Phase and Amplitude Variation in Montreal Weather

Phase and Amplitude Variation
in Montreal Weather
Jim Ramsay
McGill University

The Data
• 34 years of daily temperatures, 1961-1994
1994
inclusive
• Values are averages of daily maximum
and minimum
• 12410 observations in tenths of a degree
Celsius
• Available for Montreal and 34 other
Canadian weather stations

We know that there are two kinds of
variation in these data:
1. Amplitude variation: : day-to
to-day and
year-to
to-year variation in temperature
at events such as the depth of winter.
2. Phase variation: : the timing of these
events -- the seasons arrive early in
some years, and late in others.

Goals
• Separate phase variation from amplitude
variation by registering the series to its
strictly periodic image.
• Estimate components of variation due to
amplitude and phase variation.

Smoothing
The registration process requires that we
smooth the data two ways:
1. With an unconstrained smooth that
removes the day-to
to-day variation, but
leaves longer-term variation unchanged.
2. With a strictly periodic smooth that
eliminates all but strictly periodic trend.

Unconstrained smooth
• Raw data are represented by a B-splineB
expansion using 500 basis functions of
order 6.
• Knot about every 25 days.
• The standard deviation of the raw data
about this smooth, adjusted for degrees of
freedom, is 4.30 degrees Celsius.

Periodic smooth
• The basis is Fourier, with 9 basis functions
judged to be enough to capture most of
the strictly periodic trend for a period of
one year.
• The standard deviation of the raw about
data about this smooth is 4.74 deg C.
• Compare this to 4.30 deg C. for the
unconstrained smooth.

• Plotting the unconstrained B-splineB
smooth minus the constrained Fourier
smooth reveals some striking
discrepancies.
• We focus on Christmas, 1989. The
Ramsay’s s spent the holidays in a chalet
in the Townships, and awoke to –37 deg
C. No skiing, car dead, marooned!
• This temperature would still be cold in
mid-January, but less unusual.

Registration
• Let the unconstrained smooth be x(t) and
the strictly periodic smooth be x 0 (t).
• We need to estimate a nonlinear strictly
increasing smooth transformation of time
h(t), called a warping function, , such that
a fitting criterion is minimized.

Fitting criterion
The fitting criterion was the smallest
eigenvalue of the matrix
⎡
⎢
⎢
⎣
∫
∫
2
x () () ()
0
t dt ∫x0
t x ⎡⎣
h t ⎤⎦
dt⎤
⎥
() ()
2
x ()
0
t x⎡h t ⎤dt x ⎡h t ⎤dt
⎥
⎣ ⎦ ∫ ⎣ ⎦ ⎦
This criterion measures the extent
to which a plot of x[h(t)] against
x 0 (t) is linear, and thus whether
the two curves are in phase.

The warping function h(t)
Every smooth strictly monotone function
h(t) such that h(0) = 0 can be
represented as
t
ht () C exp wvdvdu ( )
u
= ∫ ∫
0 0
We represent unconstrained function
w(v) by a B-spline expansion. Constant
C is determined by constraint h(T) = T.

The deformation d(t) = h(t) - t
Plotting this allows us to see when the
seasons come early (negative
deformation) or late (positive
deformation).

• Mid-winter for 1989-1990 1990 arrived about
25 days early.
• The next step is to register the
temperature data by computing x*(t) =
x[h(t)]. The registered curve x*(t)
contains only amplitude variation.
• Registration was done by Matlab
function registerfd, , available by ftp from
ego.psych.mcgill
mcgill.ca/pub/ .ca/pub/ramsay/FDAfuns

Amplitude variation
• The standard deviation of the difference
between the unconstrained smooth and
the strictly periodic smooth is 2.15 C.
• The standard deviation of the difference
between the registered smooth and the
periodic smooth is 1.73 C.
• (2.15 2 – 1.73 2 )/2.15 2 = .35, the proportion
of the variation due to phase.

• The standard deviation of the raw data
around the registered smooth is 2.13 C,
compared with 2.07 C for the
unregistered smooth.
• About 10% of the total variation is due
to phase.

Conclusions
• Phase variation is an important part of
weather behavior.
• Statisticians seldom think about phase
variation, and classical time series
methods ignore it completely.
• Phase variation needs more attention, and
registration is an essential tool.