A meta extreme learning machine method for forecasting financial time series

Abstract

In the last decade, the problem of forecasting time series in very different fields has received increasing attention due to its many real-world applications. In particular, in the very challenging case of financial time series, the underlying phenomenon of stock time series exhibits complex behaviors, including non-stationary, non-linearity and non-trivial scaling properties. In the literature, a wide-used strategy to improve the forecasting capability is the combination of several models. However, the majority of the published researches in the field of financial time series use different machine learning models where only one type of predictor, either linear or nonlinear, is considered. In this paper we first measure relevant features present in the underlying process to propose a forecast method. We select the Sample Entropy and Hurst Exponent to characterize the behavior of stock time series. The characterization reveals the presence of moderate randomness, long-term memory and scaling properties. Thus, based on the measured properties, this paper proposes a novel one-step-ahead off-line meta-learning model, called μ-XNW, for the prediction of the next value x_t+1 of a financial time series $x_{t}$, t = 1, 2, 3, … , that integrates a naive or linear predictor (LP), for which the predicted value of $x_{t + 1}$ is just repeating the last value $x_{t}$, an extreme learning machine (ELM) and a discrete wavelet transform (DWT), both based on the nprevious values of $x_{t + 1}$. LP, ELM and DWT are the constituent of the proposed model μ-XNW. We evaluate the proposed model using four well-known performance measures and validated the usefulness of the model using six high-frequency stock time series belong to the technology sector. The experimental results validate that including internal estimators that are able to the capture the relevant features measured (randomness, long-term memory and scaling properties) successfully improve the accuracy of the forecasting over methods that do not include them.

Appendices

Appendix A: Sample entropy

Pincus [58] adapted the notion of ”entropy” for real-world use. In this context, entropy means order or regularity or complexity. Fix $m,N \in \mathbf {N}$ with $m\leq N$, and $r \in \mathbf {R}^{+}$. Given a time series of data $u(1), u(2),\dots , u(N)$ from measurements equally spaced in time, form the sequence of vectors$\mathbf {x}(1)$, $\mathbf {x}(2)$, …, $\mathbf {x}(N-m + 1) \in \mathbf {R}^{m}$ defined by:

$$\begin{array}{@{}rcl@{}} \mathbf{x}(i)&:=&[ u(i), u(i + 1),\dots, u(i+m-1) ]\in \mathbf{R}^{m}\\ i&=&1,2,\dots,N-m + 1 . \end{array} $$

(34)

Let d be some norm in $\mathbf {R}^{m}$, for instance:

$$\begin{array}{@{}rcl@{}} d(\mathbf{x}(i),\mathbf{x}(j)) \!&:=&\! \| \mathbf{x}(i) - \mathbf{x}(j) \|_{\infty} := \max_{1\leq k\leq m} | u(i+k-1)\\ &&\!-u(j+k-1)| \\ \!&=&\! \max\left\{ |u(i)-u(j)|, |u(i + 1)-u(j + 1)|\right.\\ &&\left., \dots , |u(i+m-1)-u(j+m-1)| \right\} . \\ \end{array} $$

(35)

Then, Pincus defines

$$ {C_{i}^{m}}(r)\!:=\! \frac{\# \left\{ j\!\in\! \mathbf{N} : j\!\leq\! N - m + 1\ \text{and}\ d(\mathbf{x}(i),\mathbf{x}(j))\!\leq\! r \right\} }{ N - m + 1 } . $$

(36)

The denominator $N-m + 1$ in (36) is the total number of segments of length m available in the signal $u(1),\dots , u(N)$, i.e. $1\leq \text {numerator}\leq N-m + 1$. Note that in the numerator of (36) the distance from the pattern (segment, template, or vector) $\mathbf {x}(i)$ must be measured w.r.t. all$N-m + 1$ patterns of length m available in the sequence $u(1),\dots , u(N)$. Only the patterns with distance $\leq r$ are counted. Thus:

$$ \frac{1}{N-m + 1}\leq {C_{i}^{m}}(r)\leq 1 . $$

(37)

We observe that, for $i\in \mathbf {N}$ fixed with $i\leq N-m + 1$:

$$\begin{array}{@{}rcl@{}} &&\# \left\{ j\in \mathbf{N} : j\leq N-m + 1\ \text{and}\ d(\mathbf{x}(i),\mathbf{x}(j))\leq r \right\} \\ &=& \sum\limits_{j = 1}^{N-m + 1} H\left( r-d\left( \mathbf{x}(i),\mathbf{x}(j)\right)\right) , \end{array} $$

where H is the Heaviside step function. Thus,

$${C_{i}^{m}}(r) = \frac{\displaystyle\sum\limits_{j = 1}^{N-m + 1} H\left( r-d\left( \mathbf{x}(i),\mathbf{x}(j)\right)\right)}{N-m + 1} . $$

Pincus further defines

$$\begin{array}{@{}rcl@{}} C^{m}(r) &:=& (N-m + 1) {\sum}_{i = 1}^{N-m + 1} {C_{i}^{m}}(r) \\ &=& {\sum}_{i = 1}^{N-m + 1} {\sum}_{j = 1}^{N-m + 1} H\left( r-d\left( \mathbf{x}(i),\mathbf{x}(j)\right)\right) , \end{array} $$

(38)

and

$$ \beta_{m} := \lim_{r\to0} \lim_{N\to\infty}\frac{\log C^{m}(r)}{\log r} . $$

(39)

In order to explain in deep the sample entropy measure, we introduce the following notation:

$$\begin{array}{@{}rcl@{}} u[1:N] :&& \text{the signal or segment}\ u(1),u(2),\dots,u(N) , \\ u[j:j+m-1] :&& \begin{array}{l} \text{the template } u(j),u(j + 1),\dots,u(j+m-1) \\ \text{of the signal } u[1:N] , \end{array} \\ \mathbf{x}(j) :&& \begin{array}{l} \text{the template }u(j),u(j + 1),\dots,u(j+m-1) \\ \text{as the }\mathit{vector}~[u(j),u(j + 1),\dots,u(j+m-1)] \in \mathbf{R}^{m} , \end{array} \\ u[j:j+m-1] \sqsubset u[1:N] :&& u[j:j+m-1]\ \text{is a \textit{template} of}\ u[1:N] . \end{array} $$

Recall that ${C_{i}^{m}}(r)$ counts those templates $\mathbf {x}(j)\equiv u[j:j+m-1]$ of the signal $u[1:N]$, whose $\|\cdot \|_{\infty }$-distance to the fixed template $\mathbf {x}(i)\equiv u[i:i+m-1]$ is $\leq r$.

Richman and Moorman [40] define:

$$\begin{array}{@{}rcl@{}} {B_{i}^{m}}(r) &:=& \frac{\displaystyle \# \left\{ \mathbf{x}_{m}(j) \sqsubset u[1:N] : 1\leq j\leq N-m ,\ j\neq i ,\ d(\mathbf{x}_{m}(j),\mathbf{x}_{m}(i))\leq r \right\} } {N-m-1} , \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} {A_{i}^{m}}(r) &:=& \frac{\displaystyle \# \left\{ \mathbf{x}_{m + 1}(j) \sqsubset u[1:N] : 1\leq j\leq N-m ,\ j\neq i ,\ d(\mathbf{x}_{m}(j),\mathbf{x}_{m}(i))\leq r \right\} } {N-m-1} . \end{array} $$

(41)

Note that there are $N-m + 1$ templates $\mathbf {x}_{m}(j)$ of length m in the signal $u[1:N]$. However, the convention of Richman-Moorman is to consider for ${B_{i}^{m}}(r)$ only $N-m$ of them, for instance, the first $N-m$ of them, thus disregarding the template $\mathbf {x}_{m}(N-m + 1)\equiv u[N-m + 1:N]\sqsubset u[1:N]$. In the case of ${A_{i}^{m}}(r)$, there are exactly $N-m$ templates of length $m + 1$ in $u[1:N]$. Then Richman and Moorman define the corresponding integral coefficients:

$$ B^{m}(r) := \frac{\displaystyle\sum\limits_{i = 1}^{N-m} {B_{i}^{m}}(r)}{N-m} , \qquad A^{m}(r) := \frac{\displaystyle\sum\limits_{i = 1}^{N-m} {A_{i}^{m}}(r)}{N-m} . $$

(42)

Thus,

$$\begin{array}{@{}rcl@{}} B^{m}(r) &=& \text{the probability that \textit{two different} templates of} \\ && \text{length }m, \text{both }\sqsubset u[1:N],\text{ will }\mathit{match~within}~ r , \\ &=& \mathscr{P}(\{ u[i,i+m-1],u[j,j+m-1]\sqsubset u[1:N] \\ && \quad\quad: i\neq j ,\ d(u[i,i+m-1], \\ &&\quad\quad u[j,j+m-1])\leq r \} ) , \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} A^{m}(r)&=& \text{the probability that \textit{two} templates of length } \\ &&m + 1. \text{both }\sqsubset u[1:N],\text{ will }\mathit{match~within}~ r , \\ &=& \mathscr{P}(\{ u[i,i+m],u[j,j+m]\sqsubset u[1:N] : \\ && i\neq j ,\ d(u[i,i+m],u[j,j+m])\leq r \} ) . \end{array} $$

(44)

“To match withinr” means here that the distance

$$\begin{array}{@{}rcl@{}} d(\text{template 1, template 2} ) &=& \| \text{template 1}-\text{template 2} \|_{\infty}\\ &&\text{is}\quad \leq r. \end{array} $$

We observe that:

$$\begin{array}{@{}rcl@{}} A^{m}(r) \!&=&\! \mathscr{P} (\{ u[i,i+m],u[j,j+m]\sqsubset u[1:N] : i\neq j ,\\ &&\quad\quad d(u[i,i+m],u[j,j+m])\leq r \} ) . \end{array} $$

Consider now the set in the argument of $\mathscr{P}$. Let us call it S:

$$\begin{array}{@{}rcl@{}} S&=& \{ u[i,i+m],u[j,j+m] \sqsubset u[1:N]\\ &&: i\neq j ,\ d(u[i,i+m],u[j,j+m])\leq r \} \\ &=& \{ u[i,i+m],u[j,j+m] \sqsubset u[1:N] \\ &&: i\neq j ,\ d(u[i,i+m-1],u[j,j+m-1])\leq r \\ && \text{and}\quad |u(i+m)-u(j+m)|\leq r \} \\ &=& \{ u[i,i+m],u[j,j+m] \sqsubset u[1:N] \\ && : i\neq j ,\ d(u[i,i+m-1],u[j,j+m-1])\leq r \} \\ &&\bigcap \{ u[i,i+m],u[j,j+m] \sqsubset u[1:N] \\ &&: i\neq j ,\ |u(i+m)-u(j+m)|\leq r \}. \end{array} $$

The measure of this set is, of course, $A^{m}(r)$:

$$\begin{array}{@{}rcl@{}} A^{m}(r) &=& \mathscr{P}(S) \\ &=& \mathscr{P} [ |u(j+m)-u(i+m)|\leq r ,\ i\neq j\ |\ i\neq j ,\\ &&|u(j+k)-u(i+k)|\leq r ,\ 0\leq k\leq m-1 ] \\ && \times\ \mathscr{P} [ \{ |u(j+k)-u(i+k)|\leq r ,\ i\neq j ,\\ && 0\leq k\leq m-1 \} ] \\ &=& \mathscr{P} [ |u(j+m)-u(i+m)|\leq r ,\ i\neq j\ |\ \\ && i\neq j ,\ |u(j+k)-u(i+k)|\leq r ,\ 0\leq k\\ &&\leq m-1 ] \times B^{m}(r). \end{array} $$

Thus,

$$\begin{array}{@{}rcl@{}} \frac{A^{m}(r)}{B^{m}(r)} \!&=&\! \mathscr{P} [ |u(j + m) - u(i+m)|\!\leq\! r \ |\ i\neq j ,\ |u(j+k)\\ &&-u(i+k)|\leq r ,\ 0\leq k\leq m-1 ]. \end{array} $$

(45)

Loosely speaking we can say that this is the conditional probability that two different templates of length $m + 1$, whose m initial points are within a tolerance r of each other, remain within the tolerance r of each other at the next point (the $m + 1$-th point of the template).

Richman and Moorman define then the statistic:

$$ \text{SampEn}(m,r,N) := -\log\frac{A^{m}(r)}{B^{m}(r)} , $$

(46)

and the corresponding parameter estimated by this statistic, as:

$$\begin{array}{@{}rcl@{}} \text{SampEn}(m,r) &:=& \lim_{N\to\infty} \text{SampEn}(m,r,N) \\ &=& \lim_{n\to\infty}\left( -\log\frac{A^{m}(r)}{B^{m}(r)}\right) . \end{array} $$

(47)

Appendix B: Wavelet transform background

Wavelet analysis is today a common tool in signal and spectral analysis, in particular because of its multiresolution and localization capabilities both in time and frequency domain. In the time series context, wavelet analysis allows the decomposition and localization, at different time and frequency scales, relevant features present on the underlying processes. The literature on wavelet theory is extensive; we mention just a few standard references: [9, 18, 51, 57]. In this section we collect the bare minimum of wavelet theory, necessary for the best understanding of the algorithms developed in the next section.

1.1 B1 Continuous wavelet transform

A function $\psi :\mathbb {R}\to \mathbb {C}$, such that $\psi \in L^{2}$ and $\|\psi \|^{2} = {\int }_{-\infty }^{\infty }|\psi (x)|^{2} dt = 1$, is called wavelet or mother-wavelet whenever $\psi$ satisfies the admissibility condition:

$$ C_{\psi}:= 2\pi{\int}_{-\infty}^{\infty} \frac{\left| \widehat{\psi}(a) \right|^{2}}{|a|} da <\infty , $$

(48)

where $\widehat {\psi }$ denotes the Fourier transform of $\psi$. Note that (48) implies that $\psi$ must have zero average, i.e., $\widehat {\psi }(0)=\frac {1}{\sqrt {2\pi }} {\int }_{-\infty }^{\infty } \psi (t) dt= 0$. When a fixed wavelet $\psi \in L^{2}$ has been selected, then the continuous wavelet-transform of a time signal $f\in L^{2}$ is defined by:

$$\begin{array}{@{}rcl@{}} \mathcal{W}\!f(a,b) &=& {\int}_{-\infty}^{\infty} f(t) \frac{1}{|a|^{1/2}} \overline{\psi\left( \frac{t-b}{a} \right)} dt,\\ && a,b\in\mathbb{R} ,\ a\neq0 , \end{array} $$

(49)

where $\overline {\psi ((t-b)/a)}$ denotes the complex conjugate of $\psi ((t-b)/a)$. Then f can be synthesized from its wavelet transform $\mathcal {W}\!f$ by means of the inversion formula: [9, (3.7), p. 67], [51, Th. 4.3, p. 81].

$$\begin{array}{@{}rcl@{}} f(t) \!&=&\! \frac{1}{C_{\psi}} \int\limits_{0\neq a\in\mathbb{R}} \int\limits_{b\in\mathbb{R}} \mathcal{W}\!f(a,b) \frac{1}{\sqrt{|a|}} \psi\!\left( \frac{t-b}{a}\right)\! \frac{da db}{|a|^{2}} ,\\ &&t\!\in\!\mathbb{R} . \end{array} $$

(50)

1.2 B2 Discrete wavelet transform

A well known approach to discrete wavelet analysis is by means of multiresolution analysis (MRA). Main references for this section are [9, Ch. 5, p. 105], [51, Ch. VII, p. 220]. MRA provides a systematic way to construct discrete wavelets. An MRA is a sequence $\left \{V_{j}\right \}_{j\in \mathbb {Z}}$ of closed subspaces of $L^{2}$ having the following properties also known as axioms: [9, Sec. 5.1, p. 106], [51, Def. 7.1, p. 220],

• (a) V_j ⊂ V_j− 1 for all $j\in \mathbb {Z}$;

• (b) $\bigcap _{j\in \mathbb {Z}}V_{j}=\{0\}$ (axiom of separation);

• (c) $\bigcup _{j\in \mathbb {Z}}V_{j}$ is dense in $L^{2}$ (axiom of completeness);

• (d) f ∈ V_j ⇔ f(∙− 2^jk) ∈ V_j;

• (e) f ∈ V_j ⇔ f(∙/2) ∈ V_j+ 1 (or, equivalently, $f(2\cdot \bullet )\in V_{j} \Leftrightarrow f\in V_{j + 1}$, obviously); and

• (f) (Blatter) There exists a function $\phi \in L^{2}\cap L^{1}$ such that $\left \{ \phi (\bullet -k) \right \}_{k\in \mathbb {Z}}$ is an orthonormal basis of $V_{0}$. This function $\phi$ is called scaling function of the MRA.

For each $j\in \mathbb {Z}$ the functions,

$$ \phi_{j,k}(t) := 2^{-j/2} \phi(2^{-j}t-k) ,\quad k\in\mathbb{Z} , $$

(51)

constitute an orthonormal basis of $V_{j}$.

Since $\phi \in V_{0}\subset V_{-1}$, $\phi$ can be represented in terms of the orthonormal basis (51) of $V_{-1}$ (i.e., $j=-1$). In fact, the inclusion $V_{0}\subset V_{-1}$ is equivalent to the existence of a sequence $\{h_{k}\}_{k\in \mathbb {Z}}$ with ${\sum }_{k\in \mathbb {Z}}|h_{k}|^{2}\leq \infty$ such that the scaling equation (52) holds: [9, p. 118, eq. (2)]

$$ \phi(t)=\sqrt{2}\sum\limits_{k\in\mathbb{Z}}h_{k}\phi(2t-k) ,\quad \text{for almost all}\ t\in\mathbb{R} . $$

(52)

The sequence $\{h_{k}\}_{k\in \mathbb {Z}}$ uniquely defines the scaling function $\phi$. Axiom (a) is an obstruction for the functions $\phi _{j,k}$ to build an orthonormal basis of $L^{2}$. One considers then an additional system of pairwise orthogonal subspaces $W_{j}$ of $L^{2}$, defined as the orthogonal complements of $V_{j}$ in $V_{j-1}$, so that $V_{j-1}=V_{j}\oplus W_{j}$, $V_{j}\perp W_{j}$, $j\in \mathbb {Z}$, having the property:

$$ f\in W_{j} \quad\Leftrightarrow\quad f\left( 2^{j}\cdot\bullet\right)\in W_{0} , $$

(53)

which is analogous to axiom (e). The orthogonal direct sum of these subspaces: [9, p. 108, (5.1)],

$$ \bigoplus_{j\in\mathbb{Z}} W_{j} \quad\text{is dense in}\quad L^{2} . $$

(54)

There is moreover a function $\psi \in W_{0}$, called the mother wavelet, such that $\{\psi (\bullet -k)\}_{k\in \mathbb {Z}}$ is an orthonormal basis of $W_{0}$, given by: [9, pp. 123-4, eqs. (16)–(19)]

$$\begin{array}{@{}rcl@{}} \psi(t)&=&\sqrt{2} {\sum}_{k\in\mathbb{Z}} g_{k} \phi(2t-k) ,\qquad g_{k}=(-1)^{k-1} \overline{h_{-k-1}} ,\\ &&k\in\mathbb{Z} , \end{array} $$

(55)

where $\overline {h_{-k-1}}$ denotes the complex conjugate of $h_{-k-1}$. Moreover, the function system $\{\psi _{j,k}\}_{j,k\in \mathbb {Z}}$ defined by: [9, p. 124, (5.13)]

$$ \psi_{j,k}(t):= 2^{-j/2} \psi\left( 2^{-j}t-k \right) , \quad j,k\in\mathbb{Z} , $$

(56)

is an orthonormal wavelet-basis of $L^{2}(\mathbb {R})$.

1.3 B.3 Algorithms

We start from (52) and (55). From (52) follows for $j,n\in \mathbb {Z}$ the identity:

$$ 2^{-j/2} \phi(2^{-j} t-n ) = 2^{-(j-1)/2} \sum\limits_{k\in\mathbb{Z}}\phi(2^{-(j-1)} t-2n-k ) , $$

(57)

which is usually written as a recursive formula “ϕ_j− 1,∙” $\to$ “ϕ_j,∙” as: [9, p. 131, eq. (3)]

$$ \phi_{j,n} = \sum\limits_{k\in\mathbb{Z}} h_{k} \phi_{j-1, 2n+k} ,\quad j,n\in\mathbb{Z} . $$

(58)

Similarly, the recursive formula “ϕ_j− 1,∙” $\to$ “ψ_j,∙” is given by: [9, p. 131, eq. (4)]

$$ \psi_{j,n} = \sum\limits_{k\in\mathbb{Z}} g_{k} \phi_{j-1,2n+k} ,\quad j,n\in\mathbb{Z} . $$

(59)

The well-known fast filter bank algorithm allows the computation of the orthogonal wavelet coefficients of a signal $f\in L^{2}$. Since at the scale j, $\{\phi _{j,n}\}_{n\in \mathbb {Z}}$ and $\{\psi _{j,n}\}_{n\in \mathbb {Z}}$ are orthonormal bases of $V_{j}$ and $W_{j}$ respectively, the projections of f ontho these spaces are given by: [51, p. 255]

$$\begin{array}{@{}rcl@{}} a_{j,n} &=& \langle f,\phi_{j,n} \rangle = {\int}_{-\infty}^{\infty} f(t) \overline{\phi\left( 2^{-j}t-n\right)} dt , \quad n\in\mathbb{Z} ,\\ d_{j,n} &=& \langle f,\psi_{j,n} \rangle = {\int}_{-\infty}^{\infty} f(t) \overline{\psi\left( 2^{-j}t-n\right)} dt , \quad n\in\mathbb{Z} . \end{array} $$

These coefficients can be calculated recursively with a cascade of discrete convolutions and subsamplings. [51, p. 255, Theorem 7.7] At the analysis or decomposition:

$$ a_{j + 1,n} = \sum\limits_{k\in\mathbb{Z}} h_{k-2n} a_{j,k} ,\qquad d_{j + 1,n} = \sum\limits_{k\in\mathbb{Z}} g_{k-2n} a_{j,k} , $$

(60)

and at the synthesis or reconstruction:

$$ a_{j,n} = \sum\limits_{k\in\mathbb{Z}} h_{n-2k} a_{j + 1,k} + \sum\limits_{k\in\mathbb{Z}} g_{n-2k} d_{j + 1,k}. $$

(61)

Appendix C: Parameter setting of forecasting models

Tables 10 and 11 show the best setting obtained by the training process in each neural network model. The activation functions correspond to Hyperbolic Tangent (Tanh), Logistic (Log), Softsign (Ssgn) and SoftPlus (Spls) and Identity (Id). The best fitting in RFB in the hidden layer, these were: Cubic (Cub) and Multiquadric (MQ). In MLP, PSN, LSTM, the setting corresponds to number of inputs, number of hidden neurons, activation function in hidden layer and activation function in output layer. ELME₁ and ENN follow the same notation previous, however, only consider an activation function on the hidden layer. In the case of SW-ELM, the notation only considers the number of inputs and hidden nodes, since the model uses a particular activation function. In the case of ARIMA model, the best settings are shown in Table 12

Table 10 The best setting of state-of-the-art models included in this study for the stocks: INT, HPQ and CSCO

Table 11 The best setting of state-of-the-art models included in this study for the stocks: VZ, MSFT and IBM

Table 12 The best setting of ARIMA for each stock

Appendix D: Computational Complexity

The forecasting models exhibited in this study were implemented in Python and R, and the execution times were measured in seconds using the modules time (Python) and tictoc (R). To ensure a sequential execution inside each individual model, or inside each individual model constituent, the number of execution threads was set to 1. More precisely, the number of threads of the OPENBLAS environmental variable was set to 1: OPENBLAS_NUM_THREADS= 1.

In Section 5.5.1 we calculate the number of elementary operations ($\mathscr{T}_{\text {WE}}$) required to find the best WE architecture. Table 13 shows the mother wavelets used for fitting the WE models and theirs corresponding indices.

Table 13 Mother wavelets

Tables 14 and 15 show the CC-values for each forecasting model and time series. The execution time for the naive model is 0.0000024594 seconds in mean for each time series.

Table 14 The computational complexity (CC) exhibited by the forecasting models included in this study for the stocks: INT, HPQ and CSCO

Table 15 The computational complexity (CC) exhibited by the forecasting models included in this study for the stocks: VZ, MSFT and IBM

Tables 16 and 17 show the CC-values and THEIL-values for ELME_k models for each time series.

Table 16 The computational complexity (CC) exhibited by the ELME models for the stocks: INT, HPQ, CSCO, VZ, MSFT and IBM

Table 17 The THEIL values exhibited by the ELME models for the stocks: INT, HPQ, CSCO, VZ, MSFT and IBM

The execution environment used to obtain the experimental results is shown in Table 18.

Table 18 Description of the execution environment