The energy cost of running depends upon speed, incline, running efficiency and body weight of the runner ^{(13)} . Other parameters like age, sex, and V̇O_{2max} are not significant ^{(7)} . However, incline and speed are very difficult to measure accurately by any method other than a treadmill or on a known circuit.

The treadmill has been used for several decades to study the physiology of running. It offers the advantage of allowing measurements under controlled laboratory conditions. Unfortunately, running on a treadmill is known to differ from a biomechanical and energetical point of view from running overground. First, air resistance is totally absent even though it can represent as much as 13% of the total expended energy ^{(17)} . Second, there are differences in the perception of running by the runner ^{(18)} . However, these differences can be considerably attenuated by training ^{(4)} . In addition, the perception and movements of the runner are affected by the size and the power of the treadmill ^{(6,10)} .

An additional problem occurring with the use of treadmill is that differences with running overground cannot be excluded. Nigg et al. ^{(16)} could only show one systematic difference: subjects planted their feet in a flatter position on the treadmill. Nelson et al. ^{(14)} observed that the amplitude of the vertical movement and the vertical speed were reduced on the treadmill. Elliott and Blansky ^{(4)} showed that at a speed above 4.8 m·s^{−1} , a decrease in stride length, an increase in stride rate, and therefore a decrease in the period of nonsupport could be observed on the treadmill.

All these differences between treadmill and overground running could have an impact on the energy cost of running. However, almost all the studies on the energy cost of running have been made in laboratory conditions. It would therefore be very useful to dispose of a tool allowing the calculation of incline and speed of a runner at a given moment and then to relate them with oxygen consumption measured by a portable device. The aim of this study was therefore to test whether accelerometric measurements could be used to accurately estimate speed and incline of overground running.

MATERIALS AND METHODS
Portable Measuring Device
The accelerometric device used in this study was developed by the Metrology Laboratory of the Swiss Federal Institute of Technology in Lausanne for a study on walking ^{(2)} . The only modification was to use a heel accelerometer which is calibrated up to 10 g (instead of 5 g).

Briefly, the device is composed of three accelerometers fixed orthogonally in the lower back by a Velcro elastic belt at the level L4-L5. These accelerometers measure anteriorposterior ("frontal") acceleration (f), vertical acceleration (v), and medial-lateral ("lateral") acceleration ^{(1)} . A fourth accelerometer measuring the frontal acceleration of the heel (h) is placed on the Achilles tendon just above the ankle.

Piezoresistive accelerometers (IC Sensor 3021, EuroSensors, London, U.K.) have been used. They have the advantage of being very accurate, miniature, solid, and inexpensive and exhibit a relatively wide frequency range (0-350 Hz) ^{(5)} . In addition, they are sensitive to constant acceleration. Thus, any modification of the position of the trunk with speed or incline should result in a modification of the gravity component in the recorded accelerations.

The four accelerometric signals resulting from the movements in the three dimension planes were recorded to determine the biomechanical parameters of human running that are necessary to obtain a good prediction of speed and incline. Gait cycle was determined by detection of the heel strike on the frontal acceleration of the heel.

Figure 1 shows a typical example of the acceleration signals obtained while running at 5.4 m·s^{−1} . As expected, a periodicity of one stride in the acceleration signal of the frontal and the vertical acceleration, of one gait cycle in the acceleration of the heel, and an inversion of the sign of the lateral acceleration at each stride can be seen.

Figure 1: Acceleration curves of a subject running on the level at 5.4 m·s^{−1} . The accelerations are given in gravitation (1 g = 9.81 m·s^{−2} ).

A spectral analysis of acceleration signals obtained during preliminary tests showed that there were no significant acceleration components above 17 Hz: over 80% of the signal's energy was under 17 Hz for all four accelerations. This is in agreement with previous work ^{(3)} . A 17-Hz low-pass filter was used to reduce the background noise.

The four signals were first amplified, then calibrated, low-pass filtered, digitized at 12 bits with a frequency of 200 Hz by a portable recorder weighing less than 300 g ^{(2)} and then stored on a 2 Mbytes SRAM memory card. The data were then transferred on a desktop computer for subsequent analysis.

Experimental Design
Voluntary subjects of normal body weight aged between 19 and 28 were studied (10 men and 10 women). All practiced sport at least once a week and most of them ran more than 10 km a week (Table 1) . Each subject gave written informed consent. The protocol was approved by the Ethical Commission of the Faculty of Medicine of the University of Lausanne.

TABLE 1: Characteristics of the subjects (mean ± SD).

Before beginning the measurements, the subjects warmed up by running for 5 min. This allowed them to get used to running while wearing the recording device at the belt. Each subject ran 18 times at various speeds (between 2.6 and 5.8 m·s^{−1} ) on tracks (80-100 m long) of different inclines: 9 different slopes ranging from −0.109 rad (−10.9%) to 0.109 rad (+10.9%). Speed was chosen by the subjects based on their physical fitness and was determined by measuring the time necessary to run the section (using a stop-watch). The subjects were asked to run as regularly as possible on the whole distance.

Of the 18 runs per subject, 12 were used to determine the relationship between the acceleration signals and speed or incline ("calibration runs"). The remaining six runs were used to test the accuracy of the predictions ("validation runs").

Data Analysis
Data analysis was done in three steps. First the accelerometric signals were described by numerical values (parameters). Then these parameters were used to calculate speed and incline with two artificial neural networks. Finally speed and incline were calculated also from the parameters using multiple regression analysis in order to simplify the method of analysis. The accuracies of these two methods of calculation were then compared.

Parameterization of accelerations. The accelerations were parameterized according to a system relying on both the physiological and statistical aspects of body acceleration. The heel acceleration was used to detect the moment when the heel touched the ground (heel strike). The gait cycle time was also calculated as the difference between successive heel strikes. The four acceleration profiles were then segmented, and the following parameters were calculated for each gait cycle: mean, median (med), variance (var), skewness (skew), and kurtosis (kur) for each acceleration (f, v, l, and h), covariance (cov) between each pair of accelerations, gait cycle time (gait), and maximal negative acceleration of the heel (hpeak).

The parameters varied with speed and incline. The correlation between these 28 parameters and speed or incline was calculated for each subject and averaged over all subjects. Ten parameters were then selected to serve as input for the neural networks. The selection was made according to their correlation with speed or incline and the accuracy of the neural networks' predictions. The parameters giving the best predictions for the regressions analysis were determined in a similar way.

Neural network structure. An artificial neural network is a group of interconnected processing elements (neurons) distributed in layers. Schematically, the neural network multiplies the input parameters by coefficients (the weights) to get an output parameter, in this case speed or incline.

Two phases can be distinguished in a neural network. First, a learning phase where in response to a given input and a given output (target), the network learns to self-adjust its weights. Second, a prediction phase, where the network applies its learned weights to the input, in order to generate the output.

The use of a neural network as an analyzing tool of gait is recent ^{(1,8,21)} . In this study, we used a two layers neural network similar to the one in our previous study on walking ^{(2)} . The inputs to each neural network were the parameters extracted for each gait cycle and the output parameter was the measured speed or incline. The learning rule to adjust the weights and biases of the neural network is based on the back-propagation algorithm.

For this study two neural networks were used, one for speed and one for incline. Twelve runs ("calibration runs") were used for the learning phase, i.e., the weights and the biases were adjusted so that the calculated speed or incline (output parameter) fitted optimally to actual speed or incline (target).

Six other runs ("validation runs"), which were not used in the learning phase, were employed to determine the quality of the predictions of the neural networks. So six speeds and inclines were calculated with the weights and biases obtained from the learning phase and then compared to the speeds and inclines measured.

Prediction analysis. For each validation run, speed, and incline have been calculated for each gait cycle using both the neural networks and multiple linear regression analysis technique. These speeds and inclines have then been averaged on all the gait cycles obtained for a given run. This has been repeated for the 120 runs measured in total (i.e., 20 subjects with 6 runs). The accuracy of the predictions was estimated in two ways: first, using the regression line between the measured and the calculated values (with the corresponding r^{2} ) and, second, using the square root of the mean square error (RMSE) = √Σ(y_{measure} −y_{calci} )^{2} /(n−1).

RESULTS
Parameter Analysis
Table 2 shows the coefficients of correlation between the 28 calculated parameters and speed or incline. Several parameters showed a strong correlation with speed, whereas for incline no parameter had a correlation coefficient higher than 0.5.

TABLE 2: Coefficients of correlation of the 28 parameters with speed and incline.

The parameters describing the frontal acceleration of the tibia were highly correlated with speed. This was expected as it is essentially the lower limbs that determine the speed of running. It is also worth noting that most parameters were also strongly correlated among themselves (data not shown). A stepwise regression analysis (Table 3) showed that almost 90% of the variability in speed was associated with four parameters: variance of the frontal acceleration of the heel (hvar), variance of the frontal acceleration (fvar), covariance between the vertical acceleration and the lateral acceleration (vlcov), and median of the frontal acceleration of the heel (vmed). One can notice that all the four measured accelerations are represented in these parameters.

TABLE 3: First 10 steps in a stepwise linear regression of the 28 parameters in function of speed; the partial r^{2} indicates the variation in speed explained by this parameter when the others are already in the model.

Correlations and stepwise regressions analysis assess only linear relationships. However, from visual inspection, it seems that there is only little departure from a linear relationship between the parameters and speed.

The correlation coefficients between incline and the various parameters were considerably lower than for speed. Therefore, a less accurate prediction for incline than for speed could be expected. However, these correlations can only describe linear relationships. Visual inspection of the plots of the parameters versus incline (data not shown) suggests that some parameters were associated with incline according to nonlinear relationships. A stepwise regression was also performed for incline (Table 4) . The partial r^{2} were very low (all below 0.25), suggesting a poor prediction of incline by multiple linear regression analysis.

TABLE 4: First 10 steps in a stepwise linear regression of the 28 parameters in function of incline; the partial r^{2} indicates the variation in incline explained by this parameter when the others are already in the model.

Neural Networks
The results of the parameter analysis were used to select the most appropriate input parameters for the neural networks. First, the parameters showing the highest correlation coefficients with speed (or incline) were tested. We also tried other configurations taking in account the correlation between the parameters. Second, we tested the first 10 parameters of the stepwise regression and finally we tested the 10 parameters used in our previous study of walking ^{(2)} . This analysis showed that the parameters previously used for walking were the one giving the best predictions. These parameters were fmed (median of the frontal acceleration), fvar, vmed, vvar (variance of the vertical acceleration), hmed (median of the frontal acceleration of the heel), fhcov (covariance between the frontal acceleration and the frontal acceleration of the heel), vhcov, lhcov (covariance between the lateral acceleration and the frontal acceleration of the heel), hpeak, and gait (gait cycle time).

To quantify the accuracy of the neural network to predict speed, the square root of the mean square error between measured and estimated speed (RMSE) was calculated for each individual. The regression coefficients between measured and estimated speed were calculated (as an index of bias) as well as the coefficient of determination r^{2} . Figure 2 shows the calculated vs measured speeds for the six validation runs of the 20 subjects. The predictions of the neural network were very accurate. The r^{2} was 0.965 and RMSE was 0.12 m·s^{−1} . The maximal difference between the measured speed and the calculated speed for the 120 tests was only 0.3 m·s^{−1} .

Figure 2: Relationship between calculated speed (by neural network) vs measured speed for 120 runs carried out by 20 subjects. The line represents the identity line.

To quantify the accuracy of the neural network to predict incline, the RMSE was calculated as well as the r^{2} . Figure 3 shows a graphical display of calculated incline versus measured incline for the six validation runs of the 20 subjects. The predictions were not as good as for speed: r^{2} was 0.937 and RMSE was 0.0142 rad (1.42% slope). The maximal error for the 120 runs was 0.038 rad (3.8% slope). Over 85% of the calculated inclines were within 0.02 rad (2% slope) of the measured incline.

Figure 3: Relationship between calculated incline (by neural network) vs measured incline for 120 runs carried out by 20 subjects. The line represents the identity line.

Based on an ANOVA we found that he prediction error for incline and speed was not associated with any of the measured characteristics of the subjects: age, height, weight, BMI, % body fat, number of training runs per week and the average distance run per week. It was also independent of speed or incline. (P values greater than 0.1).

The results shown above were obtained with neural networks that were calibrated with 2000 iterations. We next explored the effect of the number of iterations during the learning phase on the prediction error (Table 5) . As expected, r^{2} increased and the error decreased when the number of iterations increased. However, beyond 10,000 iterations, the prediction errors did not decrease further but became even worse.

TABLE 5: Error of the predictions for speed and incline for different numbers of iterations during the learning phase of the neural network.

To test the effect of sampling rate on the accuracy of the predictions, the sampling rate was decreased down to 40 Hz. This frequency is still high enough to detect frequencies up to the low-pass cut-off frequency of 17 Hz. The results were presented in Table 6 , where a significant increase in accuracy could be noted when the sampling rate increased. However, the original signal can be recorded at a sampling rate of 40 Hz and reconstructed at a sampling rate of 200 Hz, assuming that the original sampling did not violate the Nyquist sampling theory. The procedure consists of inserting new samples between the original data values by performing a low-pass interpolation ^{(9)} . As shown in Table 6 , the 200 Hz resampled signal gave almost the same results as the original signal sampled at 200 Hz.

TABLE 6: Error of the predictions for speed and incline for different sampling rates; "200 resampled" means that the recording was done at 40 Hz and then resampled by calculation at 200 Hz.

Multiple linear regression
The results obtained with the neural networks were compared with those obtained by multiple linear regression analysis using partial correlations. The three parameters that showed the best correlation with speed (or incline) by stepwise regression were used as independent variables. Adding additional parameters did not significantly improve the coefficient of determination. For the prediction of speed these parameters were hvar, fvar, and vmed. The comparison between predicted and measured speed gave a R^{2} of 0.96 and a RMSE of 0.14 m·s^{−1} . This RMSE is only 0.5% higher than that obtained with the neural network.

Incline was also predicted by multiple linear regression analysis. The best three parameters selected were lvar, lhcov, and vmean. Additional parameters did not significantly improve the coefficient of determination. The prediction power was much lower than with the neural network (R^{2} = 0.81 and RMSE = 0.0263 rad (2.63% slope).

DISCUSSION
Parameter Analysis
For the four accelerations considered, the variance had the highest linear correlation with speed (except for vertical acceleration where kurtosis showed a r slightly higher) (Table 2) . This means that the most important characteristic resulting from an increase in speed is a general increase of the amplitude of the body accelerations. Thus, to generate higher speed of movements, accelerations have to increase too. On the other hand, there is only a weak correlation between speed and mean, median or skewness for the accelerations measured in the lower back. This indicates that the symmetry of the acceleration signals is not modified with speed and that there is no change in the constant component (i.e., no change in posture).

The analysis of the correlation coefficients between the parameters and incline showed that the vmed belongs to the parameters with the highest correlations (Table 2) . This was expected as the vertical acceleration provides the energy needed to move upward. So, the steeper the incline, the more the median of the vertical acceleration will be shifted from zero. As a consequence, there will be a correlation between vmed and incline. On the other hand, there was no correlation between the fmed and incline. This suggests that there was no systematic change in the position of the trunk relative to vertical direction with changing incline. Otherwise, a component of gravity that induces a shift of the median value should have appeared in the frontal acceleration. As this was not the case, it is reasonable to infer that the vertical position of the trunk does not change with incline. Similar results were obtained by others ^{(11)} .

It can also be noted that some of the covariances between accelerations were correlated with incline, which was not always the case for speed. These associations had already been observed in our previous study on walking ^{(2)} .

Neural Networks and Multiple Regression
The neural networks allowed an accurate prediction of speed and incline: for speed, the RMSE was 0.12 m·s^{−1} , and for incline, the RMSE was 0.0142 rad (1.42% slope). The accuracy of these predictions is better than that previously observed for walking ^{(2)} . This is probably due to the calibration phase that, in previous work, was carried out on a treadmill rather than overground.

Considering the intrinsic error of measured speed, it seems difficult to expect better results with a protocol of this type. An error of 0.2 s in the time measured (by the observer) to run the section represents a relative error of 1.3%. The error due to the irregularity of speed of the subject during one run has also to be considered. This can be relatively high, because the subjects have no other indication on their speed than their own assessment. We tried to quantify this source of error for three subjects. We found a difference of almost 0.5 s (i.e., about 4%) between the time necessary to run the first half of the section versus the time necessary to run the second half. This error is not directly found in the prediction since we consider only the average speed which remains constant. Nevertheless, there is no doubt that it affects the calculation of speed, as speed is calculated for every gait cycle and then averaged on the whole section.

The prediction error decreases when the number of iterations increases but not beyond 10,000 iterations. This may be explained by the fact that the network reaches an optimum and then becomes too specific to the calibration runs, which then hinders best predictions for new runs. On the other hand, the small differences observed between the results obtained with 10,000 and 30,000 iterations can be due to a characteristic of the type of neural networks used: in the learning phase the initiation is done by the Nguyen-Widrow method which uses random numbers ^{(15)} . This means that these numbers will be different for every calculation and so logically one expects a small difference in the final biases and weights.

The number of iterations performed by the neural networks during the learning phase as well as the sampling rate of the signals are important factors determining the accuracy of the predictions of speed and incline. However, increasing the number of iterations and the sampling rate results in a rise in computation time. In addition, sampling rate has an effect on the amount of memory needed to store the signals. For instance, with 2 Mb of memory as we had in this study, at a sampling rate of 200 Hz, 22 min of measurements can be stored. Therefore a tradeoff exists between accuracy and cost in term of computation time and storage space. That is why iteration number and sampling rate have to be chosen carefully. A good compromise can be obtained with a sampling rate of 40 Hz resampled at 200 Hz and 2000 iterations. This allows almost 2 h of recording with a very high accuracy.

Multiple regression analysis also gave accurate predictions for speed. The relative increase in RMSE is 13% compared with the neural network but with a much simpler and shorter calculation method. So if maximal accuracy is not needed in the prediction of speed, a multiple regression should be adequate. It can, however, only be used if incline prediction is not needed since the error on incline almost doubled. RMSE was 0.0263 rad (2.63% slope) instead of 0.0149 rad (1.49% slope) rad with the neural network. This is likely to be due to the presence of curvilinear relationships between some parameters and incline. These relationships can be used by the neural network but not by the multiple linear regression analysis.

It is well known that a relationship exists between speed of running and the rate of energy expenditure. The magnitude of error involved in predicting energy expenditure while running from accelerometry alone remains to be explored. Obviously, the potential error in running speed determination will directly translate into an error of energy cost. Preliminary data obtained in walking subjects suggest that the uncertainty involved is around 15% for walking at a speed of 6.5 km·h^{−1} . We believe that, if these results are confirmed for running, such an error of prediction of energy expenditure from speed remains too high for research purposes. It seems more reasonable to actually measure energy expenditure while running with a portable indirect calorimeter. A number of noncalorimetric methods have been used to assess the rate of energy expenditure of various activities in free-living conditions, including heart rate monitoring ^{(19)} . Using minute-by-minute heart rate recording combined with individual regression lines between heart rate and energy expenditure, the relative error in the prediction of energy expenditure will depend upon several intrinsic and extrinsic factors in particular the stability of the regression line in a given individual. Intra-individual errors in the order of 10% have been found in 40 subjects ^{(12)} . Further studies should explore whether the combination of accelerometry with heart rate will yield more accurate estimate of energy expenditure than accelerometry alone.

In summary, the new technique presented here allows the assessment of speed of running and incline of terrain in overground conditions with little constraint from the subject. Assessment of running speed by accelerometry-or by other independent portable techniques utilizable in an unconstrained environment such as Global Positioning System ^{(20)} -can be related to the rate of energy expenditure measured by a portable indirect calorimeter, allowing to corroborate the internal validity of the vast number of experimental results previously obtained under laboratory conditions using the classical treadmill.

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