Engineering analysis: Data, uncertainty, errors

Overview

• Engineering analysis 3.1 focuses on data presentation, uncertainty, errors, and statistics.
• There are some suggested pre-requisites including, Algebra, functions, calculus

Learning Outcomes

• Presenting data with potential errors and understanding data errors/uncertainty, is crucial.
• Describe error types affecting measurements, understanding random variables, and how their variability is defined.
• Estimating expected errors via multiple data analysis or uncertainty models is also a main objective.
• An understanding 'probability density functions', estimating uncertainty for data points, and knowing how random variables combine mathematically are useful.
• Selecting error bars for plotted data and determining parameters using appropriate procedures are described.

Introduction to Data Uncertainty

• The uncertainty of a measured resistance serves as an example of data uncertainty in engineering.
• Measurements and calculations must be performed to determine the value of an unknown resistor using a voltage source, voltmeter, and ammeter, using Ohm's Law (V=IR).
• The individual uncertainties in current and voltage measurements directly impact the uncertainty in the calculated resistance value.

Sources of Errors and Uncertainty

• Several factors can affect the measurement of a quantity like current.
• Quantum noise arises from the quantum nature of current flow and exists because current is the flow of charge carried by electrons.
• There is inherent error in measuring current since the number of electrons is always an integer.
• Thermal noise comes from the material's thermal energy due to a resistor not being at absolute zero (0 Kelvin).
• Free electrons in the material are in constant random motion, even with no current flow.
• Electromagnetic interference affects measurements if stray fields cause unwanted currents in the resistor or ammeter.
• Quantisation (or digitisation) error arises when an ammeter converts an analogue current value to the nearest digital value.

Describing Errors (I) – Frequency Distributions

• Measured value can be thought of as having two components to estimate errors, the correct value and random error.
• There are two components to random error, first random error will be different for each measurement taken. The other is size and sign cannot be predicted.
• Randomness' still allows mathematical description/estimation of errors/uncertainty.
• An experiment had a true current value of 1 mAmp that is measured with two apparatuses.
• Apparatus A measurements were 1.05 mA, 1.13 mA, 0.97 mA, 1.02 mA, 0.95 mA
• Apparatus B measurements were 1.52 mA, 0.57 mA, 0.73 mA, 1.23 mA, 0.83 mA
• Estimate for current is best coming from A because of the smaller measurement values

Estimations of Variation of Data

• Quantitative estimates of variation can be made from a dataset
• Spread (or range), inter-quartile range, and standard deviation are three methods to estimate estimates of variation
• There are two approaches to calculating these frequency distribution and direct calculation from data values

Frequency Distribution

• A frequency distribution is estimated from a set of data values by counting the number of values that occur in a set of ranges
• The generated random numbers can be represented by dividing range into intervals.
• 'Bins' are what each range is sometimes described as.
• Plotting the frequency distribution is often referred to as a Histogram

Continuous and Discrete Random Variables

• To help understand data, it is helpful to understand that its values could be arbitrary values with a continuous range, or restricted to an integer number of values.
• Measurements of current are examples of continuous random variables because they can take any value.
• Discrete variables are variables such as the number of photons detected by a pixel in a camera.
• Frequency distribution can be estimated for discrete random variables similar to other numbers.

Key Points on Frequency Distribution

• The width of a frequency distribution will indicate if the variable is more uncertain. Because wider ranges of values are more uncertain. This would suggest higher uncertainty.
• Estimating uncertainty in variables becomes simply estimating the width of a frequency distribution

Probability Density Functions (PDFs)

• Besides histograms or bar charts, estimating uncertainty can sometimes be done via mathematical form for the shape of distribution.
• It is often appropriate to normalise the a frequency distribution, so it can be referred to as a probability density function, (PDF).

Continuous Random Variables

• For frequency histogram taken from data of a continuous random variable, dividing frequency by count by 𝑁∆ normalize it.
• Here, N is the amount of data values, and ∆ is the width of each bin.
• This transforms it to bars, where the area of each rectangle becomes an estimate for probability in that range, and the total area is one.

Discrete Random Variables

• Like continuous variables, frequency distribution can be normalized.
• Normalization comes from dividing each point by N, where N is the total number of data values.

PDF Properties

• The height of discrete points becomes estimate for the probability the variable is at that value; sum of all values becomes one.
• The sum over points gives an estimate for the the probability the variable lies between a pair of discrete values

Mode and Median

• For symmetrical PDF, mean value is the same as the mode and the mean.
• Mode is position of highest peak of PDF, the likely event.
• Median is the value where half the distribution area is either side.

Estimating Mode, Median and Mean From PDF

• Mode represents the peak position.
• Median should be the point where the area under PDF bisects in two.
• Mean is how easily mean value will be influenced by few relatively higher scores.

Mean of Continuous Variable X Formula

• The mean of x is determined by integral of "PDF(x) multiplied by integral of x".
• Calculating the mean in this way relies on adding values with values weighted by occurrence probability.

Calculating Functions Averaged

• This can be extended to calculate random means by using PDF(x) multiplied by integral of the square, and discrete forms where function of 'm' depends on PDF(m) multiplied by itself and summed.

Describing Errors III - Range, Interquartile Range and Standard Deviation

• Range of possible values are indicated by PDF of a random variable along with how likely different variable values are.
• The 'width' of the PDF may measure the uncertainity associating any individual data.
• Spread, interquartile range and Variance and Standard deviation are the many ways to look for these.

Spread or Range of Measurements

• Spread means the simplest measure of PDF's width that is taken to be its range over which it can be measured.

• A similar value is obtained by subtracting the smallest and largest numbers and taking their difference.

• Depending on only two data (largest and smallest values), range can be weak.

• Theoretical distributions often having no range as well, are known to extend out to infinity though probabilities are vanishingly smaller outside a certain range of values.

Interquartile Range

• Inter-quartile range, is the difference between the 25th and 75% quartiles
• The 25 percentile is the probability/area below and 75 percentile is vice versa.
• A given amount of random values are expected to occur between two ranges.
• An estimate of data is found by sorting from low to high, finding values a quarter and three quarters through the value list

Advantages and Disadvantages

• Advantage would be depending on data set. In addition of calculating for PDF.
• Exercise does have a weakness through.

Range Value Issues

• Interquartile range depends on dataset, though some points are known, it is thought there is some way it won't affect range.
• Estimations are weak.

(iii) Variance and Standard Deviation

• Using complete the PDF to form width measure is standard deviation that is root is square.
• Square Mean Deviations are Variances using brackets
• Regarding this as a constant value simplifies in the given expression.

PDF Forms and Variance/Uncertainty Calculations

• The forms for PDF can be Gaussian and Uniform

Uniform Distribution

• Distribution where PDF is flat and zero elsewhere can calculate from there.
• Distribution symmetrically around 0 (half values positive and half negative)

Quanitization Noise

• This is useful when quantifying values assigned from the nearest set values seperated by Delta V which error is like this.

Exercise 12

• A continuous random variable has PDF shown in a diagram. A Gaussian or Normal distribution.

Gaussian Distribution

• Deduction the Mean of distribution makes this PDF symmetrical, IQR indicates deviations easily.

Gaussian or Normal Distribution

• Defined where mean is distribution where sigma is the Standarad Deviation
• With this distribution, central limit theorem can also be used.
• Practically often you might need a combination of all sources/errors

Summary

• There are often sources of uncertainty
• Uncertainty is related to how much variance there is in a series if measurements to be taken.
• One can estimate both range and standard deviation though frequency distribution based on data.
• Standard Deviation is usually best width measure because it estimates the range.
• A standard deviation has integration though in some cases it is hard deduce.
• Guassian PDF has numbers in measurements with varying sources with limit thereom

Presenting Errors

• Data presentation depends from PDF/knowledge.

From set Data

A set amount measure to plot as a graph. Need to estimate with deviation of plot.

Averaging Values

• If error is averaging value as a sample. It is like standard estimate from the PDF and other.
• Combine multiple values and uncertainty to find estimate values.

Combination

Different variable numbers and uncertainty. How are values and uncertainty measured

• Rules and simple measure with this.
  1. Add with absolute variance.
  2. For Multiplications and Fraction.
  3. Power Result is special case.
• To unsure result always.

1. They need to only valid one. They value measure small.

Lines or Curve fitting

• Measurement though some alternative method for record voltage. The date can plot graph and value.

Leastsquares fit

• Normal way to do this following.
• Some first point line position.
• Find minimize to do this

Data Fit

With multiple values you can account weighted sum as well.

Conclusion

• Different weighted method to look at values. But this may vary with data points as well.

Engineering analysis 3.1 focuses on data presentation, uncertainty, errors, and statistics. It covers presenting data with potential errors, understanding data errors/uncertainty, and estimating expected errors via multiple data analysis or uncertainty models. The lesson also describes an understanding 'probability density functions'.