Referring to FIG. 7, the stage S2 of FIG. 5 may comprise sub-stages S2_1, S2_2 and S2_3. At S2_1, at least a p1-percentile P1 and a p2-percentile P2 of the sample data distribution is calculated, wherein p1 and p2 are given percentages, wherein either, in an upper control limit case, p1<p2 and p1≥70%, or, in a lower control limit case, p1>p2 and p1≤30%. At S2_2, the method may comprise calculating an edge evaluation measure based on at least one difference of percentiles P2− P1. At S2_3, a control limit is calculated based on a sum of a ps-percentile Ps of the sample data distribution and the edge evaluation measure, wherein ps is a given percentage.

The left portion and middle portion of FIG. 8 show a sample data distribution that is far from a normal distribution. For example, the sample data distribution may represent the measured position of a throttle valve during CVD (chemical vapor deposition) fabrication of semiconductor chips. However, many sampling units or sample data sources other than a throttle valve position sensor are possible. Sample data can be generated (e.g. as measured raw data and/or preprocessed measured raw data) by any sampling units, and the information provided by the sample data can be arbitrary and may encompass a wide variety of industries or technologies. The sample data represents the one or more process parameters, for example, by being based on or indicative of measured parameter values that occur during the process operation.

Optional preprocessing of measured raw data (e.g. sensor data) may include, for example, evaluating the raw data for characteristics such as mean, sigma, range, minimum, maximum, and so forth. The sample data is then the characteristics rather than the measured raw data (e.g. sensor data).

No specific assumptions or a-priory information about the sampling unit(s), which provide for the sample data, and/or the sample data are required or used. Further, the sample data may be provided by more than one sampling unit, resulting, e.g., in sample data having systematic errors and/or different bias (e.g. offsets). As shown in the left portion and middle portion of FIG. 8, the sample data distribution may, e.g., be a multimodal and/or (slightly) time variant distribution.

UCL and LCL, as shown in FIG. 8, are calculated using the edge progression evaluation method described herein. Conventional 6-sigma UCL and LCL calculated using equation 1 with X=6 would be far away from the distribution due to an overblown sigma. These conventional limits are not suitable in practice as control limits for this type of sample data distribution.

The cumulative sample data distribution is shown on the right side of FIG. 8. As mentioned above, percentiles may be defined by percentages p11, p12, p13 (lower limit case) and pu1, pu2, pu3 (upper limit case) of the normal distribution with s spacing, for example.

In general, equations 3.1 to 3.3 may be written as

CL(EEM)=Ps+D×EEM (equation 5)

wherein EEM is an edge evaluation measure. In equations 3.1 to 3.3, the EEM is calculated by a difference of (only) two percentiles. As mentioned above, scaling may be set by, e.g., D=4.

However, more than two percentiles may be used for constituting the EEM (i.e. the EEM for the upper edge case and/or the EEM for the lower edge case). For example, two differences of percentiles, namely ΔP12 and ΔP23, may be used for edge evaluation at each edge. Further, a weighting between ΔP12 and ΔP23 may be used. For example, the EEM may be expressed by

EEM=A×ΔP12+(1−A)×ΔP23 (equation 6)

with A being a weight between 0 and 1. Here, ΔP12 is referred to as the “inner” edge percentile difference and ΔP23 is referred to as the “outer” edge percentile difference (both for the upper edge case and the lower edge case). The use of two or more percentile differences for the evaluation of the edge region(s) of a sample data distribution allows for more sensitive and meaningful evaluation results.

For example, p1, p2, p3 may be chosen such that ΔP23/ΔP12=1 for a normal distribution.

A may be a constant value at least for sample data distributions meeting the condition 1/x≤(ΔP23/ΔP12)≤x, with x being equal to or greater than 5 or 10. Here, A may be the same constant value for the upper edge case and the lower edge case.

For example, A may be chosen to be approximately 0.37. It has been shown empirically by testing several hundred different distributions that evaluating the edge slope with a weight of A≈0.37 (e.g., A=0.37±20%, or ±15%, or ±10%) yields optimum control limits with respect to a tradeoff between occurrence of false alarms and too early control actions.

FIG. 8 illustrates a specific example of a (non-normal) sample data distribution with (ΔPu23/ΔPu12)=0.4 and (ΔP123/ΔP112)=0.1. Thus, the sample data distribution meets the condition 0.1≤(ΔP23/ΔP12)≤10 both for the upper and lower edge cases. For example, the control limits UCL and LCL (see middle part of FIG. 8) may then be calculated in accordance with equations 5 and 6 by

UCL=Pu2+D×(0.37×ΔPu12+(1−0.37)×ΔPu23)

LCL=P12+D×(0.37×ΔP112+(1−0.37)×ΔP123). (equation 7)

In other cases, for sample data distributions having “extreme” edge progressions meeting the condition (ΔP23/ΔP12)<1/x or (ΔP23/ΔP12)>x, with x being equal to or greater than, e.g., 5 or 8 or 10 (with 10 being the example used above), it has been found that A should no longer be chosen as a constant value. Rather, the control limit computation is improved by choosing A as a non-constant function of (ΔP23/ΔP12), written as

A
⁡
(

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

)

.

Differently stated, it has been shown empirically by testing several hundred different sample data distributions that if (ΔP23/ΔP12) is outside a range between a lower and an upper threshold, e.g. 0.1 and 10, the weight A becomes a function of the ratio itself. Equations 5 and 6 can then be written as

CL
⁡
(
EEM
)

=

Ps
+

D
×

(

A
⁡
(

Δ
⁢
P
⁢
23

Δ
⁢
P
⁢
12

)

×
Δ
⁢
P
⁢
12

 
+
 

(

1
-

A
⁡
(

Δ
⁢
P
⁢
23

Δ
⁢
P
⁢
12

)

)

 
×
Δ
⁢
P
⁢
23

)

.

(

equation
⁢

8

)

The function

A
⁡
(

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

)

starts at A (i.e. at the constant A which is, e.g., at 0.37) for

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

=

x
⁡
(

e
.
g
.

⁢
10

)

.

It then decreases to, e.g., about 0.15 at

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

=

2
⁢
0

to stay constant again thereafter, i.e. at about 0.15 for

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

>

2
⁢
0

,

for example.

In other words, it has been shown empirically by testing several hundred distributions that optimum control limits with respect to a tradeoff between too many false alarms and too early control actions can also be calculated by equations 5 and 6 for sample data distributions which are far off from normal distributions (e.g. even outside the interval

0.

1
≤

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

≤

1
⁢
0

)

.

Calculation of control limits for those sample data distributions can be done simply by gradually lowering the value of A (e.g. 0.37) in an intermediate range of

1
⁢
0

≤

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

≤

2
⁢
0

and then keeping A constant at a lower value (e.g. 0.15) again for

Δ
⁢
P
⁢
2
⁢
3

Δ
⁢
P
⁢
1
⁢
2

>

2
⁢
0

,

for example.

In addition, it has been shown that a minimum sample size of, e.g., about 100 sample data values is recommendable for the “edge evaluation method” as disclosed herein. As the method, by construction, evaluates only about 30% of the sample data, the minimum sample size may be higher than in other SPC methods with minimum sample sizes of 20-30. It has been shown empirically by testing several hundred different distributions of small sample sizes that the calculated control limits can be improved for smaller sample sizes (e.g., between 100 and 1000 sample data values) if D is optionally defined as a function of sample size n, i.e., as D(n).

FIG. 9 illustrates a number of different edge progression types of sample data distributions. The edge progression types are referred to as NORMAL, STEEP, STEP, SHARP, NEEDLE and CONSTANT.

The edge progression type NORMAL represents an edge progression of distributions which are at least somewhat similar to normal distributions with respect to their edge shape. Such distributions are characterized by ΔP12≠0 and ΔP23≠0 and ΔP10≠0, wherein P0 equals the sample data value at 50% of the cumulative sample data distribution, i.e. the median of the sample data distribution.

The NORMAL edge progression type subdivides into three subcases:

The first subcase is a Gaussian-like sample data distribution. As already mentioned, this case is characterized by ΔP12≈ΔP23≈ΔP10≈s. In this subcase the control limits may be calculated by equations 4.1 to 4.3 for the upper control limit and by equations 4.4 to 4.6 for the lower control limit or by equations 5, 6 and 7.

The second subcase is the baseline subcase which includes non-Gaussian-like sample data distributions still meeting the edge progression condition 0.1≤(ΔP23/ΔP12)≤10. The control limits can be calculated by equations 4.1 to 4.3 for the upper control limit and by equations 4.4 to 4.6 for the lower control limit or by equations 5, 6 and 7. In this case A is a constant weight of, e.g., 0.37.

The third subcase is the extreme subcase, in which (ΔP23/ΔP12) is smaller than a lower threshold (e.g. 0.1) or greater than an upper threshold (e.g. 10). The control limits can be calculated by equation 8 (with A being a function of (ΔP23/ΔP12)).

The characterization of sample data distributions in terms of their edge progression by calculating differences between percentiles and characterizing edge region(s) of the sample data distribution on the basis of the calculated differences reveals edge types in which at least one difference of percentiles is equal to (approximately) 0. These edge types are characterized by:

- STEEP: ΔP12=0, ΔP23≠0, ΔP10≠0
- STEP: ΔP12=0, ΔP23=0, ΔP10≠0
- SHARP: ΔP12≠0, ΔP23=0, ΔP10≠0
- NEEDLE: ΔP12≠0, ΔP23≠0, ΔP10=0, ΔP23/ΔP12>5
- CONSTANT: ΔP12=0, ΔP23=0, ΔP10=0.

The calculation of control limits for these types of edge progressions is not considered herein. Empirically, it has been found that the vast majority of practical sample data distributions are of the type NORMAL, while only a small portion of sample data distributions are of the edge types STEEP, STEP, SHARP, NEEDLE or CONSTANT. These non-NORMAL sample data distributions may be handled in other ways that are not part of this disclosure. They can be identified by the edge progression evaluation method and handled adequately thereafter.

FIG. 10 is a flowchart illustrating stages of an exemplary computer-implemented method of monitoring a process including edge progression characterization. FIG. 10 illustrates the discrimination between the edge progression type NORMAL and the edge progression types STEEP, STEP, SHARP, NEEDLE and CONSTANT.

At S1 a sample data distribution is obtained, compare S1 of FIGS. 5 and 7.

Classification of the edge progression into edge progression types is done at COND by checking whether ΔP12≠0 and/or ΔP23≠0 and/or ΔP10≠0.

If the condition ΔP12≠0 and ΔP23≠0 and ΔP10≠0 is met (YES), the process proceeds to S2. Alternatively, if at least one of ΔP12, ΔP23 and ΔP10 is zero (NO), the control limits of these sample data distributions are not calculated or are calculated using other procedures not described herein (S3).

Differently stated, the edge progression condition ΔP12≠0 and ΔP23≠0 may, e.g., be used as a necessary condition for the method S1, S2 to be executed. That is, the method is carried out only if ΔP12≠0 and ΔP23≠0.

FIG. 11A is a diagram representing a multimodal sample data distribution over time. The Y-axis represents the measured sample data values (in arbitrary units). The X-axis represents the time, e.g. in units of hours or days or weeks etc. The sample data is collected by, e.g., three sampling units denoted by sampling units SU1, SU2 and SU3. As apparent from FIG. 11A, the sampling units SU1, SU2, SU3 deliver sample data with systematic offsets between each other. Furthermore, for example, sampling unit SU2 does not have the same activity period as sampling units SU1 and SU3, and sampling units SU1 and SU2 have an offset from a certain point in time, for example. Multiple sample data values from a given sampling unit at a given time are represented by data points arranged in the vertical direction.

In this example 7500 data points (sample data values) have been collected. The characterization of the upper edge of the sample data distribution reveals ΔPu23/ΔPu12=0.322, and the characterization of the lower edge of the sample data distribution reveals ΔP123/ΔP112=0.217. Hence, the exemplary sample data distribution belongs to the type NORMAL and the subcase “baseline” of FIG. 9.

The upper and lower control limits UCL, LCL are calculated with equation 7, i.e. with A=0.37, D=4 and the starting point percentile Ps=Pu2=97.77%-percentile for the upper control limit and Ps=P12=2.28%-percentile for the lower control limit.

The computation of control limits yields UCL=26.8 and LCL=13.3. These control limits compare to a conventionally calculated UCL of 34 and a conventionally calculated LCL of 4. Conventional calculation was based on calculating the mean value M and the standard deviation s, modelling the obtained sample data values by a normal distribution and applying equation 1 for upper and lower control limit calculation based on the “6 sigma limit” approach.

As apparent from FIG. 11A, the upper and lower control limits UCL, LCL calculated in accordance with the disclosure are closer to the sampling data values obtained by sampling units SU1, SU2 and SU3. As a result, these control limits allow earlier control actions to be taken without undue likelihood of false alarms. On the other hand, the conventionally calculated “6 sigma limits” are not suitable for SPC for the example sample data distribution.

FIG. 11B is a diagram showing a sample data distribution with a strong edge spread. As known in the art, it is difficult to obtain meaningful control limits from such data distributions. The conventional calculation of control limits (“6-sigma limits”) leads to UCL and LCL with a false alarm rate that is far too high.

In this example 117265 data points (sample data values) have been collected. The characterization of the upper edge of the sample data distribution reveals ΔPu23/ΔPu12=14.54, and the characterization of the lower edge of the sample data distribution reveals ΔP123/ΔP112=14.96. Hence, the exemplary sample data distribution belongs to the type NORMAL and the subcase “extreme” of