Систематическая
ошибка – это смягченное выражение,
заменяющее слова «ошибка экспериментатора».
Систематические
ошибки остаются, как правило, постоянными
на протяжении всей серии опытов. Величина
их может быть и известной, и неизвестной
заранее. Например, курс шхуны «Пилигрим»8содержал неизвестную Дику Сэнду, но
известную Негоро систематическую
ошибку.
Систематические
погрешности могут быть обусловлены
различными причинами:
-
ограниченной
точностью изготовления прибора
(погрешностью прибора). Шкала линейки
может быть нанесена неточно (неравномерно);
взвешивание может производиться с
помощью неточных гирь; положение нуля
термометра может не соответствовать
нулевой температуре; капилляр термометра
может иметь разное сечение в разных
участках шкалы; стрелка амперметра
может не располагаться на нуле в
отсутствие электрического тока через
прибор; -
такие
ошибки часто возникают из-за того, что
реальная установка в чем-то отличается
от идеальной, или условия эксперимента
отличаются от предполагаемых теорией,
а поправки на это несоответствие не
делаются. Систематическая погрешность
возникает при измерении массы, если не
учитывается действие выталкивающей
силы воздуха на взвешиваемое тело и на
разновесы; при измерениях объема
жидкости или газа, если не учитывается
тепловое расширение; при калориметрических
измерениях, если не учитывается
теплообмен прибора с окружающей средой.
Другими примерами эффектов, которыми
может быть обусловлена обсуждаемая
ошибка, являются термо-ЭДС в контактах,
сопротивление подводящих проводов,
«мертвое» время счетчиков частиц; -
систематические
ошибки могут быть обусловлены также
неправильным выбором метода измерений.
Например, мы совершим такую ошибку,
определяя плотность какого-то материала
посредством измерений объема и веса
образца, если этот образец содержит
внутри пустоты, например, пузыри воздуха,
попавшие туда при отливке; -
мы
допускаем систематическую погрешность,
округляя численную величину до
какого-либо приближенного значения,
например, полагая π = 3, π = 3.1,
π = 3.14 и т. д. вместо π = 3.14159265…
При наличии скрытой
систематической погрешности результат,
приведенный с незначительной ошибкой,
будет выглядеть вполне надежным, хотя
на самом деле он является неверным.
Классическим
примером может служить опыт Милликена
по измерению элементарного электрического
заряда e. В этом
эксперименте требуется знать вязкость
воздуха. Милликен взял заниженную
величину вязкости и получил
e= (1.591 ± 0.002)∙10—
19Кл.
В настоящее же
время принято значение
e= (1.60210 ± 0.00002)∙10—
19Кл.
Долгое время
величины ряда других атомных констант,
таких, как постоянная Планка и число
Авогадро, базировались на значении
элементарного электрического заряда
e, полученном Милликеном,
и, следовательно, содержали ошибку,
превышающую 0.5 %.
Систематические
ошибкине поддаются математическому
анализу, и поэтому ихнужно выявить и
устранить. Если удается обнаружить
причину и найти величину сдвига (например,
вес вытесненного телом воздуха при
точном взвешивании), то систематическую
погрешность можно исключить введением
поправки к измеренному значению. Однако
общих рецептов и универсальных правил,
позволяющих обнаружить систематические
ошибки конкретного измерения, не
существует Выявление, оценка и устранение
таких ошибок требует опыта, догадки и
интуиции экспериментатора. Нужно
тщательно продумывать методику опытов
и придирчиво выбирать аппаратуру. Иногда
систематическую ошибку, обусловленную
измерительным прибором, можно уменьшить,
используя более точный прибор, желательно,
другого типа. Наиболее действенный
способ обнаружения систематических
ошибок – это сравнение результатов
измерений одной и той же величины,
выполненных принципиально разными
методами.
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Систематическая ошибка
- Систематическая ошибка
-
Систематическая ошибка [systematic error] — понятие математической статистики: ошибка, которая постоянно либо преувеличивает, либо преуменьшает результаты измерений (оценок наблюдаемых величин) вследствие воздействия определенных факторов, систематически влияющих на эти измерения и изменяющих их в одном направлении (например., в отличие от случайных ошибок). Оценки, лишенные систематических ошибок, называются несмещенными оценками.
Экономико-математический словарь: Словарь современной экономической науки. — М.: Дело.
.
2003.
Смотреть что такое «Систематическая ошибка» в других словарях:
-
систематическая ошибка — — [http://www.iks media.ru/glossary/index.html?glossid=2400324] систематическая ошибка Понятие математической статистики: ошибка, которая постоянно либо преувеличивает, либо преуменьшает результаты измерений (оценок наблюдаемых величин)… … Справочник технического переводчика
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систематическая ошибка — Systematic Error Систематическая ошибка Систематическая ошибка измерения это ошибка (погрешность) всегда только преувеличивающая или всегда только преуменьшающая результат измерения, стабильно, устойчиво искажающая его истинные значения.… … Толковый англо-русский словарь по нанотехнологии. — М.
-
Систематическая ошибка отбора — статистическое понятие, показывающее, что выводы, сделанные применительно к какой либо группе, могут оказаться неточными вследствие неправильного отбора в эту группу. Содержание 1 Ошибки отбора результатов 2 … Википедия
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Систематическая ошибка выжившего — (англ. survivorship bias) разновидность систематической ошибки отбора, когда по одной группе («выжившим») есть много данных, а по другой («погибшим») практически нет. Поэтому исследователи пытаются искать общие черты среди… … Википедия
-
систематическая ошибка (измерения) — вносить систематическую ошибку — [http://slovarionline.ru/anglo russkiy slovar neftegazovoy promyishlennosti/] Тематики нефтегазовая промышленность Синонимы вносить систематическую ошибку EN bias … Справочник технического переводчика
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систематическая ошибка смещения — — [Л.Г.Суменко. Англо русский словарь по информационным технологиям. М.: ГП ЦНИИС, 2003.] Тематики информационные технологии в целом EN bias error … Справочник технического переводчика
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Систематическая ошибка измерений, вызванная влиянием пола (sex bias in measurement) — С. о. и. имеет место в тех случаях, когда группы реагируют по разному на задания в тестах достижений, интеллекта или способностей, либо в др. измерительных инструментах, таких как опросники интересов. С. о. и., вызванная влиянием пола, имеет… … Психологическая энциклопедия
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Систематическая ошибка тестов, обусловленная культурными факторами (cultural bias in tests) — Между разными соц. и расовыми группами наблюдаются существенные различия в средних значениях оценок по стандартизованным тестам умственных способностей, широко применяемым при приеме в школы и колледжи, наборе в вооруженные силы и найме на работу … Психологическая энциклопедия
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систематическая ошибка результата (проверки) — 3.10. систематическая ошибка результата (проверки) Компонент ошибки результата, который остается постоянным или закономерно изменяется в ходе получения результатов проверки для одного признака. Примечание Систематические ошибки и их причины могут … Словарь-справочник терминов нормативно-технической документации
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СИСТЕМАТИЧЕСКАЯ ОШИБКА — См. постоянная ошибка. СИСТЕМАТИЧЕСКИЙ. 1. Характеризующийся отражением структуры и организационной целостности системы (1). 2. Определяемый конкретной теоретической системой (2). 3. В более широком смысле – организованный, предсказуемый,… … Толковый словарь по психологии
Systematic error occurs when an observed or calculated value deviates from the true value in a consistent way.
From: Woven Textiles, 2012
Experimental techniques
Yanqiu Huang, … Zhixiang Cao, in Industrial Ventilation Design Guidebook (Second Edition), 2021
4.3.3.2 Measurement errors
The measurement errors are divided into two categories: systematic errors and random errors (OIML, 1978).
Systematic error is an error which, in the course of a number of measurements carried out under the same conditions of a given value and quantity, either remains constant in absolute value and sign, or varies according to definite law with changing conditions.
Random error varies in an unpredictable manner in absolute value and in sign when a large number of measurements of the same value of a quantity are made under essentially identical conditions.
The origins of the above two errors are different in cause and nature. A simple example is when the mass of a weight is less than its nominal value, a systematic error occurs, which is constant in absolute value and sign. This is a pure systematic error. A ventilation-related example is when the instrument factor of a Pitot-static tube, which defines the relationship between the measured pressure difference and the velocity, is incorrect, a systematic error occurs. On the other hand, if a Pitot-static tube is positioned manually in a duct in such a way that the tube tip is randomly on either side of the intended measurement point, a random error occurs. This way, different phenomena create different types of error. The (total) error of measurement usually is a combination of the above two types.
The question may be asked, that is, what is the reason for dividing the errors into two categories? The answer is the totally different way of dealing with these different types. Systematic error can be eliminated to a sufficient degree, whereas random error cannot. The following section shows how to deal with these errors.
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EXPERIMENTAL TECHNIQUES
KAI SIREN, … PETER V. NIELSEN, in Industrial Ventilation Design Guidebook, 2001
12.3.3.11 Systematic Errors
Systematic error, as stated above, can be eliminated—not totally, but usually to a sufficient degree. This elimination process is called “calibration.” Calibration is simply a procedure where the result of measurement recorded by an instrument is compared with the measurement result of a standard. A standard is a measuring device intended to define, to represent physically, to conserve, or to reproduce the unit of measurement in order to transmit it to other measuring instruments by comparison.1 There are several categories of standards, but, simplifying a little, a standard is an instrument with a very high accuracy and can for that reason be used as a reference for ordinary measuring instruments. The calibration itself is usually carried out by measuring the quantity over the whole range required and by defining either one correction factor for the whole range, for a constant systematic error, or a correction curve or equation for the whole range. Applying this correction to the measurement result eliminates, more or less, the systematic error and gives the corrected result of measurement.
A primary standard has the highest metrological quality in a given field. Hence, the primary standard is the most accurate way to measure or to reproduce the value of a quantity. Primary standards are usually complicated instruments, which are essentially laboratory instruments and unsuited for site measurement. They require skilled handling and can be expensive. For these reasons it is not practical to calibrate all ordinary meters against a primary standard. To utilize the solid metrological basis of the primary standard, a chain of secondary standards, reference standards, and working standards combine the primary standard and the ordinary instruments. The lower level standard in the chain is calibrated using the next higher level standard. This is called “traceability.” In all calibrations traceability along the chain should exist, up to the instrument with the highest reliability, the primary standard.
The question is often asked, How often should calibration be carried out? Is it sufficient to do it once, or should it be repeated? The answer to this question depends on the instrument type. A very simple instrument that is robust and stable may require calibrating only once during its lifetime. Some fundamental meters do not need calibration at all. A Pitot-static tube or a liquid U-tube manometer are examples of such simple instruments. On the other hand, complicated instruments with many components or sensitive components may need calibration at short intervals. Also fouling and wearing are reasons not only for maintenance but also calibration. Thus the proper calibration interval depends on the instrument itself and its use. The manufacturers recommendations as well as past experience are often the only guidelines.
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Intelligent control and protection in the Russian electric power system
Nikolai Voropai, … Daniil Panasetsky, in Application of Smart Grid Technologies, 2018
3.3.1.2 Systematic errors in PMU measurements
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 ∑ aklξyl—mathematical expectation of random errors of the TE, equal to zero; ∑ aklcyl—mathematical expectation of systematic error of the TE, ωk—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.
It virtually does not exceed the threshold dk = 0.014 (a light horizontal line). Above the threshold, there is a curve of the TE discrepancy (a bold dotted line) that contains a measurement with a systematic error and a curve of nonzero mathematical expectation Δwk(t) ∈ [0.026; 0.03] (a black-blue thick line). However, the nonzero value of the calculated mathematical expectation of the TE discrepancy can only testify to the presence of a systematic error in the PMU measurements contained in this TE, but cannot be used to locate it.
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Measurements
Sankara Papavinasam, in Corrosion Control in the Oil and Gas Industry, 2014
ii Systematic or determinate error
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.
Fig. 5. Detection of a systematic error in the PMU measurements and identification of mathematical expectation of the test equation.
It virtually does not exceed the threshold dk = 0.014 (a light horizontal line). Above the threshold, there is a curve of the TE discrepancy (a bold dotted line) that contains a measurement with a systematic error and a curve of nonzero mathematical expectation Δwk(t) ∈ [0.026; 0.03] (a black-blue thick line). However, the nonzero value of the calculated mathematical expectation of the TE discrepancy can only testify to the presence of a systematic error in the PMU measurements contained in this TE, but cannot be used to locate it.
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Measurements
Sankara Papavinasam, in Corrosion Control in the Oil and Gas Industry, 2014
ii Systematic or determinate error
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 (EA) is a measure of the difference between the measured value (xi) and true or accepted value (xt) (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 (ER) is the ratio of measured value to true value and it is expressed as (Eqn. 12.6):
(Eqn. 12.6)ER=(xi−xtxt).100
Table 12.2 illustrates the absolute and relative errors for six measurements in determining the concentration of 20 ppm of an ionic species in solution.
Table 12.2. Relative and Absolute Errors in Six Measurements of Aqueous Solution Containing 20 ppm of an Ionic Species
Measured Value | Absolute Error | Relative Error (Percentage) | Remarks |
---|---|---|---|
19.4 | −0.6 | −3.0 | Experimental value lower than actual value. |
19.5 | −0.5 | −2.5 | |
19.6 | −0.4 | −2.0 | |
19.8 | −0.2 | −1.0 | |
20.1 | +0.1 | +0.5 | Experimental value higher than actual value. |
20.3 | +0.3 | +1.5 |
Systematic error may occur due to instrument, methodology, and personal error.
Instrument error
Instrument error occurs due to variations that can affect the functionality of the instrument. Some common causes include temperature change, voltage fluctuation, variations in resistance, distortion of the container, error from original calibration, and contamination. Most instrument errors can be detected and corrected by frequently calibrating the instrument using a standard reference material. Standard reference materials may occur in different forms including minerals, gas mixtures, hydrocarbon mixtures, polymers, solutions of known concentration of chemicals, weight, and volume. The standard reference materials may be prepared in the laboratory or may be obtained from standard-making organizations (e.g., ASTM standard reference materials), government agencies (e.g., National Institute of Standards and Technology (NIST) provides about 900 reference materials) and commercial suppliers. If standard materials are not available, a blank test may be performed using a solution without the sample. The value from this test may be used to correct the results from the actual sample. However this methodology may not be applicable for correcting instrumental error in all situations.
Methodology error
Methodology error occurs due to the non-ideal physical or chemical behavior of the method. Some common causes include variation of chemical reaction and its rate, incompleteness of the reaction between analyte and the sensing element due to the presence of other interfering substances, non-specificity of the method, side reactions, and decomposition of the reactant due to the measurement process. Methodology error is often difficult to detect and correct, and is therefore the most serious of the three types of systematic error. Therefore a suitable method free from methodology error should be established for routine analysis.
Personal error
Personal error occurs due to carelessness, lack of detailed knowledge of the measurement, limitation (e.g., color blindness of a person performing color-change titration), judgment, and prejudice of person performing the measurement. Some of these can be overcome by automation, proper training, and making sure that the person overcomes any bias to preserve the integrity of the measurement.
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Experimental Design and Sample Size Calculations
Andrew P. King, Robert J. Eckersley, in Statistics for Biomedical Engineers and Scientists, 2019
9.4.2 Blinding
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.
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The pursuit and definition of accuracy
Anthony J. Martyr, David R. Rogers, in Engine Testing (Fifth Edition), 2021
Systematic instrument errors
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.
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Power spectrum and filtering
Andreas Skiadopoulos, Nick Stergiou, in Biomechanics and Gait Analysis, 2020
5.10 Practical implementation
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, ωc 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
A systematic error is introduced to the signal if the correction factor is not applied. Therefore, remember to check any algorithm before using it. Let us check the correctness of the fourth-order low-pass filter that was built previously in R language. Vignette 5.2 contains the code to perform Winter’s (2009) low-pass filter in R programming language. Because the filter needs two past inputs (two data points) to compute a present filtered output (one data point), the time-series data to be filtered (the raw data) should be padded at the beginning and at the end. Additional data are usually collected before and after the period of interest.
Vignette 5.2
The following vignette contains a code in R programming language that performs the fourth-order zero-phase-shift low-pass filter from Eq. (5.37).
- 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.
- 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 (Ts=1/40 seconds) (i.e., 20 times the Nyquist frequency, fN=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.
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, Ain, 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 (f3dB=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.
The same procedure should be applied to check whether the output of the functions from closed source software used the correction factor or not. For example, using the library(signal) of the R computational software, if x is the vector that contains the raw data, then using butter() the Butterworth coefficients can be generated.
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Introduction to coal sampling
Wes B. Membrey, in The Coal Handbook (Second Edition), 2023
4.1.1 Definitions
The following definitions have been adapted from definitions given in the sampling standards.
Accuracy. A measure of the closeness of agreement between an analytical result and the true value or a reference value.
Cut. An increment taken by a sampling device typically from a conveyor belt, screen discharge, or other streams of coal.
Bias. Systematic error which leads to the average value of a series of analytical results being persistently higher or lower than the true value or a reference sample result.
Error. Difference between the measured value and the true value or a value from a reference sample result.
Increment. An amount of coal taken from a body of coal (a truck or barge, etc.) or from a stream of coal (coal on a conveyor, sizing screen or a chute, etc.) in a single operation of the sampling device.
Lot. Defined quantity of coal for which the quality is to be determined.
Particle top size is the nominal top size and is the square aperture size of the smallest sieve through which 95% of the sample passes.
Precision. A statistical term that quantifies how closely repeated experimental values agree. It usually has the value of the 95% confidence level, or 2 standard deviations from the mean of the experimental values.
Representative. A sample is representative when the sampling error, a combination of precision and accuracy, is of an acceptable level.
Sample. Quantity of coal with qualities that are representative of a larger mass (lot) for which the quality is to be determined.
Standard deviation. A measure of the spread of a set of values, equal to the square root of the variance of the results.
Sub lot. A part of a lot that is sampled and tested separately to the entire lot.
Tolerance. The maximum acceptable difference between measurement values or analytical results.
Variance. A measure of the spread of a set of values expressed as the square of the differences between the values and their mean.
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Sensors
Andrea Colagrossi, … Matteo Battilana, in Modern Spacecraft Guidance, Navigation, and Control, 2023
Quantization errors
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 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.
The same procedure should be applied to check whether the output of the functions from closed source software used the correction factor or not. For example, using the library(signal) of the R computational software, if x is the vector that contains the raw data, then using butter() the Butterworth coefficients can be generated.
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https://www.sciencedirect.com/science/article/pii/B9780128133729000051
Introduction to coal sampling
Wes B. Membrey, in The Coal Handbook (Second Edition), 2023
4.1.1 Definitions
The following definitions have been adapted from definitions given in the sampling standards.
Accuracy. A measure of the closeness of agreement between an analytical result and the true value or a reference value.
Cut. An increment taken by a sampling device typically from a conveyor belt, screen discharge, or other streams of coal.
Bias. Systematic error which leads to the average value of a series of analytical results being persistently higher or lower than the true value or a reference sample result.
Error. Difference between the measured value and the true value or a value from a reference sample result.
Increment. An amount of coal taken from a body of coal (a truck or barge, etc.) or from a stream of coal (coal on a conveyor, sizing screen or a chute, etc.) in a single operation of the sampling device.
Lot. Defined quantity of coal for which the quality is to be determined.
Particle top size is the nominal top size and is the square aperture size of the smallest sieve through which 95% of the sample passes.
Precision. A statistical term that quantifies how closely repeated experimental values agree. It usually has the value of the 95% confidence level, or 2 standard deviations from the mean of the experimental values.
Representative. A sample is representative when the sampling error, a combination of precision and accuracy, is of an acceptable level.
Sample. Quantity of coal with qualities that are representative of a larger mass (lot) for which the quality is to be determined.
Standard deviation. A measure of the spread of a set of values, equal to the square root of the variance of the results.
Sub lot. A part of a lot that is sampled and tested separately to the entire lot.
Tolerance. The maximum acceptable difference between measurement values or analytical results.
Variance. A measure of the spread of a set of values expressed as the square of the differences between the values and their mean.
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https://www.sciencedirect.com/science/article/pii/B9780128243282000169
Sensors
Andrea Colagrossi, … Matteo Battilana, in Modern Spacecraft Guidance, Navigation, and Control, 2023
Quantization errors
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.
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The Systems Approach to Control and Instrumentation
William B. Ribbens, in Understanding Automotive Electronics (Seventh Edition), 2013
Systematic Errors
One example of a systematic error is known as loading errors, which are due to the energy extracted by an instrument when making a measurement. Whenever the energy extracted from a system under measurement is not negligible, the extracted energy causes a change in the quantity being measured. Wherever possible, an instrument is designed to minimize such loading effects. The idea of loading error can be illustrated by the simple example of an electrical measurement, as illustrated in Figure 1.17. A voltmeter M having resistance Rm measures the voltage across resistance R. The correct voltage (vc) is given by
Figure 1.17. Illustration of loading error-volt meter.
(71)vc=V(RR+R1)
However, the measured voltage vm is given by
(72)vm=V(RpRp+R1)
where Rp is the parallel combination of R and Rm:
(73)Rp=RRmR+Rm
Loading is minimized by increasing the meter resistance Rm to the largest possible value. For conditions where Rm approaches infinite resistance, Rp approaches resistance R and vm approaches the correct voltage. Loading is similarly minimized in measurement of any quantity by minimizing extracted energy. Normally, loading is negligible in modern instrumentation.
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 1.17. Illustration of loading error-volt meter.
(71)vc=V(RR+R1)
However, the measured voltage vm is given by
(72)vm=V(RpRp+R1)
where Rp is the parallel combination of R and Rm:
(73)Rp=RRmR+Rm
Loading is minimized by increasing the meter resistance Rm to the largest possible value. For conditions where Rm approaches infinite resistance, Rp approaches resistance R and vm approaches the correct voltage. Loading is similarly minimized in measurement of any quantity by minimizing extracted energy. Normally, loading is negligible in modern instrumentation.
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.
In many automotive instrumentation applications, the bandwidth is purposely reduced to avoid rapid fluctuations in readings. For example, the type of sensor used for fuel-quantity measurements actually measures the height of fuel in the tank with a small float. As the car moves, the fuel sloshes in the tank, causing the sensor reading to fluctuate randomly about its mean value. The signal processing associated with this sensor is actually a low-pass filter such as is explained later in this chapter and has an extremely low bandwidth so that only the average reading of the fuel quantity is displayed, thereby eliminating the undesirable fluctuations in fuel quantity measurements that would occur if the bandwidth were not restricted.
The reliability of an instrumentation system refers to its ability to perform its designed function accurately and continuously whenever required, under unfavorable conditions, and for a reasonable amount of time. Reliability must be designed into the system by using adequate design margins and quality components that operate both over the desired temperature range and under the applicable environmental conditions.
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