Preliminary Results - Research Notebook

Block Maximum¶

In these examples, we are applying the Block Maxima (BM) method on a yearly basis. So, our block size is of one year which leaves us 62 years in total for our time series. While this is not a lot of data, we see in Figure (5) that the distribution does match one of the classical GEVD distributions. In particular, the Fréchet distribution where the shape parameter, $\kappa$ , is less than 0 (Figure (1)).

Time Series

Scatter Plot

Histogram

Madrid Daily Maximum Temperature Time Series — Figure (3): A scatter plot with boundaries for the maximum temperatures obtained using the yearly Block maxima method.

In this figure, we have different representations for the block maximum method. We already see a trend line and perhaps a hint of cyclic behaviour. In our first experiments, we see will assume a unconditional distribution however we can see that this assumption is incorrect as we can clearly see from Figure (4).

Model Metrics¶

Model	ELPD WAIC	ELPD WAIC SE	P WAIC
M0a	-157.11	4.68	0.01
M0b	-96.57	3.74	1.46
M0c	-97.18	3.87	1.30
M1a	-96.57	3.74	1.46
M1b	-96.43	3.68	1.86
M2	-97.19	5.06	1.85
M3	- 96.86	4.50	1.75

Stationary Models¶

\begin{aligned} \text{Scalar Shape}: && && \boldsymbol{\kappa}(s,t) &= \kappa_0 \\ \text{Consant Shape}: && && \boldsymbol{\kappa}(s,t) &= \kappa_0(s) \\ \end{aligned}

((1))

Static Parameters¶

\begin{aligned} \text{Location}: && && \boldsymbol{\mu}(s,t) &= \mu_0 \\ \text{Scale}: && && \boldsymbol{\sigma}(s,t) &= \sigma_0 + \sigma_2(\mathbf{s}) \\ \text{Shape}: && && \boldsymbol{\kappa}(s,t) &= \kappa_0 \\ \end{aligned}

((2))

Model	Location	Scale	Shape	100-Year RP
M0a	35.02 (0.05)	4.24 (0.03)	-0.34 (0.00)	44.81 (0.04)
M0b	39.31 (0.21)	1.47 (0.15)	-0.43 (0.08)	42.22 (0.36)
M0c	39.29 (0.19)	1.41 (0.13)	-0.34 (0.01)	42.54 (0.31)
M2	39.34 (0.17)	1.25 (0.11)	-0.30 (0.01)	42.44 (0.29)

Non-Stationary Models¶

Static Parameters¶

\begin{aligned} \text{Scale}: && && \boldsymbol{\sigma}(s,t) &= \sigma_0 + \sigma_2(\mathbf{s}) \\ \text{Shape}: && && \boldsymbol{\kappa}(s,t) &= \kappa_0 \\ \end{aligned}

((3))

Model	Scale	Shape
M1	1.40 (0.14)	-0.35 (0.01)
M3	1.24 (0.11)	-0.29 (0.01)

GMST Parameters¶

Location Parameters

\mu(t) = \ldots

((4))

Model	Historical, 0.0 [C°]	Current, 1.0 [C°]	Future, 2.0 [C°]
M1	39.09 (0.27)	39.69 (0.41)	40.30 (0.94)
M3	38.75 (0.18)	40.46 (0.20)	42.22 (0.33)

100-Year Return Level

R_{100}(t) = \ldots

((5))

Model	Historical, 0.0 [C°]	Current, 1.0 [C°]	Future, 2.0 [C°]
M1	42.27 (0.36)	42.88 (0.46)	43.50 (0.96)
M3	41.90 (0.29)	43.61 (0.32)	45.38 (0.43)