Climate modelling : investigating the relations between climatological variables using Artificial Neural Networks

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Faculty of Economics and Business

Climate Modelling

Investigating the relations between climatological variables using

Artificial Neural Networks

Author: Dylan Beijers Student ID: 10640096 Supervisors: M. Hennequin & H. Li

Bachelor’s thesis Econometrics

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Statement of Originality

This document is written by Student Dylan Beijers who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion

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1. Introduction

During the last century a warming of the Earth has been put in motion. This global warming brings along some serious consequences, such as the melting of ice at both the North Pole and the South Pole, leading to a rise of the global sea level. The questions that have entered the global debate the last few years are whether this global warming is due to human activity or due to natural forcings and whether it can be put to a halt. Investigating the relations between climatological variables can give important insights into these matters. By using statistical tests for Granger causality (Granger, 1969), the relations between the variables can be examined, in particular it can be investigated which variable is driving which.

Although the original test for Granger causality proposed by Granger (1969) provides a useful tool for investigating relations between variables, it is not able to detect nonlinear Granger causality (Baek & Brock, 1992; Hiemstra & Jones, 1994). The Granger test is designed such that it can only detect linear relationships between variables. To overcome this problem several nonlinear Granger tests have been constructed, including an Artificial Neural Network (ANN) based test constructed by Attanasio and Triacca (2011). This test can also be useful when analysing the relations between climatological variables, as Diks and Mudelsee (2000) find that most of the relations between these variables are nonlinear. Moreover, ANNs have been successfully applied before in the field of climate change studies. For instance, Attanasio and Triacca (2011) are able to detect the nonlinear relation between CO2and global temperature by

using ANNs, while this relation can not be found by Triacca (2005) when using a linear approach. Furthermore, Pasini, Lorè, and Ameli (2006) show that a neural network is able to reconstruct the time series of global temperature better than a linear model, where both models use data about natural and anthropogenic forcings.

In order to obtain insights into the dynamics of the climate system, the aim of this thesis is to investigate whether natural forcings are indeed an important driving force of global climate. For this purpose, the possible nonlinear relations between some important

climatological variables are analysed by applying the ANN based Granger test constructed by Attanasio and Triacca (2011). Moreover, an explicit model for the dependence between the climatological variables is constructed by estimating a neural network and a Vector

Autoregressive (VAR) model for each variable. The uncertainty of future climate changes can then be reduced by using the resulting models to produce out-of-sample forecasts for each of the climatological variables. The out-of-sample predictive abilities of the nonlinear models are also compared to those of the VAR models. In this way, it can be shown whether a nonlinear

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model is indeed able to better capture the dynamics between the climatological variables than a linear model.

The variables examined in this thesis include proxies for the global ice volume and the strength of formation of South Atlantic deep water. These two proxies are obtained from the Ocean Drilling Program (ODP) site 1264, containing data over the past five million years. The ODP 1264 dataset has been obtained from Bell, Jung, Kroon, Lourens, and Hodell (2014). The advantage of using this dataset instead of others is that it contains relatively many observations. Moreover, the dataset has been obtained quite recently and thus has not been the subject of many studies. Investigating the records of the ODP 1264 can possibly give some new insights into the climate dynamics. To account for the influence of natural forcings, a proxy for the solar radiation energy is added to the dataset. This proxy is obtained from Berger and Loutre (1991). Together, these three variables indicate how the global climate has changed over time and they allow for the possibility to investigate whether natural forcings are the main cause for the decrease of the global ice volume.

To investigate the relations between the variables, the dataset is divided into four distinctive climatic periods. This is necessary, as the climate dynamics might have changed over time. This approach is similar to the approach of Diks and Mudelsee (2000), although the four periods used in this thesis slightly differ from the periods they used, since the ages of the observations of the ODP 1264 dataset differ from the ages of the observations they used in their research. For each of the four periods the relations between the variables are examined using ANNs. Using this approach, it can be investigated how the climate dynamics have changed over time.

Before presenting the results, first some theoretical background is given about the variables and the methods used throughout this thesis. In the next section the variables in the used dataset are described and the significance of these variables to the climate is discussed. Moreover, the results of similar studies concerning these variables are presented. Section 3 provides a detailed theoretical background of Artificial Neural Networks, in particular the methods used in this thesis to estimate the ANNs are described. Section 4 then describes the concept of Granger causality and how ANNs can be used to test for linear and nonlinear Granger causality, followed by a description of the empirical methodology of the conducted research. Before fitting the ANNs to the data, a preliminary data analysis is given in Section 5. Section 6 presents the results of applying the ANN based Granger test, thereafter the results concerning the predictive ability of a neural network and a VAR model are presented in Section

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2. Climatological variables

The variables examined in this thesis consist of the relative carbon isotope ratio δ13_{C, the}

relative oxygen isotope ratio δ18O and insolation at 15°S. The former two are obtained from the ODP site 1264 (Bell et al., 2014), while the latter is obtained from Berger and Loutre (1991). The purpose of this section is to provide an understanding of how these proxies are important indicators of the global climate and how they relate to each other. First, the theoretical background of the variables is given. Thereafter, the relations between these variables are discussed and the results of previous studies concerning these relations are presented.

2.1 Theoretical background

The variable insolation is a proxy for the solar radiation energy the Earth receives at a given latitude, measured in watts per square meter. The values of insolation over the last five million years have been reconstructed by Berger and Loutre (1991). They made use of the dependence of insolation to the astronomical parameters of the Earth’s orbit and rotation, which in turn have been calculated by solving the Lagrangian system of the planetary point masses (1991, pp. 298-301) and the Poisson equations of the Earth-Moon system (1991, pp. 301-303).

Variations in these parameters of the Earth’s orbit and rotation causes the insolation values to vary over time, which may have its impact on global climate.

The relative oxygen isotope ratio δ18O is defined as the relative deviation of the oxygen isotope ratio in the sample from the Vienna Pee Dee Belemnite (VPDB) reference standard:

δ18O= (18O/16O)_sample− (18_O/16_O₎ VPDB (18_O/16_O₎ VPDB ×1000h. (1)

The VPDB reference standard is based on samples from the Pee Dee Formation in South Carolina and is commonly used to define δ18O. The value of δ18O serves as a proxy for the global ice volume. This can be explained due to18O being heavier than16O. Because of this, water molecules containing18O require more energy to vaporise than water molecules

containing16O. During colder periods, the vaporised water molecules containing16O become trapped in the continental ice sheets and the value of δ18O in the ocean tends to be higher (Broggy, 2011, p. 58). Information about the global ice volume can give an impression of how the climate has changed over time.

The relative carbon isotope ratio δ13C is defined similar as δ18O:

δ13C= (13_C/12_C₎ sample− (13C/12C)VPDB (13_C/12_C₎ VPDB ×1000_h. (2)

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The value of δ13_{C serves as a proxy for the strength of formation of South Atlantic deep water.}

Lisiecki (2010) states that this is due to the value of δ13C being more negative when the deep water stays longer out of contact with the surface water, indicating poor ocean circulation.

2.2 Relations between the variables

It is generally believed that changes in the Earth’s astronomical parameters, and thereby insolation, are the main driving force of climate change (Tiedemann, Sarnthein, & Shackleton, 1994). A consequence of this, is that insolation could be a Granger cause of both δ13C and δ18O. While insolation possibly Granger causes other climatological variables, the reverse of this is generally believed not to hold, as insolation is determined by the astronomical parameters of the Earth’s orbit and rotation. However, Mitrovica, Forte, and Pan (1997) state that glaciation influences some of the Earth’s astronomical parameters, such as its obliquity and moment of inertia. This essentially means that δ18O could Granger cause insolation. Moreover, δ18O could Granger cause δ13_{C, as Wang et al. (2014) state that glaciation results in a drop of the sea level}

and thereby intensifies ocean circulation.

The possible relations between insolation, δ18_{O and δ}13_{C have been the subject of a}

study conducted by Diks and Mudelsee (2000). They made use of data from the ODP site 659. Using a method of mutual information and redundancy, they find that the dependence between these variables has increased slowly over time, which is mainly due to the increase in coupling between δ13_{C and δ}18_{O. Moreover, they find a dependence between insolation and}

δ18O in period I (2000, pp. 409-410). Applying their nonlinear Granger test yields similar results.

They find that δ18_{O is a Granger cause of δ}13_{C in periods I, II and III. Moreover, they find that}

insolation is a Granger cause of δ13C in period I and of δ18O in periods I and II (2000,

pp. 412-413). These results coincide with the theoretical relations described earlier. However, Diks and Mudelsee (2000) do not find any evidence for insolation being caused by the other variables. This suggests that the effect of glaciation on the Earth’s astronomical parameters is not very strong (2000, p. 413).

Hennequin (2012) also investigated the records of the ODP site 659, but unlike Diks and Mudelsee (2000) she kept the variable dust flux in the analysis. Using the nonlinear Granger test constructed by Diks and Panchenko (2006) on the VAR-filtered variables, she does not find much evidence for insolation being a Granger cause of the other variables. Moreover, the couplings of the variables do not seem to have increased over time (2012, pp. 15-21).

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Because there is still not much known about the relations between the climatological variables, it is of high interest to investigate these possible relations further by analysing the records of the ODP site 1264. In particular, by using the ANN based Granger test for this matter, the possible nonlinear relations may be detected. In the next sections, more details are given about ANNs and how these models can be used to test for Granger causality.

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3. Artificial Neural Networks

The relations between climatological variables are generally believed to be nonlinear, which is confirmed by Diks and Mudelsee (2000). These nonlinear relations can be detected by making use of Artificial Neural Networks (ANNs). A great advantage of using ANNs for this matter, is that no assumptions have to be made about the relations between the variables. The data fully determines the functional form of the relations. Moreover, ANN regressions belong to a class of nonparametric methods, asymptotically free from the risks of misspecification (Landajo, Bilbao, & Bilbao, 2012, p. 988).

This section presents the theory needed to understand and estimate one type of neural networks, called the three-layer feedforward (TLF) neural network. First, an idea is given of how TLF neural networks work. Thereafter, the model specification of the TLF neural network is described mathematically, followed by a description of the estimation procedure used throughout this thesis.

3.1 A description of the TLF neural network

A neural network can be visualised as a network of nodes, which are spread across multiple layers. Although there exist many types of neural networks, such as networks containing cycles and feedback loops, the feedforward networks are generally used in regression analysis. A diagram of a feedforward neural network consisting of three layers is given in Figure 1. These TLF neural networks consist of one input layer, one hidden layer and one output layer. The network is able to receive input at the nodes in the input layer. These input nodes are connected to all of the nodes in the hidden layer, which in turn are connected to the output node. In this way, the input received at the input nodes can be send forward through the network, resulting in an estimate of the dependent variable. The connections between each of the nodes have numeric weights attached to them, which can be adapted to produce better estimates.

The underlying idea of a TLF neural network is that it transforms the received input at the hidden nodes, using some kind of activation function. The network uses the activation function to form a basis over the space of continuous functions. Using this approach, TLF neural networks are able to approximate any continuous function to an arbitrary accuracy, given that the number of nodes in the hidden layer is large enough. For a proof of this result, see Funahashi (1989), Hornik (1991), Hornik, Stinchcombe, and White (1989) and Kreinovich (1991). While multilayer feedforward networks with more than three layers also have this

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property, they require more parameters to be estimated and therefore require more computational effort. Moreover, the risk of overfitting increases when allowing for more hidden layers, resulting in poor predictive ability. For these reasons, TLF neural networks are used in this thesis. In the next paragraph, a more formal specification of the TLF neural network is given. Input layer Hidden layer Output layer x1 x2 x3 x4 x5 ˆy

Figure 1: Visualisation of a TLF neural network

3.2 Model specification

Assume that the dependent variable y depends on the explanatory variables x according to

yi = f(xi) +εi. (3)

Here, x denotes a column vector containing all of the explanatory variables, f(·)represents the dependence of y on x and ε is an error term. Assume further that the random process {εi} is

independent, identically distributed with E(ε_i) =0 and E(ε2_i) <∞. Moreover, the processes

{xi} and {εi} are assumed to be mutually independent. The function f(x)in (3) can then be

approximated by the following TLF neural network: ˆf(x, β) =α+ m

∑

j=1 wjσ γ’_jx+θj . (4)

Here, σ(·)represents the activation function used at the hidden nodes, m is the number of nodes in the hidden layer, γ_jis a column vector containing the weight parameters between the input nodes and the jthhidden node, wj is the weight parameter between the jthhidden node

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notation of β is used for convenience, it contains all of the free parameters of the neural network. In this thesis σ(·)is chosen to be the sigmoid function, σ(t) = ₁₊1_e−t. This function is generally chosen as the activation function (Bishop, 2006, p. 227). Unfortunately there is not a general rule to choose the number of nodes m in the hidden layer. This can be solved by estimating multiple neural networks with different values for m. The model which has the lowest out-of-sample mean squared error (MSE) is then chosen as the best model.

3.3 Estimation procedure

To estimate the parameters of the TLF neural network in (4), nonlinear least squares is used. This corresponds to minimising the residual sum of squares:

RSS(β) = T

∑

i=1 yi− ˆf(xi, β) 2 . (5)

Minimising the residual sum of squares in (5) can be done by many algorithms. Ripley (1996, pp. 158-159) states that quasi-Newton methods perform well for this purpose. One of these methods is the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, which is also used throughout this thesis. Because this algorithm makes use of the derivative of (5), it is possible that the solution found, given the initial weights, is in fact a local minimum. To account for this problem, the TLF neural network in (4) is estimated 100 times using different initial random weights each time. Of these models, the model with the best fit is chosen. This approach increases the probability of finding the global minimum. To increase the speed of the BFGS algorithm, it may be beneficial to rescale y and x to the[0, 1]interval. This is done by applying the following transformation: g(zi) = _maxzi−_z−minminzz. For more details about the BFGS algorithm, see Broyden (1970), Fletcher (1970), Goldfarb (1970) and Shanno (1970).

Before using the BFGS algorithm to estimate the TLF neural networks, the observations are first divided into two sets. The set of observations used to fit the model is called the training set. The remaining observations are included in the test set and are used to perform an

out-of-sample analysis. Usually, the size of the test set is between 10% and 30% of the size of the training set (Attanasio & Triacca, 2011, p. 105). This corresponds to a test set between 9% and 23% of the total amount of observations. In this thesis, each of the test sets consists of the last 20% of the observations of the corresponding climatic period. The out-of-sample MSE can be calculated from the test set, which may then be used for model selection and Granger causality testing.

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4. Granger causality and empirical methodology

To obtain insights into the relations between climatological variables, Granger causality provides to be a useful concept. By applying tests for Granger causality it can be investigated which variable is driving which. Although the original Granger test (Granger, 1969) can only detect linear Granger causality, other Granger tests have been constructed to detect nonlinear Granger causality. One of these tests is the ANN based Granger test constructed by Attanasio and Triacca (2011). This test is designed such that it can detect nonlinear as well as linear Granger causality. In this section, the concept of Granger causality is described, thereafter the ANN based Granger test used throughout this thesis is explained. This section concludes with a description of the empirical methodology of the conducted research.

4.1 The concept of Granger causality

Consider the two time series{Xt}and{Yt}and letFX,tandFY,t denote the information sets

containing the past observations of Xtand Ytup to and including time t respectively:

FX,t= {Xs|s ≤ t} (6)

and

F_Y,t = {Ys|s ≤ t}. (7)

Moreover, let f(Yt+1, . . . , Yt+k | FX,t, FY,t)denote the conditional joint distribution of

Yt+1, . . . , Yt+kgivenFX,tandFY,t and let f(Yt+1, . . . , Yt+k| FY,t)denote the conditional joint

distribution of Yt+1, . . . , Yt+kgivenFY,t. Then{Xt}is said to Granger cause{Yt}if the

following inequality holds for some k≥1:

f(Yt+1, . . . , Yt+k| FX,t, FY,t) 6= f(Yt+1, . . . , Yt+k| FY,t). (8)

The inequality in (8) essentially means that future values of Y given past values of X and Y are distributed differently than future values of Y given past values of Y only. Intuitively this means that better forecasts of Y can be produced by accounting for past values of X. In the next paragraph, a Granger test is described which makes use of the predictability of Y to determine whether X Granger causes Y.

4.2 ANN based Granger testing

In order to test if{Xt}is a Granger cause of{Yt}, Attanasio and Triacca (2011) suggest to

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models, one group uses the lags of both Xtand Yt, while the other group uses only the lags of

Yt. In each group, multiple models of Y are fitted on the training set. The models Attanasio and

Triacca (2011) consider are the (V)AR models and the TLF neural networks. By considering the (V)AR models, they allow for the possibility of the relations between the variables to be linear instead of nonlinear. This could also be the case when considering the relations between climatological variables, as Attanasio, Pasini, and Triacca (2012, p. 68) argue that the relations between these variables could be near-linear due to averaging. The estimated models in each group can be used to obtain forecasts of Yton the test set. The quality of the forecasts of each

model can then be measured by the MSE:

MSE= 1 N−T N

∑

t=T+1 (Yt−Yˆt)2, (9)

where T denotes the number of observations in the training set, N denotes the total number of observations and ˆYtis the forecast of Yt.

To test for Granger causality, the lowest MSE of one group is compared to the lowest MSE of the other group. Using this approach, it can be determined whether past values of X contain additional information on future values of Y. More formally, let MSE(F_X,t, F_Y,t) be the lowest MSE of the group of models that uses the lags of both Xtand Yt, while MSE(FY,t) is

defined as the lowest MSE of the group of models that uses only the lags of Yt. Then{Xt}is

said to Granger cause{Yt}if MSE(FX,t, FY,t) < MSE(FY,t) in a statistical significant way.

For this purpose, the encompassing test (ENC-T) (Clark & McCracken, 2001; Harvey, Leybourne, & Newbold, 1998) may be used. Let ˆεxy,tbe the forecast error of the model that uses

the lags of both Xtand Yt, ˆεxy,t=Yt−Yˆxy,t, and let ˆεy,tbe the forecast error of the model that

uses only the lags of Yt, ˆεy,t=Yt−Yˆy,t. Define ct = ˆεy,t(ˆεy,t−ˆεxy,t)and ¯c= _P1∑tN=T+1ct, with

P= N−T the number of observations in the test set. Moreover, let ˆσ¯cbe the estimate of the

standard deviation of ¯c, that is, it is equal to the Newey-West estimate using the Bartlett kernel if the variable c is serially correlated, otherwise it is equal to

q

1

P2 ∑tN=T+1(ct−¯c)2. The ENC-T

statistic is then defined as: ENC-T = ¯c

ˆσ¯c

. (10)

Under the null hypothesis of equal forecast accuracy, the ENC-T statistic has a limiting standard normal distribution, that is,

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where−→d denotes convergence in distribution. The null hypothesis is rejected for large values of ENC-T, giving statistical evidence of{Xt}being a Granger cause of{Yt}.

The ANN based Granger test is used in this thesis the same way as Attanasio and Triacca (2011) did to investigate the possible nonlinear relations between climatological variables.

4.3 Empirical methodology

In this paragraph, the empirical methodology of the conducted research is given. This methodology consists of two processes, the process of testing for Granger causality using ANNs and the process of generating forecasts of the time series by constructing an explicit model for each variable using ANNs.

To test for Granger causality between each of the variables, multiple models are fitted to the data. These models consist of TLF neural networks and (V)ARs. First, an univariate model is chosen for each variable in every period. This is done by fitting an AR model and multiple TLF neural networks. For each period, the univariate models with the smallest out-of-sample MSEs are chosen to be the best univariate models. Thereafter, two bivariate models are chosen for each variable in every period, using VARs and TLF neural networks. Each of these bivariate models uses the lags of one of the other variables besides the lags of the dependent variable. Again, the best bivariate model is chosen for each pair of variables by considering the

out-of-sample MSE. The out-of-sample MSEs of the univariate and bivariate models are then compared using the ENC-T test. When the MSE of the bivariate model is significantly smaller than the MSE of the corresponding univariate model, it is said that the extra variable included in the bivariate model Granger causes the variable of the univariate model.

In addition to the Granger causality analysis, an explicit model is constructed for each of the time series, which can be used to generate forecasts. For this purpose, a TLF neural network and a VAR model is chosen for each variable in every period. Besides the earlier estimated univariate and bivariate models, also a trivariate model is considered, which uses the lags of all variables as input variables. By comparing the out-of-sample MSEs of the best TLF neural networks and the best VAR models, it can be shown whether the dynamics between the climatological variables are indeed nonlinear in nature.

Before the results of the ANN based Granger test and the model construction are presented, first a description and a preliminary data analysis of the used dataset are given in the next section.

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5. Data description and preliminary analysis

Before the models are fitted to examine the relations between the variables, it is useful to describe the used dataset and to perform a preliminary data analysis of the records of the ODP 1264 and the insolation data. In this way, it is possible to obtain some first insights into the characteristics of the data.

The dataset considered in this thesis contains records from the ODP site 1264. This is a drilling program in sedimentation layers, located at the Walvis Ridge in the eastern part of the South Atlantic Ocean. The ODP 1264 contains data from over the last 5.3 million years. However, in this thesis the data from before five million years ago are omitted, due to the fact that the insolation data only spans the last five million years.

Due to sedimentation not being a constant process, the length of the time intervals between the observations differs. This is a problem when analysing time series, as most methods of analysing time series require the time intervals to be constant. This problem is partly dealt with by splitting up the dataset in four climatic periods, as is suggested by Diks and Mudelsee (2000). This approach is followed throughout this thesis, although with some minor modifications. The resulting periods, along with some details about the time intervals, are given in Table 1.

Table 1: Division of the climatic periods

Period IV Period III Period II Period I Range in ka ago 5000-4050 3600-2588 2588-892 892-0 observations 821 400 475 174 Time interval Mean 1.159 2.528 3.575 5.134 St. dev. 0.709 1.404 1.283 1.855 Coeff. of variation 0.611 0.555 0.359 0.361 Maximum 11.2 18.3 9.9 12.1 Here, 1 ka = 1000 years.

As can be seen from Table 1, the observations from 4050 ka ago till 3600 ka ago have been omitted. These observations have been left out, because the observations in this time period are sparse. Moreover, comparing the results for different periods should be done cautiously, as the time intervals are distributed differently for each period.

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Table 2 presents the descriptive statistics of the climatological variables in each period. It can be seen that the standard deviations of both δ13C and δ18O have increased over time, while the standard deviation of insolation only decreased in period I. The higher standard deviations in younger periods possibly reflects the increased variability of the climate system. The correlation matrices can give some first insights into the relations between the variables. It can be seen that insolation plays a part in most of the significant correlations. This is in line with the possibility that natural forcings are an important driving force of the climate system. On the other hand, the correlation between δ13C and δ18O appears to be significant only in period I, which might suggest that the relation between δ13C and δ18O is not as strong as Diks and Mudelsee (2000) suggested. However, this does not necessarily have to be the case, as

correlation does not imply causation. Making statements about the relationships between the climatological variables can be done by applying a causality analysis.

Table 2: Descriptive statistics of the climatological variables in every period

Period IV δ13C δ18O Insolation Period III δ13C δ18O Insolation

Mean 0.704 3.212 455.985 Mean 0.639 3.391 455.331 St. dev. 0.178 0.123 16.193 St. dev. 0.201 0.234 18.927 Skewness -1.067 0.182 0.097 Skewness -0.680 -0.187 0.186 Excess Kurtosis 2.531 2.263 -0.385 Excess Kurtosis 0.940 1.625 -0.605 Correlation matrix Correlation matrix

δ13C 1.000 δ13C 1.000

δ18O 0.017 1.000 δ18O -0.061 1.000

Insolation 0.087** -0.099*** 1.000 Insolation -0.103** -0.008 1.000

Period II δ13C δ18O Insolation Period I δ13C δ18O Insolation

Mean 0.755 3.968 456.246 Mean 0.575 4.281 455.845 St. dev. 0.218 0.290 21.338 St. dev. 0.266 0.349 18.279 Skewness -0.519 0.100 0.137 Skewness -0.107 -0.380 0.084 Excess Kurtosis 0.766 -0.338 -0.546 Excess Kurtosis 0.180 0.711 -0.651 Correlation matrix Correlation matrix

δ13C 1.000 δ13C 1.000

δ18O -0.051 1.000 δ18O -0.444*** 1.000

Insolation 0.009 0.063 1.000 Insolation -0.183** 0.056 1.000

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Before applying the causality analysis, it is useful to check if the time series of the variables are stationary. This is done by performing an Augmented Dickey-Fuller (ADF) test on each of the time series for every period. Before this test can be applied, it should be determined whether the time series contain a deterministic trend. Figure 2 gives a plot of the time series for each variable over the past five million years.

Figure 2: Time series plot of δ13C, δ18O and insolation

The plot of δ18O indicates the presence of a temporary deterministic trend, which starts in period III and ends in period II. For this reason, a deterministic trend is included in the ADF test for δ18O in period III and period II. Moreover a constant is included for each variable in each period, because the means of the variables differ significantly from zero. The lag lengths were chosen according to the Schwarz Information Criterion (SIC). The results of the ADF test are presented in Table 3.

Table 3: Results of the ADF test

period IV period III period II period I

Variable Lag Length t-Stat p-Value Lag Length t-Stat p-Value Lag Length t-Stat p-Value Lag Length t-Stat p-Value

δ13C 4 -7.084 0.000 1 -10.040 0.000 3 -5.283 0.000 1 -4.984 0.000

δ18O 3 -6.649 0.000 3 -5.486 0.000 1 -11.223 0.000 0 -8.914 0.000

Insolation 11 -13.038 0.000 7 -13.056 0.000 7 -14.018 0.000 2 -14.244 0.000

As the p-values are all equal to 0.000, there is statistical evidence that the climatological variables are (trend) stationary in each period.

The time series of δ18O still contains a deterministic trend in periods III and II, however. This deterministic trend possibly results from the overflow of18O-enriched water from the Nordic seas (Bell et al., 2014). Although neural networks should be able to deal with trends (Gorr, 1994), Zhang and Qi (2005) showed that detrending the data priorly is beneficial to obtain better forecasts. Therefore, the observations of δ18O in periods III and II were detrended. From Figure 2 it can be seen that the trend starts approximately 3500 ka ago and ends

approximately 1500 ka ago. The observations within this interval are detrended by fitting a linear time trend to this interval and then subtracting the estimated trend from the observations.

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trend at 1500 ka ago. In this way, the effect of the deterministic trend is fully eliminated from the time series. The resulting detrended δ18O variable is also used as an explanatory variable to model the other climatological variables. This can be justified, as δ18O serves as a proxy for the global ice volume. The increase of δ18O resulting from the overflow of18O-enriched water does not mean that the global ice volume had increased.

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6. Results of applying the ANN based Granger test

This section presents the results of performing the ANN based Granger causality test on the records of the ODP site 1264. First, the results of the model selection for each variable in every period are presented. Thereafter, the results of the ANN based Granger test are given and conclusions are drawn.

6.1 Selected univariate and bivariate models

To employ the ANN based Granger test, it should first be determined which univariate and bivariate models represents each variable the best. For this purpose, (V)ARs and TLF neural networks with no more than five hidden nodes were fitted. The number of hidden nodes is kept low, because allowing for more hidden nodes would result in overfitting. Moreover, the lag lengths for each of the considered models were set at one. Allowing for more lags would increase the risks of obtaining biased results due to the unequal time intervals. By only including the first lag, the unwanted variation of the unequal time intervals is kept at a minimum, as only the first difference of the age of the observations would then contribute to this variation.

Each of the models were fitted on the first 80% of the observations of the corresponding period. The remaining last 20% of the observations could then be used to calculate the

out-of-sample MSEs, which were used to determine the best models. The best univariate and bivariate models based on this MSE statistic are presented in Table 4 and Table 5 respectively, the models that were not chosen can be found in the Appendix. It is remarkable that almost in all of the cases an AR model was chosen to be the best univariate model. A TLF neural network was chosen only for insolation in period II and period I. This could indicate that each of the variables depends for a great part linearly on its lagged values.

In the bivariate case, almost every time a TLF neural network was chosen, which possibly indicates that the interactions between the variables are mainly nonlinear. The chosen models for most of the relations do differ a bit from period to period, indicating that the climate dynamics might have changed slightly over time. Furthermore, it can be seen from both Table 4 and Table 5 that the out-of-sample MSEs of the best models are higher for younger periods than for older periods, except for insolation from period II to period I. This essentially means that the variables can be less accurately predicted for the younger periods than for the older periods. This may have various reasons, such as the increased variability of the climate system, the

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length of the time intervals is higher for the more recent periods and the lags thus possess less information.

Table 4: Best univariate models

Period IV Period III Period II Period I

Variable Best Model MSE Best Model MSE Best Model MSE Best Model MSE

δ13C AR 0.020410 AR 0.025468 AR 0.027248 AR 0.042539

δ18O AR 0.009608 AR 0.050214 AR 0.062909 AR 0.115862

Insolation AR 88.487486 AR 162.179399 TLF(5) 509.050280 TLF(5) 320.869265 Here, TLF(m) denotes a TLF neural network with m hidden nodes.

Table 5: Best bivariate models

Bivariate Time Series Best Model MSE Best Model MSE Best Model MSE Best Model MSE

δ18O→δ13C TLF(1) 0.019954 VAR 0.025497 TLF(1) 0.027245 VAR 0.040555

Insolation→δ13C TLF(1) 0.019797 VAR 0.024746 TLF(1) 0.026866 VAR 0.042400

δ13C→δ18O TLF(4) 0.009046 TLF(4) 0.048593 TLF(2) 0.062417 TLF(2) 0.106780

Insolation→δ18O TLF(1) 0.009302 TLF(2) 0.048570 TLF(2) 0.061949 TLF(2) 0.114435 δ13C→Insolation TLF(3) 88.002065 TLF(1) 158.704375 TLF(1) 517.203709 TLF(3) 320.107989

δ18O→Insolation TLF(1) 88.349699 TLF(1) 160.040907 VAR 515.955151 TLF(2) 322.063615 Here, TLF(m) denotes a TLF neural network with m hidden nodes.

6.2 Results of testing for Granger causality

Now that the best univariate and bivariate models have been presented, the ANN based Granger causality analysis can be performed. This was done by using the ENC-T test, which was described above in Section 4. The results of the ANN based Granger test are shown in Table 6.

Before discussing these results, it should be mentioned that some of the significant results might have been found due to chance, as testing at, for example, a 5% level implies that on average one out of twenty tests rejects a true null hypothesis and thus leading to a false conclusion. Moreover, some Granger causality may be found between the variables, while in fact a real causal link does not exist. This may be the case when some important climatological variables have been omitted. When one of these omitted variables drives two or more variables of the ODP data, then significant Granger causality may be found between the ODP variables. It is likely that some important climatological variables have been omitted in this thesis, as Mudelsee and Stattegger (1994) state that there are at least five variables that have significant influence on the climate system, while this thesis only considers three variables. However, Granger causality still provides to be a useful concept when analysing the relations between variables, as real causality would also imply significant Granger causality.

As can be seen from Table 6, there exists evidence for some of the couplings between the variables to exist, although not for all in every period. It can be seen that there is some proof of

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the other periods, no evidence was found for this causal link, which is contrasting with the results of Diks and Mudelsee (2000). They found stronger evidence of δ18O being a Granger cause of δ13C in the periods I, II and III. The little evidence found for this causal link might suggest that the ocean circulation of the South Atlantic Ocean is little influenced by changes in glaciation. However, it does seem that the ocean circulation is influenced by natural forcings in period III, as there is evidence at the 5% level of insolation being a Granger cause of δ13C in this period. The evidence that insolation is a Granger cause of δ13C is weaker for the other periods than for period III, which is noteworthy as this might suggest that the climate dynamics have changed over time.

The evidence for insolation being a Granger cause of δ13C in period III suggests that natural forcings may be an important driving force of the global climate. This might be indeed the case, as some evidence is also found for insolation being a Granger cause of δ18O in the periods IV and II. For both periods, this relationship is only found to be significant at the 10% level. However, for period II it is not likely that the relationship was found due to chance, as Diks and Mudelsee (2000) also found this relationship in the same period. This suggests that changes in insolation are indeed a driving force of the global ice volume in period II. Another driving force of the global ice volume appears to be the ocean circulation, as δ13C Granger causes δ18O in period IV at the 10% level and in the periods III and I at the 5% level. This may be due to the heat transportation of the ocean to the ice sheets, which causes the ice sheets to melt and thus leading to a decline of the global ice volume.

The global ice volume in turn may affect some of the Earth’s astronomical parameters, according to Mitrovica et al. (1997). This would mean that insolation could be influenced by

δ18O. Some evidence for this is found in period II at the 1% level. For the periods IV, III and I,

there was less evidence in support for this theorem. The result that δ13_{C is a Granger cause of}

insolation in periods IV, III and II is harder to explain. It is likely that this causality has been found due to the presence of another astronomical parameter, which affects some of the astronomical parameters used to determine insolation and which has some effect on the intensity of the ocean circulation.

Table 6: Results of the ANN based Granger test

Bivariate Time Series ENC-T p-Value ENC-T p-Value ENC-T p-Value ENC-T p-Value

δ18O→δ13C 1.201138 0.114849 -0.109094 0.543436 0.966079 0.167002 1.306508 0.095690

Insolation→δ13C 1.258247 0.104151 2.122307 0.016906 1.066871 0.143015 0.285466 0.387643 δ13C→δ18O 1.639575 0.050547 1.651352 0.049333 0.952660 0.170381 1.844110 0.032584

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Although the results indicate that insolation is responsible for some of the changes in

δ13C and δ18O, it is not able to clarify all the changes of these two variables. This suggests that

besides natural forcings, other forcings play an important role in the global climate. One type of these forcings might be represented by the anthropogenic forcings, as Attanasio et al. (2012), Attanasio and Triacca (2011) and Pasini et al. (2006) were able to find a significant influence of these forcings on the global temperature and therefore on global climate. To determine if anthropogenic forces also influence the global ice volume and ocean circulation, it might be interesting to study the records obtained from ice cores, as these records also contain information about past concentrations of gases, such as carbon dioxide and methane, in the atmosphere.

Furthermore, the results of the ANN based Granger test indicate that there are mainly nonlinear interactions between the climatological variables, although there are also some interactions that seem to be linear in some periods. The next section investigates the nature of the interactions a bit further by also allowing for trivariate models and then comparing the forecasting abilities of the best TLF neural networks to that of the best VAR models for each of the variables.

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7. An explicit model for the climatological variables

By choosing an explicit model for each of the climatological variables, forecasts can be obtained which can indicate how the climate is going to change and thus reduce some of the uncertainty involved in the climate. First the best linear and nonlinear models are chosen amongst the univariate, bivariate and trivariate VARs and TLF neural networks. Thereafter predictions are made of the future values of the climatological variables.

7.1 The best performing models

The models that are considered consist of the univariate, bivariate and trivariate TLF neural networks and their linear counterparts. The univariate and bivariate models were already estimated in Section 6. For the trivariate models, the lag lengths were again set at one and the number of hidden nodes was kept less than or equal to five. The best linear and nonlinear models are presented in Tables 7 and 8 respectively, the trivariate models that were not chosen can be found in the Appendix.

When comparing the best linear and nonlinear models to each other, it can be seen that in ten of the twelve cases the nonlinear model performs better. This indicates that the statement of Attanasio et al. (2012), which states that the relations between the climatological variables are near-linear due to averaging, does not hold in this case. This means that the variables depend mostly nonlinearly on each other. However, for δ13C a linear model performs better in two of the four periods, indicating that the ocean circulation depends for a part linearly on the other variables.

Table 7: The best linear models

Variable Model MSE Model MSE Model MSE Model MSE

δ13C VAR(Ins) 0.020251 VAR(Ins) 0.024746 AR 0.027248 VAR(O, Ins) 0.040352 δ18O AR 0.009608 VAR(C, Ins) 0.049585 VAR(Ins) 0.062351 VAR(C, Ins) 0.112048

Insolation VAR(C, O) 88.394859 VAR(C, O) 158.348549 VAR(C, O) 513.817904 AR 352.877988 Here, VAR(x, y) denotes a VAR model which uses the lags of x and y besides the lags of the dependent variable.

C, O and Ins are abbreviations for δ13_{C, δ}18_{O and Insolation respectively.}

Table 8: The best nonlinear models

δ13C TLF(1, Ins) 0.019797 TLF(1, Ins) 0.025060 TLF(1, Ins) 0.026866 TLF(1, O, Ins) 0.045453 δ18O TLF(4, C) 0.009046 TLF(2, Ins) 0.048570 TLF(2, Ins) 0.061949 TLF(2, C) 0.106780

Insolation TLF(3, C) 88.002065 TLF(3, C, O) 157.604750 TLF(5) 509.050280 TLF(3, C, O) 290.591706 Here, TLF(m, x, y) denotes a TLF neural network with m hidden nodes and which uses the lags of x and y besides the lags of the dependent variable. C, O and Ins are abbreviations for δ13C, δ18O and Insolation respectively.

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7.2 Predicting future climate changes

The best models for period I can be used to generate forecasts about future climate changes. These models consist of a trivariate VAR model for δ13C, while a bivariate and trivariate TLF neural network perform better for δ18O and insolation respectively. The forecasts are presented in Table 9. These forecasts should be interpreted carefully, as it is unclear what year

corresponds to these forecasts, due to the unequal time intervals. The mean time interval for period I is equal to 5134 years, so it is likely that the forecasts correspond to a point of time far away in the future.

The forecasts show that the circulation of the ocean will be less intense in the future than it is currently. However, the prediction interval is quite wide, so it can not be said with great certainty whether the value of δ13C will increase or decrease. More certain statements can be made about the global ice volume. As a large part of the prediction interval of δ18O contains values larger than the current value of δ18O, it can be said with some certainty that the global ice volume will increase, leading to a drop of the global sea level. This shows that the current rise of the sea level will be neutralised in the future. The predicted value of insolation does not tell a lot about the future value of insolation, as the predicted value is just a bit smaller than the current value and the prediction interval is quite wide.

Table 9: Forecasts of the climatological variables

Variable Current Forecast ±2*√MSE

value

δ13C 1.14 0.908862 [0.507108, 1.310616] δ18O 3.39 3.953848 [3.300303, 4.607394]

Insolation 470.35 463.772344 [429.678843, 497.865846]

Although the forecasts give a bit of an idea of how the climate is going to change, they are not able to give much information about the short-term changes of the climate. For this purpose, it is necessary to analyse records of climatological variables which contain more data over a smaller time period.

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8. Conclusion

The climate of the Earth has been subject to many changes. Especially in recent years a global warming has been observed, having various consequences. Much research has been done to find out what the possible causes of this global warming could be and whether something can be done to stop it. In general it is believed that the global warming can be attributed to two kinds of forcings, namely the anthropogenic forcings and the natural forcings. The aim of this thesis was to investigate to what extent the natural forcings influence the global climate. Moreover, an explicit model was constructed to get an idea of how the global climate will change in the future. For these purposes, the relations between some key climatological variables were investigated, using data from the ODP site 1264. This dataset contains proxies for the global ice volume (δ18O) and the ocean circulation (δ13C) over the past five million years. Besides these two proxies, insolation values were included to account for the effects of natural forcings.

Since it is possible for the relations between the variables to be nonlinear, the ANN based Granger test was used, which can detect linear, as well as nonlinear couplings between the climatological variables. For the construction of an explicit model, TLF neural networks and VARs were considered. These models were evaluated by comparing their out-of-sample MSEs. The Granger causality analysis and the model selection were performed after splitting up the data in four distinctive climatic periods. This was done, because the dynamics might have changed over time. The problem of the unequal time intervals was also partly dealt with in this way.

The results of the ANN based Granger test led to numerous conclusions. For instance, it seemed that the ocean circulation influences the global ice volume, which is possibly explained due to the heat transportation of the ocean. Evidence for feedback between the global ice volume and ocean circulation was not so strong, as only for the most recent period some evidence was found for the global ice volume to influence the circulation of the ocean. Besides these causal links, insolation was found to be a driving force of the ocean circulation of the South Atlantic in one of the periods. Some evidence was also found for insolation to be a Granger cause of the global ice volume, although this result was not found for the

second-oldest period and the most recent period. These results do indicate that natural forcings are responsible for some of the changes in global climate, although it was not able to explain all of the changes in the climatological variables. For this reason, it is suspected that other forcings,

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Although insolation is generally believed to be independent of changes in the other two variables, some evidence was found for the opposite of this to be the case. While the global ice volume could possibly affect some of the Earth’s astronomical parameters and thus insolation, the effect of the ocean circulation on insolation was likely found due to the presence of an unobserved variable.

The results of constructing an explicit model showed that most of the relations that exist between the climatological variables are nonlinear in nature, only for the ocean circulation some linear dependence was found. The forecasts of the resulting models indicated that the global ice volume will rise again in the future, thereby neutralising the current rise of the global sea level. Similar statements about the intensity of the ocean circulation and insolation were harder to make, as the prediction intervals for these two variables were quite wide and the possibility of either a decrease or increase were roughly the same.

To interpret the results obtained in this thesis, some caution should be applied, as the unequal time intervals may lead to biased results. Moreover, some important variables may have been omitted in this thesis, which were possibly responsible for some of the causal links that were found in this thesis. Solving these problems may be beneficial for future research. The problem of unequal time intervals is hard to solve, although more developments are made nowadays to deal with this problem. To solve the problem of omitted variables, it might be useful to include the possible effects of anthropogenic forcings on global climate in the analysis. This can be done by using datasets obtained from ice cores, as these also include information about certain greenhouse gases.

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Appendix: Estimated models

This appendix presents most of the models that were estimated for this thesis. For the tables presenting the TLF models, only the best TLF models are shown.

Table A1: AR models

Variable MSE MSE MSE MSE

δ13C 0.020410 0.025468 0.027248 0.042539

δ18O 0.009608 0.050214 0.062909 0.115862

Insolation 88.487486 162.179399 518.174136 352.877988

Table A2: Univariate TLF models

δ13C TLF(5) 0.022993 TLF(1) 0.030636 TLF(4) 0.036992 TLF(4) 0.057032 δ18O TLF(1) 0.011889 TLF(2) 0.052620 TLF(3) 0.074726 TLF(2) 0.148935

Insolation TLF(3) 420.706335 TLF(3) 357.610768 TLF(5) 509.050280 TLF(5) 320.869265

Here, TLF(m) denotes a TLF neural network with m hidden nodes.

Table A3: Bivariate VAR models

Bivariate Time Series MSE MSE MSE MSE

δ18O→δ13C 0.020439 0.025497 0.028254 0.040555 Insolation→δ13C 0.020251 0.024746 0.027313 0.042400 δ13C→δ18O 0.009663 0.049880 0.063124 0.112375 Insolation→δ18O 0.009770 0.049836 0.062351 0.115125 δ13C→Insolation 88.460333 158.826966 517.265647 354.597838 δ18O→Insolation 88.418158 161.427663 515.955151 354.470262

Table A4: Bivariate TLF models

Bivariate Time Series Model MSE Model MSE Model MSE Model MSE

δ18O→δ13C TLF(1) 0.019954 TLF(1) 0.026030 TLF(1) 0.027245 TLF(1) 0.047147 Insolation→δ13C TLF(1) 0.019797 TLF(1) 0.025060 TLF(1) 0.026866 TLF(1) 0.045531 δ13C→δ18O TLF(4) 0.009046 TLF(4) 0.048593 TLF(2) 0.062417 TLF(2) 0.106780 Insolation→δ18O TLF(1) 0.009302 TLF(2) 0.048570 TLF(2) 0.061949 TLF(2) 0.114435 δ13C→Insolation TLF(3) 88.002065 TLF(1) 158.704375 TLF(1) 517.203709 TLF(3) 320.107989 δ18O→Insolation TLF(1) 88.349699 TLF(1) 160.040907 TLF(1) 516.005507 TLF(2) 322.063615

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Table A5: Trivariate VAR models

Variable MSE MSE MSE MSE

δ13C 0.020291 0.024765 0.028326 0.040352

δ18O 0.009844 0.049585 0.062562 0.112048

Insolation 88.394859 158.348549 513.817904 355.202065

Table A6: Trivariate TLF models

δ13C TLF(1) 0.019809 TLF(1) 0.025448 TLF(1) 0.027230 TLF(1) 0.045453 δ18O TLF(1) 0.009386 TLF(1) 0.049579 TLF(2) 0.062523 TLF(2) 0.112528

Insolation TLF(1) 88.313050 TLF(1) 157.604750 TLF(1) 513.815139 TLF(3) 290.591706 Here, TLF(m) denotes a TLF neural network with m hidden nodes.

Climate modelling : investigating the relations between climatological variables using Artificial Neural Networks

Faculty of Economics and Business