A stochastic model for the analysis of maximum daily temperature

Abstract

In this paper, a stochastic model for the analysis of the daily maximum temperature is proposed. First, a deseasonalization procedure based on the truncated Fourier expansion is adopted. Then, the Johnson transformation functions were applied for the data normalization. Finally, the fractionally autoregressive integrated moving average model was used to reproduce both short- and long-memory behavior of the temperature series. The model was applied to the data of the Cosenza gauge (Calabria region) and verified on other four gauges of southern Italy. Through a Monte Carlo simulation procedure based on the proposed model, 10⁵ years of daily maximum temperature have been generated. Among the possible applications of the model, the occurrence probabilities of the annual maximum values have been evaluated. Moreover, the procedure was applied for the estimation of the return periods of long sequences of days with maximum temperature above prefixed thresholds.

Appendices

Appendix 1: Estimation of the number of harmonics

The hypothesis, H _0 , μ   :  μ _Y , 1 = μ _Y , 2 =  …  = μ _Y , M  = μ _Y , can be verified by using the statistics

$$ {S}_V^2={s^{\mathit{{\prime\prime}}}}_Y^2/{s^{\mathit{\prime}}}_Y^2 $$

(A1)

with

$$ {s^{\mathit{\prime}}}_Y^2=\frac{1}{K_{*}-M}{\displaystyle \sum_{m=1}^M\left({N}_m-1\right){s}_{Y,m}^2} $$

(A2)

$$ {s^{\mathit{{\prime\prime}}}}_Y^2=\frac{1}{M-1}{\displaystyle \sum_{m=1}^M{N}_m}{\left({m}_{Y,m}-{m}_Y\right)}^2 $$

(A3)

where ${s}_{Y,m}^2$ is the sample variance of the data referred to the m ^th class, m _Y , m is the sample mean of the data referred to the m ^th class, and m _Y is the mean of the whole sample $y\left({i}_{k_{*}}\right)$. The statistics ${S}_V^2$ is approximately distributed according to a Fisher variance-ratio law v ²(f ₁, f ₂) with f ₁ = M − 1 and f ₂ = K _* − M degrees of freedom. For a significance level α _SL, the null hypothesis cannot be refused if ${S}_V^2<{v}_{1-{\alpha}_{SL}}^2\left({f}_1,{f}_2\right)$, where ${v}_{1-{\alpha}_{SL}}^2\left({f}_1,{f}_2\right)$ is the 1 − α _SL percentile of the ν² distribution.

The hypothesis ${H}_{0,{\sigma}^2}\kern0.5em :\kern0.5em {\sigma}_{Y,1}^2={\sigma}_{Y,2}^2=\dots ={\sigma}_{Y,M}^2={\sigma}_Y^2$ can be verified through the Bartlett’s test (Snedecor and Cochran 1989), based on the statistics

$$ {S}_B^2=\frac{1}{c_B}\left[\left({K}_{*}-M\right) \ln {s^{\mathit{\prime}}}_Y^2-{\displaystyle \sum_{m=1}^M\left({N}_m-1\right) \ln {s}_{Y,m}^2}\right] $$

(A4)

where

$$ {c}_B=1-\frac{1}{3\left(M-1\right)}\left(\frac{1}{K_{*}-M}-{\displaystyle \sum_{m=1}^M\frac{1}{N_m-1}}\right) $$

(A5)

The statistics ${S}_B^2$ is approximately distributed according to a χ ²(f ₁) law, with f ₁ = M − 1 degree of freedom. With a significance level equal to α _SL, the hypothesis cannot be rejected if ${S}_B^2<{\chi}_{1-{\alpha}_{SL}}^2\left({f}_1\right)$, where ${\chi}_{1-{\alpha}_{SL}}^2\left({f}_1\right)$ is the 1 − α _SL percentile of the χ ²distribution.

The smallest values of n _h , μ and ${n}_{h,{\sigma}^2}$, for which both the hypotheses H _0 , μ and ${H}_{0,{\sigma}^2}$ cannot be rejected, detect the number of harmonics ${\widehat{n}}_{h,\mu }$ and ${\widehat{n}}_{h,{\sigma}^2}$ to be used in the truncated Fourier expansion for the functions μ _T (i) and ${\sigma}_T^2(i)$.

Appendix 2: Monte Carlo procedure

The Monte Carlo simulation procedure, used in this work to generate the daily maximum temperature t _max (i) series, can be schematized as follows:

• 1. By using L’Ecuyer random generator, a sequence υ(i), with i = 1 , 2 ,  .  .  .  , L _s , of random number uniformly distributed on the interval (0,1) is created.

• 2. The sequence υ(i) is transformed into a sequence ε(i) of random numbers, distributed according to a normal law, with zero mean and variance ${\sigma}_{\varepsilon}^2$, through the Box and Müller technique.

• 3. According to an $\mathrm{ARMA}\;\left(\widehat{p},\widehat{q}\right)$ model, initialized with u(i) = 0 and ε(i) = 0 for i ≤ 0, a sequence of numbers is generated,

  $$ \begin{array}{cc}\hfill u(i)={\displaystyle \sum_{k_p=1}^{\widehat{p}}{\widehat{\varphi}}_{k_p}}u\left(i-{k}_p\right)+{\displaystyle \sum_{k_q=0}^{\widehat{q}}{\widehat{\psi}}_{k_q}}\varepsilon \left(i-{k}_q\right)\hfill & \hfill i=1,2,\dots, {L}_s\hfill \end{array} $$

  (B1)

• 4. By using the series development of the ${\left(1-B\right)}^{-\widehat{d}}$ operator, the sequence u(i) is transformed into a number sequence z(i) corresponding to the $\mathrm{FARIMA}\kern0.5em \left(\widehat{p},\widehat{d},\widehat{q}\right)$ model with zero mean and unit variance

  $$ \begin{array}{cc}\hfill z(i)=\frac{1}{\varGamma \left(\widehat{d}\right)}{\displaystyle \sum_{s=0}^{s_{\max }}\frac{\varGamma \left(\widehat{d}+s\right)}{s!}}u\left(i-s\right)\hfill & \hfill i={s}_{\max }+1,\dots, {L}_s\hfill \end{array} $$

  (B2)

  where

  s _max is fixed so that $\varGamma \left(\widehat{d}+{s}_{\max }+1\right)/\left({s}_{\max }+1\right)\kern0.5em !<\xi {\displaystyle {\sum}_{s=0}^{s_{\max }}\varGamma \left(\widehat{d}+s\right)/s!}$, with ξ = 10⁻⁴.

  The value of the variance ${\sigma}_{\varepsilon}^2$ is fixed in order to obtain ${\sigma}_Z^2=1$. If a $\mathrm{FARIMA}\kern0.5em \left(1,\widehat{d},0\right)$ model is employed, the value for ${\sigma}_{\varepsilon}^2$ is

  $$ {\sigma}_{\varepsilon}^2=\frac{\varGamma^2\left(1-\widehat{d}\right)}{\varGamma \left(1-2\widehat{d}\right)}\cdot \frac{1+{\widehat{\varphi}}_1}{{}_2F_1\left(1,1+\widehat{d},1-\widehat{d};{\widehat{\varphi}}_1\right)} $$

  (B3)

  where Γ (.) and ₂ F ₁(.) indicate the complete gamma function and the hypergeometric function, respectively.

• 5. The sequence z(i) = s _max + 1 ,  .  .  .  , L _s is transformed into the sequence y(i) = s _max + 1 ,  .  .  .  , L _s , by using the inverse function of the unbounded Johnson transformation

  $$ \begin{array}{cc}\hfill y(i)=\widehat{\alpha}+\widehat{\beta} \sinh \left[\frac{z(i)-\widehat{\eta}}{\widehat{\theta}}\right]\hfill & \hfill i={s}_{\max }+1,\dots, {L}_s\hfill \end{array} $$

  (B4)

  or the inverse function of the bounded Johnson transformation

  $$ \begin{array}{cc}\hfill y(i)=\frac{\widehat{\alpha}+\left(\widehat{\alpha}+\widehat{\beta}\right) \exp \left[\frac{z(i)-\widehat{\eta}}{\widehat{\theta}}\right]}{1+ \exp \left[\frac{z(i)-\widehat{\eta}}{\widehat{\theta}}\right]}\hfill & \hfill i={s}_{\max }+1,\dots, {L}_s\hfill \end{array} $$

  (B5)

• 6. The sequence of daily maximum temperature t _max(i) = s _max + 1 ,  .  .  .  , L _s is obtained as

  $$ \begin{array}{cc}\hfill {t}_{\max }(i)={\mu}_T(i)+{\sigma}_T(i)y(i)\hfill & \hfill i={s}_{\max }+1,\dots, {L}_s\hfill \end{array} $$

  (B6)

  where

  $$ {\mu}_T(i)=\frac{1}{2}{\widehat{a}}_{\mu, 0}+{\displaystyle \sum_{j=1}^{n_{h,\mu }}\left[{\widehat{a}}_{\mu, j} \cos \left(\frac{2\pi \kern0.5em j}{D}i\right)+{\widehat{b}}_{\mu, j} \sin \left(\frac{2\pi \kern0.5em j}{D}i\right)\right]} $$

  (B7)
  $$ {\sigma}_T(i)={\left\{\frac{1}{2}{\widehat{a}}_{\sigma^2,0}+{\displaystyle \sum_{j=1}^{n_{h,{\sigma}^2}}\left[{\widehat{a}}_{\sigma^2,j} \cos \left(\frac{2\pi \kern0.5em j}{D}i\right)+{\widehat{b}}_{\sigma^2,j} \sin \left(\frac{2\pi \kern0.5em j}{D}i\right)\right]}\right\}}^{1/2} $$

  (B8)