• Tidak ada hasil yang ditemukan

The thresholds adaptive probability assignment function

IV. Weld Defect Detection via Sensor Fusion under Uncertainty

4.3 Proposed probability assignment

4.3.2. The thresholds adaptive probability assignment function

Probability distribution functions were estimated by using good-welded signals. The best-fitted probability distribution function was selected in accordance with the lowest value of NlogL(Negative log likelihood). The best-fitted probability distribution function is divided into three intervals corresponding the probability of 2.5%(lower control limit), 50%(midpoint), 97.5%(upper control limit) respectively.

The probability assignment was conducted based on the generated thresholds from the training process.

0 𝐡𝑒𝑙(𝐴) 𝑃𝑙𝑠(𝐴) 1

Belief of 𝐴 (Support interval)

Belief of ¬𝐴 (unsupported interval) Uncertain interval

Plausibility interval

Figure 4-7 A best-fitted probability distribution function of plasma intensity signal

We assumed that the value between lower and upper control limit tends to be the normal state based on the estimated probability model. If current sensor signal level is larger than the upper control limit or smaller than the lower control limit, the probability of being a normal state decreased. Chen and Aicklein (Chen and Aickelin, 2006) implemented an anomaly detection system to evidence theory.

They adopted standard sigmoid function to their anomaly detection algorithm to analyze the Wisconsin Breast Cancer Dataset (WBCD), which is widely used dataset for benchmarking of fault detection methods (Goldstein and Uchida, 2016). They utilized the function as a basic probability assignment function(activation function).

We defined a modified version of the dual sigmoid function(equation (8) and (9)) to control the slope of the probability function for gradual changes around the thresholds. The exact equation of the proposed adaptive probability assignment function is:

π‘€π‘π‘Žπ‘“(𝐻) = {(1 + β…‡π‘ π‘™π‘œπ‘π‘’πΆπ‘œπ‘’π‘“π‘“βˆ—(π‘‘π‘Žπ‘‘π‘Žπ‘π‘œπ‘–π‘›π‘‘βˆ’π‘ˆπΆπΏ))βˆ’1 , π‘‘π‘Žπ‘‘π‘Žπ‘π‘œπ‘–π‘›π‘‘ β‰₯ π‘šπ‘–π‘‘π‘π‘œπ‘–π‘›π‘‘

(1 + β…‡ βˆ’π‘ π‘™π‘œπ‘π‘’πΆπ‘œπ‘’π‘“π‘“βˆ—(π‘‘π‘Žπ‘‘π‘Žπ‘π‘œπ‘–π‘›π‘‘βˆ’πΏπΆπΏ))βˆ’1, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ (8)

where

π‘ π‘™π‘œπ‘π‘’πΆπ‘œπ‘’π‘“π‘“ =π‘ˆπΆπΏ + 𝐿𝐢𝐿

π‘ˆπΆπΏ βˆ’ 𝐿𝐢𝐿 (9)

It illustrated that the probability of a hypothesis which is a current state is normal. A slopeCoeff controls and adjusts the proposed probability assignment function by changing the slope of the graph

Plasma intensity (V)

Probability density (%)

Midpoint UCL

LCL

near the thresholds. The difference between upper and lower control limit is getting larger, slopeCoeff is getting larger, which result in the steeper slope of the probability function nearby the upper and lower control limits. A simple comparison of the probability assignment function with respect to the different slopeCoeff is illustrated in Figure 4-8. It illustrated how the coefficient adjusts and controls the slope of the probability assignment functions.

Figure 4-8 A comparison of proposed probability assignment function with respect to the magnitudeCoeff: (a) 1, (b) 3, (c) 5 , (d) 20

Figure 4-9 An example of the adaptive probability assignment function of upper control limit:

23.4V, lower control limit: 11.7V, and midpoint:17.3V Plasma intensity (V)

Probability of normal state

LCL Midpoint UCL

(a)

(b)

Traditionally, a binary decision approach based on the statistical analysis is shown in the Figure 4-9 (a). It implies that the decision model classifies the current weld status as a defect state if the current voltage is out of the thresholds range between the LCL and UCL. However, the binary categorical classification was unable to handle the uncertainty in terms of the in-tolerance failure problem.

To solve the problem, we defined a new probability assignment function by assigning the gradually decreasing probability of normal state nearby the thresholds. A graphical illustration of the probability assignment function with thresholds(in this case of the plasma signal, upper control limit: 23.4V, lower control limit: 11.7V, and midpoint:17.3V) is shown in Figure 4-9 (b).

The proposed probability assignment function was quite reasonable to consider the uncertainty around the thresholds. We further improved the proposed probability assignment function by adding the information of the estimated probability density function from chapter 4.3.1. The weighted summation of the two probability information(The estimated probability density function and The proposed probability assignment function) was used to assign the probability of the normal state. The weight(Ξ±) was selected as an arbitrary value from the pre-experiments. The weight decided how much the amount of information from the estimated probability density function would be adjusted to the proposed probability assignment function. By applying the weight, the probability assignment function contains richer information in terms of skewness of the original training data.

Figure 4-10 A comparison of the adjusted probability assignment function with respect to the different weights(Ξ±): β€œΞ±=0” implies that there is no further information of the estimated

probability density function Plasma intensity (V)

Probability of normal state

The detailed equation for the adjusted probability assignment function is described as equation (10).

π‘€π‘ π‘’π‘›π‘ π‘œπ‘Ÿ(𝐻) = 𝛼 βˆ— 𝑔𝑓𝑖𝑑(𝐻) + (1 βˆ’ 𝛼) βˆ— π‘€π‘π‘Žπ‘“(𝐻) (10)

Further, we have to assign a value to the set {𝐻, ¬𝐻} to express the ignorance of the sensor and the possibility that might be uncertain to determine whether the current state is normal or not;

𝑀(𝐻, ¬𝐻) β†’ [0, 1].

Figure 4-11 A general guideline to define basic probability assignments of dual threshold interval(a) and modified probability assignment of 𝑴(𝑯, ¬𝑯)

A general guideline to define the 𝑀(𝐻, ¬𝐻) is shown in Figure 4-11(a)(Siaterlis and Maglaris, 2004). The intuition behind this β€˜general guideline’ is that the decision beliefs should be treated with an uncertainty at the transient state even though the thresholds provide a quite certain decision.

From the Siaterlis and Maglaris research, we also modified the guidelines for assigning the ignorance as shown in Figure 4-11(b). The level of uncertainty given by 𝑀(𝐻, ¬𝐻) gradually increases or decreases at the transient state of the graph(see Figure 4-11(b)), denoting the sensor's uncertainty related to the value of 𝑀(𝐻).

We followed the equation that 𝑀(𝐻) + 𝐻(¬𝐻) + 𝐻(𝐻, ¬𝐻) = 1 to assign a probability of 𝐻(¬𝐻).

Algorithm 2. Threshold adaptive probability assignment Require:

𝑒𝑐𝑙𝑆 = π‘’π‘π‘π‘’π‘Ÿ π‘π‘œπ‘›π‘‘π‘Ÿπ‘œπ‘™ π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ π‘ π‘’π‘›π‘ π‘œπ‘Ÿ 𝑆 𝑙𝑐𝑙𝑆= π‘™π‘œπ‘€π‘’π‘Ÿ π‘π‘œπ‘›π‘‘π‘Ÿπ‘œπ‘™ π‘™π‘–π‘šπ‘–π‘‘ π‘œπ‘“ π‘ π‘’π‘›π‘ π‘œπ‘Ÿ 𝑆 π‘šπ‘π‘†= π‘šπ‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ π‘ π‘’π‘›π‘ π‘œπ‘Ÿ 𝑆

π‘₯𝑆𝑖= π‘–π‘‘β„Ž π‘‘π‘Žπ‘‘π‘Žπ‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ π‘ π‘’π‘›π‘ π‘œπ‘Ÿ 𝑆

π‘ π‘™π‘œπ‘π‘’πΆπ‘œπ‘’π‘“π‘“π‘†= π‘šπ‘Žπ‘”π‘›π‘–π‘‘π‘’π‘‘π‘’ π‘π‘œπ‘’π‘“π‘“π‘–π‘π‘–π‘’π‘›π‘‘ π‘œπ‘“ π‘ π‘’π‘›π‘ π‘œπ‘Ÿ 𝑆 𝛼𝑆= π‘€π‘’π‘–π‘”β„Žπ‘‘ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ π‘ π‘’π‘›π‘ π‘œπ‘Ÿ 𝑆

𝑀𝑆𝑖(𝐻) = π‘π‘Ÿπ‘œπ‘π‘œπ‘ π‘’π‘‘ π‘π‘Ÿπ‘œπ‘π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ π‘‘π‘–π‘ π‘‘π‘Ÿπ‘–π‘π‘’π‘‘π‘–π‘œπ‘› π‘Žπ‘ π‘ π‘–π‘”π‘›π‘šπ‘’π‘›π‘‘ π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘› π‘œπ‘“ π‘ π‘’π‘›π‘ π‘œπ‘Ÿ 𝑆 π‘€π‘–π‘‘β„Ž π‘–π‘‘β„Ž π‘‘π‘Žπ‘‘π‘Žπ‘π‘œπ‘–π‘›π‘‘ π‘€β„Žπ‘’π‘› β„Žπ‘¦π‘π‘œπ‘‘β„Žπ‘’π‘ π‘–π‘  𝐻 𝑖𝑠 π‘‘π‘Ÿπ‘’π‘’

1: for S = 1 : number of sensors

2: for i = 1 : number of data point on a stitch 3: if π‘₯𝑆𝑖β‰₯ π‘šπ‘π‘† then

4: 𝑀𝑆_π‘π‘Žπ‘“_𝑖(𝐻) = 1

1+exp(π‘ π‘™π‘œπ‘π‘’πΆπ‘œπ‘’π‘“π‘“π‘†βˆ—(π‘₯π‘†π‘–βˆ’π‘ˆπΆπΏπ‘†))

5: else

6: 𝑀𝑆_π‘π‘Žπ‘“_𝑖(𝐻) =1+exp(βˆ’π‘ π‘™π‘œπ‘π‘’πΆπ‘œπ‘’π‘“π‘“1 π‘†βˆ—(π‘₯π‘†π‘–βˆ’πΏπΆπΏπ‘†))

7: end if

8: 𝑀𝑆_π‘π‘Žπ‘“_𝑖(𝐻) = 1

1+exp(π‘ π‘™π‘œπ‘π‘’πΆπ‘œπ‘’π‘“π‘“π‘†βˆ—(π‘₯π‘†π‘–βˆ’π‘ˆπΆπΏπ‘†))

9: 𝑀𝑆𝑖(𝐻) = π›Όπ‘†βˆ— 𝑔𝑓𝑖𝑑(π‘₯𝑆𝑖|πœƒ) + (1 βˆ’ 𝛼𝑆) βˆ— 𝑀𝑆_π‘π‘Žπ‘“_𝑖(𝐻) 8: if 𝑀𝑆𝑖(𝐻) β‰₯ 0.5 then

9: 𝑀𝑆𝑖(𝐻, ¬𝐻) = 1 βˆ’ 𝑀𝑆𝑖(𝐻) 10: else

11: 𝑀𝑆𝑖(𝐻, ¬𝐻) = 𝑀𝑆𝑖(𝐻) 12: end if

13: 𝑀𝑆𝑖(βˆ…) = 0

14: 𝑀𝑆𝑖(¬𝐻) = 1 βˆ’ 𝑀𝑆𝑖(𝐻) βˆ’ 𝑀𝑆𝑖(𝐻, ¬𝐻) 15: end for

16: end for

Unduh sekarang "In-Process Laser Weldi..."

Garis besar

Dokumen terkait