# Multiparametric Monitoring in Equatorial Tomato Greenhouses (II): Energy Consumption Dynamics
## Appendix A Time Series Notation and M-Mode Plots for Different Runs. The energy consumption signals in WSN nodes are temporal random processes, since they are defined as time functions at some observation interval, and it is not possible to describe exactly their waveform as it will be observed in the future before performing another experiment. The recorded temporal signals are defined as continuous stochastic processes, and we can denote each of these processes as \(\varphi(t)\)[57]. In general terms, the mathematical notation assigned to an acquired consumption signal is \(x(t)\), and each signal generated to \(j^{th}\) realizations is defined as \(x_{j}(t)\), with \(j=1,2,...,R\), where \(R\) is the total number of realizations. All signals are observed in a time interval given by \((t_{i},t_{f})\) with duration \(T\), and each of these measured signals is denoted as  
\[x_{j}(t)= \begin{cases}x(t)&\text{if }t\in(t_{i},t_{i}+T)\\ 0&\text{otherwise}\end{cases}  \tag{1}\]  
The temporal evolution of a stochastic process in a sample period \(T_{s}\) is analyzed using discrete-time series, which are also called _sampled signals_. The sampling process transforms a band-limited continuous-time signal into a series of amplitude values at discrete-time instants [58]. Sampled signals can be obtained with the classical impulse- and uniform-sampling technique,which multiplies the input signal by a train of unit impulses in which the intervals between consecutive impulses are the equidistant sampling times, as described by  
\[\Delta_{j}(t\mid T_{s})=x_{j}(t)\cdot\sum_{n=-\infty}^{\infty}\delta(t-nT_{s}) \tag{10}\]  
where the Dirac's delta function or continuous-time unitary-impulse function is denoted as \(\delta(t)\)[59]. Note that \(x_{j}(t)\) is limited in duration, so that \(\Delta_{j}(t\mid T_{s})\) has a limited set of \(N\) possibly non-zero samples. Based on the description of process \(\varphi(t)\) in the time domain, we can obtain its estimated average from the set of samples available for the \(j\) realizations by  
\[\tilde{\Delta}_{j}(t\mid T_{s})=E[\Delta_{j}(t\mid T_{s})] \tag{11}\]  
In addition, for a signal \(x(t)\), we can define an M-mode by defining a convenient second dimension \(d\), so that \(\Gamma(t,d)\) denotes this representation in general terms. For instance, if the second dimension is the index corresponding to the \(j^{th}\) realization, we can define the M-mode of the sampled process as a index-temporal and simultaneous representation of the realizations of the stochastic process as follows,  
\[X(j,t)=x_{j}(t) \tag{12}\]  
where the convention of capital notation is used for the M-mode of the represented signal. Probability Density Function and Bootstrap AveragingThe study of the probability density function (pdf) of a continuous random process \(\varphi(t)\) is relevant because it quantifies the certainty and the uncertainty of the results obtained in each realization. The value of this function is positive throughout the domain [60]. Since \(x_{j}(t)\) is a continuous random signal, we can define the pdf of its amplitude as \(f_{x_{j}}(x)\), and it represents the overall distribution of its amplitudes, integrated with respect to time and then independently from it. M-mode plots were also used in this work to represent jointly the estimated pdf of each realization, as denoted by:  
\[F(j,f_{x_{j}})=f_{x_{j}}(x) \tag{13}\]  
where again the convention of capital is used for the M-mode of the pdf. If we want to obtain the overall estimation of process \(\varphi(t)\), denoted as \(f_{x}(x)\), we can average in the available realizations, as denoted by  
\[\tilde{f}(x)=E_{j}[f_{x_{j}}(x)] \tag{14}\]  
Bootstrap resampling can provide us with nonparametric estimation of a variety of CI for different estimations with non-trivial analytical estimation equations. By using bootstrap resampling, we generate \(b=1,...,B\) resamples of the empirical pdf by sampling with replacement of the process time-samples, which is denoted as \(f_{x_{j}}^{*}(x,b)\).