Phase and Amplitude Phase and Amplitude Variation in Montreal Variation in Montreal

WeatherWeather

Jim RamsayJim Ramsay

McGill UniversityMcGill University

The DataThe Data

34 years of daily temperatures, 34 years of daily temperatures, 1961-1994 inclusive1961-1994 inclusive

Values are averages of daily Values are averages of daily maximum and minimummaximum and minimum

12410 observations in tenths of a 12410 observations in tenths of a degree Celsiusdegree Celsius

Available for Montreal and 34 other Available for Montreal and 34 other Canadian weather stationsCanadian weather stations

We know that there are two kinds of We know that there are two kinds of variation in these data:variation in these data:

1.1. Amplitude variationAmplitude variation: day-to-day : day-to-day and year-to-year variation in and year-to-year variation in temperature at events such as the temperature at events such as the depth of winter.depth of winter.

2.2. Phase variationPhase variation: the timing of : the timing of these events -- the seasons arrive these events -- the seasons arrive early in some years, and late in early in some years, and late in others.others.

GoalsGoals

Separate phase variation from Separate phase variation from amplitude variation by amplitude variation by registeringregistering the series to its strictly periodic the series to its strictly periodic image.image.

Estimate components of variation Estimate components of variation due to amplitude and phase due to amplitude and phase variation.variation.

SmoothingSmoothing

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

1.1. With an unconstrained smooth that With an unconstrained smooth that removes the day-to-day variation, removes the day-to-day variation, but leaves longer-term variation but leaves longer-term variation unchanged.unchanged.

2.2. With a strictly periodic smooth that With a strictly periodic smooth that eliminates all but strictly periodic eliminates all but strictly periodic trend.trend.

Unconstrained smoothUnconstrained smooth

Raw data are represented by a B-Raw data are represented by a B-spline expansion using 500 basis spline expansion using 500 basis functions of order 6.functions of order 6.

Knot about every 25 days.Knot about every 25 days. The standard deviation of the raw The standard deviation of the raw

data about this smooth, adjusted for data about this smooth, adjusted for degrees of freedom, is 4.30 degrees degrees of freedom, is 4.30 degrees Celsius.Celsius.

Periodic smoothPeriodic smooth

The basis is Fourier, with 9 basis The basis is Fourier, with 9 basis functions judged to be enough to functions judged to be enough to capture most of the strictly periodic capture most of the strictly periodic trend for a period of one year.trend for a period of one year.

The standard deviation of the raw The standard deviation of the raw about data about this smooth is 4.74 about data about this smooth is 4.74 deg C.deg C.

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

Plotting the unconstrained B-spline Plotting the unconstrained B-spline smooth minus the constrained smooth minus the constrained Fourier smooth reveals some Fourier smooth reveals some striking discrepancies. striking discrepancies.

We focus on Christmas, 1989. The We focus on Christmas, 1989. The Ramsay’s spent the holidays in a Ramsay’s spent the holidays in a chalet in the Townships, and awoke chalet in the Townships, and awoke to –37 deg C. No skiing, car dead, to –37 deg C. No skiing, car dead, marooned!marooned!

This temperature would still be cold This temperature would still be cold in mid-January, but less unusual.in mid-January, but less unusual.

RegistrationRegistration

Let the unconstrained smooth be Let the unconstrained smooth be x(t)x(t) and the strictly periodic smooth be and the strictly periodic smooth be xx00(t).(t).

We need to estimate a nonlinear We need to estimate a nonlinear strictly increasing smooth strictly increasing smooth transformation of time transformation of time h(t), h(t), called a called a warping functionwarping function, such that a fitting , such that a fitting criterion is minimized. criterion is minimized.

Fitting criterionFitting criterion

The fitting criterion was the smallest eigenvalue of the matrix

20 0

20

x t dt x t x h t dt

x t x h t dt x h t dt

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 The warping function h(t)h(t)Every smooth strictly monotone Every smooth strictly monotone

function function h(t)h(t) such that such that h(0) = 0h(0) = 0 can can be represented asbe represented as

0 0

( ) exp ( )t u

h t C w v dvdu We represent unconstrained function w(v) by a B-spline expansion. Constant C is determined by constraint h(T) = T.

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

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

Mid-winter for 1989-1990 arrived Mid-winter for 1989-1990 arrived about 25 days early.about 25 days early.

The next step is to register the The next step is to register the temperature data by computing temperature data by computing x*(t) x*(t) = x[h(t)]. = x[h(t)]. The registered curve The registered curve x*(t) x*(t) contains only amplitude variation.contains only amplitude variation.

Registration was done by Matlab Registration was done by Matlab function registerfd, available by ftp function registerfd, available by ftp fromfrom

ego.psych.mcgill.ca/pub/ramsay/ego.psych.mcgill.ca/pub/ramsay/FDAfunsFDAfuns

Amplitude variationAmplitude variation

The standard deviation of the difference The standard deviation of the difference between the unconstrained smooth and between the unconstrained smooth and the strictly periodic smooth is 2.15 C.the strictly periodic smooth is 2.15 C.

The standard deviation of the difference The standard deviation of the difference between the registered smooth and the between the registered smooth and the periodic smooth is 1.73 C.periodic smooth is 1.73 C.

(2.15(2.1522 – 1.73 – 1.7322)/2.15)/2.1522 = .35, the proportion = .35, the proportion of the variation due to phase.of the variation due to phase.

The standard deviation of the raw The standard deviation of the raw data around the registered smooth data around the registered smooth is 2.13 C, compared with 2.07 C for is 2.13 C, compared with 2.07 C for the unregistered smooth.the unregistered smooth.

About 10% of the total variation is About 10% of the total variation is due to phase.due to phase.

ConclusionsConclusions

Phase variation is an important part Phase variation is an important part of weather behavior.of weather behavior.

Statisticians seldom think about Statisticians seldom think about phase variation, and classical time phase variation, and classical time series methods ignore it completely.series methods ignore it completely.

Phase variation needs more Phase variation needs more attention, and registration is an attention, and registration is an essential tool.essential tool.