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Onthepredictionofpersistentprocessesusingtheoutputofdeterministic

models

Hristos Tyralis

*

and Demetris Koutsoyiannis

Department of Water Resources and Environmental Engineering, School of Civil

Engineering, National Technical University of Athens, Heroon Polytechniou 5, 157 80

Zographou, Greece

*

Corresponding author, montchrister@gmail.com

Abstract: A problem frequently met in engineering hydrology is the forecasting of

hydrologic variables conditional on their historical observations and the hindcasts and

forecasts of a deterministic model. On the contrary, it is a common practice for

climatologists to use the output of general circulation models (GCMs) for the prediction

of climatic variables despite their inability to quantify the uncertainty of the predictions.

Here we apply the well-established Bayesian Processor of Forecasts (BPF) for

forecasting hydroclimatic variables using stochastic models through coupling them with

GCMs. We extend the BPF to cases where long-term persistence appears, using the

Hurst-Kolmogorov process (HKp, also known as fractional Gaussian noise) and we

investigate analytically its properties. We apply the framework to calculate the

distributions of the mean annual temperature and precipitation stochastic processes for

the time period 2016-2100 in the United States of America conditional on historical

observations and the respective output of GCMs.

Keywords: Bayesian Processor of Forecasts; fractional Gaussian noise; general

circulation model; Hurst-Kolmogorov; hydroclimatic prediction; hydrological statistics

1. Introduction

1.1 Uncertainty in deterministic models in hydrological science

Recently, various studies regarding the prediction of hydrologic variables based on

stochastic models have been published. To mention some of them, Koutsoyiannis et al.

(2008b) proposed a stochastic model for the prediction of the Nile flow a month ahead.

On larger time scales, Koutsoyiannis et al. (2007) proposed a stochastic framework to

calculate future climatic uncertainties conditional on historic observations, while Tyralis

and Koutsoyiannis (2014) solved this problem using a Bayesian framework. Engineering

2

hydrologists frequently use stochastic models for the prediction of hydrologic variables,

whereas the climatologists focus on deterministic models (General Circulation Models,

GCMs) (Koutsoyiannis et al. 2008a). While it is true that deterministic models

incorporate knowledge of the climatic mechanisms expressed through deterministic

equations, they are not appropriate to quantify the uncertainty of predictions.

Consequently, climatologists have recently started reconsidering their approach,

introducing stochastic models in climate science (Macilwain 2014), while earlier

Schneider (2002) set a debate on how and when to assign probabilities to future

projections of the GCMs, simultaneously expressing some concerns about their absence

in specific cases.

Estimating uncertainties of forecasted geophysical variables using information from

deterministic models is frequently met in the hydrological science and in particular in

rainfall-runoff modelling (e.g. Montanari and Grossi 2008, Wang et al. 2009, Zhao et al.

2011, Smith et al. 2012, Pokhrel et al. 2013, Zhao et al. 2015a and others). The Bayesian

Forecasting System (BFS) and its extensions in a series of papers (Krzysztofowicz

1999b, 2001, 2002, Krzysztofowicz and Maranzano 2004) is a primary tool for

estimating uncertainties in rainfall-runoff modelling. Another interesting tool for

quantifying uncertainties is the Bayesian Processor of Forecasts (BPF) introduced in

Krzysztofowicz (1985) and compared with the BFS in Krzysztofowicz (1999a). The BPF

“combines a prior distribution, which describes the natural uncertainty about the

realization of a hydrologic process, with a likelihood function, which describes the

uncertainty in categorical forecasts of that process, and outputs a posterior distribution of

the process, conditional upon the forecasts” (Krzysztofowicz 1985). It is mostly used for

weather forecasting and while it is a general algorithm, which can be applied to any

distribution and dependence pattern of the process, it has been investigated solely for

independent or Markov dependent variables (e.g. Krzysztofowicz 1999a, Krzysztofowicz

and Evans 2008, Chen et al. 2013). The term “Bayesian” refers to the use of the Bayes

theorem, however the BPF does not use full Bayesian statistics. Consequently, the

parameter uncertainty (Montanari and Koutsoyiannis 2012) is not considered in the

model.

A frequent approach for modelling mean annual geophysical time series is the

implementation of the Hurst-Kolmogorov stochastic process (HKp) (also known as

Fractional Gaussian Noise, e.g. Koutsoyiannis 2002, 2003, 2006b, Koutsoyiannis and

3

Montanari 2007). The investigation of big geophysical data sets has confirmed the HK

behaviour of geophysical variables in the annual time scale (Fatichi et al. 2012,

Iliopoulou et al. 2016, Markonis and Koutsoyiannis 2016). The HK process is suitable for

modelling the variability observed in geophysical time series, and not only because it

can model the HK behaviour. Specifically, while it is stationary (for the benefits of using

stationary models see Koutsoyiannis and Montanari 2014, Montanari and Koutsoyiannis

2014), it can model higher variations of the observed time series unlike the Markovian

models. Thus, it can model observed trends (Koutsoyiannis 2006a) and it does not

underestimate uncertainties of the forecasted variable (Tyralis and Koutsoyiannis

2014).

1.2 General Circulation Models

The Coupled Model Intercomparison Project Phase 5 (CMIP5) includes GCMs, which

contain historical runs, i.e. simulations of the past forced by observed atmospheric

composition changes and time-evolving land cover (Taylor et al. 2012). Each historical

run is extended with a projection of the climate driven by concentration or emission

scenarios consistent with the representative concentration scenarios (RCPs, Hibbard et

al. 2007, Moss et al. 2010). The evaluation of GCMs for reproducing the past has been

studied extensively with varying results, depending on the examined variable (usually

temperature and precipitation), time scale of the variable, statistic or parameter of

interest, region and the time-period. Most studies include comparisons with

observations, re-analysis data, satellite data or all (Koutsoyiannis et al. 2008a,

Anagnostopoulos et al. 2010, Santer et al. 2013, Sheffield et al. 2013a, 2013b, Xu et al.

2013, Koutroulis et al. 2015, Nasrollahi et al. 2015, Aloysius et al. 2016, Matthes et al.

2016), visualizations (Potter et al. 2009) and even comparisons between the models

themselves (Johnson and Sharma 2009). However, Notz (2015) points out that the direct

comparison of model simulations with observations allows for limited inferences about

the deficiencies of the model.

Of practical interest and gaining place in the literature is the quantification of the

uncertainties of the GCMs projections (Katz 2002) whose sources are measurement

errors, variations of the geophysical processes and model structure according to Katz

(2002) or internal variability, scenario and model uncertainties (e.g. Hawkins and

Sutton 2009). Therefore, it is apparent that we cannot consider raw projections as a

4

product, which we can use without further processing. Significant part of the literature

has been devoted to the quantification of the uncertainties (Hawkins et al. 2014,

Woldemeskel et al. 2014, Tian et al. 2015, Zhao et al. 2015b) and their partition

(Hawkins and Sutton 2009, 2011, Yip et al. 2011, Ylhäisi et al. 2015, Hewitt et al. 2016)

usually to internal, scenario and model uncertainties. Beyond what narrowly concerns

the climate science there is a discussion on the uncertainty attributed to human

behaviour, which seems not quantifiable. Consequently, the use of scenarios is proposed

(Dessai and Hulme 2004) to consider the human behaviour with the use of RCPs. There

is also a discussion on the potential of the reduction of uncertainties (Hawkins and

Sutton 2009, 2011) while Knutti and Sedláček (2013) conclude that the progress in

terms of narrowing uncertainties is too limited. An overview of methods to evaluate

uncertainty of deterministic models, not only in the climate science, is presented in

Uusitalo et al. (2015).

The limitations in reducing uncertainties are primarily due to the internal climate

variability (Knutti and Sedláček 2013); thus the development of methods, which are

based on GCMs and simulate the local weather (e.g. Groves et al. 2008) gain place in

practical applications of the GCMs. While future climate is still projected based on single

GCM outputs (Maloney et al. 2014), combining multiple models for future projections is

proposed as an alternative for skilful climate predictions (Smith et al. 2009, Chowdhury

and Sharma 2011, Strobach and Bel 2015). However, Pirtle et al. (2010) claim that the

quality of analyses based on multiple models cannot be evaluated, while Kundzewicz

and Stakhiv (2010) mention that the spread of outcomes of the GCMs is incorrectly used

as a type of uncertainty analysis. The so-called “bias corrections” refers to another group

of methods, which are used to improve the projections of GCMs through (a posteriori)

increasing the agreement between GCM outputs and observations. However, this

procedure is artificial and is criticized for hiding the uncertainty rather than reducing it

(Ehret et al. 2012).

1.3 On the proposed framework

It seems that the arsenal of methods to improve the GCMs projections and quantify their

forecasting uncertainty (mainly use of multiple models and “bias correction”) is

inadequate. In the present study, we propose using the BPF, which is based on solid

scientific foundation, i.e. the concept of conditional stochastic independence (de Finetti

5

1974, Krzysztofowicz 1985). Hence, it can be an appropriate alternative. Here we apply

the BPF to quantify the uncertainties of the forecasts of mean annual temperature and

precipitation. We model the variables of interest with the HKp, which we assume is the

prior distribution that describes the natural uncertainty about the realization of the

process. The deterministic forecasts of the process are the GCMs outputs, while the BPF

outputs the posterior distribution of the process conditional on the GCMs outputs and

the realization of the process.

The posterior distribution depends initially on the fitted HKp but eventually (and in a

determinative manner) on the agreement of the GCM output with the observations. The

model uses six parameters. The three parameters of the HKp are estimated from the

observations. The degree of agreement of the GCM with the observations is determined

by three parameters, estimated when fitting the model using a common period of

observations and GCMs output. The two fittings are performed independently. As a

result, the application of the BPF avoids the artificial improvement of the model (e.g.

“bias correction” and related methods), while the natural variability of the process is

modelled using a well-established stochastic model. Furthermore, uncertainties are

quantified using a single output of the model, while the human influence is modelled

through the selection of a single scenario. Finally, we avoid to narrow uncertainty.

Instead, uncertainties are presented as they are, i.e. without reducing them artificially.

The theoretical contribution of the present study is the application of the BPF to

processes with more complicated dependence structure compared to the Markovian

model. We apply the BPF to the HKp, which results in posterior multivariate normal

distributions. We apply the framework to the mean annual temperature and total annual

precipitation in a large area, the contiguous part of the United States of America, while

we show whether and how a purely probabilistic forecast could be improved by using a

deterministic forecast.

2. Methods

In this section, we present the BPF, the definition of the HKp and the application of the

BPF to normal stationary stochastic processes and as a case study to the HKp. In the next

sections we use the Dutch convention for notation, according to which random variables

and stochastic processes are underlined (Hemelrijk 1966).

6

2.1 The Bayesian Processor of Forecasts

Let x

2(1:(n1+n2))

be a geophysical process which we wish to forecast and x

1(1:(n1+n2))

be its

forecast given by a deterministic model. The respective time periods (in discrete time,

denoted through the integers n

1

and n

2

) for each variable are presented in Figure 1. We

assume that x

2(1:n1)

denotes the observed (historical) values of the time series, while

x

1(1:(n1+n2))

and x

2(1:(n1+n2))

are the stochastic processes which represent in stochastic

terms the deterministic model and the geophysical process respectively, defined in eqs.

(1), (2).

x

1(1:(n1+n2))

:= (x

11

, …, x

1n1

, x

1(n1+1)

, …, x

1(n1+n2)

)

T

(1)

x

2(1:(n1+n2))

:= (x

21

, …, x

2n1

, x

2(n1+1)

, …, x

2(n1+n2)

)

T

(2)

Figure 1. Time periods for the BPF data input and output. The prediction time period

refers to the distribution of y

4

|y

3

, x

1

. y

3

and y

4

are defined in eqs. (7) and (8).

To shorten the equations used in the subsequent sections we use the notation of eqs.

(3)-(8), in which we remove the time indexes.

x

1

:= x

1(1:(n1+n2))

(3)

x

2

:= x

2(1:(n1+n2))

(4)

y

1

:= (x

11

, x

12

, …, x

1n1

)

T

: n

1

×1 (5)

y

2

:= (x

1(n1+1)

, …, x

1(n1+n2)

)

T

: n

2

×1 (6)

y

3

:= (x

21

, …, x

2n1

)

T

: n

1

×1 (7)

y

4

:= (x

2(n1+1)

, …, x

2(n1+n2)

)

T

: n

2

×1 (8)

Henceforth, y

1

will be called deterministic hindcast. The BPF is based on the

fundamental eqs. (9) and (10), which exploit the concept of conditional stochastic

independence (for intuitive explanations of the BPF the reader is referred to de Finetti

1974 and Krzysztofowicz 1985):

7

f

n

(x

11

, x

12

, …, x

1n

|x

21

, x

22

, …, x

2n

) =

∏

i = 1

n f

i

(x

1i

|x

21

, x

22

, …, x

2n

) ∀ n ∊ {1, …, n

1

+n

2

} (9)

f

i

(x

1i

|x

21

, x

22

, …, x

2n

) = f

i

(x

1i

|x

2i

) ∀ i, n ∊ {1, …, n

1

+n

2

} (10)

The deterministic forecasts are independent on each other conditional on the

observations according to eq. (9) (Krzysztofowicz 1985), while each forecast depends

only on the parallel observation according to eq. (10). Eqs. (9) and (10) combined result

in

f

n

(x

11

, x

12

, …, x

1n

|x

21

, x

22

, …, x

2n

) =

∏

i = 1

n f

i

(x

1i

|x

2i

) ∀ n ∊ {1, …, n

1

+n

2

} (11)

Given an observation of x

2

, the distribution of x

1

is determined by eqs. (9) and (10).

The purpose of the BPF is to find the distribution of y

4

conditional on y

3

and x

1

, which is

given by

h(y

4

|y

3

, x

1

) = f(x

1

|y

3

, y

4

) g(y

3

, y

4

) / ξ(y

3

, x

1

) (12)

where both g( ) and ξ( ) denote (joint) distributions (more precisely, probability

densities).

As proved in Appendix A, h can be simulated using the equation:

h(y

4

|y

3

, x

1

) ∝ f(y

2

|y

4

) g(y

4

|y

3

) (13)

Consequently, to simulate from h we must calculate the distributions f and g.

2.2 Α normal stationary stochastic process in the Bayesian Processor of

Forecasts

Let x

2

denote a normal stationary stochastic process (Wei 2006, p. 10) with parameters

μ, σ, ρ

i,j

, given by:

μ := E[x

2n

] ∀ n ∊ {1, …, n

1

+n

2

} (14)

σ

2

:= Var[x

2n

] ∀ n ∊ {1, …, n

1

+n

2

} (15)

ρ

i,j

:= ρ

|i−j|

∀ i, j ∊ {1, …, n

1

+n

2

} (16)

The joint distribution of x

2

is multivariate normal with constant mean μ and

autocovariance matrix Σ given by eq. (17). Furthermore, the joint distributions of y

3

, y

4

and all subsets of x

2

are also multivariate normal, with the same mean and

autocovariance matrix given by extracting respective parts of Σ. The proofs of the results

of Section 2.2 are given in Appendix A.

Σ = σ

2

[ρ

i,j

] ∀ i, j ∊ {1, …, n

1

+n

2

} (17)

8

The autocovariance matrix Σ can be partitioned in the following way:

Σ = σ

2

R

11

R

12

R

21

R

22

(18)

where the dimensions of the matrices are: R

11

: n

1

×n

1

, R

21

: n

2

×n

1

, R

12

: n

1

×n

2

, R

22

: n

2

×n

2

.

Then the distribution of y

4

|y

3

is given by:

g(y

4

|y

3

) = Ν(Μ

1

, Λ

1

) (19)

where N denotes the multivariate normal distribution and

Μ

1

:= μ

2

+ R

21

R

−1

11

(y

3

− μ

1

) (20)

Λ

1

:= σ

2

(R

22

− R

21

R

−1

11

R

12

) (21)

μ

1

:= (μ, …, μ)

Τ

, n

1

×1 (22)

μ

2

:= (μ, …, μ)

Τ

, n

2

×1 (23)

An intuitive modelling of the relationship between x

1n

and x

2n

is given by the

distribution (24) (e.g. Krzysztofowicz 1999a).

f(x

1n

|x

2n

) = Ν(q

n

, σ

2

e

) ∀ n ∊ {1, …, n

1

+n

2

} (24)

where

q

n

:= ax

2n

+ b ∀ n ∊ {1, …, n

1

+n

2

} (25)

Eq. (24) means that the deterministic forecast x

1n

can be modelled as a linear function of

the observation x

2n

. Thus, the level of the deterministic forecast depends on the level of

the observation, while its variation is modelled by a constant parameter, regardless of

the level of x

2n

. Given eqs. (9), (10) and (24) we prove in Appendix A that the

distribution of y

4

conditional on y

2

is:

f(y

4

|y

2

) = N((y

2

− b

2

)/a, (σ

e

/a)

2

I

n2

) = N(M

2

, Λ

2

) (26)

where

M

2

:= (y

2

− b

2

)/a (27)

Λ

2

:= (σ

e

/a)

2

I

n2

(28)

b

2

:= (b, …, b)

Τ

, n

2

×1 (29)

Combining eqs. (19) and (26) we prove in Appendix A that the joint distribution of

the future process of interest, given the historical observations and the deterministic

forecast, is:

h(y

4

|y

3

, x

1

) = N(M, Λ) (30)

9

where

Λ

−1

= (1/σ

2

) (R

22

− R

21

R

−1

11

R

12

)

−1

+ (a/σ

e

)

2

I

n2

(31)

M = Λ Λ

−1

1

M

1

+ (a/σ

2

e

) Λ (y

2

− b

2

) (32)

2.3 Estimation of the parameters of the Bayesian Processor of Forecasts

The parameters of the BPF are μ, σ, ρ

|i−j|

defined in eqs. (14)-(16) and a, b, σ

2

e

defined in

eqs. (24) and (25). The parameters μ, σ, ρ

|i−j|

can be estimated from fitting the joint

distribution of y

3

to y

3

. In the next sections, we will use the maximum likelihood

estimator. The parameters a, b, σ

e

can be estimated from the linear regression of x

1n

on

x

2n

over the time period {1, …, n

1

}. Figure 1 depicts the fitting periods.

2.4 Distinct fitting periods and other special cases

Sometimes the periods of fitting of the normal stationary model to estimate the

parameters μ, σ, ρ

|i−j|

and fitting of the linear model to estimate the parameters a, b, σ

e

do

not coincide. In such cases, the parameters can be estimated in distinct periods. For

example, in Figure 2, we assume that the deterministic model has already used

information from the historical observations to adjust the hindcast, therefore the {1, …,

n

1

} period cannot be used for the linear model fitting. However, the period of

observations {1, …, n

1

+n

2

} can be used for fitting the normal stationary model. In such

cases the intersection of the deterministic forecast period and the historical

observations {n

1

+1, …, n

1

+n

2

} can be used for fitting the linear model. We present the

distributions of interest and the proofs in Appendix B.

Figure 2. Time periods for the BPF data input and output and the related periods for

model fitting in the case of distinct periods.

In cases that the geophysical process is nonnegative (e.g. precipitation), the modelling

framework should be adapted to truncated variables. The necessity for doing this

appears when the coefficient of variation of the process is high (so that the probability of

10

getting a negative value from the normal distribution is not negligible). For such cases,

the BPF can be extended to include truncated normal distributions (Horrace 2005).

2.5 Hurst-Kolmogorov process

The model of interest for x

2

is the HKp, as explained in Section 1.1. The HKp is a three-

parameter normal stationary stochastic process in discrete time. Its parameters μ and σ

are defined by eqs. (14) and (15), while its parameter H is defined by eq. (33) (Tyralis

and Koutsoyiannis 2011):

ρ

k

:= Corr[x

t

, x

t + k

] = |k + 1|

2H

/ 2 + |k − 1|

2H

/ 2 − |k|

2H

, k = 0, 1,… (33)

We use the maximum likelihood estimator to estimate μ, σ and H simultaneously, as

proposed in Tyralis and Koutsoyiannis (2011) while the estimator is implemented in the

R package HKprocess (Tyralis 2016).

2.6 Investigation for various values of the parameters of the Bayesian Processor

of Forecasts

For specific values of the parameters of the BPF and in particular of the linear model, we

can understand its behaviour in extreme cases. We present the proofs of the results of

Section 2.6 in Appendix C. A similar investigation is presented in Krzysztofowicz (1985).

If σ

e

= 0, i.e., the deterministic model is perfect, then the BPF forecast prediction

interval is 0, while the BPF forecast is equal to the deterministic forecast (see eqs. (C.4)

and (C.5)). When a = 0 (see eq. 25), then the deterministic forecast does not improve the

BPF forecast. Then the BPF forecast is equal to the forecast of the stochastic process (see

eq. (C.8)). This problem has already been solved in Tyralis and Koutsoyiannis (2014),

who also employed a Bayesian treatment of the parameters of the stochastic process.

Intuitively, high values of σ

e

will result in high uncertainties. Furthermore negative

values of a will result in BPF forecasts with inverse trends compared to the

deterministic forecasts.

We can also assess the quality of the deterministic model using the sufficient

characteristic defined in Krzysztofowicz (1987) and the informativeness score defined

in Krzysztofowicz (1992, 2010). The sufficient characteristic (SC) and the

informativeness score (IS), defined respectively by eqs. (34) and (35) summarize the

information contained in the parameters a and σ:

SC := |

a| / σ

e

(34)

11

IS := ((SC / (1/

σ

))

−2

+ 1)

−1/2

(35)

Krzysztofowicz (2010) proved that:

r = sign(a) IS (36)

where r is the Pearson’s r defined by

r := Corr[y

1

, y

3

] (37)

For an intuitive explanation of the SC and the IS the reader is referred to

Krzysztofowicz (2010). In brief, the sufficiency characteristic is interpretable as a

“signal-to-noise ratio”, with |a| being the measure of signal, and σ being the measure of

noise, while the posterior variance depends on the SC. The SC ranges in the interval

[0, ∞] and the IS ranges in the interval [0, 1]. Higher values of both parameters imply a

more informative deterministic model and lower posterior variance. For the perfect

deterministic model we have SC = ∞ and IS = 1, while for a completely uninformative

deterministic model we have SC = 0 and IS = 0.

Normal stationary stochastic processes have finite 1

st

and 2

nd

order moments,

therefore μ and σ defined in eqs. (14) and (15) are finite. Subsequently the results

presented in Appendix C can be generalized using the SC and IS parameters. For

instance, a = 0 implies SC = 0 and IS = 0, while σ

e

= 0 implies SC = 1 and IS = ∞.

Furthermore the SC and the IS can be estimated from different samples, e.g. as in Figure

2. In such case |r| ≠ IS (Krzysztofowicz 2010), and both parameters provide different

information. Further investigations using simulations for special (artificially designed)

cases will be presented in Section 4.1.

3. Data

We apply the BPF to instrumental temperature and precipitation data, which we

aggregated on the annual time scale, and to the GCM projections, which we used as

deterministic forecasts.

3.1 Temperature data

We use monthly temperature data from the unadjusted version 3 of the Global Historical

Climatology Network-Monthly (GHCN-M) temperature dataset (Lawrimore et al. 2011).

The GHCN-M includes mean monthly temperatures observed in a large number of

stations, which cover the earth surface. We choose the stations for latitude in the

interval [25°, 50°] and longitude in the interval [−125°, −65°] (USA region).

12

Furthermore, we consider all monthly values in the time period 1916-2015, while we

exclude all stations with more than 12 missing values. We impute missing values using a

seasonal Kalman filter as implemented in the R package zoo (Zeileis and Grothendieck

2005). A number of 362 stations, depicted in Figure 3, remained after this procedure.

Figure 3. Map of locations for the 362 stations with temperature data (dots). Thiessen

polygons for each station within the convex hull of the stations are also depicted.

We used the Albers equal-area conic projection to map the data onto a flat plane and

perform all subsequent calculations. However, all map visualizations in the figures of the

manuscript are presented in an equirectangular map projection. After defining the

convex hull of the 362 stations, we defined all Thiessen polygons corresponding to each

station. The Thiessen (also known as Voronoi or Dirichlet) tessellation is computed by

functions in the spatstat and deldir R packages (Baddeley et al. 2015, Turner 2016

respectively) according to the second (iterative) algorithm of Lee and Schacter (1980).

The mean annual temperature in the convex hull for the time period 1916-2015 is

computed using the Thiessen polygon method.

3.2 Precipitation data

We use daily precipitation data from the Global Historical Climatology Network (GHCN,

Menne et al. 2012a, 2012b). The initial dataset included time series with missing or

flagged (i.e. data of low quality for reasons explained in Menne et al., 2012a) values. We

choose the stations with latitude in the interval [25°, 50°] and longitude in the interval

[−125°, −65°] (USA region). We processed the dataset according to a briefly described in

Appendix D sequence of actions. The locations of the 319 stations, which remained after

the selection procedure, are depicted in Figure 4.

13

Figure 4. Map of locations for the 319 stations with precipitation data (dots). Thiessen

polygons for each station within the convex hull of the stations are also depicted.

The definition of the convex hull of the stations and the methodology for the Thiessen

polygons and the calculation of the spatial average precipitation over the convex hull are

same as those described in Section 3.1 for temperature.

3.3 GCM data

By GCM data we mean the GCM outputs for monthly temperature and precipitation from

the CMIP5 experiment, which involves more than 50 GCMs modelled by 20 modelling

groups (Taylor et al. 2012). Each model comes with its own spatial grid resolution. The

models used in the present study and the variables of interest are presented in Table 1.

Each GCM in Table 1 includes a simulation of the recent past (1850-2005) (historical

run) and a future projection (2006-2100) forced by the representative concentration

pathway 6.0 (RCP6). The RCP6 experiment represents a high concentration pathway in

which stabilization of the radiative forcing at 6.0 Wm

−2

occurs around 2100 and then

forcing remains fixed (Masui et al., Meinshausen et al. 2011, Fig. 4). Most of the models

have multiple ensemble members. Here we use the ensemble member r1i1p1 for each

model.

14

Table 1. CMIP5 models acronyms, modelling groups and institutes, and variable of

interest. The model outputs were downloaded from

https://pcmdi.llnl.gov/search/cmip5/.

Model Name Temperature Precipitation Modelling Centre (or Group) Institute ID

GISS-E2-H

NASA Goddard Institute for Space Studies NASA GISS

GISS-E2-R

NASA Goddard Institute for Space Studies NASA GISS

HadGEM2-AO

National Institute of Meteorological

Research/Korea Meteorological

Administration

NIMR/KMA

IPSL-CM5A-LR

Institut Pierre-Simon Laplace IPSL

IPSL-CM5A-MR

Institut Pierre-Simon Laplace IPSL

MIROC5

Atmosphere and Ocean Research Institute

(The University of Tokyo), National Institute

for Environmental Studies, and Japan Agency

for Marine-Earth Science and Technology

MIROC

MIROC-ESM

Japan Agency for Marine-Earth Science and

Technology, Atmosphere and Ocean Research

Institute (The University of Tokyo), and

National Institute for Environmental Studies

MIROC

MIROC-ESM-CHEM

Japan Agency for Marine-Earth Science and

Technology, Atmosphere and Ocean Research

Institute (The University of Tokyo), and

National Institute for Environmental Studies

MIROC

MRI-CGCM3

Meteorological Research Institute MRI

NOAA GFDL GFDL-CM3

NOAA Geophysical Fluid Dynamics

Laboratory

NOAA GFDL

NOAA GFDL GFDL-ESM2G

NOAA Geophysical Fluid Dynamics

Laboratory

NOAA GFDL

NOAA GFDL GFDL-ESM2M

NOAA Geophysical Fluid Dynamics

Laboratory

NOAA GFDL

NorESM1-M

Norwegian Climate Centre NCC

NorESM1-ME

Norwegian Climate Centre NCC

We extract GCM grid data corresponding to points within the respective convex hulls

defined in Figure 3 and Figure 4 and to the time period 1916-2100. Two examples of the

Thiessen polygons formed from the GCMs points within the convex hull defined in

Figure 3, are presented in Figure 5. The methodology for aggregating the temperature

and precipitation over the convex hull is presented in Section 3.1.

Figure 5. Temperature (left) and precipitation (right) Thiessen polygons for each grid

centre point (dot) for the GISS-E2-H model within the convex hull of the stations.

4. Application

In Section 4, we present the results of the model presented in Section 2.2 to controlled

simulation data (for testing) and data of Section 3 (for prediction).

15

4.1 Framework testing using simulations

We test the performance of the BPF on simulated series with n

1

= 100 and n

2

= 50. The

aim is to show the performance of the BPF even in extreme conditions. In Table 2, we

present the types of simulated time series to which we applied the BPF. In Table 3, we

present the estimated parameters of the BPF. Additionally we present the Pearson’s r of

x

1(1:100)

and x

2(1:100)

and the respective values of the SC and the IS. In all examined cases

we use the same simulated time series x

2(1:100)

, therefore the parameter σ has a common

value. Thus in all cases, the SC and IS provide the same amount of information.

Table 2. Simulated time series presented in the Figures of Section 4.1.

Case Figure Variable Simulation

1 Figure 6 (top) x

1

HKp with μ = 0, σ = 0.40, H = 0.50 with added trend = 0.01

x

2

HKp with μ = 5, σ = 2, H = 0.70

2 Figure 6 (bottom) x

1

Equal to x

2

of case 1 in the period 1-100. Linear trend = 0.50 with starting point

equal to (x

2(100)

of case 1 + 0.50) in the period 101-150

x

2

Equal to

x

2

of case 1

3 Figure 7 (top) x

1

Equal to x

2

of case 1 in the period 1-100. Linear trend = 0.10 with starting point

equal to (x

2(100)

of case 1 + 0.10) in the period 101-150. In the resulting time

series we add an HKp with μ = 0, σ = 1, H = 0.50

x

2

Equal to x

2

of case 1

4 Figure 7 (bottom) x

1

Equal to x

2

of case 1 in the period 1-100. Linear trend = 0.10 with starting point

equal to (x

2(100)

of case 1 + 0.10) in the period 101-150. In the resulting time

series we add an HKp with μ = 0, σ = 1, H = 0.50 and we shift it up 5 units

x

2

Equal to x

2

of case 1

5 Figure 8 (top) x

1

HKp with μ = 5, σ = 2, H = 0.50 in the period 1-100. Linear trend = 0.10 with

starting point equal to (x

1(100)

of the present case + 0.10) in the period 101-150

x

2

Equal to x

2

of case 1

6 Figure 8 (middle) x

1

Equal to x

1

of case 6 in the period 1-100. Linear trend = 0.50 with starting point

equal to (x

1(100)

of case 6 + 0.50) in the period 101-150

x

2

Equal to

x

2

of case 1

7 Figure 8 (bottom) x

1

HKp with μ = 5, σ = 2, H = 0.50 in the period 1-100. Linear trend = 0.50 with

starting point equal to (x

1(100)

of the present case + 0.50) in the period 101-150

x

2

Equal to x

2

of case 1

Table 3. Estimates of the BPF parameters defined in eqs. (14)-(16), (24), (25), (33) for

the cases of Table 2. r is defined in eq. (37) and is estimated using sample Pearson’s r of

x

1(1:100)

and x

2(1:100)

. SC and IS are defined in eqs. (34) and (35) and estimated by

substituting σ, a, σ

e

with their estimates. Cases with higher IS have a better ranking.

Case Figure μ σ H a b σ

e

r SC IS ranking

1 Figure 6 (top) 4.63 1.93 0.67 −0.05 0.81 0.54 −0.16 0.08 0.16 4

2 Figure 6 (bottom) 4.63 1.93 0.67 1.00 0.00 0.00 1.00 ∞ 1.00 1

3 Figure 7 (top) 4.63 1.93 0.67 0.99 0.03 1.11 0.86 0.89 0.86 3

4 Figure 7 (bottom) 4.63 1.93 0.67 1.07 4.57 0.97 0.91 1.11 0.91 2

5 Figure 8 (top) 4.63 1.93 0.67 −0.02 4.92 1.90 −0.02 0.01 0.02 6

6 Figure 8 (middle) 4.63 1.93 0.67 −0.02 4.92 1.90 −0.02 0.01 0.02 7

7 Figure 8 (bottom) 4.63 1.93 0.67 −0.06 6.41 1.95 −0.06 0.03 0.06 5

Figure 6 (top) shows the results of the application assuming: (a) x

2

follows a HKp, (b)

a linear deterministic forecast model and (c) the deterministic forecast is of low quality

(a is almost equal to 0 and IS is low). Pearson’s r is related to a and both are slightly

16

negative. Therefore, the influence of the deterministic forecast on the probabilistic

forecast is negligible.

Figure 6. 95% prediction intervals produced by the BPF for the case of a time series

(green) simulated from a HKp, when the deterministic model (blue) is of low quality

(top) and perfect (bottom). The mean is equal to the estimated μ of the HKp model fitted

to the observations of the period 1-100. The BPF is fitted on the period 1-100 and

predicts for the period 101-150. The characteristics of the simulated time series are

presented in Table 2, while the estimated parameters of the BPF are shown in Table 3.

In the test application of Figure 6 (bottom) the assumptions are radically different.

Here again x

2

follows a HKp, but the deterministic hindcast is assumed to be perfect

(zero error and is = 1). Furthermore, the deterministic forecast is assumed to be a huge

linear trend. Because of the perfect performance of the deterministic model in the past

the BPF forecast fully complies with the deterministic model forecast (it is exactly equal

and 95% prediction interval has zero length in each time step). One could view this case

as model testing in nonstationary conditions, because at time 100 the deterministic

model fully changes its behaviour, yielding the huge linear trend, which did not appear

before. Even though the BPF framework is founded on a fully stationary setting, it

perfectly captures the assumed nonstationary behaviour. The reason is that the

deterministic model is found to behave very well in the past; had it behaved badly, the

probabilistic forecast (BPF) would disregard the linear trend and would be similar to

that in Figure 6 (top).

17

In the application depicted in Figure 7 (top), the deterministic hindcast is almost

perfect (with a small, nonzero, error and a high IS). Here again (as in Figure 6 (bottom)),

the BPF forecast is strongly influenced by the deterministic forecast; however, now

some prediction intervals of nonzero size appear. In the application, depicted in Figure 7

(bottom) the deterministic hindcast and forecast of Figure 7 (top) have been shifted up.

However, the BPF forecast has not changed. This shows that the BPF is invariant under

the mean change, which is a desirable property. The meaning of this that if the

deterministic model has a systematic bias, however high, the BPF framework

automatically removes it.

Figure 7. 95% prediction intervals produced by the BPF for the case of a time series

(green) simulated from a HKp with μ = 5, σ = 2 and H = 0.7, when the deterministic

model (blue) is almost perfect and varies a bit around the observations (top), or is

shifted up (bottom). The mean is equal to the estimated μ of the HKp model fitted to the

observations of the period 1-100. The BPF is fitted in the period 1-100 and predicts for

the period 101-150. The characteristics of the simulated time series are presented in

Table 2, while the estimated parameters of the BPF are shown in Table 3.

In the application depicted in Figure 8 (top), the deterministic hindcast is of low

quality. Furthermore, a is slightly negative. In this case, the deterministic forecast is

increasing while the BPF is slightly decreasing, which is reasonable because of the

negative a. In Figure 8 (middle) the deterministic hindcast and observations are equal to

those of Figure 8 (top). However, Figure 8 (middle) differs from Figure 8 (top) in that

the deterministic forecast in the period {101, …, 150} increases faster resulting in a

faster decrease of the BPF forecast. In Figure 8 (bottom), a is even more negative,

18

resulting in an even higher decrease of the BPF forecast. Finally the IS provides a

ranking of the models (cases) in terms of their informativeness (from the highest to the

lowest) which in the examined cases is 2, 4, 3, 1, 7, 5 and 6.

Figure 8. 95% prediction intervals produced by the BPF for the case of a time series

(green) simulated from a HKp with μ = 5, σ = 2 and H = 0.7. The deterministic model

(blue) is a simulated HKp with equal parameters, but of low quality in hindcast and a

slight linear trend in forecast (top) or high trend (middle) and more negative correlation

(bottom). The mean is equal to the estimated μ of the HKp model fitted to the

observations of the period 1-100. The BPF is fitted in the period 1-100 and predicts for

the period 101-150. The characteristics of the simulated time series are presented in

Table 2, while the estimated parameters of the BPF are shown in Table 3.

Overall, all testing experiments indicate an ideal performance of the BPF framework

in all cases, even the most extreme ones and those with huge nonstationary trends. The

methodology presented complies with the simple truth of the scientific method that

model predictions for the future are taken into account insofar models comply with

evidence from data of the past. Also, the methodology complies with the blueprint by

Montanari and Koutsoyiannis (2012) insofar it takes a deterministic model and

incorporates it into a stochastic framework, thus converting the deterministic into

19

stochastic predictions. If the deterministic model is good, the final stochastic prediction

highly relies on it. If the model is bad, it is almost automatically discarded.

4.2 Case studies

In Section 4.2, we present the application of the BPF to the data of Section 3. We present

two variants of the BPF, which are described in Sections 2.2 (Figure 1) and 2.4 (Figure 2)

and the respective fitting and forecasting periods in Figure 9. In the case of Figure 9

(top), the GCM historical runs have already been adjusted using information from the

observations, therefore using the time period 1916-2005 would use the same

information twice. Instead, in the case of Figure 9 (bottom), the fitting of the BPF linear

model in the time period 2006-2015 is based on a forecast with the assumptions of the

RCP6 experiment regarding the emissions scenario which have not been checked. In

both cases, we used the HKp to model the observations.

Figure 9. BPF fitting and predicting time periods. The fitting period is defined as the

period of the historical run (top) or the intersection of the historical observations and

the RCP4.5 time periods (bottom). The prediction period succeeds the fitting period and

extends to the year 2100.

We present the results in Figures 10-15. In all figures the mean of the observations, is

equal to the maximum likelihood estimate of μ as given in Section 2.5 for the fitting time

period. While we examined all GCMs of Table 1, we present here in detail two of them,

i.e. the GISS-E2-H and the MRI-CGCM3 along with summary information for all results.

Figure 10 shows the prediction of the mean annual temperature in the USA for the two

GCMs when the fitting time-period is 1916-2005. In the case of the GISS-E2-H model the

forecasted increase is equal to 0.8 °C while the 95% prediction interval is 1.8 °C wide. In

20

the case of the MRI-CGCM3 model, the forecasted increase is negligible while the 95%

prediction interval is again 1.8 °C wide. In Figure 11, the prediction intervals for the

fitting time-period 2006-2015 indicate a mean increase in the annual temperature equal

to 1.4 °C and 0.9 °C for both models respectively, while the respective prediction

intervals are 2.0 °C wide.

Figure 10. 95% prediction intervals of the mean annual temperature in the USA

produced by the BPF for the case of Figure 9 (top). The fitting time period is 1916-2005,

while the deterministic models are ensembles from the GISS-E2-H (top) and MRI-CGCM3

(bottom) models. The mean of the observations is equal to the maximum likelihood

estimate of μ in Section 2.5 for the fitting time period.

21

Figure 11. 95% prediction intervals of the mean annual temperature in the USA

produced by the BPF for the case of Figure 9 (bottom). The fitting time period is 2006-

2015, while the deterministic models are ensembles from the GISS-E2-H (top) and MRI-

CGCM3 (bottom) models.

Figures 12 and 13 depict similar results for the annual precipitation in the USA. In

particular, in Figure 12, which shows the results for the fitting time-period 1916-2005,

we observe a negligible increase of the annual precipitation while the 95% prediction

intervals are 200 mm wide. In Figure 13, where the fitting time-period is 2006-2015, we

observe an insignificant and a mean annual increase of 120 mm respectively, while the

respective 95% prediction intervals are 220 and 140 mm wide.

22

Figure 12. 95% prediction intervals of the annual precipitation in the USA produced by

the BPF for the case of Figure 9 (top). The fitting time period is 1916-2005, while the

deterministic models are ensembles from the GISS-E2-H (top) and MRI-CGCM3 (bottom)

models.

Figure 13. 95% prediction intervals of the annual precipitation in the USA produced by

the BPF for the case of Figure 9 (bottom). The fitting time period is 2006-2015, while the

deterministic models are ensembles from the GISS-E2-H (top) and MRI-CGCM3 (bottom)

models.

23

Figures 14 and 15 display the results for all models of Table 1. In particular, they

show the forecasted mean annual temperatures and annual precipitations for both

method variants defined in Figure 9. Furthermore, the graphs include the envelopes of

all 95% prediction intervals. In Figure 14, we observe an envelope of the mean annual

temperature 5.8 °C wide when the fitting time-period is 1916-2005 and an envelope

8.8 °C wide when the fitting time-period is 2006-2015. In the former case, the

temperature increase is centred around 2.5 °C for the year 2100, while in the latter case

the mean annual change seems to be negligible, while the overall shape of the graph

could be called a “Bayesian thistle”. Regarding the precipitation, we observe in Figure 15

envelopes 270 and 330 mm wide for the fitting time-periods 1916-2005 and 2006-2015

respectively. The forecasted increase in precipitation is negligible in the former case,

while it is approximately equal to 50 mm for the year 2100 in the latter case.

Figure 14. Prediction intervals of the mean annual temperature in the USA produced by

the BPF. The GCM medians correspond to all GCMs of Table 1. The prediction quantiles

are the envelopes of all 95% prediction intervals of the GCMs of Table 1 produced by the

BPF. The fitting time period is 1916-2005 (top, corresponds to Figure 9 (top)) and 2006-

2015 (bottom, corresponds to Figure 9 (bottom)).

24

Figure 15. Prediction intervals of the annual precipitation in the USA produced by the

BPF. The GCM medians correspond to all GCMs of Table 1. The prediction quantiles are

the envelopes of all 95% prediction intervals of the GCMs of Table 1 produced by the

BPF. The fitting time period is 1916-2005 (top, corresponds to Figure 9 (top)) and 2006-

2015 (bottom, corresponds to Figure 9 (bottom)).

5. Conclusions

The aim of this paper is to probabilistically predict the future evolution of a normal

stationary stochastic process used to model a geophysical variable conditional on

historical observations of the variable and hindcasts and forecasts of the variable

produced by a deterministic model. To this end, we apply the Bayesian Processor of

Forecasts (BPF) to the data of interest. The BPF has previously been applied to

independent variables or Markovian processes. Here, we extend its use to include any

normal stationary stochastic processes and we present an application to the special case

of the Hurst-Kolmogorov process.

We investigate the properties of the BPF and test its performance using simulated

time series. We show that the influence of the deterministic forecast increases when

there is a good fitting of the deterministic model to the historical observations. Indeed,

when this fitting is perfect, the BPF forecast is equal to the deterministic forecast. In

contrast, when this fitting is insufficient, the forecast depends on the observations and

the stochastic model and not on the deterministic model. Furthermore, even if the

25

stochastic model is stationary, the BPF can incorporate changes, which can be attributed

to non-stationarity.

The BPF is applied to the mean annual temperature and annual precipitation in the

time period 1916-2005 in the USA. The GCMs (the historical and the RCP6 scenarios) are

used as deterministic models. Using the estimated BPF parameters, we probabilistically

forecast the mean annual temperature and annual precipitation until the year 2100. The

results are sensitive to the choice of the fitting period between the observations and the

deterministic forecast and the choice of the GCM model. Regarding the temperature the

overall results show increasing temperature when the fitting period is the intersection

of the data time period and the historical scenario, while the temperature remains

unchanged when the fitting period is the intersection of the data time period and the

RCP6 scenario. In both cases, the envelopes of the 95% prediction intervals for each

GCM model are significantly wide (5.8 °C and 8.8 °C respectively). Regarding the

precipitation, the deterministic models had negligible effect in improving the forecast of

the stochastic model, regardless of the fitting period.

We emphasize that the estimation of the stochastic model parameters should better

be performed using only data that were not used in the GCM fitting/tuning, i.e. for the

period after 2006. This would correspond to the so-called split-sample technique

(Klemeš 1986), which avoids possible model overfitting on the available data and thus

artificially good performance. This corresponds to model fitting period after 2006. The

applications with this variant of the methodology showed that the uncertainty of the

forecasts increases considerably and practically result in total neglect of the GCM

predictions regarding for both temperature and precipitation. Finally the inclusion of

the uncertainty in a fully Bayesian setting, also considering the uncertainty of

parameters, would result in even higher uncertainties of the forecasted variables.

Funding information: The authors received no funding for this research, which was

performed for scientific curiosity.

Acknowledgement: We acknowledge two anonymous reviewers whose suggestions as

well as comments, both positive and negative, helped to improve the manuscript. We

acknowledge the World Climate Research Programme's Working Group on Coupled

Modelling, which is responsible for CMIP, and we thank the climate modelling groups

(listed in Table 1 of this paper) for producing and making available their model outputs.

26

For CMIP the U.S. Department of Energy's Program for Climate Model Diagnosis and

Intercomparison provides coordinating support and led development of software

infrastructure in partnership with the Global Organization for Earth System Science

Portals.

AppendixA TheBayesianProcessorof Forecastsapplied tonormalstationary

stochasticprocesses

Here we prove the results presented in Section 2.2. Overall, we prefer to use techniques

typically met in the Bayesian statistics literature, such as proportionality of the

distributions, and avoid to calculate integrals. For example, Marty et al. (2015) used

these techniques when they examined the Bayesian processor of output, while

Krzysztofowicz (1985) preferred the other way when he examined the BPF.

Let x

1

and x

2

and their subsets y

1

, y

2

, y

3

, y

4

be defined as follows:

x

1(1:(n1+n2))

:= (x

11

, …, x

1n1

, x

1(n1+1)

, …, x

1(n1+n2)

)

T

, deterministic forecast (A.1)

x

2(1:(n1+n2))

:= (x

21

, …, x

2n1

, x

2(n1+1)

, …, x

2(n1+n2)

)

T

, observations (A.2)

x

1

:= x

1(1:(n1+n2))

(A.3)

x

2

:= x

2(1:(n1+n2))

(A.4)

y

1

:= (x

11

, x

12

, …, x

1n1

)

T

: n

1

×1 (A.5)

y

2

:= (x

1(n1+1)

, …, x

1(n1+n2)

)

T

: n

2

×1 (A.6)

y

3

:= (x

21

, …, x

2n1

)

T

: n

1

×1 (A.7)

y

4

:= (x

2(n1+1)

, …, x

2(n1+n2)

)

T

: n

2

×1 (A.8)

Then the conditional independence mentioned in Section 2.1 is defined by

f

n

(x

11

, x

12

,…, x

1n

|x

21

, x

22

,…, x

2n

) :=

∏

i = 1

n f

i

(x

1i

|x

21

, x

22

, …, x

2n

) ∀ n ∊ {1, …, n

1

+n

2

} (A.9)

f

i

(x

1i

|x

21

, x

22

, …, x

2n

) := f

i

(x

1i

|x

2i

) ∀ i, n ∊ {1, …, n

1

+n

2

} (A.10)

which results in

f

n

(x

11

, x

12

, …, x

1n

|x

21

, x

22

, …, x

2n

) =

∏

i = 1

n f

i

(x

1i

|x

2i

) ∀ n ∊ {1, …, n

1

+n

2

} (A.11)

Hence,

h(y

4

|y

3

, x

1

) = f(x

1

|y

3

, y

4

) g(y

3

, y

4

) / ξ(y

3

, x

1

) ⇒ (A.12)

h(y

4

|y

3

, x

1

) ∝ f(x

1

|y

3

, y

4

) g(y

3

, y

4

) ⇒ (A.13)

27

h(y

4

|y

3

, x

1

) ∝ f(y

1

, y

2

|y

3

, y

4

) g(y

3

, y

4

) ⇒ (A.14)

h(y

4

|y

3

, x

1

) ∝ f(y

2

|y

4

) g(y

3

, y

4

) ⇒ (A.15)

h(y

4

|y

3

, x

1

) ∝ f(y

2

|y

4

) g(y

4

|y

3

) g(y

3

) ⇒ (A.16)

h(y

4

|y

3

, x

1

) ∝ f(y

2

|y

4

) g(y

4

|y

3

) (A.17)

Equation (A.17) proves eq. (13).

We define the parameters of the normal stationary stochastic process used to model

the observations with eqs. (A.18)-(A.28). Matrices Σ, R and their submatrices are

symmetric Toeplitz positive definite matrices (Golub and Van Loan 1996, p.193). This

facilitates their handling using the Levinson or related algorithms (e.g. Tyralis and

Koutsoyiannis 2011). Consequently,

μ := E[x

2n

] ∀ n ∊ {1, …, n

1

+n

2

} (A.18)

σ

2

:= Var[x

2n

] ∀ n ∊ {1, …, n

1

+n

2

} (A.19)

μ

1

:= (μ, …, μ)

Τ

, n

1

×1 (A.20)

μ

2

:= (μ, …, μ)

Τ

, n

2

×1 (A.21)

Σ := σ

2

[ρ

i,j

] ∀ i, j ∊ {1, …, n

1

+n

2

} (A.22)

ρ

i,j

= ρ

|i−j|

∀ i, j ∊ {1, …, n

1

+n

2

} (A.23)

Σ =

Σ

11

Σ

12

Σ

21

Σ

22

(A.24)

Σ

11

: n

1

×n

1

, Σ

21

: n

2

×n

1

, Σ

12

: n

1

×n

2

, Σ

22

: n

2

×n

2

(A.25)

Σ = σ

2

R

11

R

12

R

21

R

22

(A.26)

R

11

: n

1

×n

1

, R

21

: n

2

×n

1

, R

12

: n

1

×n

2

, R

22

: n

2

×n

2

(A.27)

Σ

11

= σ

2

R

11

, Σ

21

= σ

2

R

21

, Σ

12

= σ

2

R

12

, Σ

22

= σ

2

R

22

(A.28)

Since we model x

2

with a multivariate normal distribution, the distribution of y

4

conditional on y

3

is given by (Eaton 2007, p.116),

g(y

4

|y

3

) = N(μ

2

+ Σ

21

Σ

−1

11

(y

3

− μ

1

), Σ

22

− Σ

21

Σ

−1

11

Σ

12

) (A.29)

which can be written as

g(y

4

|y

3

) = Ν(Μ

1

, Λ

1

) (A.30)

where

Μ

1

:= μ

2

+ R

21

R

−1

11

(y

3

− μ

1

) (A.31)

28

Λ

1

:= σ

2

(R

22

− R

21

R

−1

11

R

12

) (A.32)

whereas eq. (A.32) denotes the Schur complement (Horn and Zhang 2005).

If the distribution of x

1n

conditional on x

2n

is given by

f(x

1n

|x

2n

) := Ν(q

n

, σ

2

e

) ∀ n ∊ {1, …, n

1

+n

2

} (A.33)

then, using eq. (A.11) and the properties of the product of normal distributions

(Bromiley 2014) we find:

q

n

:= ax

2n

+ b ∀ n ∊ {1, …, n

1

+n

2

} (A.34)

f(y

2

|y

4

) =

∏

n = n

1

+1

n

1

+n

2

Ν(q

n

, σ

2

e

) ⇒ (A.35)

f(y

2

|y

4

) = N(Q, V) (A.36)

where

Q := (q

n1+1

, …, q

n1+n2

)

T

, n

2

×1 (A.37)

V := σ

2

e

I

n2

, n

2

×n

2

(A.38)

However, in the Bayesian setting, y

2

is known while the distribution of interest is that

of y

4

|y

2

. Therefore, eq. (A.36) is transformed to eq. (A.45) in which y

4

is the random

variable and y

2

is a value, after some algebraic manipulations:

b

2

:= (b, …, b)

Τ

, n

2

×1 (A.39)

f(y

4

|y

2

) ∝ exp(−(1/2) (y

2

− ay

4

− b

2

)

T

V

−1

(y

2

− ay

4

− b

2

)) ⇒ (A.40)

f(y

4

|y

2

) ∝ exp(−(a

2

/2 σ

2

e

) (y

4

– (y

2

− b

2

)/a)

T

I

−1

n2

(y

4

– (y

2

− b

2

)/a)) ⇒ (A.41)

f(y

4

|y

2

) ∝ exp(−(1/2) (y

4

– (y

2

− b

2

)/a)

T

(a/σ

e

)

2

I

−1

n2

(y

4

– (y

2

− b

2

)/a)) ⇒ (A.42)

f(y

4

|y

2

) ∝ exp(−(1/2) (y

4

– (y

2

− b

2

)/a)

T

((σ

e

/a)

2

I

n2

)

–1

(y

4

– (y

2

− b

2

)/a)) ⇒ (A.43)

f(y

4

|y

2

) = N((y

2

− b

2

)/a, (σ

e

/a)

2

I

n2

) ⇒ (A.44)

f(y

4

|y

2

) = N(M

2

, Λ

2

) (A.45)

where,

M

2

:= (y

2

− b

2

)/a (A.46)

Λ

2

:= (σ

e

/a)

2

I

n2

(A.47)

The distribution of y

4

|y

3

, x

1

in eq. (A.17) is normal, i.e.,

h(y

4

|y

3

, x

1

) = N(M, Λ) (A.48)

29

because it is proportional to the product of the two normal distributions (A.30) and

(A.45) (Bromiley 2014). Its parameters are given by eqs. (A.50) and (A.54) after the

following manipulations:

Λ

−1

:= Λ

−1

1

+ Λ

−1

2

⇒ (A.49)

Λ

−1

= (1/σ

2

) (R

22

− R

21

R

−1

11

R

12

)

−1

+ (a/σ

e

)

2

I

n2

(A.50)

Λ

−1

M = Λ

−1

1

M

1

+ Λ

−1

2

M

2

⇒ (A.51)

M = Λ (Λ

−1

1

M

1

+ Λ

−1

2

M

2

) ⇒ (A.52)

M = Λ Λ

−1

1

M

1

+ Λ (a/σ

e

)

2

((y

2

− b

2

)/a) ⇒ (A.53)

M = Λ Λ

−1

1

M

1

+ (a/σ

2

e

) Λ (y

2

− b

2

) (A.54)

AppendixB TheBayesianProcessorofForecastsappliedtodistinctfitting

periods

Here we repeat the procedure of Appendix A but for distinct fitting periods. The time

period {1, …, n

1

+n

2

+ n

3

} is divided in three subperiods {1, …, n

1

}, {n

1

+1, …, n

1

+n

2

},

{n

1

+n

2

+1, …, n

1

+n

2

+ n

3

}. The processes of interest are x

1

and x

2

and their subsets y

1

, y

2

,

y

3

, y

4

, y

5

, y

6

defined as:

x

1((n1+1):(n1+n2+n3))

:= (x

1(n1+1)

, …, x

1(n1+n2)

, x

1(n1+n2+1)

, …, x

1(n1+n2+n3)

)

T

(B.1)

x

2(1:(n1+n2+n3))

:= (x

21

, …, x

2n1

, x

2(n1+1)

, …, x

2(n1+n2)

, x

2(n1+n2+1)

, …, x

2(n1+n2+n3)

)

T

(B.2)

x

1

:= x

1((n1+1):(n1+n2+n3))

(B.3)

x

2

:= x

2(1:(n1+n2+n3))

(B.4)

y

1

:= (x

1(n1+1)

, …, x

1(n1+n2)

)

T

: n

2

×1 (B.5)

y

2

:= (x

1(n1+n2+1)

, …, x

1(n1+n2+n3)

)

T

: n

3

×1 (B.6)

y

3

:= (x

21

, …, x

2n1

)

T

: n

1

×1 (B.7)

y

4

:= (x

2(n1+1)

, …, x

2(n1+n2)

)

T

: n

2

×1 (B.8)

y

5

:= (x

2(n1+n2+1)

, …, x

2(n1+n2+n3)

)

T

: n

3

×1 (B.9)

y

6

:= (x

21

, …, x

2(n1+n2)

)

T

: (n

1

+n

2

)×1 (B.10)

Then the conditional independence mentioned in Section 2.1 is defined by

f

n

(x

11

, x

12

,…,x

1n

|x

21

,x

22

,…,x

2n

) :=

∏

i = 1

n f

i

(x

1i

|x

21

, x

22

, …, x

2n

) ∀ n ∊ {1,…,n

1

+n

2

} (B.11)

30

f

i

(x

1i

|x

21

, x

22

, …, x

2n

) := f

i

(x

1i

|x

2i

) ∀ i, n ∊ {1, …, n

1

+n

2

} (B.12)

which result in

f

n

(x

11

, x

12

, …, x

1n

|x

21

, x

22

, …, x

2n

) =

∏

i = 1

n f

i

(x

1i

|x

2i

) ∀ n ∊ {1, …, n

1

+n

2

} (B.13)

Hence,

h(y

5

|y

3

, y

4

, x

1

) = f(x

1

|y

3

, y

4

, y

5

) g(y

3

, y

4

, y

5

) / ξ(y

3

, y

4

, x

1

) ⇒ (B.14)

h(y

5

|y

3

, y

4

, x

1

) ∝ f(x

1

| y

3

, y

4

, y

5

) g(y

3

, y

4

, y

5

) ⇒ (B.15)

h(y

5

|y

3

, y

4

, x

1

) ∝ f(y

1

, y

2

| y

3

, y

4

, y

5

) g(y

3

, y

4

, y

5

) ⇒ (B.16)

h(y

5

|y

3

, y

4

, x

1

) ∝ f(y

2

|y

5

) g(y

3

, y

4

, y

5

) ⇒ (B.17)

h(y

5

|y

3

, y

4

, x

1

) ∝ f(y

2

|y

5

) g(y

5

|y

3

, y

4

) g(y

3

, y

4

) ⇒ (B.18)

h(y

5

|y

3

, y

4

, x

1

) ∝ f(y

2

|y

5

) g(y

5

|y

3

, y

4

) (B.19)

We define the parameters of the normal stationary stochastic process used to model

the observations through the following equations:

μ := E[x

2n

] ∀ n ∊ {1, …, n

1

+n

2

+n

3

} (B.20)

σ

2

:= Var[x

2n

] ∀ n ∊ {1, …, n

1

+n

2

+n

3

} (B.21)

μ

1

:= (μ, …, μ)

Τ

, (n

1

+n

2

)×1 (B.22)

μ

2

:= (μ, …, μ)

Τ

, n

3

×1 (B.23)

Σ := σ

2

[ρ

i,j

] ∀ i, j ∊ {1, …, n

1

+n

2

+n

3

} (B.24)

ρ

i,j

= ρ

|i−j|

∀ i, j ∊ {1, …, n

1

+n

2

+n

3

} (B.25)

Σ =

Σ

11

Σ

12

Σ

21

Σ

22

(B.26)

Σ

11

: (n

1

+n

2

)×(n

1

+n

2

), Σ

21

: n

3

×(n

1

+n

2

), Σ

12

: (n

1

+n

2

)×n

3

, Σ

22

: n

3

×n

3

(B.27)

Σ = σ

2

R

11

R

12

R

21

R

22

(B.28)

R

11

: (n

1

+n

2

)×(n

1

+n

2

), R

21

: n

3

×(n

1

+n

2

), R

12

: (n

1

+n

2

)×n

3

, R

22

: n

3

×n

3

(B.29)

Σ

11

= σ

2

R

11

, Σ

21

= σ

2

R

21

, Σ

12

= σ

2

R

12

, Σ

22

= σ

2

R

22

(B.30)

Since we model x

2

with a multivariate normal distribution, the distribution of y

5

conditional on y

3

and y

4

is given by (Eaton 2007, p.116)

g(y

5

|y

3

, y

4

) = N(μ

2

+ Σ

21

Σ

−1

11

(y

6

− μ

1

), Σ

22

− Σ

21

Σ

−1

11

Σ

12

) ⇒ (B.31)

which can be written as

31

g(y

5

|y

3

, y

4

) = Ν(Μ

1

, Λ

1

) (B.32)

where

Μ

1

:= μ

2

+ R

21

R

−1

11

(y

6

− μ

1

) (B.33)

Λ

1

:= σ

2

(R

22

− R

21

R

−1

11

R

12

) (B.34)

If the distribution of x

1n

conditional on x

2n

is given by

f(x

1n

|x

2n

) = Ν(q

n

, σ

2

e

) ∀ n ∊ {1, …, n

1

+n

2

} (B.35)

then, using eq. (B.13) and the properties of the product of normal distributions

(Bromiley 2014), we find

q

n

:= ax

2n

+ b ∀ n ∊ {1, …, n

1

+n

2

} (B.36)

f(y

2

|y

5

) =

∏

n = n

1

+n

2

+1

n

1

+n

2

+n

3

Ν(q

n

, σ

2

e

) ⇒ (B.37)

f(y

2

|y

5

) = N(Q, V) (B.38)

where

Q = (q

n1+n2+1

, …, q

n1+n2+n3

)

T

, n

3

×1 (B.39)

V = σ

2

e

I

n3

, n

3

×n

3

(B.40)

However, in the Bayesian setting y

2

is given while the distribution of interest is that of

y

5

|y

2

. Therefore, eq. (B.38) is transformed to eq. (B.47) in which y

5

is the random

variable and y

2

is a value, after some algebraic manipulations.

b

2

:= (b, …, b)

Τ

, n

3

×1 (B.41)

f(y

5

|y

2

) ∝ exp(−(1/2) (y

2

− ay

5

− b

2

)

T

V

−1

(y

2

− ay

5

− b

2

)) ⇒ (B.42)

f(y

5

|y

2

) ∝ exp(−(a

2

/2 σ

2

e

) (y

5

– (y

2

− b

2

)/a)

T

I

−1

n3

(y

5

– (y

2

− b

2

)/a)) ⇒ (B.43)

f(y

5

|y

2

) ∝ exp(−(1/2) (y

5

– (y

2

− b

2

)/a)

T

(a/σ

e

)

2

I

−1

n3

(y

5

– (y

2

− b

2

)/a)) ⇒ (B.44)

f(y

5

|y

2

) ∝ exp(−(1/2) (y

5

– (y

2

− b

2

)/a)

T

((σ

e

/a)

2

I

n3

)

–1

(y

5

– (y

2

− b

2

)/a)) ⇒ (B.45)

f(y

5

|y

2

) = N((y

2

− b

2

)/a, (σ

e

/a)

2

I

n3

) ⇒ (B.46)

f(y

5

|y

2

) = N(M

2

, Λ

2

) (B.47)

M

2

:= (y

2

− b

2

)/a (B.48)

Λ

2

:= (σ

e

/a)

2

I

n3

(B.49)

The distribution of y

5

|y

3

, y

4

, x

1

in eq. (B.19) is normal, i.e.

32

h(y

5

|y

3

, y

4

, x

1

) = N(M, Λ) (B.50)

because it is proportional to the product of the two normal distributions (B.32) and

(B.47) (Bromiley 2014). Its parameters are given by eqs. (B.52) and (B.56) according to

the following manipulations:

Λ

−1

:= Λ

−1

1

+ Λ

−1

2

(B.51)

Λ

−1

= (1/σ

2

) (R

22

− R

21

R

−1

11

R

12

)

−1

+ (a/σ

e

)

2

I

n3

(B.52)

Λ

−1

M = Λ

−1

1

M

1

+ Λ

−1

2

M

2

⇒ (B.53)

M = Λ (Λ

−1

1

M

1

+ Λ

−1

2

M

2

) ⇒ (B.54)

M = Λ Λ

−1

1

M

1

+ Λ (a/σ

e

)

2

((y

2

− b

2

)/a) ⇒ (B.55)

M = Λ Λ

−1

1

M

1

+ (a/σ

2

e

) Λ (y

2

− b

2

) (B.56)

AppendixC InvestigationoftheBPFforvariousvaluesofitsparameters

From eq. (A.49) we obtain

Λ = Λ

2

(I

n2

+ Λ

−1

1

Λ

2

)

−1

(C.1)

while from eqs. (A.52) and (C.1) we obtain

M = Λ

2

(I

n2

+ Λ

−1

1

Λ

2

)

−1

Λ

−1

1

M

1

+ Λ

2

(I

n2

+ Λ

−1

1

Λ

2

)

−1

Λ

−1

2

M

2

(C.2)

In the case where

σ

e

= 0 (C.3)

using eqs. (C.1) and (C.2), we obtain:

M = M

2

(C.4)

Λ = 0

n2

(C.5)

In the case where

a = 0 (C.6)

we find that

f(y

4

|y

2

) = constant (C.7)

Hence, from eqs. (A.17) and (A.30) we obtain

h(y

4

|y

3

, x

1

) = N(M

1

, Λ

1

) (C.8)

33

AppendixD Precipitationdata

Here we present the sequence of steps to aggregate the precipitation from the daily to

the annual scale. This sequence is reproduced from Tyralis et al. (2017) who use the

same dataset and procedures.

A. Flagged values were considered as missing values.

B. Months with a percentage of recorded values higher than 0.83 (i.e. with more

than 25/30 or 26/31 daily observations) are considered good, while months with a

percentage of recorded values less than 0.34 (i.e. equal or less than 10/30 and 10/31

daily observations) are considered of poor quality. The reason for the differentiation is

that we first aggregate to the monthly time scale and then to the annual time scale. Thus

even if all values in a month are missing we can fill the monthly value after the first

aggregation as described in step C.

B1. Missing values within months with observed values more than 83% are filled

using linear interpolation.

B2. All values within months with observed values less than 34% were considered

as missing.

B3. For the rest of the months the missing values were filled in using linear

interpolation and then these months were considered as missing. The reason is

explained in step D.

C. Missing months corresponding to steps B2 and B3 (the latter after the

substitution with missing values) were filled in using a seasonal Kalman filter,

implemented in the R package zoo (Zeileis and Grothendieck 2005).

D. Mean monthly values for months in which both steps B3 and C (i.e. months with

missing values more than 34% and less than 83%) were applied, were calculated with

the mean of monthly values of steps B3 and C.

E. From the mean monthly values we obtained the mean annual values.

F. Finally we discarded annual time series if one of the following constraints was

satisfied:

F1. Two or more missing years.

34

F2. Hurst parameter estimate H

˄ ≥ 0.95, mean annual rainfall μ

˄ ≥ 3000 mm, standard

deviation of annual rainfall σ

˄ ≥ 750 mm, coefficient of variation of annual rainfall c

˄

v

≥

0.8. These constraints on the estimated parameters were justified from a preliminary

analysis, which showed that higher values were outliers.

F3. Four or more years with less than 60% of observed daily values.

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