## 1. Introduction

## 2. Zeroth Algorithm

${x}_{1}$ 1.046794 ${x}_{2}$ 1.049179 ${x}_{3}$ 1.039641 ${x}_{4}$ 1.046794 ${x}_{5}$ 1.049179 | ${x}_{6}$ 1.042025 ${x}_{7}$ 1.030103 ${x}_{8}$ 1.061101 ${x}_{9}$ 1.046794 ${x}_{10}$ 1.056332 |

## 3. New Method

Algorithm 1 The RTDP method in pseudocode. | |

Require:$\mathbf{x},\text{}m,\text{}{\delta}_{max},\text{}{N}_{p},\text{}{N}_{msp}$ | |

Ensure:$\left|\mathbf{x}\right|>m\ast {\delta}_{max},{N}_{msp}\le {N}_{p}$ | |

$\mathbb{Y}\leftarrow \left\{\right\}$ | |

for i = 1 to ${N}_{p}$do | ▹ tries ${N}_{p}$ $RTDP$s |

$\delta \leftarrow $ random vector of length m containing random integer numbers from 1 to ${\delta}_{max}$ | |

$\tau \leftarrow ({\delta}_{1},\text{}{\delta}_{1}+{\delta}_{2},\text{}{\delta}_{1}+{\delta}_{2}+{\delta}_{3},\text{}\dots ,\text{}{\delta}_{1}+{\delta}_{2}+\cdots +{\delta}_{m})$ | |

$RTDP\leftarrow ({\tau}_{1},{\tau}_{2},{\tau}_{3},\dots ,{\tau}_{m})$ | ▹$RTDP$ is actually a cumulative sum of $\delta $ |

for k = 1 to $\left|\mathbf{x}\right|-m\ast {\delta}_{max}$do | ▹ goes through all possible subsequencies ${\mathbf{x}}_{k}$ in $\mathbf{x}$ |

${\epsilon}_{k}\leftarrow \parallel {\mathbf{x}}_{last}-{\mathbf{x}}_{k}\parallel $ | ▹ each distance ${\epsilon}_{k}$ between ${\mathbf{x}}_{k}$ and ${\mathbf{x}}_{last}$ is stored |

end for | |

${k}_{min}\leftarrow arg{min}_{k}{\epsilon}_{k}$ | ▹ finds the k for which ${\mathbf{x}}_{k}$ is closest to ${\mathbf{x}}_{last}$ |

${y}_{i}\leftarrow {y}_{best}\leftarrow {x}_{t-{k}_{min}}$ | ▹ assumed best prediction made by this $RTDP$ |

${\epsilon}_{i}\leftarrow {\epsilon}_{min}\leftarrow {\epsilon}_{{k}_{min}}$ | ▹ distance of the closest subsequence ${\mathbf{x}}_{{k}_{min}}$ |

$\mathbb{Y}\leftarrow \mathbb{Y}\cup ({y}_{i},{\epsilon}_{i})$ | ▹ adds results from this $RTDP$ to the overall result set |

end for | |

${\mathbb{Y}}_{\epsilon}\leftarrow {\mathrm{sort}}_{\epsilon}\left(\mathbb{Y}\right)$ | ▹ ranking all $RTDP$s predictions by assumed accuracy $\epsilon $ |

${\widehat{y}}_{t}\leftarrow ({y}_{1}+{y}_{2}+{y}_{3}+\cdots +{y}_{{N}_{msp}})/{N}_{msp}$ | ▹ averaging the best ${N}_{msp}$ predictions |

${x}_{1}$ 1.046794 ${x}_{2}$ 1.049179 ${x}_{3}$ 1.039641 ${x}_{4}$ 1.046794 ${x}_{5}$ 1.049179 | ${x}_{6}$ 1.042025 ${x}_{7}$ 1.030103 ${x}_{8}$ 1.061101 ${x}_{9}$ 1.046794 ${x}_{10}$ 1.056332 | ${x}_{11}$ 1.022949 ${x}_{12}$ 1.022949 ${x}_{13}$ 1.027718 ${x}_{14}$ 1.027718 ${x}_{15}$ 1.020565 | ${x}_{16}$ 0.999104 ${x}_{17}$ 1.011027 ${x}_{18}$ 1.008642 ${x}_{19}$ 1.013411 ${x}_{20}$ 1.025334 |

$\mathit{k}$ | 1 | 2 | 3 | 4 | 5 |

${y}_{k}={x}_{21-k}$ | 1.025334 | 1.013411 | 1.008642 | 1.011027 | 0.999104 |

${\epsilon}_{k}$ | 0.052459 | 0.050074 | 0.114456 | 0.081073 | 0.090611 |

$\mathit{R}\mathit{T}\mathit{D}\mathit{P}$ | (2,4,5,8,11) | (1,2,4,5,6) | (3,4,7,8,10) | (3,6,9,12,13) | (2,4,7,10,13) |

y | 1.013411 | 1.025334 | 1.025334 | 1.008642 | 1.013411 |

$\epsilon $ | 0.050074 | 0.057228 | 0.052459 | 0.054843 | 0.059612 |

## 4. Comparison

## 5. Results

## 6. Conclusions and Future Work

## Supplementary Materials

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACF | Autocorrelation function |

ANN | Artificial neural networks |

ARIMA | Auto-regressive integrated moving average |

KNN | k-nearest neighbor |

PACF | Partial autocorrelation function |

RF | Random forest |

RMSE | Root mean square error |

RTDP | Random time delays pattern |

XGB | Extreme gradient boosting |

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**Figure 1.**A simplified example of consumption aggregation. Energy consumption of 3 nodes running different jobs at the same time. Regular patterns merged together give a regular pattern, but the pattern is more complex, with a smaller degree of predictability.

**Figure 2.**Power energy consumption time series. This is the normalized measured power from the infrastructure of the IT4Innovations [3] supercomputer. The measured timerange is from 1:00 p.m., 2 November to 9:00 p.m., 5 November 2017.

**Figure 3.**Results of a series of predictions designed to empirically determine the optimal parameters of the RTDP method. The time series of power energy consumption, shown in Figure 2, was used to calculate these results. The numbers at the nodes represent the value of ${\delta}_{max}$. The number of patterns and the number of the most successful patterns were set to ${N}_{p}=30$ and ${N}_{msp}=21$ for the whole series. The RTDP method is (in this case) most accurate when the parameters ${\delta}_{max}=5$ and $m=25$ are set.

**Figure 4.**Comparison of the prediction accuracy waveforms of the methods used with the new prediction method RTDP. The moving RMSE was calculated as the RMSE of a moving 300 samples wide window.

Method | Parameters |
---|---|

ANN 1 | $\eta =0.1$, 3 layers of 15 neurons each, $maxerror=0.01$ |

ANN 2 | $\eta =0.1$, 3 layers of 15 neurons each, $maxerror=0.02$ |

ARIMA(0,1,2) | $p=0,\text{}d=1,\text{}q=2$ |

ARIMA(8,1,6) | $p=8,\text{}d=1,\text{}q=6$ |

KNN | $k=5,\text{}N=40$ |

RF | $ntree=13,\text{}mtry=19$ |

RTDP | ${\delta}_{max}=5,\text{}m=25,\text{}{N}_{p}=30,\text{}{N}_{msp}=21$ |

XGB | $nrounds=22,\eta =0.23,minweight=20,maxdepth=1,\gamma =0$ |

Zeroth | $m=31,\text{}\tau =1,\text{}\epsilon =0.151$ |

**Table 2.**The ranked results are summarized here by the total RMSE and also by the total runtime taken to calculate the predictions of the entire time series of supercomputer power consumption.

Method | Total RMSE [-] | Method | Total Run-Time [s] | |
---|---|---|---|---|

RTDP | 0.02719 | Zeroth | 23 | |

ARIMA(8,1,6) | 0.02722 | RTDP | 42 | |

ARIMA(0,1,2) | 0.02738 | ARIMA(0,1,2) | 58 | |

XGB | 0.02773 | KNN | 3240 | |

RF | 0.02836 | XGB | 4515 | |

Zeroth | 0.03231 | ARIMA(8,1,6) | 4714 | |

KNN | 0.03350 | RF | 7250 | |

ANN 1 | 0.03414 | ANN 2 | 25,501 | |

ANN 2 | 0.03841 | ANN 1 | 56,549 |

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