Statistical & Uncertainty Analysis Yeghiazarian v2..." Common problem in summing infinite series...

Statistical & Uncertainty Analysis UC 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

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

n 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

n  Error ¨  numerical error ¨  data uncertainty in measurement error

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

4

Types of Error n  General Error (cannot blame computer) :

¨  Blunders n  human error

¨  Formulation or model error n  incomplete mathematical model

¨  Data uncertainty n  limited to significant figures in physical measurements

n  Numerical Error:

¨  Round-off error (due to computer approximations) ¨  Truncation error (due to mathematical approximations)

5

6

n  Gare Montparnasse, Paris, 1895 n  Accident causes:

¨  Faulty brake ¨  Engine driver trying to make up lost time

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

Errors associated with both calculations & measurements can be characterized in terms of: •  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

n  Occur because digital computers have a limited ability to represent numbers

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

n  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 n  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”

11

- Overflow Overflow +

1.797693134862316 x 10308 -1.797693134862316 x 10308

2.225073858507201 x 10-308 -2.225073858507201 x 10-308

15 significant figures

15 significant figures

As good as it gets on our PCs …

“Hole” on either side of zero

0

Underflow

For 64-bit, IEEE double precision format systems 12

Implications of Finite Number of bits (1) n  Range

¨ Finite range of numbers a computer can represent

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

¨ Underflow error – smaller than computer can handle n  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) n  Precision

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

14

Round-Off Error and Common Arithmetic Operations

n  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)

n  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 and Large Computations

16

Particular Problems Arising from Round-Off Error (1) n  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)

n  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) n  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

n  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 n  Occur when exact mathematical formulations are

represented by approximations

n  Example: Taylor series

19

Taylor series expansions where h = xi+1 - xi

4)4(

3)3(

2"

'1 )(

!4)()(

!3)()(

!2)())(()()( hxfhxfhxfhxfxfxf iii

iii ++++≅+

3)3(

2"

'1 )(

!3)()(

!2)())(()()( hxfhxfhxfxfxf ii

iii +++≅+

2"

'1 )(

!2)())(()()( hxfhxfxfxf 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:

Figure 4.6, Chapra, p. 93

= 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

n  Sum of ¨  round-off error and ¨  truncation error

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

due to subtractive cancellation or large numbers of computations)

¨  truncation error ↓ n  Point of diminishing returns is when

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

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

22

Control of Numerical Errors n  Experience and judgment of engineer

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

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

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

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

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Measurements & Uncertainty

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Errors as Uncertainties

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

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

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

n  In this section, words error and uncertainty are used interchangeably

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Inevitability of Uncertainty

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

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

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

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

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Measuring Length with Ruler

27


28


29

Best estimate of length = 82.5 mm Probable range: 82 to 83 mm We have measured the length to the nearest millimeter

Note: markings are 1 mm apart

How To Report & Use Uncertainties

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

of quantity x is stated as (measured value of x) = xbest ± Δx

n Δx is called uncertainty, or error, or margin of error

n Δx is always positive

30

Basic Rules About Uncertainty

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

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

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

31

Basic Rules About Uncertainty

n  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

n  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

Addition and Subtraction x = a + b - c

Multiplication and Division x = a * b /c

Exponentiation (no ± in b) x = ab

Logarithm Base 10: x = log10 a

Base e : x = ln a

Antilog Base 10: x = 10a

Base e : x = e a

!

"x

= "a

2+"

b

2+"

c

2+ ...

!

"x

= x"a

a

#

$ %

&

' ( 2

+"b

b

#

$ %

&

' ( 2

+"c

c

#

$ %

&

' ( 2

+ ...

!

"x

= x #b #"a

a

$

% &

'

( )

!

"x

= 0.434 #"a

a

$

% &

'

( )

!

"x

="a

a

#

$ %

&

' (

!

"x

= 2.303# x #"a

!

"x

= x #"a

Statistics & Curve Fitting

34

Curve Fitting n  Could plot points and

sketch a curve that visually conforms to the data

n  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 PT4.1, C

hapra

Curve Fitting and Engineering Practice n  Two typical applications when fitting experimental

data:

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

n  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 …

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

n  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 terminology ¨ descriptive statistics

for talking about sets of data

Basic Terminology Data from TABLE 13.3

Maximum? Minimum?

6.775 6.395

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 = 24 n – 1 = 23

Residual? yyi−

Use Descriptive Statistics To Characterize Data Sets: Location of center of distribution of the data

¨ Arithmetic mean ¨ Median (midpoint of data, or 50th percentile) ¨ Mode (value that occurs most frequently)

Degree of spread of the data set ¨ Standard deviation ¨ Variance ¨ Coefficient of variation (c.v.)

Arithmetic Mean

ny

y i∑=

6.6244.158==y

Data from TABLE 13.3

Standard Deviation

1)(

1

2

−

−=

−= ∑

nyy

nSs it

y

097133.0124

217000.0=

−=ys


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

Variance

1/)(1)(

122

22

−

−=

−

−=

−=

∑ ∑

∑

nnyy

nyy

nSs

ii

ity

009435.0124

217000.02 =−

=ys


Coefficient of Variation (c.v.)

%47.1%1006.6

097133.0c.v. ==

%100c.v.ysy=


c.v. = standard deviation / mean Normalized measure of spread

Figure 12.4, Chapra

Histogram of data

For a large set of data, histogram can be approximated by

a smooth, symmetric bell-shaped curve

→ normal distribution


Confidence Intervals

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

n  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 = 6.395 ans = 6.775

>>range=max(s)-min(s) range = 0.38

>>mean(s), median(s), mode(s)

ans = 6.6 ans = 6.61 ans = 6.555

>>var(s), std(s)

ans = 0.0094348 ans = 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!!! distribution of residuals is large

Curve Fitting Techniques

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

n  Interpolation (not covered) ¨ Polynomial ¨ Splines

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

Can reduce the distribution of the residuals if use curve-

fitting techniques such as linear least-squares regression

Figure 13.8b, Chapra

Linear Least-Squares Regression

n  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)

n  Mathematical expression for a straight line:

y = a0+ a1x + e

intercept slope error or residual

Least-Squares Regression: Important Statistical Assumptions

n  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

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

n  The y values for a given x must be normally distributed

Residuals in Linear Regression

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

n  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

!Sr!a0

= "2 (yi# " a0 " a1xi )

!Sr!a1

= "2 [(yi# " a0 " a1xi )]xi

210

1

)( i

n


=

n  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

n  Set derivatives = 0

n  Will result in a minimum Sr

n  Can be expressed as

0)][(2

0)(2

101

100

=−−−=∂

∂

=−−−=∂

∂

∑

∑

iiir

iir

xxaayaS

xaayaS

210

10

0

0

iiii

ii

xaxaxy

xaay

∑∑∑

∑∑∑

−−=

−−=


210

10

0

0

iiii

ii

xaxaxy

xaay

∑∑∑

∑∑∑

−−=

−−= ( )

iiii

ii

yxaxax

yaxna

∑∑∑

∑∑

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

=+

12

0

10

n  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

yxyxna

ii

iiii

102

2

1 and −=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−

=

∑ ∑

∑∑∑

( )

iiii

ii

yxaxax

yaxna

∑∑∑

∑∑

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

=+

12

0

10

n  Finally, can solve these normal equations for a0 and a1

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

∑∑

∑∑∑

ii

i

ii

i

yxy

aa

xxxn

1

02

Improvement Due to Linear Regression


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


=

2

1

)( yySn

iit −=∑

=



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

SSSr −

=2

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

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

n  Used for comparison of several regressions

n  Value of zero represents no improvement

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

“Goodness” of Fit

Figure 12.9, Chapra

small residual errors r2 → 1

large residual errors r2 << 1

t

rt

SSSr −

=2

Linearization of Nonlinear Relationships n  What to do when relationship

is nonlinear?

n  One option is polynomial regression

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

Figure 14.1, Chapra

Linearization Transformation Examples n  Exponential model: n  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

n  Take ln of both sides to linearize data:

Figure 13.11, Chapra

xey 11

βα=

xy 11lnln βα +=

Linearization Transformation Examples

n  Power model:

n  Widely applicable in all fields of engineering

n  Take log of both sides to linearize data:

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βα xy =

Figure 13.11, Chapra 22 logloglog αβ += xy

Linearization Transformation Examples n  Saturation-growth-rate model

n  Used to characterize population growth under limiting conditions or enzyme kinetics

n  To linearize, invert equation to give:

xxy+

=3

3 βα

Figure 13.11, Chapra

33

3 111αα

β+⎟⎠

⎞⎜⎝

⎛=xy

Linear Regression with MATLAB n  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

n  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 n  Extend the linear least-

squares procedure to fit data to a higher order polynomial

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

n  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) n  Cases in engineering where nonlinear models –

models that have a nonlinear dependence on their parameters – must be fit to data

n  For example,

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

n  Most convenient to do this with optimization

errorxa eeaxf +−= − )1()( 1

0

More Statistics: Comparing 2 Means

69

n  depends on mean and amount of variability

n  can tell there is a difference when variability is low

n  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 the square of the standard deviation


Standard Error of difference between means

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

71

n  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").

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

n  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.

n  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 significant (even given the variability).


What About More Than 2 Sets Of Data?

72

n  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

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

!

F = variation between groups

variation within groups

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

Steps for ANOVA

73

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

n  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)

n  Choose risk level, alpha (=0.05 is a typical level) n  Check if assumptions reasonable (previous slide) n  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

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

n Collect this info from data set:

ANOVA calculations


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

ANOVA calculations


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

n  F ≥ F (1- alpha)(k-1, N-k)

n  If yes, then more evidence against H0, reject H0

The End

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Statistical & Uncertainty Analysis Yeghiazarian v2..." Common problem in summing infinite series...

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