En la Ley Notarial.

Our goal is to examine the extreme fluctuations of Xt(s) relative to the estimated

mean and standard deviation. To this end, we consider the time series

Yt(s) :=

Xt(s) − ˆµ(t)(s)

ˆ

η(t)(s) , t = 1, 2, . . . , (4.4)

where ˆµ(t) and ˆη(t) are obtained as in Section 4.3.1 for the station s. The time series {Yt(s)} is standardized to have zero mean and unit variance marginal distributions.

Figure 4.3 left shows a plot of the standardized time series and also the auto co- variance plot for the station, Ann Arbor. The auto-covariance plots reveal that the standardized time series is fairly stationary and can be suitable for analysis.

We explore next the empirical distribution of the standardized time series {Yt(s)}.

It turns out that the standard Normal model offers a fairly adequate approximation to the time series for most of the stations s. Indeed, in Figure 4.4, we show nor- mal quantile-quantile plots for the empirical distribution of {Yt(s)} for two stations,

Faulkton North West, SD and Pasadena, CA. Qualitatively, the QQ-plots for all other stations look nearly identical. While the standardization does not remove periodic dependencies and non-stationarity, it puts the temperature fluctuations in different seasons across different stations on the same scale.

Figure 4.3: Standardized daily time series for Ann Arbor, MI. Left and right panels indicate the standardized time series, Yt(s) and its corresponding auto-

−4 −2 0 2 4 −4 −2 0 2 4 PASADENA , CA x Quantiles y Quantiles 1−1 line regression line 95% confidence bands −3 −2 −1 0 1 2 3 −4 −2 0 2 4 FAULKTON 1 NW , SD x Quantiles y Quantiles 1−1 line regression line 95% confidence bands

Figure 4.4: Normal quantile-quantile plots for standardized weekly min Yt(s). Left panel

and right panels correspond to stations Faulkton North West, SD and Pasadena, CA re- spectively.

4.3.3 Defining heat waves

In order to define heat waves, we first define the minimum of the standardized time series Yt(s) over a span of ∆ = 7 days as:

Zk,∆(s) = min

t=k,...,k+∆−1Yt(s) (4.5)

We have a heatwave event at location s starting on day k if

Zk,∆(s) > U (u0, s, ∆) (4.6)

where u0 is the intensity level, s is the station and ∆ is size of the window. With ∆

fixed the heat wave is defined as: if

where u0 is a quantile level and

F_s−1(u0) = inf{x ∈ R : Fs(x) ≥ u0} (4.8)

with Fs denoting the empirical cumulative distribution function of Zk(s). Therefore,

values of Zk(s) well above its extreme quantiles correspond to the occurrence of a

heat wave event.

To explore the spatial impact of the so-defined heat waves, we define

Ak(u0) :=

Z

D

I(Zk(s) > Fs−1(u0))ds, (4.9)

which is the total area of the sites s in region D experiencing a heatwave during week k. Since the area Ak(u0) is bounded above by the total area of the region D, we

consider

Qk(u0) :=

Ak(u0)

|D| ∈ [0, 1],

which is the proportion of the region D experiencing a heatwave of intensity level u0

during week k.

For the stations s = s1, s2, · · · , s` ∈ D, the series Zk(s) and Fs−1(u0) is determinis-

tic. However, for the computation of the integral in (4.9), the series Zk(s) and Fs−1(u0)

needs to be evaluated at all values of s ∈ D. This is facilitated by using thin plate splines smoothing approach where a surface is fit to the values Zk(si), i = 1, · · · , `,

with some error allowed at each si, i = 1, · · · , `. At every iteration, a station is

omitted from the estimation of the fitted surface and the mean error is found. This procedure is repeated over a range of values of the smoothing parameter and the value that minimizes the mean error is taken to give the optimum smoothing (also

called minimizing the generalized cross validation criterion). Chapter 12 in Wilson and Mair (2004) and Section 2.4 in Tait et al. (2006) explain the thin spline interpo- lation when applied to rainfall data. The tps function in fields (2018) package of R automatically applies this methodology to produce an integral approximation to the quantity (4.9). 1920 1940 1960 1980 2000 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Proportion of US area affected: u0 = 0. 95

Time (years) Propor tion 1920 1940 1960 1980 2000 0.0 0.1 0.2 0.3

Proportion of US area affected: u0 = 0. 99

Time (years)

Propor

tion

Figure 4.5: Distribution of the time series Qk(u0) for varying values of the quantile

level u0 for the period 1911-2010. Extreme events in the series have been

marked with red.

Figure 4.5 gives a plot of the time series Qk(u0) for two different values of u0 viz

0.95 and 0.99. As is evident from the plot, depending on value of the quantile u0,

the extremes of the time series Qk goes on changing. For example, the second largest

value for Qk(0.95) is recorded for December 31, 2010 in contrast to the second largest

value for Qk(0.95) which is recorded on November 28, 1998. The largest value for

both Qk(0.95) and Qk(0.99) is however on the same date, December 5, 1939. Figure

4.6 gives a plot of the spatially interpolated time series Zk(s) corresponding to the top

record events in the time series Qk(u0), u0 = 0.95, 0.99. If one were to interpret these

plots, the weeks starting on December 5, 1939 and November 28, 1998 experienced a heat wave of intensity 0.95 over 63% and 61% of the territory in US respectively.

Figure 4.6: Thin spline interpolated time series Zk(s) for s ∈ D on time points cor-

responding to extreme values of Qk(s)

At least 11%−spatial coverage

Day of the year

Frequency

0 100 200 300

0

10

30

At least 14%−spatial coverage

Day of the year

Frequency 0 100 200 300 0 5 15 25

At least 19%−spatial coverage

Day of the year

Frequency 0 100 200 300 0 5 10 15

At least 31%−spatial coverage

Day of the year

Frequency 0 100 200 300 0 2 4 6

Figure 4.7: Histogram of the seasonal distribution of Qk > p for varying values of

proportion p.

heat-waves is. In this direction, we consider histogram of

{(k mod 365)|Qk(u0) > p}

for u0 = 0.95. The histograms corresponds to nothing but the daily distribution of

extreme events with at least 100p% of spatial coverage . Figure 4.7 clearly demon- strates that with increase in proportion p, only the most extreme heat wave events present themselves. For an intensity level of 0.95, it seems spatially extreme heatwave events tend to occur less often in the summer than during the other seasons. This hypothesis is further corroborated under Sections 4.4.4 and 4.5.3.