The AMVD exclusively uses the acceleration signals to carry out the distinction of (in)activity periods. A sliding window is applied over the signal in which the variance of the acceleration is computed. The figure of merit of the detection algorithm is computed as follows, (1) V ( n ) = 1 N ∑ k = 1 N ‖ a k − a ¯ n ‖ 2 < γwhere n is the frame at instant n, i.e., the content of the sliding window at instant n, a_k is the acceleration vector at instant k, a(x00304)_n is the mean of the acceleration of the frame at instant n, N is the length of the frame and γ is the predefined threshold that characterizes the decision based on the resultant value of the figure of merit.

The AMD is also solely based on the acceleration signals. The magnitude of the gravity acceleration vector is subtracted from the magnitude of the acceleration vector which is computed at every instant. The figure of merit used as the input of the classifier can be computed as (2) V ( n ) = 1 σ a 2 N ∑ k = 1 N ( ‖ a k ‖ − g ) 2 < γwhere g is the magnitude of the gravity acceleration (1 g or 9.8 m/s²) and σ a 2 is the variance of the acceleration signal noise that is used as a scaling factor to make the threshold less sensitive to noise.

On the other hand, the ARED, uses only the angular rate signals as the input. The squared magnitude of the angular rate vector at each instant is compared with a predefined threshold. This can be expressed in the following way (3) V ( n ) = 1 σ ω 2 N ∑ k = 1 N ‖ ω k ‖ 2 < γwhere ω_k is the angular rate vector at instant k and σ ω 2 is the variance of the angular rate noise signal, which is also used as a scaling factor.

The SHOD uses both acceleration and angular rate signals. Its goal is to increase the precision of the previous detectors by taking into consideration that there might be instants where human body movement presents angular rate but no acceleration and vice versa. The figure of merit used as the input of the classifier is (4) V ( n ) = 1 N ∑ k = 1 N ( 1 σ a 2 ‖ a k − g a ¯ n ‖ a ¯ n ‖ ‖ 2 + 1 σ ω 2 ‖ ω k ‖ 2 ) < γ

The FRD was developed by Veltink [17] as a preprocessing step for a simple classifier of Activities of Daily Life (ADL). The operating principle of the detector is very simple. The frame of the input signal is first high-pass filtered, then rectified and finally low-pass filtered. Therefore, its figure of merit can be expressed as (5) V ( n ) = LPE { RECT [ HPF ( n ) ] } < γIn their work the tangential acceleration is used as the input of the detector but, as seen later, the performance of the detector can be improved by using other inputs, such as the magnitude of the acceleration, the magnitude of the angular rate, or a linear combination of both.

The LTSD computes the Long Term Spectral Envelope (LTSE) of the signal. Let x(k) be the sensor signal which is segmented into frames with a certain degree of overlapping and X(l, n) its amplitude spectrum for the l band at frame n. The N-order long-term spectral envelope can be computed as (6) LTS E N ( l , n ) = max { X ( l , n + j ) } j = − N j = + NThe figure of merit used as the input of the classification process for each frame can be obtained by applying (7) V ( n ) = 10 log 10 ( 1 N FFT ∑ l = 0 N FFT − 1 LTS E 2 ( l , n ) N 2 ( l ) ) < γwhere N_FFT = 512 in our case is the resolution of the Fast Fourier Transform and N(l) is the average noise spectrum magnitude for the band l, (l = 0, 1,…, N_FFT − 1). For further information about the definition of the LTSD see [21].

The FSD is similar to the LTSD, but instead of computing the Long Term Spectral Envelope, it uses the spectrum of each frame in which the input signal is divided. Its expression is as follows (8) V ( n ) = 10 log 10 ( 1 N FFT ∑ l = 0 N FFT − 1 X 2 ( l , n ) N 2 ( l ) ) < γwhere again, N_FFT is the resolution of the Fast Fourier Transform, N(l) is the average noise spectrum magnitude for the band l and X(l,n) is the spectrum of the input signal for the band l at frame n.

The MBGT algorithm is based on computing the distance between two distributions, which are indirectly specified by means of two sample sets. Consider that we have a buffer which is filled with the last N sensor readings. Instead of splitting the sample buffer into two equal parts, and testing for difference between them, the MBGT algorithm considers all possible pairs of indices (i, j), such that 1≤i<j≤N, which split the sample frame into two adjacent windows α_ij−1 and α_j,N, where α_a,b = {x_a,x_b} is a window that contains all samples from index a to index b, i.e., the starting points of the first and second window respectively. Once we have divided the frame into two sub-windows, we can compute the average Euclidean distance between two points included in the pair (i,j) applying the following expression (9) C i , j = ∑ k = i j − 1 ∑ l = j N d k , l ( j − i ) ( N − j + 1 )where d_k,l is the Euclidean distance between points k and l.

The overall figure of merit of the detection algorithm is the maximum C_i,j computed over all the possible frame splits, which is (10) V MBGTD = max 1 ≤ i < j ≤ N C i , j < γ

Further information about the algorithm and about how to implement it so its complexity is suitable for practical applications can be found in [25].

The Memory-Based Cumulative Sum Detector is a variation of the well-known Cumulative Sum (CUSUM) algorithm first proposed in [26] and explained in depth in [27]. The CUSUM algorithm accumulates the log-likelihood of the current reading with respect to the distributions, before (p_θ0(x_i)) and after (p_θ1(x_i)), which is the hypothesized change point. The procedure is as follows (11) g k = S k − m k , S k = ∑ i = 1 k s i s i = ln p θ 1 ( x i ) p θ 0 ( x i ) , m k = min 1 ≤ j ≤ k S jThe change point will be detected when g_k ≤ h where h is an empirically set threshold.

However, the CUSUM algorithm can only be applied when both the distributions (p_θ0(x_i)) and (p_θ1(x_i)) are known. The MBCD solves this drawback by estimating both pre-change and post-change distributions via Parzen kernel density estimates [28] by using the following expression (included here forclarity), (12) p ⌢ θ ( x k ) = 1 N ( 2 λ 2 π ) 1 2 ∑ i = 1 N e 1 2 ( ‖ x 1 − x k ‖ / λ ) 2where λ stands for the standard deviation of the distribution and N is the total number of samples included in the frame.

Once we know how to estimate the distributions of each one of the sub-windows, we can proceed to compute the log-likelihood ratio as follows (13) S i . j = ∑ l = j N log 1 N − j + 1 ∑ k = j N w l , k 1 j − 1 ∑ k = i j − 1 w l , kwhere w_l,k is a kernel weight for the pair of samples (x_l, x_k) computed using Parzen's approximation.

The general figure of merit of the algorithm is (14) V MBCD = max 1 ≤ i < j ≤ N S i , j < γ

Further information about the algorithm and how to reduce its computational complexity can also be found in [25].

All the presented algorithms have been implemented following the structure which is explained by the steps mentioned below.

Finally, the following list clarifies the corresponding inputs and outputs of every presented algorithm:

Input:

At the start of the simulations we need to generate the synthetic signals and to that purpose we use the signal synthesizer. The signal synthesizer has been built to generate acceleration-like and angular rate-like signals coming from five different basic activities: walking, sitting on a chair and standing up, laying on a bed and standing up, running and jumping. Two more general activities have been implemented. The first one includes no acceleration and shows a constant angular rate and the second one includes no angular rate and shows a constant acceleration period. Although this may look like an unrealistic activity, there exist instants of time where this may happen. Thus, we have included them to ensure that the detector is as much robust as possible. The intensity of each activity, i.e., frequency, amplitude, and also the length of each activity period are set randomly every time the synthesizer is called. The sensing axis that we want to be parallel to the gravity vector can also be set. In addition, random noise is added to the signals in order to get a better approximation of real sensor signals. The magnitude of the acceleration is set to be 1g at every static instant. The acceleration and angular rate ranges can also be set according to the level of expected acceleration that will be present in the exercise that we are simulating, and also to simulate similar ranges to those of commercial MEMS inertial sensors.

Figure 2 shows the synthesized signals for the following activity sequence: walking, laying down-standing up, walking, sitting down-standing up, running, no angular rate, jumping, walking, laying down-standing up, no acceleration. The depicted binary activity marker is normalized to the signal magnitude to allow visibility.

Once the signal synthesizer is set, a Monte Carlo simulation of N repetitions can be performed. At every repetition a new set of signals is synthesized. Then, an optimization routine based on a grid search procedure is called for every algorithm. We opted for a grid search procedure since we observed that convergence of the objective functions depended highly on the initial values of the parameters to be optimized and this was causing the optimizers to stop in local minima that were far from the optimal values. The optimizer performs a sweep through the different parameters of each method, for example, window length and threshold for magnitude-based and memory-based methods and window length, frame shift and threshold for the spectrum-based methods. For every parameter configuration, the accuracy and correlation coefficient are computed. After the sweep, we extract the maximum values of the statistics and also store the value of the parameters for which they maximize. At the end of the Monte Carlo simulation, we obtain the average value of every statistic and the average value of the optimal configuration parameters for each one of the eight methods. Figure 3 depicts a diagram showing the steps to be followed during the theoretical simulation.

Spectrum-based and memory-based methods can be computed using different combinations of sensor inputs. We have used four different combinations: the magnitude of the acceleration; the magnitude of the angular rate; and the sum and product of both acceleration and angular rate magnitudes. Proceeding this way, we will be able to determine which of the sensor combinations offers the best performance.

Tables 1–6 show the average Accuracy and Correlation coefficient values, as well as the associated parameter values, for each one of the detection methods put into a Monte Carlo simulation of N = 500 runs. Therefore, all the eight methods have been tested using a set composed of 500 synthetic signals. Figure 4 shows the average Accuracy values obtained from the optimization procedure, when sweeping values of window size and threshold, searching for maximum accuracy.

Three male healthy subjects (179.33±4.04 cm, 72.33±7.09 kg, 25±1 years) wearing two Wagyromag units placed at the hip and the ankle respectively performed twice a circuit composed of the following activities: walk 20 m, stop, sit down-stand up, stop, run 20 m, stop, jump 5 times, stop, and lay downstand up. A total of 96 signals were gathered (3 acceleration axes + 3 angular rate axes)×2 IMUs×4 subjects× 2 runs) and used as the input for all detection algorithms.

The (in)activity markers were set manually by visually inspecting the gathered signals.

Like in the theoretical simulations, an analogous optimization procedure was carried out using the real dataset in order to obtain the average maximum Accuracy and Correlation coefficient values and their associated algorithm configuration parameters. By doing this we aimed to verify those results previously obtained from the theoretical simulation and check for possible differences. Tables 7–12 show the average values obtained when the detectors are applied to the 96 signals.

Figure 6 shows the average output of the optimization process when maximizing the accuracy and using the MBGTD. Figures 7 and 8 show the input (product of acceleration and angular rate magnitudes) and output (figure of merit) of the AMVD and the LTSD for a set of gathered signals when subject number 1 is following the activity circuit wearing the IMU at the ankle. Binary activity markers have been normalized to the input amplitude in order to allow visibility in the same plot.