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3 . An engineer is choosing between systems to predict a signal x 2 Rd . For a prediction y 2 Rd of the signal
An engineer is choosing between systems to predict a signal x Rd For a prediction
y Rd of the signals values at times i D ; ; : : : ; d the engineer considers three loss
measures. The first takes a threshold t and evaluates
Lx; y D t
d
dX
i D
fjyi xi j t g
where fg D if its argument is true and zero otherwise. The second and the third are
the mean absolute error MAE
Lx; y D
d
dX
iD
jyi xi j D
d ky xk
and the root mean square RMS error
Lx; y D rmsy x D
v
u
u
t
d
dX
i D
yi xi D
r
d ky xk
respectively. The engineer claims that
Lx; ya
Lx; yb
Lx; y:
abonus Justify inequality a
bbonus Justify inequality b
cbonus There are m different possible prediction systems which produce prediction vectors
y; y; : : : ; ym Rd The engineer will choose one of the losses Lk and for the
choice k f; ; g will use the system j satisfying
j D arg min
j f;:::;mg
Lk x; yj :
The engineer only cares that the predictions satisfy jyi xi j : the majority of
the time. Which loss most reflects the engineers preferences? Why
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