Securing Anomaly Detection for Process-Based Time Series

Figures & data

Fig. 1. Proprietary System, Water Pumped into Leaking Reservoir. Variables, a1 through a5, labeled with Red Arrows.

TABLE I Description of Simulation Variables, Proprietary System

Name	Variable Description
a1 $({\mathrm{m}}^3/\mathrm{s})$	Volumetric flow through “UpstreamValve” block
a2 $({\mathrm{m}}^3/\mathrm{s})$	Volumetric flow through “DownstreamValve” block
a3 $(\mathrm{W})$	Power of flow from “Reservoir” block
a4 $(\mathrm{kg}/\mathrm{s})$	Mass flow through “pipe” block
a5 $(\mathrm{bar})$	Relative pressure in “Reservoir” block

TABLE II Description of Simulation Variables, Generic System

Name	Variable Description
b1 $({ }^{\circ }\mathrm{C})$	Fluid temperature, “pipe8” outlet
b2 $(\mathrm{kJ})$	Enthalpy flow, “pipe6” outlet
b3 $(\mathrm{N})$	Force of friction, “pipe6” outlet
b4 $(\mathrm{g}/\mathrm{c}{\mathrm{m}}^3)$	Fluid density, “pipe4” outlet
b5 $(\mathrm{bar})$	Differential pressure, “valve1”

Fig. 2. Generic System, Series of Heated Pipes. Variables, b1 through b5, labeled with Red Arrows.

Fig. 3. Proprietary System Modified for the Prominent Anomaly Case. Anomalous Valves Highlighted in Gray as “AnomalousValve1” through “AnomalousValve4” whose openings are determing by “AnomalousLeakage1” through “AnomalousLeakage4.”

TABLE III Details of Pipe Leakage Anomalies, Prominent Anomaly Case

Anomaly Description	Maximum Mass Flow (kg/s)	Duration, Start Times (s)	Number of Instances
AnomValve1 is open	1000	20 (100, 900)	2
AnomValve2 is open	400	75 (1200, 1400, 1600)	3
AnomValve3 is open	1000	100 (1000)	1
AnomValve4 is open	400	200 (400)	1

Fig. 4. Prominent Anomaly Case, Power of Flow through the Reservoir.

Fig. 5. Generic Data, Temperature of Fluid at Pipe 8.

Fig. 6. Prominent Anomaly Case, Masked Power of Flow through the Reservoir.

Fig. 7. Prominent Anomaly Case, volumetric flow through “DownstreamValve” versus power of flow from “Reservoir.”

Fig. 8. Comparison of power of flow through reservoir across experiments. Left, Prominent Anomaly Case (restatement of Fig. 4). Right, Subtle Anomaly Case.

Fig. 9. Subtle Anomaly Case, Volumetric Flow through the Downstream Valve vs. Power of Flow through the Reservoir.

Fig. 10. Subtle Anomaly Case, Volumetric Flow through the Downstream Valve Colored in Black, Volumetric Flow through AnomValve1 Colored in Red.

Fig. 11. Proprietary System Modified for the Multi-Regime Anomaly Case. Controller Set Point Highlighted in Orange, Secondary Pump Highlighted in Blue, Anomalous Valves Highlighted in Gray.

TABLE IV Description of Anomalous Regimes

Label	Description	Time (s)	Proportion (%)	Number of Instances
0	“PrimaryPump” operating normally, “SecondaryPump” is off, all “AnomalousValves” closed	5 to 105, 405 to 800, 1000 to 1795,	25.8	3
1	“PrimaryPump” and “SecondaryPump” operating normally, all “AnomalousValves” closed	3795 to 5000	24.1	1
2	“PrimaryPump” failure	105 to 405	6	1
3	At least one “AnomalousValve” is open	800 to 1000, 2223 to 2905, 3205 to 3795	29.5	3
4	“SecondaryPump” operating, at least one “AnomalousValve” is open, and “PrimaryPump” failure	2905 to 3205	6	1
5	Postprocess noise addition	1795 to 2223	8.6	1

TABLE V Classification Accuracy of Each Experiment, Secs. IV, V, and VI

		Neural Network		k-Nearest Neighbor		Support Vector Machine
Experiment	Data	Original Data	Masked Data	Original Data	Masked Data	Original Data	Masked Data
Prominent Anomaly	Train	100 ± 0	100 ± 0	100 ± 0	100 ± 0	98.94 ± 0.05	98.98 ± 0.04
	Test	99.92 ± 0.11	99.86 ± 0.12	99.82 ± 0.13	99.82 ± 0.17	99.02 ± 0.40	98.91 ± 0.26
Subtle Anomaly	Train	95.93 ± 7.56	95.43 ± 8.29	100 ± 0	100 ± 0	63.96 ± 0.75	64.58 ± 0.56
	Test	95.39 ± 8.28	95.24 ± 8.67	98.91 ± 0.28	98.88 ± 0.39	63.47 ± 2.34	62.12 ± 1.29
Multi-Regime Anomaly	Train	99.41 ± 0.32	99.39 ± 0.32	100 ± 0	100 ± 0	94.61 ± 0.05	94.70 ± 0.06
	Test	99.31 ± 0.45	99.17 ± 0.41	99.87 ± 0.08	99.90 ± 0.07	94.46 ± 0.21	94.66 ± 0.43

TABLE VI Classification Accuracy of Randomized Labels, Multi-Regime Case

		Neural Network		k-Nearest Neighbor		Support Vector Machine
Experiment	Data	Original Data	Masked Data	Original Data	Masked Data	Original Data	Masked Data
Multi-regime anomaly	Train	99.41 ± 0.32	99.39 ± 0.32	100 ± 0	100 ± 0	94.61 ± 0.05	94.70 ± 0.06
	Test	99.31 ± 0.45	99.17 ± 0.41	99.87 ± 0.08	99.90 ± 0.07	94.46 ± 0.21	94.66 ± 0.43
Partially randomized labels	Train	79.22 ± 0.43	79.40 ± 0.24	100 ± 0	100 ± 0	75.94 ± 0.06	75.59 ± 0.13
	Test	78.83 ± 0.92	78.74 ± 0.86	79.47 ± 0.72	79.58 ± 0.76	75.08 ± 0.62	75.37 ± 1.09
Fully randomized labels	Train	19.24 ± 0.34	19.56 ± 0.40	100 ± 0	100 ± 0	17.95 ± 0.13	18.13 ± 0.14
	Test	17.05 ± 0.68	16.76 ± 1.15	16.67 ± 1.18	16.16 ± 0.58	16.28 ± 0.51	16.53 ± 0.78

TABLE A.I Description of Variables in Eqs. (1)(1) $y_P\left(\mathit{x},\mathit{\alpha }\right)\approx \sum _{\mathit{i}=1}^r{\psi }_i^P\left(\mathit{x}\right){\varphi }_i^P\left(\mathit{\alpha }\right)$(1) , (2)(2) $y_G\left(x^{\prime },{\alpha }^{\prime }\right)\approx {\displaystyle \sum _{\mathit{i}=1}^r{\psi }_i^G\left(x^{\prime }\right)}{\varphi }_i^G\left({\alpha }^{\prime }\right)$(2) , and (3)(3) $\begin{array}{c}{y}_D\left(\mathit{x}{ }^{\mathrm{\prime }},\mathit{\alpha }\right)=\sum _{\mathit{i}=1}^r{\psi }_i^G\left(\mathit{x}{ }^{\mathrm{\prime }}\right){\varphi }_i^P\left(\mathit{\alpha }\right)\end{array}.$(3)

Variable	Description
$\mathit{x}$	General form of the temporal/spatial variable within the proprietary data [e.g., at $\mathit{x}=\left(5,9\right)$ may refer to five units across a one-dimensional space and at $\mathit{t}=9$ s into a given process].
$\mathit{\alpha }$	General form of the process parameters associated with the proprietary data. Intuition is similar to the state variables of a dynamic system (i.e., underlying parameters informing the behavior of the data with respect to its typical form). Depending on the problem, $\mathit{\alpha }$ may involve physical values (e.g., temperature, pressure, etc.) or material conditions (type, enrichment, etc.).
$x^{\prime }$	General form of the temporal/spatial dependence of the generic data.
${\alpha }^{\prime }$	General form of the process parameters associated with the generic data.
$y_P\left(\mathit{x},\mathit{\alpha }\right)$	The observed value of the proprietary data at some $\mathit{x}$ and $\mathit{\alpha }.$
$y_G\left(\mathit{x}{ }^{\mathrm{\prime }},\mathit{\alpha }{ }^{\mathrm{\prime }}\right)$	The observed value of the generic data at some $x^{\prime }$ and ${\alpha }^{\prime }.$
$y_D\left(\mathit{x}{ }^{\mathrm{\prime }},\mathit{\alpha }\right)$	The observed value of the masked data at some $x^{\prime }$ and $\mathit{\alpha }.$ Note the masked data do not reflect a real system and carry the same inference as $y_P$ associated with $\mathit{\alpha }$ but do not give any information about $\mathit{x}$ or $y_P\left(\mathit{x},\mathit{\alpha }\right).$
${\varphi }^P\left(\mathit{\alpha }\right)$	Proprietary inferential metadata; a subset of the data in response to some $\mathit{\alpha }$ that is relevant to inferential problems (e.g., regression, condition monitoring, etc.).
${\psi }^P\left(\mathit{x}\right)$	Proprietary fundamental metadata; a subset of the dataset in response to a temporal/spatial value of $\mathit{x}$. While this gives some description of the data’s shape and behavior (i.e., periodic, parabola, etc.), it is not referring to an individual variable or snippet of the data.
${\varphi }^G\left(\mathit{\alpha }{ }^{\mathrm{\prime }}\right)$	Generic inferential metadata; a subset of the generic dataset in response to some parameter ${\alpha }^{\prime }$ relevant to inferential content. This subset is discarded within DIOD.
${\psi }^G\left(\mathit{x}{ }^{\mathrm{\prime }}\right)$	Generic fundamental metadata; a subset of the generic dataset in response to temporal/spatial value of $\mathit{x}{ }^{\mathrm{\prime }}$. This informs the overall behavior of the generic data but is NOT an individual variable or section of the data. This will inform the shape/behavior of the masked data after using Eqs. (1)(1) $y_P\left(\mathit{x},\mathit{\alpha }\right)\approx \sum _{\mathit{i}=1}^r{\psi }_i^P\left(\mathit{x}\right){\varphi }_i^P\left(\mathit{\alpha }\right)$(1) , (2)(2) $y_G\left(x^{\prime },{\alpha }^{\prime }\right)\approx {\displaystyle \sum _{\mathit{i}=1}^r{\psi }_i^G\left(x^{\prime }\right)}{\varphi }_i^G\left({\alpha }^{\prime }\right)$(2) , and (3)(3) $\begin{array}{c}{y}_D\left(\mathit{x}{ }^{\mathrm{\prime }},\mathit{\alpha }\right)=\sum _{\mathit{i}=1}^r{\psi }_i^G\left(\mathit{x}{ }^{\mathrm{\prime }}\right){\varphi }_i^P\left(\mathit{\alpha }\right)\end{array}.$(3) .