# Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

_{in}+ I

_{begin}− T

_{out}− I

_{end}, where T is a transfer and I is an inventory. The main quantitative assessment of safeguards effectiveness is the measurement error standard deviation of the material balance, which can be related to the probability of detecting a specified loss of NM for specified false alarm probability [4].

## 3. Method

#### 3.1. Data Description

_{R}and σ

_{A}, respectively. Such simulated data represents a discrete time series of measurements (volume or mass) of aqueous solutions at a reprocessing facility [4,5,6,7,8].

_{R}range from approximately 0.001 (0.1%) to 0.03. Typical values for σ

_{A}depend on the measurement units and for the range of T values we consider, σ

_{A}ranges from approximately 0.1 to 2 [5,10]. This paper uses nominal values of σ

_{A}= 1 and σ

_{R}= 0.015 for additive and relative errors, respectively; other values for σ

_{R}and σ

_{A}are also used when indicated in Section 4. Notice that there is no transform of Equation (1) that converts the measurement error model to purely additive error.

_{A}= 1 and σ

_{R}= 0.015 and clearly includes four abrupt events over two tank cycles; each cycle has a receipt and a shipment. Because this data is simulated, the exact start and stop time indices of the events are known, which is useful because the change point estimates of event detection algorithms can then be compared to the known true values. Therefore, this data set and variations of it are used throughout this paper. For Figure 1, the data is generated at constant intervals. The first rise spans 10 data points (somewhat abrupt), while the second rise spans only 1 data point (very abrupt); the first drop spans 10 data points and the second drop spans 20 data points (more gradual).

#### 3.2. Wavelets

#### 3.3. Lag-One Differencing Change Detection

_{1}and t

_{2}if $\text{\Delta}=\frac{\left|M\left({t}_{2}\right)-M\left({t}_{1}\right)\right|}{c}>h$, where c is a scale factor and h is a statistically determined threshold value used to differentiate events from noise [7]. The scale factor c can be defined as the empirical standard deviation of the ∆ values. Alternatively, if multiple events are present and c is intended to characterize the typical change rate during non-events, then c can be defined as a robust estimate of the standard deviation (see Section 3.4) of the ∆ values during non-events. If multiple sub-events occur over a contiguous set of time points, the sub-events are concatenated together to form a single event, such as a shipment that occurs over multiple time steps. This “lag-one differencing” option is simple and it is adequate for detecting abrupt events; however, it can fail to detect gradual events [7] as we will show.

#### 3.4. Wavelet Change Detection

_{jk}to locate change points. If a wavelet decomposition coefficient │w

_{jk}│ is above a predetermined “critical value”, it is assumed that the coefficient corresponds to a change point in the data. The optimal critical value depends on characteristics of the original data. As an example, Figure 3 shows example simulated data with a gradual receipt and shipment, and the corresponding wavelet coefficients at various resolution levels. Resolution level 1 is the finest resolution, corresponding to lag-one differencing. Resolution level 2 is the next finest resolution, and so on.

**Figure 3.**Simulated scaled volumes for one tank cycle and the wavelet coefficients at each of five resolutions. Horizontal threshold lines at ±5 are included. Resolution 4 or 5 work well for this example and for similar examples because the threshold can be selected to find only one outlier in the receipt change region and one outlier in the shipment change region. Plot (

**a**) is the scaled volume versus the time index. Plots (

**b**)–(

**f**) are the corresponding wavelet coefficients at 5 different resolutions.

^{j}for j ∈ ℤ). SM data is plentiful so it is easy to break the data into dyadic sections. All simulated datasets have dyadic length (usually 256, 512, or 1024) to keep the analysis simple. Wavelet change detection can be done at varying resolution levels. For our SM data, using the Haar-based wavelet, we find it is easiest to detect abrupt events at low resolution numbers (which correspond to high resolution), and to detect gradual events at higher resolution numbers, as illustrated in Figure 4. In Figure 4, the first tank cycle has an abrupt receipt that is most easily detected at low resolution number and the second tank cycle has a gradual receipt that is most easily detected at high resolution number. The coefficients w1–w4 are wavelet coefficients at resolutions 1 to 4. Figure 4 is similar to Figure 8 in [15].

**Figure 4.**Haar-wavelet based change detection for two tank cycles. In the first cycle, there is an abrupt receipt and gradual shipment. In the second cycle, there is a gradual receipt and gradual shipment. Wavelet coefficients w1–w4 can be used alone or together for change detection.

_{jk}that correspond to change regions, we use a simulation-based approach to estimate the distribution of wavelet coefficients and to find an effective resolution j for change-region identification. We standardize w

_{jk}to approximately unit standard deviation by dividing by a robust estimate (that down weights outliers) of the standard deviation ${\widehat{\sigma}}_{{w}_{j}}$ (we used the function cov.mve in R [26] to calculate a robust estimate ${\widehat{\sigma}}_{{w}_{j}}$). The time index k is inferred to be an outlier at resolution j, if $\frac{{w}_{jk}}{{\widehat{\sigma}}_{{w}_{jk}}}\ge a$ for threshold a that is also chosen by simulation. The appropriate a value varies, depending on the resolution level. We found by extensive simulation using variations of SM data such as in Figure 1 and Figure 4 that a moderate resolution of 4 is adequate for both abrupt and gradual SM events.

## 4. Comparison of Change-Point Detection Methods in Simulated SM Data

#### 4.1. Finding the Correct Number of Change Points and the Approximate Times of Each Change Point

_{R}and σ

_{A}values unless the correct number of change points is known. For example, Figure 5 illustrates that breakpoints finds only two of the four change points in the first tank cycle data such as in Figure 1. The Bayesian information criterion (BIC) is used in breakpoints to infer the number of change points. The BIC is defined as BIC = −2log(maxlikelihood) + plog(n), where maxlikelihood is the maximum likelihood (corresponds to the usual least squares regression estimate in the case of Gaussian data) and p is the number of model parameters to penalize for the number of breakpoints. The model having the smallest BIC is preferred, and changes in BIC of less than approximately 10 are commonly assumed to be noise. Therefore, Figure 5 shows that breakpoints selected only two of four change points, failing to detect a slope change within each transfer. In addition, breakpoints is slow to run on 1024 data points despite its use of a relatively fast dynamic programming algorithm to try many options for the number of change points and locations.

**Figure 5.**An example of the Bayesian information criterion (BIC) for breakpoints applied to data such as the first tank cycle in Figure 1. The true number of change points is four.

_{t}at each time step t were known exactly, then rescaling M(t) by dividing by $\sqrt{{\sigma}_{A}^{2}+{T}_{t}^{2}{\sigma}_{R}^{2}}$ would convert the time series to a unit variance time series with purely additive error. Because T

_{t}is unknown, it must be estimated using ${\widehat{T}}_{t}$, which we obtained using an initial Haar-based wavelet transform. Because of estimation error in ${\widehat{T}}_{t}$ the scaled M(t) has only approximately unit variance, and departures from unity tend to occur near the change points. All quantitative results are for the option that first scales M(t) to approximately unit variance prior to performing change detection; however, whether to transform the simulated SM data to approximately unit variance by first performing an initial smoothing made very little difference in the ability of Haar wavelets to find approximate change point regions.

#### 4.2. Refining the Initial Estimate of the Time of Each Change Point

#### 4.3. Qualitative and Quantitative Simulation Results

_{A}and σ

_{R}in Equation (1) increase in going from subplots (a) to (d) in Figure 6, the gradual receipt and shipment regions become very difficult to detect. Recall that Figure 4 illustrated the potential of Haar-wavelet based change detection for a gradual receipt and gradual shipment. Comparing Figure 7 to Figure 6, note that Haar-wavelet based change region detection performs much better than the lag-one differencing. More quantitatively, using repeated sets of 1000 simulations of data as in Figure 3 with a gradual receipt and gradual shipment, we found that the breakdown σ

_{A}and σ

_{R}values (the σ

_{A}and σ

_{R}values at or above which the method to find each change region results in finding the wrong number of change regions) are approximately 1 and 0.03 for lag-one differencing and are approximately 8 and 0.12 for Haar-wavelet based change detection, which are both well above the values anticipated for our SM application. Of course the breakdown point depends on the rate of change of T(t) during each change region. But this example is representative of SM data, demonstrating a large advantage of a wavelet-based method to locate approximate change-point regions. We therefore have implemented Haar wavelet based change detection using high resolution to find approximate change point regions in sections of typically 512 or 1024 observations.

**Figure 6.**The scaled lag-one change for increasing values of σ

_{A}and σ

_{R}in Equation (1). Note that the abrupt shipment is easily detected but that the more gradual receipt is less easily detected. The two true change regions are indicated with vertical dashed lines. The values σ

_{A}and σ

_{R}in Equation (1) increase in going from subplots (

**a**) to (

**d**) in Figure 6, beginning with σ

_{A}= 0.5 and σ

_{R}= 0.005 in (

**a**).

**Figure 7.**The scaled change arising from the Haar-wavelet coefficient monitoring for increasing values of σ

_{A}and σ

_{R}in Equation (1) in subplots (

**a**–

**d**). Note that both the gradual shipment and the gradual receipts are easily detected for (

**a**–

**d**) as σ

_{A}and σ

_{R}increase. The two true change regions are indicated with vertical dashed lines.

_{A}(1.0) and σ

_{R}(0.015) for a tank cycle similar to that in Figure 1, with a gradual receipt and gradual shipment, both occurring over 10 time steps. We verified by repeating the set of 1000 simulations that the RMSEs are repeatable across sets of 10

^{4}simulations to within approximately ±0.01 or less.

**Table 1.**The root mean squared errors (RMSEs) for change point detection with and without refinement using the optimize function using 10

^{4}simulations.

Event | Breakpoints without refinement | Optimize to refine | Multi-Scale wavelets to refine |
---|---|---|---|

Receipt Start | 1.0 | 0.14 | 0.83 |

Receipt Stop | 5.3 | 0.22 | 1.7 |

Ship Start | 3.9 | 0.23 | 1.0 |

Ship Stop | 2.0 | 0.15 | 0.81 |

^{4}simulations. Notice that the optimize option leads to very good (low) RMSE and the multiscale wavelet option is also quite effective, without having been tuned to this change detection problem (see the Supplemental Information). Using optimize, the start of the receipt was estimated as t = 499, 500, or 501 (the true was 500 in all 10

^{4}simulations) with relative frequency 0.003, 0.995, and 0.002, respectively, so the RMSE is very small (0.14). The RMSE value of 1.7 for the receipt using multi-scale wavelets to refine the initial estimate from breakpoints can probably be reduced by more careful tuning of the multiscale wavelet approach (see the Supplemental Information). Estimation behavior without and with refinement was similar for the shipment. Qualitatively, we expect lower RMSE at the start of the receipt and at the end of the shipment because relative errors have smaller impact at the lower volumes.

## 5. Conclusion

_{jk}approximately follow a known distribution that can be assessed using simulation as we did, a simple outlier test can identify the w

_{jk}associated with change regions. Haar wavelets performed extremely well at finding change regions, much better, for example, than the lag-one-differencing option in terms of breakdown, which is the additive and relative error standard deviation at or above which the method fails to find the correct number of change regions. Therefore, all methods we consider first locate an approximate change “region” using wavelet coefficients. Then, we use breakpoints to find the approximate start and stop index in the change region and recommend refining the estimates from breakpoints by isolating each breakpoint and applying optimize as shown in the Supplemental Information. Another refinement method that uses multi-scale wavelets was also evaluated as shown in the Supplemental Information. Because of the excellent RMSE results resulting from using optimize to refine the initial estimates from breakpoints, we did not attempt to optimize the multi-scale wavelet refinement option, but leave such optimization to future work.

_{A}and σ

_{R}in Equation (1). Quantitatively, our key summary is the RMSEs in Table 1 where refinement of the each change point location initial estimate reduces the RMSEs.

## Conflict of Interest

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Longo, C.; Burr, T.; Myers, K.
Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards. *Axioms* **2013**, *2*, 271-285.
https://doi.org/10.3390/axioms2020271

**AMA Style**

Longo C, Burr T, Myers K.
Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards. *Axioms*. 2013; 2(2):271-285.
https://doi.org/10.3390/axioms2020271

**Chicago/Turabian Style**

Longo, Claire, Tom Burr, and Kary Myers.
2013. "Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards" *Axioms* 2, no. 2: 271-285.
https://doi.org/10.3390/axioms2020271