Statistically Derived Rate-of-Change Oil Analysis Limits and Alarms

Noria Corporation Drew Troyer, Noria Corporation
Tags: oil analysis

Using statistics to set oil analysis alarms and limits is a powerful and time-saving tool that allows the analyst to focus attention on those machines that are believed to be in trouble. Typically, this approach is used to set level limits, which act as flags or trip wires alerting the analyst to nonconforming results. Level limits simply mean setting a maximum or minimum acceptable level for oil analysis parameters such as iron, copper, lead, etc.

In setting statistically derived level limits, it is usually assumed that that the data is normally distributed (follows the familiar bell-shaped distribution curve), allowing caution and critical levels to be identified as a function of the mean and standard deviation of the data. For a large enough data set, this is generally the case. For more on using statistics to set level alarms, see “Use Statistical Analysis to Create Wear Debris Alarm Limits,” by Jonathon Sowers, in the November-December 2001 issue of Practicing Oil Analysis magazine.

This article illustrates how these same powerful statistical principles can be applied to the calculation of precise rate-of-change alarms, which can offer far greater sensitivity when interpreting historical oil analysis data.

Calculating Rate-of-Change Alarms
Rate-of-change, as the name implies, focuses not on the actual measured level of the parameter, but rather the rate at which the parameter is moving over a given period of time. With some work, rate-of-change analysis can be much more sensitive than level-based alarms because the analyst tracks movement of the target parameter relative to time, miles or cycles.

Rate-of-change analysis is an evaluation of the slope of the data (Figure 1). In this model, the slope is simply the rise divided by the run, or the change since the last reading.

For example, if a machine generates 100 ppm of iron during a 100-hour run period, the slope is one (1.0) and it can be said that the machine has an iron production rate of 1 ppm/hour. If the same machine generates 160 ppm of iron during the next 100-hour run period, the slope is 1.6, or an iron production rate of 1.6 ppm/hr. Similarly, if the iron level, for some reason decreases by 100 ppm during the next 100-hour run period, the slope is negative one (-1.0). So, slope calculations can be either positive or negative, depending on whether the measured parameter goes up or down.

The first stage in a statistical analysis is to calculate the mean. However, because the slope can be either positive or negative, this can create a problem when calculating the average using the standard formula for the arithmetic mean. To circumvent this problem, a slightly modified formula using the absolute value of the slope is required. To determine the absolute value of an observation, simply disregard the sign of the slope, so that a slope of +1.0 or -1.0 is considered to have the same absolute value of 1.0.

Using the absolute value to calculate the average and standard deviation seems intuitively unsettling to many people. The natural reaction is that the analyst is interested in determining if the target parameter is moving up or down, so the sign has meaning. However, keep in mind that statistical alarms are simply a mathematical tool, designed to help the analyst differentiate normal variations from an abnormal event. Once the alarm sounds, so to speak, the mathematical tool prompts one to investigate and determine the nature of the problem, including the direction of movement.

For the purpose of setting oil analysis alarms, it is also necessary to calculate the standard deviation from the target oil analysis parameter slope observations. While the mean defines what might be considered a normal reading for a given parameter, the standard deviation is simply a way of determining the probability that the parameter will vary from the arithmetic mean. Simply put, the larger the standard deviation, the wider the expected variation in the slope. A large standard deviation will result in a wider range of acceptable values, but with less precision as to what can be considered normal.

The calculation of the standard deviation for rate-of-change slopes observations is exactly like the standard formula except, again, one must use the absolute value of the observation.

Some machines will normally have increasing levels for some oil analysis parameters, such as the iron concentration in an unfiltered gearbox. In these instances, rate-of-change may be the only reasonable way to judge a machine’s condition because the level is in a constant state of flux. The statistically derived rate-of-change method accounts for this fluctuation. The table in Figure 2 identifies a clearly escalating iron level, which when applying level limits, may indicate cause for concern. However, the corresponding rate-of-change trend plot is relatively flat because the change per hour remains fairly stable at just under 0.1 ppm per hour, indicating that nothing significant has changed with this gearbox.

Figure 2

Going one stage further, apply the statistical model to calculate the lower and upper caution and critical limits based on the mean and standard deviation of the slopes. The limits in Figure 2 were set at ± one and two standard deviations from the mean for caution and critical respectively. As the figure illustrates, only the 1002- and 2240-hour readings are cautionary, and in fact only the 1002-hour reading is a high caution, indicating a potential problem.

Setting alarms at the one and two standard deviation level is very conservative. Setting alarms at ± two and three standard deviations for caution and critical respectively is much more liberal, and requires higher confidence in the data, typically as a result of a larger data set.

For statistical analysis, the larger the data set, the greater the precision to which limits can be set. The rule of thumb has always been that 30 observations is the point at which the sample begins to accurately estimate the population mean. Don’t wait until you have 30 observations to begin, simply recognize tht your confidence will grow as the data set becomes larger. For smaller data sets, a more sophisticated, but not terribly difficult, approach can be used to define the confidence interval using Student’s t tables. This is beyond the scope of this article, but can be applied in much the same way as the simple normal population approach outlined here.

Statistically derived rate-of-change oil analysis limits are very effective and easy to apply. Often, these alarms are more sensitive than simple level limits, but are most effective when used in conjunction with level limits. Here are some final tips for making this strategy work: