Систематическая ошибка измерения это

The systematic errors caused by the errors of the instrument transformers that exceed the class of their accuracy are constantly present in the measurements and can be identified by considering some successive snapshots of measurements. The TE linearized at the point of a true measurement, taking into account random and systematic errors, can be written as:

(25)wky¯=∑l∈ωk∂w∂ylξyl+cyl=∑aklξyl+∑aklcyl

where ∑ a_klξ_yl—mathematical expectation of random errors of the TE, equal to zero; ∑ a_klc_yl—mathematical expectation of systematic error of the TE, ω_k—a set of measurements contained in the kth TE.

The author of Ref. [28] suggests an algorithm for the identification of a systematic component of the measurement error on the basis of the current discrepancy of the TE. The algorithm rests on the fact that systematic errors of measurements do not change through a long time interval. In this case, condition (17) will not be met during such an interval of time. Based on the snapshots that arrive at time instants 0, 1, 2, …, t − 1, t…, the sliding average method is used to calculate the mathematical expectation of the TE discrepancy:

(26)Δwkt=1−αΔwkt−1+αwkt

where 0 ≤ α ≤ 1.

Fig. 5 shows the curve of the TE discrepancy (a thin dotted line) calculated by (26) for 100 snapshots of measurements that do not have systematic errors.

Fig. 5. Detection of a systematic error in the PMU measurements and identification of mathematical expectation of the test equation.

To define systematic error, one needs to understand ‘accuracy’. Accuracy is a measure of the closeness of the data to its true or accepted value. Figure 12.3 illustrates accuracy schematically.⁴ Determining the accuracy of a measurement is difficult because the true value may never be known, so for this reason an accepted value is commonly used. Systematic error moves the mean or average value of a measurement from the true or accepted value.

FIGURE 12.3. Difference between Accuracy and Precision in a Measurement.⁴

Reproduced with permission from Brooks/Cole, A Division of Cengage Learning.

Systematic error may be expressed as absolute error or relative error:

•

The absolute error (E_A) is a measure of the difference between the measured value (x_i) and true or accepted value (x_t) (Eqn. 12.5):

(Eqn. 12.5)EA=xi−xt

Absolute error bears a sign:

•

A negative sign indicates that the measured value is smaller than true value and

•

A positive sign indicates that the measured value is higher than true value

The relative error (E_R) is the ratio of measured value to true value and it is expressed as (Eqn. 12.6):

(Eqn. 12.6)ER=(xi−xtxt).100

Systematic errors can arise because either the participants or the researchers have particular knowledge about the experiment. Probably the best known example is the placebo effect, in which patients’ symptoms can improve simply because they believe that they have received some treatment even though, in reality, they have been given a treatment of no therapeutic value (e.g. a sugar pill). What is less well known, but nevertheless well established, is that the behavior of researchers can alter in a similar way. For example, a researcher who knows that a participant has received a specific treatment may monitor the participant much more carefully than a participant who he/she knows has received no treatment. Blinding is a method to reduce the chance of these effects causing a bias. There are three levels of blinding:

1.

Single-blind. The participant does not know if he/she is a member of the treatment or control group. This normally requires the control group to receive a placebo. Single-blinding can be easy to achieve in some types of experiments, for example, in drug trials the control group could receive sugar pills. However, it can be more difficult for other types of treatment. For example, in surgery there are ethical issues involved in patients having a placebo (or sham) operation.²

2.

Double-blind. Neither the participant nor the researcher who delivers the treatment knows whether the participant is in the treatment or control group.

3.

Triple-blind. Neither the participant, the researcher who delivers the treatment, nor the researcher who measures the response knows whether the participant is in the treatment or control group.

Typical systematic errors (Fig. 19.2C) include the following:

1.

Zero errors—the instrument does not read zero when the value of the quantity observed is zero.

2.

Scaling errors—the instrument reads systematically high or low.

3.

Nonlinearity—the relation between the true value of the quantity and the indicated value is not exactly in proportion; if the proportion of error is plotted against each measurement over full scale, the graph is nonlinear.

4.

Dimensional errors—for example, the effective length of a dynamometer torque arm may not be precisely correct.

As suggested by Winter (2009), to cancel the phase shift of the output signal relative to the input that is introduced by the second-order filter, the once-filtered data has to filtered again, but this time in the reverse direction. However, at every pass the −3dB cutoff frequency is pushed lower, and a correction is needed to match the original single-pass filter. This correction should be applied once the coefficients of the fourth-order low-pass filter are calculated. Nevertheless, it should be also checked whether functions of closed source software use the correction factor. If they have not used it, the output of the analyzed signal will be distorted. The format of the recursive second-order filter is given by Eq. (5.36) (Winter, 2009):

(5.36)yk=α0χk+α1χk−1+α2χk−2+β1yk−1+β2yk−2

where y are the filtered output data, x are past inputs, and k the kth sample.The coefficients α0,α1,α2,β1, and β2 for a second-order Butterworth low-pass filter are computed from Eq. (5.37) (Winter, 2009):

(5.37)ωc=tanπfcfsCK1=2ωcK2=ωc2K3=α1K2α0=K21+K1+K2α1=2α0α2=α0β1=K3−α1β2=1−α1-K3

where, ω_c is the cutoff angular frequency in rad/s, fc is the cutoff frequency in Hz, and fs is the sampling rate in Hz. When the filtered data are filtered again in the reverse direction to cancel phase-shift, the following correction factor to compensate for the introduced error should be used:

(5.38)C=(21n−1)14

where n≥2 is the number of passes. For a single-pass C=1, and no compensation is needed. For a dual pass, (n=2), a compensation is needed, and the correction factor should be applied. Thus, the ωc term from Eq. (5.37) is calculated as follows:

(5.39)ωc=tan(πfcfs)(212−1)14=tan(πfcfs)0.802

1.

The first step is to create a sine (or equally a cosine) wave with known amplitude and known frequency. Vignette 5.3 is used to synthesize periodic digital waves. Let us create a simple periodic sine wave s[n] with the following characteristics:

a.

Amplitude A=1 unit (e.g., 1 m);

b.

Frequency f=2 Hz;

c.

Phase θ=0 rad;

d.

Shift a₀=0 unit (e.g., 0 m).

Vignette 5.3

The following vignette contains a code in R programming language that synthesizes periodic waveforms from sinusoids.

Let us choose an arbitrary fundamental period T₀=2 seconds, which corresponds to a fundamental frequency of f0=1/T0=0.5 Hz. Now, knowing the fundamental frequency, the fourth harmonic that corresponds to a sine wave with frequency of f=2 Hz will be chosen. The periodic sine wave s[n] will be sampled at Fs=40 Hz (T_s=1/40 seconds) (i.e., 20 times the Nyquist frequency, f_N=2 Hz). The sine wave will be recorded for a time interval of t=2 s, which corresponds to N=80 data points. Thus, and because ω₀=2πf₀, we have:

s[n]=sin(2ω0nTs)

which means that the fourth harmonic has frequency f=2

 

Hz. Fig. 5.14A shows the sine wave created. The first and last 20 data points can be considered as extra points (padded). Additional data at the beginning and end of the signal are needed for the next steps because the filter is does not behave well at the edges. Thus, the signal of interest starts at 0.5

 

seconds and ends at 1.5

 

seconds, which corresponds to N=40 data points.

Figure 5.14. (A) Example of a low-pass filter (cutoff frequency=2 Hz) applied to a sine wave sampled at 40 Hz, with amplitude equal to 1 m, and frequency equal to 2 Hz. (B) The signal interpolated by a factor of 2, and filtered with cutoff frequency equal to the frequency of the sine wave (cutoff frequency=2 Hz). (C) Since the amplitude of the filtered signal has been reduced by a ratio of 0.707, the low-pass filter correctly attenuated the signal. The power spectra of the original and reconstructed signal are shown.

2.

An extra, but not mandatory, step is to interpolate the created sine wave in order to increase the temporal resolution of the created signal (Fig. 5.14B). Of course, when a digital periodic signal is created from scratch, like we are doing using the R code in the vignettes, we can easily sample the signal at higher frequencies. However, if we want to use real biomechanical time series data, that have already been collected, a possible way to increase its temporal resolution is by using the Whittaker–Shannon interpolation formula. With the Whittaker–Shannon interpolation a signal is up-sampled with interpolation using the sinc() function (Yaroslavsky, 1997):

(5.40)s(x)=∑n=0N−1αnsin(π(xΔx−n))Nsin(π(xΔx−n)/N)

The Whittaker–Shannon interpolation formula can be used to increase the temporal resolution after removing the “white” noise from the data. Without filtering, the interpolation results in a noise level equal to that of the original signal before sampling (Marks, 1991). However, the interpolation noise can be reduced by both oversampling and filtering the data before interpolation (Marks, 1991). An alternative, and efficient, method is to run the DFT, zero-pad the signal, and then take the IDFT to reconstruct it. Vignette 5.4 can be used to increase the temporal resolution by a factor of 2, which corresponds to a sampling frequency of 80 Hz.

Vignette 5.4

The following vignette contains a code in R programming language that runs the normalized discrete sinc() function, and the Whittaker–Shannon interpolation function.

3.

The third step is to filter the previously created sine wave with the fourth-order zero-phase-shift low-pass filter, setting the cutoff frequency equal to the sine wave frequency f=2 Hz. Vignette 5.2 is used for step 3. To cancel any shift (i.e., a zero-phase-shift filter) n=2 passes must be chosen. The interpolated signal has a sampling rate of 80 Hz.

4.

The fourth step is to investigate the frequency response of the filtered sine wave. The frequency response of the Butterworth filter is given by Eq. (5.41)

(5.41)|AoutAin|=1(1+fsf3dB)2n

where the point at which the amplitude response, Aout, of the input signal, s[n], with frequency, f, and amplitude, A_in, drops off by 3dB and is known as the cutoff frequency, f3dB. When the cutoff frequency is set equal to the frequency of the signal (f_3dB=f), the ratio should be equal to 0.707, since:

(5.42)|AoutAin|=12≈0.707

Fig. 5.14C shows the plots of the filtered and interpolated sine wave. Since the ratio of the maximum value of the filtered sine wave to the original sine wave ratio=0.707, the created fourth-order zero-phase-shift low-pass filter works properly. Without the correction factor the amplitude reduces nearly to half (0.51), indicating that the coefficients of the filter needed correction. Fig. 5.15 also shows an erroneously filtered signal. You can try to create Fig. 5.15 by yourself.

Figure 5.15. Example of a recursive low-pass filter applied to a sine wave with amplitude equal to 1 m and cutoff frequency equal to the frequency of the sine wave. Since the amplitude of the filtered signal has been reduced by a ratio of 0.707, the low-pass filter correctly attenuated the signal. However, the function without the correction factor reduced the amplitude by nearly one-half (0.51), indicating that the coefficients need correction.

Quantization error is a systematic error resulting from the difference between the continuous input value and its quantized output, and it is like round-off and truncation errors. This error is intrinsically associated with the AD conversion that maps the input values from a continuous set to the output values in a countable set, often with a finite number of elements. The quantization error is linked to the resolution of the sensor. Namely, a high-resolution sensor has a small quantization error. Indeed, the maximum quantization error is smaller than the resolution interval of the output, which is associated to the least significant bit representing the smallest variation that can be represented digitally:

LSB=FSR2NBIT

where FSR is the full-scale range of the sensor, and NBIT is the number of bits (i.e., the resolution) used in the AD converter to represent the sensor’s output. Quantization errors are typically not corrected, and the discrete values of the output are directly elaborated by the GNC system, which is designed to operate on digital values.

Fig. 6.9 shows a convenient model block to simulate quantization errors.

Figure 6.9. Quantization error model.

(71)vc=V(RR+R1)

However, the measured voltage v_m is given by

(72)vm=V(RpRp+R1)

where R_p is the parallel combination of R and R_m:

(73)Rp=RRmR+Rm

Another significant systematic error source is the dynamic response of the instrument. Any instrument has a limited response rate to very rapidly changing input, as illustrated in Figure 1.18. In this illustration, an input quantity to the instrument changes abruptly at some time. The instrument begins responding, but cannot instantaneously change and produce the new value. After a transient period, the indicated value approaches the correct reading (presuming correct instrument calibration). The dynamic response of an instrument to rapidly changing input quantity varies inversely with its bandwidth as explained earlier in this chapter.

Figure 1.18. Illustration of instrument dynamic response error.