Reason behind using Manhattan distance

Suppose there are 3 data points:

1 reference data point:

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

And 2 data points in the database:

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (Euclidean distance = 3.87, Squared Euclidean distance = 15, Manhattan distance = 15)

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] (Euclidean distance = 4, Squared Euclidean distance = 16, Manhattan distance = 4)

If noise changes a single 0 into a 4, it will affect Euclidean distance 4x times higher than Manhattan distance. Euclidean distance will pick the first, Manhattan distance will pick the second.

this is a good demonstration! euclidean is sensitive to outliners and prefer the averagely non-bad one rather than some good point with some dimensions being noise.

Shouldn´t you be adding that +1 to the x value before squaring?

Euclidean distance = sqrt( (x+1)^2 )

Manhattan distance = | x+1 |

my case is noise in another dimension ;)

however if noise is added to the main dimension,

it will be

sqrt((1 + x)^2 + 1)

vs

|1 + x | + 1

and if we put two curves together (shifted so that tey intersects on x=0)

http://robowiki.net/w/images/5/5a/C3BD3E15-EEB6-4F63-826F-7C1F5E54A78E.gif

euclidean looks terrible with large noise in one dimension, and manhattan looks robust.