phase and amplitude variation in montreal weather
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Phase and Amplitude Variation in Montreal Weather. Jim Ramsay McGill University. The Data. 34 years of daily temperatures, 1961-1994 inclusive Values are averages of daily maximum and minimum 12410 observations in tenths of a degree Celsius

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Presentation Transcript
the data
The Data
  • 34 years of daily temperatures, 1961-1994 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
slide4
We know that there are two kinds of variation in these data:
  • Amplitude variation: day-to-day and year-to-year variation in temperature at events such as the depth of winter.
  • Phase variation: the timing of these events -- the seasons arrive early in some years, and late in others.
goals
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
Smoothing

The registration process requires that we smooth the data two ways:

  • With an unconstrained smooth that removes the day-to-day variation, but leaves longer-term variation unchanged.
  • With a strictly periodic smooth that eliminates all but strictly periodic trend.
unconstrained smooth
Unconstrained smooth
  • Raw data are represented by a B-spline 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
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.
slide11
Plotting the unconstrained B-spline smooth minus the constrained Fourier smooth reveals some striking discrepancies.
  • We focus on Christmas, 1989. The Ramsay’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
Registration
  • Let the unconstrained smooth be x(t) and the strictly periodic smooth be x0(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
Fitting criterion

The fitting criterion was the smallest eigenvalue of the matrix

This criterion measures the extent to which a plot of x[h(t)] against x0(t) is linear, and thus whether the two curves are in phase.

the warping function h t
The warping function h(t)

Every smooth strictly monotone function h(t) such that h(0) = 0 can be represented as

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
The deformation d(t) = h(t) - t

Plotting this allows us to see when the seasons come early (negative deformation) or late (positive deformation).

slide18
Mid-winter for 1989-1990 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.ca/pub/ramsay/FDAfuns

amplitude variation
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.
  • (2.152 – 1.732)/2.152 = .35, the proportion of the variation due to phase.
slide21
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
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.
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