# Time Series Data Analysis of Wireless Sensor Network Measurements of Temperature

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## Abstract

**:**

## 1. Introduction

## 2. Previous Work

## 3. Temperature Data from Springbrook WSN Deployment

## 4. Accuracy versus Sampling Interval

## 5. Repeating for Another Data Series

## 6. Time Series Analysis of Random Processes

#### 6.1. Time Series and Stochastic Process

#### 6.2. Time Series Model Development Strategy

#### 6.2.1. Model Specification

#### 6.2.2. Parameter Estimation

_{i}and θ

_{i}are estimated using a Least Square (LSE) or Maximum Likelihood (ML) estimator. These parameters can then be used to estimate future values of the series

#### 6.2.3. Model Diagnostics

#### 6.2.4. Time Series Forecasting

## 7. Forecasting Experiments

#### 7.1. Structural Analysis of Time Series

#### 7.2. Model Order Selection

#### 7.3. Forecasting

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 13.**Detail of RMSE versus Prediction Horizon for Different Predictors with 95% confidence interval for ARIMA60.

Sampling Interval (Mins) | RMSE Linear | MAE Linear | RMSE Cubic | MAE Cubic | 99% Linear | 99% Cubic |
---|---|---|---|---|---|---|

10 | 0.0884 | 0.0528 | 0.0852 | 0.0519 | 0.3250 | 0.2893 |

15 | 0.1097 | 0.0664 | 0.1088 | 0.0669 | 0.4000 | 0.4037 |

20 | 0.1166 | 0.0755 | 0.1228 | 0.0793 | 0.4200 | 0.4496 |

30 | 0.1527 | 0.0937 | 0.1531 | 0.0962 | 0.5800 | 0.5709 |

45 | 0.1865 | 0.1152 | 0.1921 | 0.1190 | 0.6867 | 0.7410 |

60 | 0.2224 | 0.1335 | 0.2330 | 0.1430 | 0.8425 | 0.8753 |

90 | 0.2439 | 0.1566 | 0.2507 | 0.1629 | 0.9133 | 0.8774 |

120 | 0.2646 | 0.1720 | 0.2893 | 0.1882 | 0.9425 | 1.0206 |

240 | 0.3297 | 0.2161 | 0.3290 | 0.2215 | 1.2758 | 1.2189 |

Sampling Interval (Mins) | RMSE Linear | MAE Linear | RMSE Cubic | MAE Cubic | 99% Linear | 99% Cubic |
---|---|---|---|---|---|---|

10 | 0.1746 | 0.0941 | 0.1751 | 0.0960 | 0.6740 | 0.6366 |

15 | 0.2085 | 0.1164 | 0.2185 | 0.1211 | 0.7554 | 0.8286 |

20 | 0.2342 | 0.1360 | 0.2487 | 0.1459 | 0.8862 | 0.9436 |

30 | 0.2723 | 0.1588 | 0.2846 | 0.1693 | 1.0099 | 1.0027 |

45 | 0.3664 | 0.2029 | 0.3694 | 0.2087 | 1.2578 | 1.3131 |

60 | 0.4655 | 0.2498 | 0.4635 | 0.2493 | 1.5781 | 1.5309 |

90 | 0.5837 | 0.3093 | 0.5762 | 0.3033 | 1.9658 | 1.8047 |

120 | 0.6057 | 0.3836 | 0.5840 | 0.3663 | 2.1344 | 2.0859 |

240 | 0.9780 | 0.6687 | 0.8121 | 0.5515 | 3.0073 | 2.7782 |

Sampling Rate (Minutes) | Fitted Models |
---|---|

5 | ARIMA(3,1,1) |

10 | ARIMA(2,1,2) |

15 | ARIMA(1,1,3) |

20 | ARIMA(1,1,3) |

30 | ARIMA(2,1,1) |

60 | ARIMA(1,1,0) |

120 | ARIMA(3,1,1) |

Forecast | Simple Models | ARIMA Models Sampling Intervals (Minutes) | |||||||
---|---|---|---|---|---|---|---|---|---|

Time (Mins) | Zero Diff | Same Diff | 5 | 10 | 15 | 20 | 30 | 60 | 120 |

5 | 0.33 | 0.49 | 0.33 | 0.17 | 0.13 | 0.13 | 0.11 | 0.12 | 0.12 |

10 | 0.48 | 0.78 | 0.45 | 0.45 | 0.38 | 0.39 | 0.35 | 0.34 | 0.34 |

15 | 0.59 | 1.07 | 0.51 | 0.51 | 0.51 | 0.51 | 0.45 | 0.44 | 0.46 |

20 | 0.62 | 1.36 | 0.53 | 0.54 | 0.54 | 0.63 | 0.54 | 0.54 | 0.57 |

30 | 0.91 | 2.02 | 0.75 | 0.76 | 0.77 | 0.90 | 0.86 | 0.84 | 0.85 |

60 | 1.56 | 4.05 | 1.07 | 1.07 | 1.06 | 1.29 | 1.17 | 1.39 | 1.33 |

120 | 3.32 | 8.47 | 2.50 | 2.52 | 2.52 | 2.71 | 2.58 | 2.80 | 2.48 |

Forecast | Simple Models | ARIMA Models Sampling Intervals (Minutes) | |||||||
---|---|---|---|---|---|---|---|---|---|

Time (Mins) | Zero Diff | Same Diff | 5 | 10 | 15 | 20 | 30 | 60 | 120 |

5 | 0.24 | 0.27 | 0.21 | 0.11 | 0.08 | 0.08 | 0.07 | 0.08 | 0.09 |

10 | 0.35 | 0.49 | 0.32 | 0.32 | 0.26 | 0.26 | 0.23 | 0.21 | 0.22 |

15 | 0.45 | 0.65 | 0.38 | 0.38 | 0.38 | 0.36 | 0.31 | 0.29 | 0.31 |

20 | 0.52 | 0.87 | 0.42 | 0.42 | 0.42 | 0.46 | 0.40 | 0.37 | 0.43 |

30 | 0.73 | 1.31 | 0.58 | 0.58 | 0.59 | 0.63 | 0.63 | 0.55 | 0.62 |

60 | 1.27 | 2.60 | 0.82 | 0.82 | 0.81 | 0.85 | 0.82 | 0.90 | 0.96 |

120 | 2.71 | 5.71 | 1.91 | 1.94 | 1.98 | 2.03 | 1.97 | 2.03 | 1.74 |

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**MDPI and ACS Style**

Bhandari, S.; Bergmann, N.; Jurdak, R.; Kusy, B.
Time Series Data Analysis of Wireless Sensor Network Measurements of Temperature. *Sensors* **2017**, *17*, 1221.
https://doi.org/10.3390/s17061221

**AMA Style**

Bhandari S, Bergmann N, Jurdak R, Kusy B.
Time Series Data Analysis of Wireless Sensor Network Measurements of Temperature. *Sensors*. 2017; 17(6):1221.
https://doi.org/10.3390/s17061221

**Chicago/Turabian Style**

Bhandari, Siddhartha, Neil Bergmann, Raja Jurdak, and Branislav Kusy.
2017. "Time Series Data Analysis of Wireless Sensor Network Measurements of Temperature" *Sensors* 17, no. 6: 1221.
https://doi.org/10.3390/s17061221