbm as the second chord - that's common enough to have its own name (the "backdoor cadence"). But using the melodic minor scale, IV still fits.

The third line is just that Phrygian half cadence again from the first line, but brought to a close with an authentic cadence (V-I) at the end of the line.

Hi NoteBoat- Thank you so~~ much for your kind, great and clear explanation. It's very interesting to know Phrygian half cadence and backdoor cadence too. :D