1. Phase and Amplitude Variation in Montreal Weather Jim Ramsay McGill University

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

4. We know that there are two kinds of variation in these data: 1. Amplitude variation: day-to-day and year-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.

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

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

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

9. 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 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. 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. Compare this to 4.30 deg C. for the unconstrained smooth.

11. Plotting the unconstrained B-spline smooth minus the constrained Fourier smooth reveals some striking discrepancies. 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! 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. This temperature would still be cold in mid-January, but less unusual.

13. Registration Let the unconstrained smooth be x(t) and the strictly periodic smooth be x 0 (t). 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. 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.

14. 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 x 0 (t) is linear, and thus whether the two curves are in phase.

15. 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.

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

18. Mid-winter for 1989-1990 arrived about 25 days early. 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. 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 Registration was done by Matlab function registerfd, available by ftp fromego.psych.mcgill.ca/pub/ramsay/FDAfuns

20. 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 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. 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. (2.15 2 – 1.73 2 )/2.15 2 =.35, the proportion of the variation due to phase.

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

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