Transcript accuracy

Lecturer: Hongqiangli
Email:[email protected]
Unit Thirteen
Measurement Errors and Accuracy
Basic Concepts and Terms
A measurable quantity is a property of phenomena, bodies, or substances that
can be defined qualitatively and expressed quantitatively. Measurable
quantities are also called physical quantities.
Measurement is the process of determinating the value of a physical quantity
experimentally with the help of special technical means called measuring
instruments.
The value of a physical quantity is the product of a number and a unit adapted
for these quantities.
The true value of a measurand is the value of the measured physical quantity,
which, being known, would ideally reflect, both qualitatively and quantitatively,
the corresponding property of the object.
We shall use the term uncertainty to characterize the inaccuracy of a
measurement result, whereas the term error is used to characterize the
components of the uncertainty.
Basic Concepts and Terms
The measurement error is the deviation of the result of measurement from the
true value of the measurable quantity, expressed in absolute or relative form.
If A is the true value of the measurable quantity and ˜A is the result of
measurement, then the absolute error of measurement is ζ = ˜A − A.
The absolute error is usually identified by the fact that it is expressed in the
same units as the measurable quantity. Absolute error is a physical quantity,
and its value may be positive, negative, or even given by an interval that
contains that value. One should not confuse the absolute error with the absolute
value of that error. For example, the absolute error −0.3 mm has the absolute
value 0.3.
The relative error is the error expressed as a fraction of the true value of the
measurable quantity ε = (˜A − A)/A. Relative errors are normally given as
percent and sometimes per thousand (denoted by ‰).
Measurement Error
测量误差
Measurement error may be defined as the difference between the true value
and the measured value of the quantity.
Systematic errors 系统误差
Random errors 随机误差
What Causes Measurement Errors?
Now that we know the types of measurement errors
that can occur, what factors lead to errors when we
take measurements? We can separate this category
into 2 basic categories: instrument and operator
errors.
instrument errors
仪器误差
operator errors
操作误差
Instrument Errors
Some basic information that usually comes with an instrument is:
accuracy
range
response time
sensitivity
accuracy - this is simply a measurement of how accurate is a measurement
likely to be when making that measurement within the range of the
instrument. For instance a mercury thermometer that is only marked off in
10th's of a degree can really only be measured to that degree of accuracy.
Instrument Errors
Some basic information that usually comes with an instrument is:
accuracy
range
response time
sensitivity
range - instruments are generally designed to measure values only within a
certain range. This is usually a result of the physical properties of the
instruments, such as instrument mass or the material used to make the
instrument. For instance a cup anemometer that measures wind speed has a
minimum rate that is can spin and thus puts a limit on the minimum wind
speed it can measure.
Instrument Errors
Some basic information that usually comes with an instrument is:
accuracy
range
response time
sensitivity
response time - if an instrument is making measurements in changing
conditions every instrument will take time to detect that change. This again is
often associated with the physical properties of the instrument. For instance a
mercury thermometer taken from room temperature and put into boiling
water will take some time before it gets to 100 oC. Reading the thermometer
too early will give an inaccurate observation of the temperature of boiling
water.
Instrument Errors
Some basic information that usually comes with an instrument is:
accuracy
range
response time
sensitivity
sensitivity - many instruments are have a limited sensitivity when detecting
changes in the parameter being measured. For instance some cup
anemometers, because of their mass cannot detect small wind speeds. The
problem gets the worse as the anemometer gets heavier.
Operator Errors
These errors generally lead to systematic errors and sometimes cannot be
traced and often can create quite large errors.
Example－－Measurement Location Errors
Data often has errors because the instrument making the measurements was
not placed in an optimal location for making this measurement. A good
example of this, is again associated with measurements of temperature. Any
temperature measurement will be inaccurate if it is directly exposed to the sun
or is not properly ventilated. In addition, a temperature device place too close
to a building will also be erroneous because it receives heat from the building
through radiation.
Quality Indicator: Precision of measurement
Precision is the degree of repeatability (or closeness) that repeated
measurements of the same quantity display, and is therefore a means of
describing the quality of the data with respect to random errors.
Quality Indicator: Accuracy of measurement
Accuracy is the degree of closeness (or conformity) of a measurement to
its true value.
Quality Indicator: Reliability of measurement
reliability=precision+accuracy
mean percentage error (MPE)
mean absolute percentage error (MAPE)
mean bias error (MBE)
mean absolute bias error (MABE)

1 k  H ic  H im
MPE   
100 
k i 1  H im


1 k  H ic  H im
MAPE  
100 


k i 1  H im

1 k
MBE   H ic  H im 
k i 1
1 k
MABE   H ic  H im
k i 1
root mean square error (RMSE).
k
RMSE 
 H
i 1
 H im 
2
ic
k
where Him is the ith measured value, Hic is the ith calculated value and n is the
total number of the observations.
Linear association
Correlation can be used to summarize the amount of linear association
between two continuous variables x and y
If there is a strong linear association between the two variables, then the
points lie nearly in a straight line, like this:
A positive association between the x and y variables (i.e. an increase in x is
accompanied by an increase in y) is shown by the scatterplot having a
positive slope. Similarly, a strong negative association (i.e. an increase in x
is accompanied by a decrease in y) is shown by points with a negative slope.
The strength of linear association, is summarized by the correlation coefficient,
defined as:
标准差
Example - Calculation of R for students' heights and weights
R=0.28
R=0.89
R=0.72
R=0.99
How about this ?
Engineers are increasingly being asked to monitor or evaluate the efficiency of
a process or the performance of a device.
1. Measurement Errors, Accuracy, and Precision
three kinds of errors
how to characterize measurements and instrumentation as being of high or
low precision
2. Estimating Measurement Uncertainty
multi-sample experiments
single-sample experiments
In this case we refer to an "uncertainty distribution" rather than a
"frequency distribution".
Frequently used words and phrases:
uncertainty
不确定性
true value
真值
recording errors
记录误差
accidental or random errors
measurement system
mean value
平均值
probability density function
distribution function
measurement error
测量误差
measured value
测量值
systematic or fixed errors 系统误差
随机误差
测量系统
frequency distribution
频率分布
概率密度函数
分布函数
discrete probability distribution
离散型概率分布
continuous probability densities
连续型概率密度
conditional probability 条件概率 Law of Large Numbers 大数定律
Central Limit Theorem 中心极限定律
1. Results are often derived from the combination of values determined
from a number of individual measurements.
be derived from…: 从…中得到
2. Unfortunately, every measurement is subject to error, and the degree to
which this error is minimized is a compromise between the (overall)
accuracy desired and the expense required to reduce the error in the
component measurements to an acceptable value.
be subject to…: 受…支配
compromise:妥协 折衷
3. Good engineering practice dictates that an indication of the error or
uncertainty should be reported along with the derived results.
dictate: 要求 规定
indication: 指标
4. Implicit in this assumption is that the worst-case errors will occur
simultaneously and in the most detrimental fashion.
implicit: 暗示
the worst-case error: 最大误差 detrimental: 有害的
5. A more realistic estimate of error was presented by Kline and
McClintock based on single-sample uncertainty analysis.
single-sample uncertainty analysis: 单样本不确定性分析
6. The following discussion is meant to provide an insight into
measurement uncertainty rather than a rigorous treatment of the
theoretical basis.
insight into…: 对…的洞察力或深入的理解
rigorous:严格的
7. The errors that occur in an experiment are usually categorized as
mistakes or recording errors, systematic or fixed errors, and accidental
or random errors.
be categorized as: 可分为
systematic or fixed errors: 系统误差
recording errors: 记录误差
accidental or random errors: 随机误差
8. Systematic errors may result from incorrect instrument calibrations
and relate to instrument accuracy (the ability of the instrument to
indicate the true value).
instrument calibration: 仪表刻度 instrument accuracy: 仪表精度
indicate: 指示 显示
9. Random errors cause readings to take random values on either side of
some mean value. They may be due to the observer or the instrument
and are revealed by repeated observations.
mean value: 平均值
reveal: 显现 显示
10. In measurement systems, accuracy generally refers to the closeness
of agreement of a measured value and the true value.
accuracy: 准确性
refer to: 指的是
11. All measurements are subject to both systematic (bias) and random
errors to differing degrees, and consequently the true value can only
be estimated.
differing degree: 不同程度
true value: 真值
12. To illustrate the above concepts, consider the case shown in Fig. 13-1,
where measurements of a fixed value are taken over a period of time.
illustrate: 举例说明
case: 例子 案例
13. If we further grouped the data into ranges of values, it would be
possible to plot the frequency of occurrence in each range as a
histogram.
grouped into: 按…分类
histogram: 柱状图 直方图
frequency of occurrence : 出现频率
14. Figure 13.3 is often referred to as a plot of the probability density
function, and the area under the curve represents the probability that a
particular value of x (the measured quantity) will occur.
probability density function: 概率密度函数
area: 面积
15. The total area under the curve has a value of 1, and the probability
that a particular measurement will fall within a specified range (e.g.,
between x1 and x2 ) is determined by the area under the curve
bounded by these values.
fall within: 落入
bounded by: 受…限制 以…为界
histogram
16. Figure 13-3 indicates that there is a likelihood of individual
measurements being close to xm , and that the likelihood of obtaining
a particular value decreases for values farther away from the mean
value, xm.
likelihood: 可能性
17. The frequency distribution shown in Fig. 13-3 corresponds to a
Gaussian or normal distribution curve, the form generally assumed to
represent random measurement uncertainty.
frequency distribution: 频率分布
Gaussian or normal distribution: 高斯分布 正态分布
18. There is no guarantee that this symmetrical distribution, indicating an
equal probability of measurements falling above or below the mean
value, xm , will occur, but experience has shown that the normal
distribution is generally suitable for most measurement applications.
guarantee: 保证 证明
symmetrical distribution: 对成分布
unsymmetrical: 非对称的
19. In analyzing these results we may apply standard statistical tools to
express our confidence in the determined value based on the
probability of obtaining a particular result.
statistical tool: 统计工具
confidence: 信心 信任
20. If experimental errors follow a normal distribution, then a widely
reported value is the standard deviation, σ. There is a 68% (68.27%)
probability that an observed value x will fall within ± σ of xm (Fig.
13-3).
standard deviation: 标准偏差 observed value: 观测值 测量值
21. In reporting measurements, an indication of the probable error in the
result is often stated based on an absolute error prediction [e.g., a
temperature of 48.3±0.1℃ (based on a 95% probability)] or on a
relative error basis [e.g., voltage of 9.0 V ±2% (based on a 95%
probability)].
indication: 指标
state: 表达 陈述
absolute error: 绝对误差
relative error: 相对误差
22. Based on the previous discussions we may characterize
measurements and instrumentation as being of high or low precision.
characterize as: 描述为…
23. The low-precision measurements have a wider distribution and are
characterized by a greater standard deviation, ±σlp, compared with
the high-precision measurements, ±σhp.
low-precision measurement: 低精度测量
high-precision measurement: 高精度测量
24. Therefore, in the absence of bias or systematic error, the mean of a
large sample of low-precision measurements theoretically indicates
the true value.
in the absence of: 缺少 mean: 平均值
a large sample: 大量样本
25. The previous discussion, illustrating the concept of random
measurement error, has not addressed the effects of systematic or
fixed (bias) errors.
address：指出
26. In reality, even though a high probability exists that an individual
measurement will be close to the mean value, there is no guarantee
that the value of the mean of the large sample of measurements will be
the true value (Fig. 13-5).
exist: 存在
27. An example of a source of systematic error is a pressure gauge needle
that is bent (i.e., not zeroed).
pressure gauge needle: 压力表指针
bent: 弯曲的 偏离的
zero: 指零
28. Multi-sample experiments are those in which, for a given set of the
independent experimental variables, the readings are taken many
times.
multi-sample experiment: 多样本实验 set: 系列
independent experimental variable: 独立测量变量
29. If we could repeat our tests many times, with many observers and a
variety of instruments, we could apply statistics to determine the
reliability of the results as in the previously discussed methods.
statistics: 统计学
30. Single-sample experiments are those in which, for a given set of
experimental conditions, the readings are taken only once. These are
typical in engineering, where financial or time constraints limit the
number of repetitions of a particular test.
single-sample experiment: 单样本实验
31. The experimenter also expresses the degree of confidence in the
stated uncertainty based on "odds," in a manner analogous to the
standard deviation.
odds: 几率 set: 系列
analogous to: 类似于