Comparison of methods for the automated anaerobic threshold determination
for by P.Kvaca[1]

This chapter is devoted to the comparison of the derived method with two other methods for the automated Anaerobic Threshold determination. Firstly, the background of these methods will be introduced. Then both methods will be mathematically described. Next, the comparison on the sample of 10 healthy men will be presented. Lastly the comparison of the V-Slope Method and derived method will be mentioned.

Automation Of the At Determination in Linear Regressive Model

Two methods for the comparison with the derived method were applied. Algorithms of these methods are derived from the widespread methodology employed in automated software systems for the exercise test evaluation. However, contrary to generally used procedure of the Anaerobic Threshold determination these algorithms remove the problem of physician experience during evaluation.

This problem consists in the request on a physician who has to divide measured points into the two groups according to his experience and his subjective opinion. (The problem of division in previous chapter was discussed in detail.) After the division two linear regressive lines are constructed. Their intersection is regarded as an anaerobic threshold. This construction is from the Figure 3.1 obvious.

Two criterions for the objective evaluation are required [1]:
  • [All combination of group division of measured points has to be executed.
  • The best solution is selected according to chosen criterion, which has to be unambiguous.

The first part of algorithm is same for both chosen criterions. The measured data into the two groups are divided gradually point by point. The calculation of regressive coefficient for two regressive lines is accomplished.

The first line has k points and the second one n-k points. The regressive coefficients of the first and second line from the equations are determined. The first group has two points and the second one has the rest of points during the first step of algorithm.

The next step is the derivation of line intersection. It is derived by the comparison of two equations. The x-coordinate and the y-coordinate from the equation are determined.

The set of solutions after the calculation of all combinations is obtained. Now the optimal solution could be selected. Two criterions are utilized [2]. The algorithms of these criterions in the next section are discussed. The essential condition for the evaluation, which stands for results of both methods, is non-acceptance of solutions situated beyond the first and the last point of measurement.

Figure: Model of Two Regressive Lines

Sum of Squared Errors Minimum

The sum of squared errors of the linear regressive model is applied in the first criterion. It is derived from the following deliberation. The linear regressive method interlaces all selected points with the minimal magnitude of the sum of squared errors. Consequently the magnitude for different divisions of measured points is diverse. Therefore the selection of the best group division can be accomplished with respect to the minimum of the sum of squared error. The procedure firstly calculates the differences between the regressive linear model and the actual measured points. Because of sign elimination each difference is squared. Then these squared differences are summed. Lastly the minimal sum of all division combinations is selected. The mathematical description in the equation is expressed. The equation search for what division k is the sum Sk minimal. This division is regarded as optimal and the determined intersection as an Anaerobic Threshold is considered. The method either for concave or convex function can be used.

Break Angle Maximum

The angle of two regressive lines is applied in the second criterion. The angle is denoted a in the Figure. The method is based on the deliberation that the break of ventilatory curve could be as maximal as possible. Therefore the selection of the best group division of measured points is accomplished according to the maximum of the angle containing linear regressive lines. The angle from the following procedure is detected. The angle containing line y1 and yP from the angle containing y2 and yP is subtracted. Mentioned lines have the angle included in their tangents B2 and B1 respectively. Hence arc cotangents of these tangents B2 and B1 can be used. Only the positive values of tangents are important from the physiological point of view. The negative values could have showed either the false group division or pathological reactions. The equation search for what division k is the angle a maximal. This division is regarded as an optimal and the determined intersection as an Anaerobic Threshold is considered. The absolute value of the difference is applied because of non-sensitivity for eventual negative signs. Then the method for the concave or for the convex function can be used.

Comparison Results

The comparison with the clinical study was carried out. The clinical study with volunteers aged from 15 to 43 years (average 23.3) and the following workload protocol was applied - firstly 5-minute workload 1 W/kg, then gradual increase of 0.25 to 0.5 W/kg each minute according to the physical constitution of the tested volunteer. The Anaerobic Threshold ended after a subjective patient’s distress or the breach of the selected cycle pace. The spiroergometric relation showing the exponential character [2] was selected because of testing objectivity. The reason is that the linear methods are independent of the trend character. They can work with the concave or convex function. However, relations correspond with the exponential regressive model only if they have the concave character. The second reason of the selection is the practical applicability of the parameter for the prescription of physical activity. The gold basis for the prescription is Heart Rate (HR). Therefore the relation Minute Ventilation as the function of Heart Rate was selected.


The Exponential Regressive Method shows the smallest variance. The band of determined Anaerobic Threshold's varies from 30 beats/minute. Contrary to linear regressive methods it shows on average the smallest Anaerobic Threshold values and the smallest Standard Deviations. The information about the sum of squared errors is important, because it informs us about the success during the regressive interlacing. The Exponential Regressive Method interlaces most excellent the selected spiroergometric relation than the other methods and ERM overcomes the Linear Regressive Model of any used criterion. The variance of ratio Anaerobic Threshold and HR maximum ranges from 72.78% to 84.65%, that means 12% bandwidth.

The Linear Regressive Method applying the Sum of Square Error Minimum criterion shows the greatest variance. The band of determined Anaerobic Threshold's varies from 70 beats/minute. This method has the largest Standard Deviation. The sum of squared errors is smaller than the second linear method - BAM, but the sum squared errors is greater then the ERM. The variance of ratio Anaerobic Threshold and HR maximum ranges from 56,67% to 97,19%, that means 40% bandwidth.

The second linear regressive method applying the criterion of Break Angle Maximum shows on average the greatest Anaerobic Threshold values and smaller Standard Deviations then the first linear method - SSEM. The band of determined Anaerobic Threshold's varies from 50 beats/minute. The greatest sum squared errors means the worst interlacing of regressive model, e.g. it is three-times grater then the Exponential Regressive Method. The variance of ratio Anaerobic Threshold and HR maximum ranges from 70,56 % to 95,74%, that means 25% bandwidth.


Both linear methods tend to solutions at limits of range (beginnings and ends). Exponential Regressive Method determines the Anaerobic Threshold lower than other methods and it does not tend to determine extreme values of the Anaerobic Threshold.


[1] Kvaca P., J. Radvansky: “Methods for Automated Anaerobic Threshold Determination.” Med Sport Boh Slov, 8/2 (1999), 51-57. [2] Kvaca, P., Radvansky., J., Cermak, M.: "Determinaton of Anaerobic Threshold from Spiroergometric Parameters - Method for Computer Implementation." Med. Sport. Boh. Slov., Vol. 7(1), pp. 14-19: 1998 ©2007