Statistical & Uncertainty Analysis UC Summer REU Programs Instructor: Lilit Yeghiazarian, PhD Environmental Engineering Program 1

Statistical & Uncertainty AnalysisUC Summer REU Programs

Instructor: Lilit Yeghiazarian, PhD Environmental Engineering Program

1

Instructor

Dr. Lilit Yeghiazarian

Environmental Engineering Program

Office: 746 Engineering Research Center (ERC)

Email: [email protected]

Phone: 513-556-3623

2

Textbooks

Applied Numerical Methods with MATLAB for Engineers and Scientists, 3rd edition, S.C. Chapra, McGraw-Hill Companies, Inc., 2012

An Introduction to Error Analysis:

The Study of Uncertainties in Physical Measurements, 2nd editions, J.R. Taylor, University Science Books, Sausalito, CA

3

Outline for today

Error numerical error data uncertainty in measurement error

Statistics & Curve Fitting mean standard deviation linear regression t-test ANOVA

4

Types of Error

General Error (cannot blame computer) :

Blunders human error

Formulation or model error incomplete mathematical model

Data uncertainty limited to significant figures in physical measurements

Numerical Error:

Round-off error (due to computer approximations)

Truncation error (due to mathematical approximations)

5

6

Gare Montparnasse, Paris, 1895

Accuracy and Precision

Figure 4.1, Chapra

(a) inaccurate and imprecise

(b) accurate and imprecise

(c) inaccurate and precise

(d) accurate and precise

Note: Inaccuracy = bias Imprecision = uncertainty

7

• Accuracy: how closely a computed/measured value agrees with true value

• Precision: how closely individual computed/measured values agree with each other

Accuracy and Precision

Figure 4.1, Chapra

(a) inaccurate and imprecise

(b) accurate and imprecise

(c) inaccurate and precise

(d) accurate and precise

Note: Inaccuracy = bias Imprecision = uncertainty

8

• Inaccuracy: systemic deviation from truth

• Imprecision: magnitude of scatter

Error, Accuracy and Precision

9

In this class we refer to Error as collective term to represent both inaccuracy and imprecision of our predictions

Round-off Errors

Occur because digital computers have a limited ability to represent numbers

Digital computers have size & precision limits on their ability to represent numbers

Some numerical manipulations highly sensitive to round-off errors arising from

mathematical considerations and/or performance of arithmetic operations on computers

10

Computer Representation of Numbers

Numerical round-off errors are directly related to way numbers are stored in computer

The fundamental unit whereby information is represented is a word

A word consists of a string of binary digits or bits

Numbers are stored in one or more words, e.g.,

-173 could look like this in binary on a 16-bit computer:

(10101101)2=27+25+23+22+20=17310

off “0” on “1”

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- Overflow Overflow +

1.797693134862316 x 10308-1.797693134862316 x 10308

2.225073858507201 x 10-308-2.225073858507201 x 10-308

15 significantfigures

15 significantfigures

As good as it gets on our PCs …

“Hole”on eitherside of

zero

0

Underflow

For 64-bit, IEEE double precision format systems12

Implications of Finite Number of bits (1)

Range Finite range of numbers a computer can represent

Overflow error – bigger than computer can handle For double precision (MATLAB and Excel): >1.7977 x 10308

Underflow error – smaller than computer can handle For double precision (MATLAB and Excel): <2.2251 x 10-308

Can set format long and use realmax and realmin in MATLAB to test your computer for range

13

Implications of Finite Number of bits (2)

Precision

Some numbers cannot be expressed with a finite number of significant figures, e.g., π, e, √7

14

Round-Off Error andCommon Arithmetic Operations

Addition Mantissa of number with smaller exponent is modified so both

are the same and decimal points are aligned Result is chopped Example: hypothetical 4-digit mantissa & 1-digit exponent computer 1.557 + 0.04341 = 0.1557 x 101 + 0.004341 x 101 (so they have same exponent)

= 0.160041 x 101 = 0.1600 x 101 (because of 4-digit mantissa)

Subtraction Similar to addition, but sign of subtrahend is reversed Severe loss of significance during subtraction of nearly equal

numbers → one of the biggest sources of round-off error in numerical methods – subtractive cancellation

15

Even though an individual round-off error could be small, the cumulative effect over the course of a large computation can be significant!!

Large numbers of computations Computations interdependent Later calculations depend on results of earlier ones

Round-Off Error andLarge Computations

16

Particular Problems Arising from Round-Off Error (1) Adding a small and a large number

Common problem in summing infinite series (like the Taylor series) where initial terms are large compared to the later terms

Mitigate by summing in the reverse order so each new term is comparable in size to the accumulated sum (add small numbers first)

Subtractive cancellation Round-off error induced from subtracting two nearly equal

floating-point numbers Example: finding roots of a quadratic equation or parabola Mitigate by using alternative formulation of model to minimize

problem 17

Particular Problems Arising from Round-Off Error (2) Smearing

Occurs when individual terms in a summation are > summation itself (positive and negative numbers in summation)

Really a form of subtractive cancellation – mitigate by using alternative formulation of model to minimize problem

Inner Products Common problem in solution of simultaneous linear algebraic

equations Use double precision to mitigate problem (MATLAB does this

automatically)

nni

n

ii yxyxyxyx

...22111

18

Truncation Errors

Occur when exact mathematical formulations are represented by approximations

Example: Taylor series

19

Taylor series expansionswhere h = xi+1 - xi

4)4(

3)3(

2"

'1 )(

!4

)()(

!3

)()(

!2

)())(()()( h

xfh

xfh

xfhxfxfxf iii

iii

3)3(

2"

'1 )(

!3

)()(

!2

)())(()()( h

xfh

xfhxfxfxf ii

iii

2"

'1 )(

!2

)())(()()( h

xfhxfxfxf i

iii

)()( 1 ii xfxf

))(()()( '1 hxfxfxf iii

0th

1st

2nd

3rd

4th

Taylor series widely used to express functions in an approximate fashion

Taylor’s Theorem: Any smooth function can be approximated as a polynomial

20

Each term adds more information:

Fig

ure

4.6

, Ch

ap

ra, p

. 9

3

= 1.2

e.g., f(x) = - 0.1x4 - 0.15x3 - 0.5x2 - 0.25x + 1.2 at x = 1

≈ 1.2 – 0.25(1) = 0.95

≈ 1.2 – 0.25(1) –(1.0/(1*2))*12 = 0.45

= 1.2 – 0.25(1) – (1.0/(1*2))*12

– (0.9/(1*2*3))*13 = 0.3

21

Total Numerical Error

Sum of round-off error and truncation error

As step size ↓, # computations round-off error (e.g. due

to subtractive cancellation or large numbers of computations)

truncation error ↓ Point of diminishing returns is when

round-off error begins to negate benefits of step-size reduction

Trade-off here Figure 4.10, Chapra, p. 104

22

Control of Numerical Errors Experience and judgment of engineer

Practical programming guidelines: Avoid subtracting two nearly equal numbers Sort the numbers and work with the smallest numbers first

Use theoretical formulations to predict total numerical errors when possible (small-scale tasks)

Check results by substituting back in original model and see if it actually makes sense

Perform numerical experiments to increase awareness Change step size or method to cross-check Have two independent groups perform same calculations

23

Measurements & Uncertainty

24

Errors as Uncertainties

Error in scientific measurement means the inevitable uncertainty that accompanies all measurements

As such, errors are not mistakes, you cannot eliminate them by being very careful

The best we can hope to do is to ensure that errors are as small as reasonably possible

In this section, words error and uncertainty are used interchangeably

25

Inevitability of Uncertainty

Carpenter wants to measure the height of doorway before installing a door

First rough measurement: 210 cm If pressed, the carpenter might admit that the

height in anywhere between 205 & 215 cm For a more precise measurement, he uses a

tape measure: 211.3 cm How can he be sure it’s not 211.3001 cm? Use a more precise tape?

26

Measuring Length with Ruler

27


28


29

Best estimate of length = 82.5 mmProbable range: 82 to 83 mmWe have measured the length to the nearest millimeter

Note: markings are 1 mm apart

How To Report & Use Uncertainties

Best estimate ± uncertainty In general, the result of any measurement

of quantity x is stated as

(measured value of x) = xbest ± Δx Δx is called uncertainty, or error, or margin

of error Δx is always positive

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Basic Rules About Uncertainty

Δx cannot be known/stated with too much precision; it cannot conceivably be known to 4 significant figures

Rule for stating uncertainties: Experimental uncertainties should almost always be rounded to one significant figure

Example: if some calculation yields Δx=0.02385, it should be rounded to Δx=0.02

31

Basic Rules About Uncertainty

Rule for stating answers: The last significant figure in any stated answer should usually be of the same order of magnitude (in the same decimal position) as the uncertainty

Examples: The answer 92.81 with uncertainty 0.3 should be

rounded as 92.8 ± 0.3 If the uncertainty is 3, then the answer should be

rounded as 93 ± 3 If the uncertainty is 30, then the answer should be

rounded as 90 ± 30 32

Propagation Of Uncertainty

33

Statistics & Curve Fitting

34

Curve Fitting Could plot points and

sketch a curve that visually conforms to the data

Three different ways shown:

a) Least-squares regression for data with scatter (covered)

b) Linear interpolation for precise data

c) Curvilinear interpolation for precise data

Figure P

T4.1,

Chapra

Curve Fitting and Engineering Practice

Estimation of intermediate numbers from tables in design handbooks → interpolation

Trend analysis – use pattern of data to make predictions: Imprecise or “noisy” data → regression (least-squares) Precise data → interpolation (interpolating polynomials)

Hypothesis testing – compare existing mathematical model with measured data Determine unknown model coefficient values … or … Compare predicted values with observed values to test

model adequacy

You’ve Got a Problem …

Wind tunnel data relating force of air resistance (F) to wind velocity (v) for our friend the bungee jumper

The data can be used to discover the relationship and

find a drag coefficient (cd), i.e., As F , v Data is not smooth, especially

at higher v’s If F = 0 at v = 0, then the

relationship may not be linear

Figure 13.1, Chapra

Figure 13.2, Chapra

especially if you are this guy

How to fit the “best” line or curve to these data?

Before We Can Discuss Regression Techniques … We Need To Review

basic terminologydescriptive statistics

for talking about sets of data

Basic Terminology

Data from TABLE 13.3

Maximum?Minimum?

6.7756.395

Range?6.775 - 6.395 = 0.380

Individual data points, yi

y1 = 6.395 y2 = 6.435

↓ y24 = 6.775

Number of observations?Degrees of freedom?

n = 24n – 1 = 23

Residual? yy i

Use Descriptive Statistics To Characterize Data Sets:

Location of center of distribution of the dataArithmetic mean Median (midpoint of data, or 50th percentile)Mode (value that occurs most frequently)

Degree of spread of the data setStandard deviationVarianceCoefficient of variation (c.v.)

Arithmetic Mean

n

yy i

6.624

4.158y


Standard Deviation

1

)(

1

2

n

yy

n

Ss it

y

097133.0124

217000.0

ys


St: total sum of squares of residuals between data points and mean

Variance

1

/)(

1

)(

122

22

n

nyy

n

yy

n

Ss

ii

ity

009435.0124

217000.02

ys


Coefficient ofVariation (c.v.)

%47.1%1006.6

097133.0c.v.

%100c.v.y

sy


c.v. = standard deviation / meanNormalized measure of spread

Fig

ure

12

.4, C

ha

pra

Histogram of data

For a large set of data,histogram can beapproximated by

a smooth, symmetricbell-shaped curve

→ normal distribution


Confidence Intervals

If a data set is normally distributed, ~68% of the total measurements will fall within the range defined by

Similarly, ~95% of the total measurements will be encompassed by the range

yy sysy to

yy sysy 2to2

Descriptive Statistics in MATLAB>>% s holds data from Table 13.2>>s=[6.395;6.435;6.485;…;6.775]

>>[n,x]=hist(s)n = 1 1 3 1 4 3 5 2 2 2

x = 6.414 6.452 6.49 6.528 6.566 6.604 6.642 6.68 6.718 6.756

>>min(s), max(s)

ans = 6.395ans = 6.775

>>range=max(s)-min(s)

range = 0.38

>>mean(s), median(s), mode(s)ans = 6.6ans = 6.61ans = 6.555

>>var(s), std(s)

ans = 0.0094348ans = 0.097133

n is the number of elements in each bin; x is a vector specifying the midpoint of each bin

Back to the Bungee Jumper Wind Tunnel Data … is the mean a good fit to the data?

Figure 13.8a, Chapra

velocity, m/s Force, N10 2520 7030 38040 55050 61060 122070 83080 1450

mean: 642

Figure 12.1, Chapra

Figure 13.2, Chapra

not very!!! distributionof residuals is large

Curve Fitting Techniques

Least-squares regression Linear Polynomial General linear least-squares Nonlinear

Interpolation (not covered) Polynomial Splines

Figure 12.8, Chapra, p. 209Figure 12.8, Chapra, p. 209

Can reduce the distributionof the residuals if use curve-

fitting techniques such aslinear least-squares regression

Figure 13.8b, Chapra

Linear Least-Squares Regression

Linear least-squares regression, the simplest example of a least-squares approximation is fitting a straight line to a set of paired observations:

(x1, y1), (x2, y2), …, (xn, yn)

Mathematical expression for a straight line:

y = a0+ a1x + e

intercept slopeerror or residual

Least-Squares Regression: Important Statistical Assumptions

Each x has a fixed value; it is not random and it is known without error, this means that

x values must be error-free regression of y versus x is not the same as the

regression of x versus y

The y values are independent random variables and all have the same variance

The y values for a given x must be normally distributed

Residuals in Linear Regression

Regression line is a measure of central tendency for paired observations (i.e., data points)

Residuals (ei) in linear regression represent the vertical distance between a data point and the regression line

Figure 13.7, Chapra

How to Get the “Best” Fit:

Minimize the sum of the squares of the residuals between the measured y and the y calculated with the (linear) model:

210

1

2,

1,

1

2

)(

)(

i

n

ii

modeli

n

imeasuredi

n

iir

xaay

yy

eS

Yields a unique line for a given dataset

How do we compute the best a0 and a1?

210

1

)( i

n

iir xaayS

One way is to use optimization techniques since looking for a minimum (more common for nonlinear

case)… or …

Another way is to solve the normal equations for a0 and a1 according to the derivation in the next few

slides

Derivation of Normal Equations Used to

Solve for a0 and a1

210

1

)( i

n

iir xaayS

First, differentiate the sum of the squares of the residuals with respect to each unknown coefficient

Derivation of Normal Equations Used to Solve for a0 and a1 - continued

Set derivatives = 0

Will result in a minimum Sr

Can be expressed as

0)][(2

0)(2

101

100

iiir

iir

xxaaya

S

xaaya

S

210

10

0

0

iiii

ii

xaxaxy

xaay


210

10

0

0

iiii

ii

xaxaxy

xaay

iiii

ii

yxaxax

yaxna

12

0

10

Realizing ∑a0 = na0, we can express these equations as a set of two simultaneous linear equations with two unknowns (a0 and a1) called the normal equations:

Normal Equations:


xaya

xxn

yxyxn

a

ii

iiii

102

2

1 and

iiii

ii

yxaxax

yaxna

12

0

10

Finally, can solve these normal equations for a0 and a1

ii

i

ii

i

yx

y

a

a

xx

xn

1

02

Improvement Due to Linear Regression

Figure 12.8, Chapra, p. 209

mean of the dependentvariable

best-fit linesy

sy/xFigure 12.8, Chapra, p. 209


best-fit linesy

sy/x

Spread of data around the mean of the dependent variable

Figure 13.8, Chapra

Spread of data around the best-fit line

The reduction in spread in going from (a) to (b), indicated by bell-shape curves, represents improvement due to linear regression

Improvement Due to Linear Regression

210

1

)( i

n

iir xaayS

2

1

)( yySn

iit

Figure 12.8, Chapra, p. 209


best-fit linesy

sy/xFigure 12.8, Chapra, p. 209


best-fit linesy

sy/x

Coefficient of determination quantifies improvement or error reduction due to describing data in terms of a straight line rather

than an average value

Total sum of squares around the mean of dependent variable

Sum of squares of residuals around the best-fit

regression line

t

rt

S

SSr

2

St - Sr quantifies error reduction due to using line instead of mean

Normalize by St because scale sensitive → r 2 = coefficient of determination

Used for comparison of several regressions

Value of zero represents no improvement

Value of 1 is a perfect fit, the line explains 100% of data variability

“Goodness” of Fit

Figure 12.9, Chapra

small residual errorsr2 → 1

large residual errorsr2 << 1

t

rt

S

SSr

2

Linearization of Nonlinear Relationships

What to do when relationship is nonlinear?

One option is polynomial regression

Another option is to linearize the data using transformation techniques Exponential Power Saturation-growth-rate

Figure 14.1, Chapra

Linearization Transformation Examples Exponential model:

Used to characterize quantities that increase (+β1) or decrease

(-β1) at a rate directly proportional to their own magnitude, e.g., population growth or radioactive decay

Take ln of both sides to linearize data:

Figure 13.11,

Chapra

xey 11

xy 11lnln

Linearization Transformation Examples

Power model:

Widely applicable in all fields of engineering

Take log of both sides to linearize data:

22

xy

Figure 13.11,

Chapra

22 logloglog xy

Linearization Transformation Examples Saturation-growth-rate model

Used to characterize population growth under limiting conditions or enzyme kinetics

To linearize, invert equation to give:

x

xy

33

Figure 13.11,

Chapra

33

3 111

xy

Linear Regression with MATLAB Polyfit can be used to determine the slope

and y-intercept as follows:

>>x=[10 20 30 40 50 60 70 80];>>y=[25 70 380 550 610 1220 830 1450];>>a=polyfit(x,y,1)a = 19.4702 -234.2857

Polyval can be used to compute a value using the coefficients as follows:

>>y = polyval(a,45)y = 641.8750

use 1 for linear (1st order)

Polynomial Regression

Extend the linear least-squares procedure to fit data to a higher order polynomial

For a quadratic (2nd order polynomial), will have a system of 3 normal equations to solve instead of 2 as for linear

For higher mth-order polynomials, will have a system of m+1 normal equations to solve

Figure 14.1, Chapra

Data not suited for linear least-squares regression

Nonlinear Regression (not covered)

Cases in engineering where nonlinear models – models that have a nonlinear dependence on their parameters – must be fit to data

For example,

Like linear models in that we still minimize the sum of the squares of the residuals

Most convenient to do this with optimization

errorxa eeaxf )1()( 1

0

More Statistics: Comparing 2 Means

69

depends on mean and amount of variability

can tell there is a difference when variability is low

use t-test to do this mathematically

http://www.socialresearchmethods.net/kb/stat_t.php

The t-test Is A Ratio Of “Signal To Noise”

70

Remember, variance (var) is just thesquare of the standard deviation


Standard Error of difference between means

How It Works ... Once You Have A t-value

71

Need to set a risk level (called the alpha level) – a typical value is 0.05 which means that five times out of a hundred you would find a statistically significant difference between the means even if there was none (i.e., by "chance").

Degrees of freedom (df) for the test = sum of # data points in both groups (N) minus 2.

Given the alpha level, the df, and the t-value, you can use a t-test table or computer program to determine whether the t-value is large enough to be significant.

If calculated t is larger than t (alpha, N-2) in table, you can conclude that the difference between the means for the two groups is different (even given the variability).


What About More Than 2 Sets Of Data?

72

ANOVA = Analysis of Variance commonly used for more than 2, if ... k samples are random and independently selected treatment responses are normally distributed treatment responses have equal variances

ANOVA compares variation between groups of data to variation within groups, i.e.,

Source: http://www.uwlax.edu/faculty/baggett/Math_145/HANDOUTS/anova.pdf

Steps for ANOVA

73

Define k population or treatment means being compared in context of problem

Set up hypotheses to be tested, i.e., H0: all means are equal

Ha: not all means are equal (no claim as to which ones not)

Choose risk level, alpha (=0.05 is a typical level) Check if assumptions reasonable (previous slide) Calculate the test statistic ... pretty involved ...see next page!

Note: usually use a computer program, but is helpful to know what computer is doing ... can do simple problems by hand

Source: http://www.uwlax.edu/faculty/baggett/Math_145/HANDOUTS/anova.pdf

ANOVA calculations

74Source: http://www.uwlax.edu/faculty/baggett/Math_145/HANDOUTS/anova.pdf

Collect this info from data set:

ANOVA calculations


Fill out a table like this to compute the F ratio statistic:

ANOVA calculations


Now what do we do with F statistic ? Compare it to an F distribution like we did with the t-test This time we need

alpha df of numerator (k-1) df of denominator (N-k)

to look up F (1- alpha)(k-1, N-k)

This time we need to compare F ≥ F (1- alpha)(k-1, N-k)

If yes, then more evidence against H0, reject H0

The End

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Documents

Statistical & Uncertainty Analysis UC Summer REU Programs Instructor: Lilit Yeghiazarian, PhD Environmental Engineering Program 1